optimal allocation without money: an engineering...

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Optimal Allocation Without Money An EngineeringApproachItai Ashlagi Peng Shi

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Published online in Articles in Advance 19 Aug 2015

httpdxdoiorg101287mnsc20152162

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MANAGEMENT SCIENCEArticles in Advance pp 1ndash20ISSN 0025-1909 (print) ISSN 1526-5501 (online) httpdxdoiorg101287mnsc20152162

copy 2015 INFORMS

Optimal Allocation Without MoneyAn Engineering Approach

Itai Ashlagi Peng ShiMassachusetts Institute of Technology Cambridge Massachusetts 02139 iashlagimitedu pengshimitedu

We study the allocation of heterogeneous services to agents with incomplete information and withoutmonetary transfers Agents have private multidimensional utilities over services drawn from commonly

known priors The social plannerrsquos goal is to maximize a potentially complex public objective For tractabilitywe take an ldquoengineeringrdquo approach in which we solve a large-market approximation and convert the solutioninto a feasible finite-market mechanism that still yields good results We apply this framework to real data fromBoston to design a mechanism that assigns students to public schools in order to maximize a linear combinationof utilitarian and max-min welfare subject to capacity and transportation constraints We show how to opti-mally solve a large-market formulation with more than 868 types of students and 77 schools and we translatethe solution into a finite-market mechanism that significantly outperforms the baseline plan chosen by the cityin terms of efficiency equity and predictability

Data as supplemental material are available at httpdxdoiorg101287mnsc20152162

Keywords market design priors assignment school choice optimal design large-market approximationHistory Received May 22 2014 accepted November 30 2014 by Martin Lariviere operations management

Published online in Articles in Advance

1 IntroductionIn many settings goods or services are allocated toagents without monetary transfers Examples includethe allocation of seats in public schools spaces in col-lege dorms or courses and positions in medical res-idency programs Social planners have multifacetedconcerns such as social welfare equity and systemcosts

One example of such a problem was faced by thecity of Boston in the 2012ndash2013 school assignmentreform Prior to 2013 seats in Boston Public Schools(BPS) had been allocated as follows The city wasdivided into three zones and each family rankedschools within a menu of choices that consisted ofschools within the familyrsquos zone and schools withinone mile of the familyrsquos home A centralized algo-rithm then allocated seats based on familiesrsquo rankinglists priorities and lotteries These large menus ofschools led to unsustainably high busing costs repre-senting approximately 10 of the total school boardbudget (Russell and Ebbert 2011) In 2012 the cityused simulation to choose a new plan from a shortlist of proposals to decrease the choice menus whilemaintaining equity among neighborhoods Althoughthe proposals in the short list were ad hoc the tech-niques in this paper can be used to design the optimalallocation in a systematic way

When agentsrsquo preferences are publicly known theallocation is merely an optimization problem This

study addresses how to allocate services when agentsrsquopreferences are only privately known In our modelthere are multiple kinds of services and agentshave private multidimensional utilities over the ser-vices drawn from commonly known priors whichmay depend on agentsrsquo observable information Sincemechanism design with multidimensional valuationshas traditionally been difficult especially for generalobjectives we adopt an engineering approach wefirst find the optimal mechanism for a large-marketformulation then we convert it into a feasible finite-market mechanism and we evaluate it using data-driven simulation

In the large-market formulation there are finitelymany types of agents and a continuum of agents ofeach type Types in this paper represent the publicinformation of agents For example in school choicea type may represent students from a certain neigh-borhood of a certain race or of a certain socioeco-nomic status We require mechanisms to be incentivecompatible and Pareto optimal among agents of thesame type and we refer to such mechanisms as validmechanisms Whereas agents of the same type aretreated symmetrically agents of different types maybe differentiated The goal is to find a valid mech-anism that maximizes the social plannerrsquos objectivewhich can be fairly general

We first characterize all valid mechanisms Undermild assumptions on the utility priors we show that

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Ashlagi and Shi Optimal Allocation Without Money2 Management Science Articles in Advance pp 1ndash20 copy 2015 INFORMS

any valid mechanism can be described as a collectionof competitive equilibria with equal income (CEEI)More precisely agents of each type are given ldquovir-tual pricesrdquo for probabilities of each service and theallocation can be interpreted as giving agents oneunit of ldquovirtual moneyrdquo and allowing them to ldquopur-chaserdquo their preferred probabilistic bundle of services(a related mechanism was introduced by Hyllandand Zeckhauser 1979) Virtual prices may vary acrosstypes but agents within the same type are given thesame prices

In many contexts only relative preferences are elic-ited but not preference intensities (For example inBoston families submit preference rankings insteadof utilities) Such mechanisms are called ordinal asopposed to cardinal mechanisms which have no infor-mation requirements Under mild regularity assump-tions we show that any valid ordinal mechanismcan be described as ldquolottery-plus-cutoffrdquo each agentreceives a lottery number drawn from the uniform dis-tribution between 0 and 1 For each service and eachtype there is a ldquolottery cutoffrdquo and an agent is ldquoadmit-tedrdquo to a service if her lottery number is below the cut-off Finally each agent is allocated her most preferredservice to which she is admitted

These structural results provide insights into thetypes of mechanisms observed in practice In manybusiness schools course allocation is made by a bid-ding process in which students are given a number ofpoints (virtual currency) and the highest bidders areassigned seats1 This is conceptually similar to CEEIIn many public school districts2 students submitpreference rankings for schools and a centralizedmechanism uses submitted preferences predefinedpriorities and random lottery numbers to determinethe assignment3 Given ex post lottery number cutoffsat each school for each priority class of students thisis analogous to the lottery-plus-cutoff mechanism

These theorems also simplify the computation ofthe optimal mechanism since we no longer have tooptimize over allocation functions but over a finitenumber of prices or cutoffs To illustrate we com-pute the optimal large-market ordinal mechanism inan empirically relevant setting We encode optimalcutoffs by an exponentially sized linear program andwe show that the dual can be solved efficiently whenagentsrsquo utilities are based on a multinomial-logit dis-crete choice model

To demonstrate the relevance of our large-marketformulation we employ it to optimally design school

1 See eg Soumlnmez and Uumlnver (2010) and Budish and Cantillon(2012)2 Examples include Boston New York City New Orleans andSan Francisco3 For more information see Abdulkadiroglu et al (2006 2009)

choice in Boston We use the same amount of busingas in the mechanism chosen by the city referred tohere as the baseline mechanism while optimizing alinear combination of utilitarian welfare and max-minwelfare All the analyses use real data from BPS

We first define a finite-market formulation of theproblem which resembles one the city faced duringthe 2012ndash2013 reform We define a large-market approx-imation and compute the optimal (large-market) cut-offs We use these cutoffs to design the correspondingmenus and priorities and use the deferred acceptance(DA) algorithm (Gale and Shapley 1962) to producea feasible finite-market mechanism that can be seenas ldquoasymptotically optimalrdquo4 We evaluate this mecha-nism in the finite-market setting and show by simula-tion that it significantly improves on the baseline in allaspects It improves utilitarian welfare by an amountequivalent to decreasing studentsrsquo average distance toschool by 005 miles and it improves max-min welfareby about 205 miles These are significant gains becausethe baseline improves over the most naiumlve plan in util-itarian welfare by only 006 miles and in max-min wel-fare by 107 miles so our solution effectively doublesthe gains Furthermore our mechanism improves stu-dentsrsquo chances of getting their first choice by 15

Our methodology may also be applied to otherallocation problems requiring consideration of agentsrsquopreferences Potential examples include housing lot-teries in which the social planner may want tooptimize welfare and fairness subject to various dis-tributional constraints business school course bid-ding in which an optimal cardinal mechanism maybe used in place of current bidding schemes and theassignment of internships or relocation opportunitiesWe leave detailed explorations for future work

11 Related LiteratureOur work connects four strands of previous researchThe first is the matching literature which tradition-ally focuses on designing mechanisms that satisfycertain properties such as Pareto efficiency an appro-priate notion of fairness and strategy-proofness (seeeg Roth and Sotomayor 1990 Abdulkadiroglu andSoumlnmez 2013) Hylland and Zeckhauser (1979) studycardinal mechanisms that achieve Pareto efficiencyand propose the CEEI mechanism which also arisesin our characterization of valid cardinal mechanismsBogomolnaia and Moulin (2001) study ordinal mecha-nisms that satisfy ordinal efficiency an ordinal notionof Pareto optimality that we adopt (except we con-sider ordinal efficiency within each type of agent)

4 This solution is asymptotically optimal in the sense that if the mar-ket scales up with independent copies of itself the finite-marketformulation converges to the large-market formulation and thefinite-market solution also converges to the large-market optimum

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Ashlagi and Shi Optimal Allocation Without MoneyManagement Science Articles in Advance pp 1ndash20 copy 2015 INFORMS 3

Abdulkadiroglu and Soumlnmez (2003) Abdulkadirogluet al (2009) and (2006) apply matching theory toschool choice and their works have been influentialin the adoption of strategy-proof ordinal mechanismsover nonstrategy-proof alternatives in cities such asNew York Boston Chicago New Orleans and SanFrancisco More recent works on school choice studynuances in tie breaking (Pathak and Sethuraman 2011Erdil and Ergin 2008 Ashlagi and Shi 2014) incorpo-rating partial preference intensities (Abdulkadirogluet al 2015) and ldquocontrolled school choicerdquo whichaddresses implementing constraints such as diversity(Kominers and Soumlnmez 2015 Echenique and Yenmez2015 Ehlers et al 2014) Unlike our work this liter-ature (and the matching literature in general) rarelyassumes priors on agent utilities (especially asymmet-ric priors) and rarely optimizes a global objective Ourwork can be viewed as bridging the matching and themechanism designauction literature5

Another strand is optimal mechanism design with-out money in the finite-market framework6 Cur-rently analytical difficulties restrict positive results tosingle-dimensional valuations Miralles (2012) solvesfor the optimal cardinal mechanism for two servicesand many bidders with a symmetric utility prior Heshows that under certain regularity conditions theoptimal allocation can be described as CEEI He fur-ther shows that this can be implemented using a com-bination of lottery insurance and virtual auction inwhich agents use probabilities of the less desirablegood as virtual currency to bid for the more desir-able good However the analysis requires reducingthe valuation space to a single dimension by tak-ing the ratio of the utilities for the two services soit does not generalize to more than two servicesOur work shows stronger results by leveraging alarge-market approximation Hartline and Roughgar-den (2008) Hoppe et al (2009) Condorelli (2012)and Chakravarty and Kaplan (2013) study modelsin which agents cannot pay money but may ldquoburnmoneyrdquo or exert effort to signal their valuation Oneinsight from their work is that if a utility prior doesnot have a thick tailmdashor more precisely if it satisfiesa monotone hazard rate conditionmdashthen requiringagents to exert effort is unnecessary and a lottery willmaximize the social welfare However their resultscannot be easily extended to a multidimensional set-ting which is more realistic when multiple types ofservices are offered

A third strand is optimal school districting whichstudies giving each family a designated school by

5 See eg Myerson (1981) who models agentsrsquo preferences withBayesian priors and Budish (2012) who compares matching andstandard mechanism design He emphasizes the absence of hetero-geneous priors and global objectives in the matching literature6 For a survey see Schummer and Vohra (2007)

home location Representative works include Clarkeand Surkis (1968) Sutcliffe et al (1984) and Caroet al (2004) These papers use linear or mixed-integeroptimization to optimize a global objective whichmay take into account distances between studentsand schools racial balance capacity utilization geo-graphic contiguity and uncertainties in future studentpopulation We complement this literature by com-bining optimization with the elicitation of familiesrsquoprivate preferences thus bringing together the best ofcentral planning and the market

In terms of techniques our paper builds on previouslarge-market models with a continuum of agents Con-tinuum agent models often simplify the analysis overa finite model leading to stronger and cleaner resultsSuch models are common in the industrial organi-zation literature (Tirole 1988) Azevedo and Leshno(2015) study matching markets where each side of themarket has preferences over the other (in our case ser-vices do not have preferences over individual agents)They focus on characterizing stable matchings and donot consider global optimization In the operationsmanagement literature our engineering approach ofsolving for a large-market optimum and using it toconstruct a feasible solution has appeared before butmost previous applications have been related to queu-ing7 Our paper illustrates the power of this approachwhen applied to the allocation problem

2 The Large-Market EnvironmentA social planner needs to allocate services to a contin-uum of agents There is a finite set T of agent typesand a mass nt of agents for each type t isin T In contrastto the convention in mechanism design our notionof ldquotyperdquo does not denote agentsrsquo private informa-tion but rather their public information8 There is afinite set S of services and every agent must be allo-cated exactly one service (outside options can be rep-resented by an additional ldquonull servicerdquo) Allocationsmay be probabilistic so the set of possible allocationsfor each agent is the probability simplex9

atilde=

p isinS2 p ge 01sum

s

ps = 1

0

Agents of type t have private utilities for ser-vices which are distributed according to a com-monly known continuous measure Ft over utilityspace U =S Each u isinU is a utility vector in which

7 See eg Maglaras and Zeevi (2005) and Perry and Whitt (2009)8 For example in school choice the type may correspond to the stu-dentrsquos neighborhood in course allocation the type may correspondto the studentrsquos major9 It suffices to study assignment probabilities because by theBirkhoff-von Neumann theorem assignment probabilities can bedecomposed as a random lottery over deterministic assignments

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each component denotes utility for a service For anymeasurable subset A sube U Ft4A5 denotes the mass oftype t agents who have utilities in A Since the totalmass for type t is nt the total measure Ft4U5 equals nt

The social planner must design a mechanism toelicit agentsrsquo preferences and to decide on the allo-cation The goal is to maximize a certain objectiveMechanisms can either be cardinal or ordinal cardinalmechanisms allow unrestricted preference elicitationwhereas ordinal mechanisms allow only elicitationof relative rankings but not preference intensitiesWe study these separately in sectsect3 and 4

3 Cardinal MechanismsStated formally a cardinal mechanism x is a collection ofallocation rules xt one for each type t where each xt isa mapping from reported utilities to a possible alloca-tion xt2 U rarratilde and is measurable with respect to Ft 10

For agents to submit their true preferences safely werequire the allocation rule to be incentive compatible11

An allocation rule is incentive compatible if it is in theagentrsquos best interest to report her true utility vector

u isin arg maxuprimeisinU

u middot xt4uprime5

0

We further require allocation rules to be Pareto effi-cient within type which implies that agents of the sametype cannot benefit from trading allocation probabili-ties Formally stated xt is Pareto efficient within typeif no other function xprime

t2 U rarratilde exists that has the sameaverage allocation

int

Uxprime

t4u5 dFt =int

Uxt4u5 dFt1

is weakly preferred by all agents

u middot xprime

t4u5ge u middot xt4u51

and is strictly preferred by a positive measure ofagents AsubeU such that Ft4A5 gt 0

Pareto efficiency within type can be viewed as aldquostabilityrdquo criterion our formulation implicitly as-sumes that the social planner treats agents withina given type symmetrically and cannot discriminatebased on the exact identity of the agent Therefore itmay be unreasonable to enforce no trading of alloca-tions for agents of the same type This may also beinterpreted as foreseeing possible Pareto-improvingtrades and internalizing them in the mechanism

Observe that our requirement of Pareto efficiencywithin type still allows agents of different types to

10 If we did not work within a large-market environment the allo-cation rule would also be a function of othersrsquo reports11 If one assumes agents play in a Bayesian Nash equilibrium as-suming incentive compatibility is without loss of generality

benefit from trading allocations12 This allows ourmechanism to differentiate freely with respect to typesif this improves the objective For school choice thiscorresponds to differentiating students with respect tohome location Unless otherwise noted all subsequentmentions of Pareto efficiency refer to within type

We call an allocation rule valid if it is both incen-tive compatible and Pareto efficient within type anda mechanism is valid if all its allocation rules arevalid For now we allow the objective function W4x5to arbitrarily depend on all the allocation rules xtrsquosThis generality allows the objective to incorporatemany considerations such as agentsrsquo welfare capac-ity constraints and differential costs To illustrate howsuch considerations can be modeled observe that theexpected utility of an agent of type t is

vt =1nt

int

Uu middot xt4u5 dFt0

Thus one can incorporate social welfare by includingsum

t ntvt in the objective function As another exam-ple the total amount of service s allocated is qs =sum

t

int

Uxts4u5 dFt We can model a hard capacity limit

ms for service s by setting W4x5 to be negative infin-ity when qs gt ms Alternatively one can model asmooth penalty for exceeding capacity by subtractinga penalty term C4max801 qs minusms95 from the objectivewhere C4 middot 5 is a convex cost function

31 Characterization of Valid Allocation RulesWe show that under mild regularity conditions on Ft any incentive-compatible and Pareto-efficient alloca-tion rule xt corresponds to a CEEI of an artificial cur-rency This means that for each service s there exists aldquopricerdquo as isin 4017 (possibly infinite) for buying unitsof probability using units of the artificial currencyand the allocation is what agents would buy with oneunit of artificial currency when offered probabilitiesof various services at these prices

xt4u5 isin arg maxpisinatilde

8u middot p2 a middot p le 190

Figure 1 illustrates a CEEI with three services Theprice vector a is the same for all agents of the sametype but it may differ across types

This result implies that the search for the optimalmechanism can be restricted to a search over the setof price vectors for each type For each type the spaceof price vectors is only S-dimensional as opposed tothe space of allocation rules which is the space of allfunctions xt2 U rarratilde

Since every agent must be assigned we gain nouseful information if an agentrsquos utilities were all

12 If full Pareto efficiency were desired one could model the situa-tion with one type or use weighted social welfare as the objective

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Figure 1 (Color online) Illustration of CEEI with S = 3

Budget line

(010) (001)

(100)

u

ax(u)

Allowable allocations

Notes The triangle represents the space of possible allocations atilde Theshaded region is 8p isin atilde2 a middot p le 19 This represents the ldquoallowable alloca-tionsrdquo for this type which is the convex hull of 8xt 4u59 An agent of this typecould obtain allocations in the interior of this region by randomizing overseveral reports For utility vector u the agent receives an allocation p thatmaximizes expected utility u middot p subject to the price of p a middot p not exceedingthe budget of one

shifted by the same additive constant To eliminatethis degree of freedom let D = 8u isinn2 u middot 1 = 09 Thisis the normal subspace for the all-ones vector and itrepresents the directions in which utility reports areinformative Given a preference report u we call theprojection of u onto D the relative preference Givenany set AsubeD we define U4A5 to be the subset of Uwhose projections onto D are in A

The regularity condition we impose on Ft is thata priori an agentrsquos relative preference could withpositive probability take any direction in D This canbe interpreted as the central planner not being able torule out any relative preferences a priori

Definition 1 (Full Relative Support) Let D bethe set of relative preferences Define the set of nor-malized relative preferences D = 8d isin D1 d = 19where middot is the Euclidean norm This is a spherein 4S minus 15-dimensional space and can be endowedwith the topology of a 4Sminus25-sphere This induces atopology on the set of cones13 C subeD by defining C asopen if and only if C cap D is open in D DistributionFt has full relative support if for every nonempty opencone C subeD Ft4U4C55 gt 0

Theorem 1 Consider a given type and suppose thatits utility distribution F over U is continuous and hasfull relative support Then any incentive-compatible andPareto-efficient allocation rule can be supported as CEEIwith some price vector a isin 4017S

Note that because of our large-market environmentthis theorem implicitly assumes that the allocationrule treats agents symmetrically and is nonatomicmeaning that no individual agent can affect the

13 A cone is a set C in which x isinC implies x isinC for all isin 4015

market because in evaluating Pareto efficiency weconsider only positive masses of agents

The full proof of this theorem contains fairly tech-nical steps and is deferred to the appendix But weprovide the main idea here incentive compatibilityin our setting implies that the set Xt = 8xt4u59 lies onthe boundary of its convex hull and that xt4u5 maxi-mizes the linear function u middot x over this convex hullThis yields a correspondence between an incentive-compatible allocation rule xt and a convex set Anyincentive-compatible allocation rule maps to a uniqueconvex set and any convex set corresponds to anincentive-compatible rule which is given by the opti-mal solution of maximizing the linear functional u middot xover the set Denote by Xt the convex set that corre-sponds to the allocation rule xt

Now any convex set can be specified by a family ofsupporting hyperplanes If there is a unique support-ing hyperplane that intersects Xt in the interior of theunit simplex we are done since that hyperplane canbe represented by a price vector If there are at leasttwo such hyperplanes then we show that there is aldquotrading directionrdquo d by which some positive measureof agents may prefer allocations in the direction d andothers would prefer allocations in the direction minusdthus producing a Pareto-improving trade contradict-ing Pareto efficiency The existence of such positivemeasures of agents to carry out the trade is guaran-teed by the full relative support assumption whichcan be interpreted as a ldquoliquidityrdquo criterion

Similar results have previously appeared in the lit-erature but with different assumptions They do notimply our result Aumann (1964) shows conditionsin which any Pareto-efficient allocation is supportedby equilibrium prices although not necessarily fromequal incomes His analysis crucially depends on theunboundedness of the space of allocations which inour cases is the bounded unit simplex Zhou (1992)and Thomson and Zhou (1993) show that under cer-tain regularity conditions any ldquostrictly envy-freerdquo andPareto-efficient allocation is supported as CEEI How-ever their strictly envy-free condition is stronger thanour incentive compatibility condition Their analysisalso requires the allocation space to not contain itsboundary which is not true in our case since our unitsimplex is closed

Our result has similarities to the second welfare the-orem which states that any Pareto-efficient allocationcan be sustained by a competitive equilibrium Animportant difference is that in the second welfare the-orem14 the budgets of agents may need to be different

14 The second welfare theorem is traditionally stated with locallynonsatiated utilities which is not satisfied when agents can be allo-cated only one good and utilities are not transferable See MirallesAsensio and Pycia (2014) for a version of the welfare theorem inthis setting

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and possibly depend on their preferences Howeverin our case we can give all agents a common budgetindependent of their utilities

4 Ordinal MechanismsOrdinal mechanisms cannot elicit preference intensi-ties but only relative rankings Let ccedil be the set ofpermutations of S corresponding to preferences rank-ings over services An ordinal mechanism is a collectionof ordinal allocation rules xt one for each type t eachof which is a mapping between preference rankingsand allocations xt2 ccedilrarratilde

Denote by U45 sube U the set of utility vectors thatare consistent with the ranking in the sense thatutilities for services are ranked according to the per-mutation

u415 gtu425 gt middot middot middotgtu4S50

Let Ft45= Ft4U455 be the measure of agents of type tthat adhere to the strict preference ranking

An allocation rule xt is incentive compatible if truth-telling maximizes utility That is for every u isinU45

xt45 isin arg max primeisinccedil

u middot xt4prime5

0

An allocation rule xt is ordinal efficient within type if nocoalition of agents of this type can trade probabilitiesand all improve in the sense of first-order stochasticdominance Precisely speaking xt is ordinal efficientif no another function xprime

t2 ccedilrarratilde exists with the sameaverage allocation

int

ccedilxprime

t dFt =int

ccedilxt dFt1

but xprimet always first-order stochastically dominates xt

which means that for every isin ccedil and for every 1 le

k le Sksum

j=1

xprime

t4j545ge

ksum

j=1

xt4j5451

and the inequality is strict for some k and some ofpositive measure Ft45 gt 0

The social plannerrsquos goal is to optimize an arbi-trary function W4x5 subject to each allocation rule xt

being valid Note again that we require only ordinalefficiency within type so the mechanism has the flex-ibility to tolerate potential Pareto-improving tradesbetween different types if this improves the objectiveAll subsequent mentions of ordinal efficiency refer towithin type

41 Characterization of Valid OrdinalAllocation Rules

We show that any valid ordinal allocation rulecan be represented as a lottery-plus-cutoff mecha-nism agents are given lottery numbers distributed as

Figure 2 (Color online) Illustration of Lottery-Plus-Cutoff

1

a1

a2

x3(1) = a3 ndash a1

a3

a4

2 3

0

12

34

1

1 3 2 2 1 3 4

Notes The vertical axis represents lottery numbers which are uniformly dis-tributed from 0 to 1 The dotted lines are lottery cutoffs for this type Thecolumns represent various preference reports For preference report 1 theallocation probability for service 3 is the difference a3 minusa1 which representslottery numbers for which the agent is not admitted to her first choice ofservice 1 but is admitted to her second choice of service 3

Uniform60117 and each service has a lottery cutoff foreach type An agent is ldquoadmittedrdquo to a service if andonly if her lottery number does not exceed the cut-off and the agent chooses her most preferred serviceamong those that she is admitted to This is illustratedin Figure 2 In mathematical terms this can be statedas follows

Definition 2 An ordinal allocation rule x2 ccedil rarr atildeis lottery-plus-cutoff if there exist ldquocutoffsrdquo as isin 60117such that

x4k545=k

maxj=1

a4j5 minuskminus1

maxj=1

a4j50

For additional insights we provide here an equiva-lent formulation of lottery-plus-cutoff which we callrandomized-menus-with-nested-menus define probabili-ties p11 0 0 0 1 pS and menus of services M11 0 0 0 1MSsuch that S =M1 supeM2 supe middot middot middot supeMS An ordinal alloca-tion rule is randomized-menus-with-nested-menus ifeach agent is offered menu Mk with probability pk Tosee that this is equivalent to lottery-plus-cutoff rela-bel the services in increasing order of cutoffs a1 le

a2 le middot middot middot le aS Define a0 = 0 The one-to-one mappingis Mk = 8k1 0 0 0 1 S9 and pk = ak minus akminus1

Analogous to the full relative support assumptionin the cardinal setting we require a regularity condi-tion on the utility distribution which says that everyordinal ranking isinccedil is possible

Definition 3 (Full Ordinal Support) DistributionFt satisfies full ordinal support if Ft45 gt 0 for everypreference ranking isinccedil

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Theorem 2 Consider a given type and suppose thatits utility prior F induces strict preference rankings withprobability 1 and has full ordinal support Let x45 beany incentive-compatible and ordinal-efficient allocationrule then x45 is lottery-plus-cutoffs for some cutoffsa isin 60117S

As with the cardinal case because of how we set upthe large-market environment this theorem implic-itly assumes that the allocation rule is symmetric andnonatomic

The proof is given in Appendix C15 The intuitionbehind it is similar to Theorem 1 In the cardinal caseincentive-compatible allocation rules are associatedwith arbitrary convex subsets of atilde In the ordinal caseinstead of a convex set we have an associated poly-matroid More precisely for any incentive-compatibleordinal allocation rule xt the set 8xt9 is the vertexset of the base polymatroid of a monotone submodu-lar function f and any monotone submodular f in-duces an incentive-compatible allocation rule Usingthis characterization we show that subject to incen-tive compatibility the full ordinal support conditionimplies that unless the allocation rule is a lottery-plus-cutoff some agents can trade and obtain allocationsthat first-order stochastically dominate their currentallocations

This result can be seen as a generalization of pre-vious characterizations of ordinal efficiency in largemarkets Bogomolnaia and Moulin (2001) prove thatin a finite market every ordinal efficient mechanismis a variant of their ldquoprobabilistic serialrdquo mechanism(Their formulation only has one type of agent) Cheand Kojima (2011) show that random serial dicta-torship the simple mechanism of ordering agentsrandomly and having them pick their best remain-ing choices iteratively is asymptotically equivalentto the probabilistic serial in a large market Liu andPycia (2013) further show that in a large marketall asymptotically efficient symmetric and asymp-totically strategy-proof ordinal mechanisms coincidewith the probabilistic serial in the limit One can viewour lottery-plus-cutoff mechanism (or randomized-menus-with-nested-menus) as a generalization of theprobabilistic serial but allowing arbitrary prioritiza-tion based on agentsrsquo types so our result can beviewed as a generalization to allow heterogeneousagent types

42 Comparing Cardinal and Ordinal MechanismsIntuitively in a cardinal setting agents of the sametype can trade probabilities for various services at dif-ferent ratios hence expressing their preferences fornot only which services they value but also how much

15 A similar theorem appears in Ashlagi and Shi (2014) but thesetting is slightly different here and we give a different proof

they value each In an ordinal setting agents of thesame type can also trade probabilities but they musttrade services one-for-one hence they can expressonly preference rankings The value of a cardinalmechanism over an ordinal mechanism lies in its abil-ity to differentiate agents with extreme preferenceintensities Thus if agentsrsquo preferences for various ser-vices are of similar intensities then we would expectordinal mechanisms to perform well compared withcardinal ones If preferences however exhibit extremedifferences in intensities then one would expect car-dinal mechanisms to outperform ordinal ones

In Appendix C3 we give an example showing thatextreme differences in preference intensities may leadto an arbitrarily large ratio between the optimal socialwelfare achievable from a valid cardinal mechanismand that achievable from a valid ordinal mechanism16

5 Empirical Application PublicSchool Assignment

Now we demonstrate how our techniques can beapplied to a real-world problem and yield empiri-cally relevant results The problem is based on the2012ndash2013 Boston school assignment reform whichconsidered a list of potential plans and used simula-tion to evaluate them on a portfolio of metrics In thissection we ask the reverse question Given the objec-tive and constraints what is the optimal plan Moreprecisely we seek a plan that uses the same trans-portation budget as the baseline plan chosen by thecity but is optimized for efficiency and equity as mea-sured by utilitarian welfare and max-min welfare Wewill also improve predictability (chances of getting atop choice) in the process

Although we use real data the problem presentedhere has been simplified for conceptual clarity17 Werecognize that to produce implementable recommen-dations the precise objective and constraints mustbe scrutinized and debated by all stakeholders andconstituents which has not yet taken place Hencethe purpose of this section is not to provide concretepolicy recommendations for Boston but to provide aproof of concept of our methodology applied to a real-world setting

We first give a finite-market formulation of theproblem This is an asymmetric multidimensionalmechanism design problem with complex objectiveand constraints for which an exact optimum remains

16 Independently Pycia (2014) provides another example17 In the actual problem faced by the city committee one needsto consider continuing students specialized programs for EnglishLanguage Learners and disabled students and special preferencefor students who have older siblings already attending a schoolIn this paper we ignore these complications However all of themcould be accommodated in principle

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elusive As a baseline we describe the actual planadopted by BPS which can be seen as an intuitiveheuristic solution to the original problem To applythe techniques in this paper we define a large-marketapproximation of the finite-market formulation andsolve for the large-market optimum We then definea finite-market analog to this large-market solutionwhich is a feasible solution to the finite-market for-mulation and is asymptotically optimal in the sensethat it becomes the large-market optimum as thefinite-market formulation is scaled up We comparethis ldquoasymptotically optimalrdquo solution to the baselinequantify the improvements and discuss insights

51 Finite-Market Formulation

511 Student Population and School CapacitiesApproximately 4000 students each year apply to BPSfor kindergarten 2 (K2) the main entry grade for ele-mentary school The social planner (the city in thiscase) is charged with designing an assignment sys-tem for K2 that is efficient equitable and respectful ofcertain capacity budget and institutional constraintsThe social planner partitions Boston into 14 neigh-borhoods Based on historical data the social plannermodels the number of K2 applicants from each neigh-borhood as the product of two normally distributedrandom variables The first represents the overallnumber of applications and has mean 4294 and stan-dard deviation 115 reflecting the historical distribu-tion in years 2010ndash2013 (four years of data)18 The sec-ond represents the proportion of applicants from eachneighborhood The mean and standard deviation foreach neighborhood is estimated using historical dataand is shown in Table B1 in Appendix B

Each of the 14 neighborhoods is broken downfurther into geocodes which partition the city into868 small contiguous blocks Since home location isthe only allowable way to differentiate students weuse geocodes as agent types As an approximation weuse each geocodersquos centroid as the reference locationfor all students in that geocode Given the numberof students from each neighborhood we assume thatthese students are distributed among the geocodes ofthat neighborhood according a multinomial distribu-tion with probabilities matching the historic averagesfrom years 2010ndash2013

Each student is to be assigned to one of 77 schoolsFor each school s there is a capacity limit ms which isthe number of seats available for K2 students More-over for a certain set of schools Sc sube S which we callcapacity schools there is additional capacity availablefor students who live in the schoolrsquos catchment region

18 These statistics are for the first round of BPS applications onlyThis is the main round and represents over 80 of all applicants

We assume that the catchment region of a capacityschool s isin Sc is exactly the geocode for which s is theclosest capacity school For capacity school s the limitms applies only to students outside of its catchmentregion and we assume that it can accept an unlimitednumber of students inside its catchment region Thisguarantees that even if no capacity is available else-where every student can at least be assigned to theclosest capacity school19 There are 19 capacity schoolsdistributed across the city

The distribution of students across geocodes andthe capacities of schools are plotted in Figure B1panel (a) The locations of the capacity schools arealso shown The capacities are the actual numbersfrom 2013

512 Utility Prior for Students Using historicalchoice data the social planner estimates the followingutility prior for student i in geocode t and school s

uis = s minus Distancets +Walkts +is1

where Distancets and Walkts are from the data andthe coefficients s and are to be estimated fromhistorical choices using maximum likelihood

The coefficient s represents ldquoschool qualityrdquo whichencapsulates all common propensities that familiesobserve before choosing a school (eg facilities testscores teacher quality school environment) The vari-able Distancets is the distance from geocode t toschool s in miles estimated using walking distanceaccording to Google Maps The model is normalizedso that the coefficient of Distancets is minus1 which allowsus to measure utility in the unit of miles so oneadditional unit of utility can be interpreted as theequivalent of moving a school one mile closer Thevariable Walkts is an indicator for whether geocode tis within the walk zone of school s which is an approx-imately one-mile radius around The variable repre-sents additional utility for walk-zone schools as theseschools are in the immediate neighborhood and stu-dents can avoid busing is represents unobservableidiosyncratic preferences assumed to be independentand identically distributed (iid) standard Gumbeldistributed20 and represents the strength of theidiosyncratic preference

19 This is only approximately true in reality Although BPS has com-mitted to expanding the capacity schools as needed by adding newteachers and modular classrooms in reality there are hard spaceconstraints and BPS uses a more complicated ad hoc system toguarantee that every student is assigned This is called ldquoadministra-tive assignmentrdquo which is based on distance and many other fac-tors In BPS literature capacity schools were later renamed ldquooptionschoolsrdquo20 The Gumbel distribution is chosen because it makes the modeleasy to estimate via maximum likelihood as the likelihood functionhas a closed-form expression

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Table 1 Parameters of the Random Utility Model Estimated Using2013 Choice Data Using Grade K1 and K2 NoncontinuingRegular Education Students

Parameter Value Interpretation

s 0ndash629 Quality of schools For a school of atildeq additionalquality holding fixed other components a studentwould be willing to travel atildeq miles farther Thesevalues are graphically displayed in Figure B1panel (b) We normalize the smallest value to be 0

086 Additional utility for going to a school within the walkzone

188 Standard deviation of idiosyncratic preference

Notes We use K1 data in fitting the utility model as well since families facesimilar choices in the two grades and more data allow greater precisionThe values can be interpreted in units of miles (how many miles a student iswilling to travel for one unit of this variable)

We plot the school qualities in Figure B1 panel (b)and tabulate the other coefficients in Table 1 Al-though more sophisticated demand models are possi-ble in this exercise we treat the above as the ldquotruerdquoutility prior

513 Objective and Constraints The social plan-nerrsquos objective W is to maximize a linear combinationof the average utility and the utility of the worst-offneighborhood (we also call these terms the utilitarianwelfare and the max-min welfare respectively)

W = sum

t

(

wtvt + 41 minus5mintprime

vtprime

)

0

Here vt is the expected utility of a student fromgeocode t wt is the proportion of students who livein geocode t (taking the expected number of stu-dents from each geocode and normalizing so thatthe weights sum to 1) and is a parameter spec-ifying the desired trade-off between efficiency andequity Note that = 1 represents maximizing theaverage expected utility = 0 represents maximizingthe expected utility for the worst-off geocode and =

005 represents an equal weighting of the twoIn addition to capacity constraints the social plan-

ner faces a busing constraint Since busing is neededonly for students outside the one-mile walk zonethe amount of busing needed for a student fromgeocode t to school s is

Bts =

Distancets if geocode t is not in school srsquoswalk zone1

0 otherwise

Suppose that the social planner budgets C miles ofbusing per student in expectation then the busingconstraint21 is

sum

t

wtBtspts leC1

21 In reality busing cost is much more complicated involving rout-ing the number of buses used the kinds of buses used and legal

where pts is the probability that a random student ingeocode t is assigned to school s22

The institutional constraint is that the social plan-ner must design an ordinal mechanism that is incen-tive compatible is ldquoex post Pareto efficientrdquo within ageocode23 and requires only eliciting preference rank-ings rather than preference intensities The reason weconsider only ordinal mechanisms is that the systemhas been ordinal from 1988 to 2012 so families areused to gathering and submitting preference rank-ings The reason we require incentive compatibilityis that in 2006 a nonincentive-compatible mechanismwas rejected by the Boston School Committee in favorof an incentive-compatible one and since then BPShas committed to having a mechanism that allowsfamilies to submit truthful preferences without con-cern that it might negatively affect their chances24

The reason we require ex post Pareto efficiency is tomitigate public discontent as students in the samegeocodes are likely to compare their assignments withone another

These objectives and constraints along with theabove assumptions on the student population theschool capacities and the utility prior define a con-crete allocation problem We call this the finite-marketformulation

52 Baseline Actual ImplementationOne feasible solution to the finite-market formulationis the actual plan adopted by BPS after the reformwhich is called the Home Based plan It is based on aproposal by Shi (2013) although there are significantdeviations An input to this plan is a classification ofBPS schools into four tiers according to standardizedtest scores with Tier I being the best and Tier IV beingthe worst Each studentrsquos choice menu consists of anyschool within one mile of his or her residence plus acertain number of the closest schools of various typesas well as some idiosyncratic additions For details ofthe Home Based plan see Appendix A In this paperwe call this the baseline

requirements for more expensive door-to-door busing for certainspecial education students We leave finer modeling of transporta-tion costs in Boston for future work22 Note that this is a ldquosoftrdquo budget constraint in that it needs to besatisfied only in expectation We use this rather than a hard budgetconstraint because typically in school board operations the initialbudget is only a projection but may be revised later if needed23 This means that students in the same geocode should not beable to trade assignments with one another and all improve in util-ity Note the difference from the definition of Pareto efficiency inour large-market formulation in sect3 which considers the trading ofprobabilities not just assignments As the market is scaled up thedistinction between these definitions disappears24 For more details of that reform see Abdulkadiroglu et al (2006)

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The priority structure is as follows25 One of the14 neighborhoods is East Boston In the assignmentplan East Boston students get priority for East Bostonschools and non-East Boston students get priority fornon-East Boston schools26 We encode this using aux-iliary variable hts which is an indicator for whethergeocode t and school s are both in East Boston or bothoutside of East Boston

Having defined the menus and priorities the plancomputes the assignment by the algorithm presentedbelow

521 Deferred Acceptance Algorithm withMenus and Priorities Given menus for each studentand priority hts for geocode (type) t to school s definea studentrsquos score to school s as is = ri minus hts where riis a Uniform60117 random variable independentlydrawn across the students Compute the assignmentas follows

Step 1 An arbitrary student i applies to her topchoice s within her menu

Step 2 School s tentatively accepts the studentStep 3 If this acceptance causes the schoolrsquos ca-

pacity to be exceeded then the school finds the ten-tatively accepted student with the highest (worst)score and bumps that student out This school is thenremoved from that studentrsquos choice ranking and thestudent applies to the next preferred choice within thestudentrsquos menu

Step 4 Iterate Steps 1ndash3 until all unassigned stu-dents have applied to all their choices27

It is well known that this algorithm does notdepend on the order of studentsrsquo application in Step 1and that the induced mechanism is strategy-proofwhich means that it is a dominant strategy for all stu-dents to report their truthful preference rankings28

We simulated the baseline plan 10000 times accord-ing to the assumptions described in sect51 and wetabulated the planrsquos transportation burden averageexpected utility expected utility of the worst-offgeocode and predictability in Table 3 under the col-umn ldquoBaselinerdquo

53 Defining the Large-Market ApproximationTo apply our methodology we first define a large-market approximation to the finite-market formulation

25 The actual plan also contains priorities for continuing studentsstudents with siblings in a school and students on a wait list fromprevious rounds Since we do not model these complexities ourpriority structure is simpler26 The reason is that East Boston and the rest of Boston are separatedby water and connected only by a few bridges and tunnels socommuting may be inconvenient27 Note that if all students include the closest capacity school intheir rankings then in the end all students will be assigned28 See Roth and Sotomayor (1990) and Abdulkadiroglu and Soumlnmez(2003)

We do this by replacing each student with a contin-uum of infinitesimal students of mass 1 Instead ofa stochastic mass of students from each geocode weapproximate the scenario with a deterministic massnt of students setting nt to be the expected value

For the transportation budget we use 006 mileswhich is just under the 0063 used in the plan chosenby the city the baseline (see Table 3)

This yields exactly the environment in sect4 sincethe capacity constraints and the busing constraint canbe incorporated into the objective function by set-ting regions of constraint violation to negative infin-ity Moreover ex post Pareto efficiency is equivalentto ordinal efficiency in the large-market setting29

54 Computing the Large-Market OptimumIn this section we show how to compute the large-market optimum in the ordinal case when the utilitypriors take a multinomial-logit form Our techniquecan be applied more generally but we maintain theschool choice language for ease of exposition

Definition 4 Utility prior Ft is multinomial logit ifthe utilities can be written as

uis = uts + btis1

where bt gt 0 and uts are given parameters and isvalues are iid standard Gumbel distributed

This structure arises from multinomial logit dis-crete choice models which in practice are widely usedbecause of their tractability Our utility prior in sect512takes this form

The lottery-plus-cutoff characterization in sect41 im-plies that we can optimize on the T S cutoffs How-ever if one were to formulate the optimization interms of the cutoffs the resulting optimization isnonlinear Moreover the expected utilities have dis-continuous first derivatives in the cutoffs at pointswhere the cutoffs cross It turns out that it is sim-pler to work with the equivalent characterizationof randomized-menus-with-nested-menus from sect41This offers menu M sube S to an agent of geocode twith probability zt4M5 and agents pick their mostpreferred school from their menus For each type themenus that are offered with positive probability areall nested within one another (If M1 and M2 are bothoffered to agents of geocode t with positive probabil-ity then either M1 subeM2 or M2 subeM1)

For a menu of schools M sube S we abuse nota-tion slightly and let vt4M5 denote the expected utilityof the best school in this menu for a student fromgeocode t

vt4M5=1nt

int

UmaxsisinM

us dFt4u50

29 For works that study this equivalence see Che and Kojima (2011)and Liu and Pycia (2013)

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Let pt4s1M5 denote the probability that a studentfrom geocode t would choose school s to be the mostpreferred in menu M sube S

pt4s1M5=

s isin arg maxsprimeisinM

usprime

∣

∣

∣

∣

u sim Ft

0

Let zt4M5 denote the probability a student fromgeocode t is shown menu M sube S Let Ts denotethe set of geocodes for which the capacity limit forschool s applies30 If Ft is multinomial logit then wehave vt4M5 = bt log4

sum

sisinM exp4utsbt55 and pt4s1M5 =

exp4utsbt5sum

sprimeisinM exp4utsprimebt5By Theorem 2 to have ordinal efficiency within

type the menus must be nested However it is dif-ficult to formulate this as a constraint We will fornow consider the following relaxed linear program(LP) which ignores nestedness It will turn out thatthe nested structure will automatically be satisfied atoptimality

(LargeMarketLP)

max W = sum

t1M

wtvt4M5zt4M5+ 41 minus5y

st yminussum

M

vt4M5zt4M5le 0 forall t isin T 1

sum

M

zt4M5= 1 forall t isin T 1

sum

tisinTs1M

ntpt4s1M5zt4M5lems forall s isin S1

sum

s1 t1M

ntpt4s1M5Btszt4M5leC1

zt4M5ge 0 forall t isin T 1 M sube S0

There are 2S minus 1 possible menus M sube S the numberof variables of this LP is exponential in S Howeverbecause of the multinomial-logit structure the dualcan be solved efficiently and the menus given positiveprobabilities will be nested This implies an efficientalgorithm to solve for the optimal cutoffs

Theorem 3 Suppose that utility distributions Ftrsquos areall multinomial logit and gt 0 and weights wt gt 0 forall t then an optimal solution to the exponentially sized(LargeMarketLP) can be found in time polynomial inT and S Moreover at any optimum if zt4M15 gt 0 andzt4M25 gt 0 then either M1 subeM2 or M2 subeM1

The full proof is in Appendix C but we give anoverview here In the dual there are shadow prices forthe budget and capacity constraints which togetherdefine an ldquoallocation costrdquo from geocode t to school sThe multinomial logit structure implies that if we put

30 For capacity schools this is all geocodes not in the catchmentregion for other schools this is all geocodes

Table 2 Performance in the Large-Market Formulation of the OptimalPlans Under Various Choices of

= 1 = 005 = 0

Utilitarian welfare 7078 7066 7039Max-min welfare 2052 7039 7039

a school s in a menu we would always include allthe schools with a lower allocation cost as well Giventhe shadow prices one can find the optimal menu ofeach geocode simply by deciding how many schools toinclude The larger the menu the greater the expectedutility but the greater the expected allocation costThe importance we put on the utility of a geocode tdepends on its weight wt in the utilitarian welfareand its dual variable in the equity constraint whichis higher for the worst-off geocodes Since the menusare easy to compute given the prices we only haveto solve a convex optimization problem involving theprices which can be done efficiently Moreover sinceall the possible optimal menus for each type are nestedwithin one another by the above structure the menusthat are given positive probability in the optimal pri-mal solution will be nested

From the optimal solution to (LargeMarketLP) weget the optimal cutoffs for each school s to eachgeocode t2 ats =

sum

M3s zt4M5We evaluate the optimal plan in the large-market

formulation using three settings of in the objective = 1 (utilitarian welfare) = 0 (max-min welfare)and = 005 (equal weighting of the two) We tabulatethe results in Table 2 As can be seen setting = 005yields near optimal utilitarian welfare and max-minwelfare so for the remainder of this paper we use= 005

55 Converting the Optimal Large-MarketMechanism to a Feasible Finite-MarketMechanism

To convert the optimal large-market mechanism to afeasible finite-market mechanism we simply use thedeferred acceptance (DA) algorithm in sect521 and usethe optimal large-market cutoffs ats estimated usingprevious yearsrsquo data to guide the menus and priori-ties for students of geocode t define their menu to beschools for which the cutoff ats is positive and theirpriority for school s to be simply hts = ats We thenassign using these menus and priorities as in sect521We call the resulting mechanism the DA analog to theoptimal large-market mechanism

The DA analog is ldquoasymptotically optimalrdquo in thefollowing sense in the limit in which the studentsand school capacities are duplicated with many inde-pendent copies the finite-market environment con-verges to the large-market approximation and theDA analog converges to the large-market optimum

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Table 3 Evaluating a Variety of Plans in the Finite-Market Formulation Using 10000 Independent Simulations

Minimum Baseline ApproximateOpt1 ApproximateOpt2

Miles of busingstudent 0035 0064 0071 0063Average expected utility 6031 6095 7062 7049Expected utility of the worst-off geocode 2086 4053 7005 7002 getting top choice 0066 0064 0080 0079 getting top-three choice 0088 0085 0094 0093

This is because the independent copies wash awaythe stochasticity in the number of students fromeach geocode Moreover in this limit a student i isassigned to school s in the DA analog if and only ifher score is negative which implies the same assign-ment probabilities as in the large-market optimum

Observe that the DA analog is incentive compatibleand ex post Pareto efficient within each geocode It isincentive compatible because deferred acceptance isstrategy-proof for the students when the menus andscores are exogenous (in our case they are calculatedfrom previous yearsrsquo preference submissions) Ex postPareto efficiency within geocode follows from priori-ties ats being the same for students of each geocodeand from our using for each student the same randomnumber ri at all schools31

56 Numerical ResultsLet ApproximateOpt1 denote the DA analog to theoptimal large-market mechanism with = 005 anda busing budget of 0060 miles We simulate 10000times and compare its performance to the baseline inTable 3 It turns out that this plan evaluated in thefinite-market formulation uses 0071 miles of busingwhich exceeds the 0063 miles of the baseline Thus welet ApproximateOpt2 denote the DA analog with =

005 and a busing budget of 0050 In the finite-marketsimulations ApproximateOpt2 stays within the busingbudget of the baseline and significantly improves it interms of the average utility the utility of the worst-off geocode and the probability of students gettinginto their top or top-three choices To give a frameof reference for the magnitude of the improvementwe also evaluate what we call the most naiumlve planwhich has no priorities and includes in the menufor each geocode only the capacity school and schools

31 To see this consider all students of a given geocode and all theschools they are assigned to (counting multiplicity of seats) fromdeferred acceptance Consider the student with the best (smallest)random number ri out of these Of these schools this student mustbe assigned to her most preferred (This follows from the studenthaving a lower score than all other students from this geocode to allschools) Hence this student would not take part in any improve-ment cycle with other students from this geocode Considering thepartial market without this student and the assigned seat and pro-ceeding by induction we rule out all improvement cycles with stu-dents from this geocode

with zero transportation costs (schools in the walkzone)

As shown in Table 3 whereas the baseline uses0029 miles of additional busing per student overthe most naiumlve it improves the average utility by0064 miles and the utility of the worst-off geocodeby 1067 miles However ApproximateOpt2 uses fewermiles of transportation while improving the averageutility over the most naiumlve by 1018 miles (almost dou-bling the gains) and improving the expected utilityof the worst-off geocode by 4016 miles (more thandoubling the gains) It also significantly improves stu-dentsrsquo chances of getting into their top or top threechoices

To gain further insight we compute for each schoolits ldquoavailability areardquo in each plan This is the totalarea of the geocodes for which this school shows upin the menu We then divide schools into quartilesby their quality s with Q1 being the best and Q4being the worst We compare the average availabilityareas for different quality quartiles in the baseline andApproximateOpt2 in Figure 3 As shown in the tableApproximateOpt2 promotes lower-quality schools byoffering them to larger areas The intuition is thatthe higher-quality schools already have high demandfrom nearby areas so it is more efficient in terms oftransportation to restrict access to them to the nearby

Figure 3 (Color online) Comparison of Availability Areas for Schoolsof Various Quality in the Baseline and ApproximateOpt2

Q1

Q2

Q3

Q4

0 10 20

BaselineApproximateOpt2

Average availability area (sq miles)

Qua

rtile

s by

infe

rred

qua

lity

(Q1

is b

est)

30 40 50

Note Here Q1 is the best quartile in quality Q4 is the worst

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areas However to compensate the students who donot live near such schools the plan offers them manylower-quality schools in hopes that the students willhave high idiosyncratic personal affinity for themInterestingly the baseline mimics the same behaviorfor Q1 Q2 and Q3 schools but not for Q4 schoolswhich makes sense in retrospect because some of theQ4 schools may be at risk of closure

6 DiscussionThis paper studies the allocation of goods and ser-vices to agents without monetary transfers The socialplanner has priors over agentsrsquo utilities and is inter-ested in maximizing a public objective The approachin this paper sacrifices exact analysis by taking a con-tinuum approximation to gain the analytical tractabil-ity needed to handle large-scale applications withcomplex objectives and many types of agents and ser-vices In some sense the thrust of this paper is totake mechanism design further into the engineeringrealm focusing on tractable and useful approxima-tions rather than complex exact analysis

In the large-market environment we character-ize incentive-compatible mechanisms that are Paretooptimal within each type and we show how to com-pute the optimal ordinal mechanism when the utilityprior follows a multinomial-logit structure One openproblem is the efficient computation of the optimalordinal mechanism with other utility priors as wellas the efficient computation of the optimal cardinalmechanism32

In our empirical exercise we use past demand pat-terns to inform the city on how to allocate the limitedamount of busing between various neighborhoodsto maintain efficient and equitable access to schools(Limits on busing are needed because the costs areborne by tax payers) If one were to implement thisin practice the optimization of menus and priori-ties should not be done more often than once every5ndash10 years in order to maintain the predictability ofchoice options for families

One difficulty in implementing optimal mechanismdesign without money is in estimating utility dis-tributions With transfers estimation becomes sim-pler because the social planner may infer willingnessto pay from past transactional data However with-out money it is harder to infer preferences espe-cially preference intensities The demand modelingused in our empirical application assumes a particularfunctional form but the underlying utilities are not

32 Our current techniques reduce this to a polynomial-sized nonlin-ear program and we have not found any interesting distributionsfor which the structure significantly simplifies

observable so one may question the modelrsquos valid-ity A fundamental question is whether behavior canindeed be captured with such models33 Neverthelessif a utility model can be estimated the methods usedin this paper can be used to compute the optimalmechanism

Supplemental MaterialSupplemental material to this paper is available at httpdxdoiorg101287mnsc20152162

AcknowledgmentsThe authors thank Itay Fainmesser Steve Graves and OumlzalpOumlzer for helpful discussions I Ashlagi acknowledges theresearch support of the National Science Foundation [GrantSES-1254768]

Appendix A Details of the Home Based PlanIn the Home Based plan implemented in 2014 a studentrsquoschoice menu is the union of the following sets

bull any school within one mile straight-line distancebull the closest two Tier I schoolsbull the closest four Tier I or II schoolsbull the closest six Tier I II or III schoolsbull the closest school with Advanced Work Class (AWC)34

bull the closest Early Learning Center (ELC)35

bull the three closest capacity schools36

bull the three citywide schools which are available to every-one in the city

Furthermore for students living in parts of RoxburyDorchester and Mission Hill their menu includes the Jack-sonMann K8 School in AllstonndashBrighton

Appendix B Additional Tables and FiguresTable B1 shows the forecasted proportion of studentsapplying from each neighborhood A big picture view ofthe distribution of supply and demand for schools and ofinferred school quality in Boston is given in Figure B1panel (a) and Figure B1 panel (b) respectively

33 One project that examines the validity of utility models in Bostonschool choice compared with an alternative model based on mar-keting or salience is Pathak and Shi (2015) in which the authorsuse various methods to predict how families choose schools afterthe 2012ndash2013 reform and evaluate the prediction accuracy ofutility-based models34 AWC is a full-time invited program that provides an acceleratedacademic curriculum35 ELCs are extended-day kindergartens36 Recall that capacity schools are those which BPS has committedto expanding capacity as needed to accommodate all students Inthe 2014 implementation of the Home Based plan for elementaryschools capacity schools are exactly the Tier IV schools

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Figure B1 (Color online) Diagrams Showing the (a) Distribution of Students and Capacities of Schools and (b) Estimates of s (Inferred Qualityof Schools)

(a) Supply and demand (b) School quality

Lyon k-8

Edison Baldwin

WinshipJacksonMann

Gardner

Lyon k-8

Edison Baldwin

Winship

JacksonMann

Gardner

Bradley

Kennedy Patrick

Guild

BradleyGuild

East boston EEC

AdamsEliot

QuincyBlackstone

Hurley Condon

HarvardKent Otis

MarioODonnellWarrenPrescott

TobinOrchard Gardens

Perkins PerryTynan

West ZoneHaleELC Dudley ClapJ F KennedyHigginsonLewis

RussellHaynesWinthropCurley

MendellTrotter

KingEverett

MatherHollandMission HillManning

Lyndon

Kilmer

Beethoven

Conley

MozartBates

Sumner

BTULee

UP Acad Dor

HendersonMurphy

Kenny

Taylor

Young Achievers

EllisonParksMattahunt

ChittickE Greenwood

Grew

Channing

Roosevelt

Dever

S GreenwoodHolmes

EllisHernandez

PhilbrickHaley

Eliot Adams

East Boston EEC

Otis

WarrenPrescott

HarvardKent Kennedy PatrickODonnell

MarioUmana

Roosevelt

Channing

Grew

Beethoven

Kilmer

LyndonMozart

Bates

Sumner

Young AchieversTaylor

MattahuntEllisonParksConley

E GreenwoodChittick

BTU

ManningMission Hill

Curley

Tobin

HurleyBlackstone

Condon

Quincy

Tynan

PerryPerkinsOrchard Gardens

ClapDudleyHale

West Zone ELCJ F KennedyHigginsonLewis

RussellMendellTrotter

EverettDever

MatherKingEllis

Hernandez

Holland

UP Acad Dor

Lee

S GreenwoodHolmesMurphy

Henderson

Kenny

WinthropHaynes

PhilbrickHaley

Notes In (a) each blue circle represents a geocode with its area proportional to the expected number of students from that geocode Each yellow circlerepresents a school with its area proportional to the number of K2 seats available The capacity schools are shaded The distribution of students is based onthe 2010ndash2013 average The capacities are based on data from 2013 In (b) the size of the circle is proportional to the estimated s based on the 2013 datawith higher-quality schools having larger circles

Table B1 Means and Standard Deviations of the Proportion of K2Applicants from Each Neighborhood Estimated UsingFour Years of Historical Data

Neighborhood Mean Standard deviation

AllstonndashBrighton 000477 000018Charlestown 000324 000024Downtown 000318 000039East Boston 001335 000076Hyde Park 000588 000022Jamaica Plain 000570 000023Mattapan 000759 000025North Dorchester 000522 000047Roslindale 000771 000048Roxbury 001493 000096South Boston 000351 000014South Dorchester 001379 000065South End 000475 000022West Roxbury 000638 000040

Appendix C Omitted Proofs

C1 Characterization for Cardinal Mechanisms

Proof of Theorem 1 The proof uses a series of lemmasFor clarity of exposition we first show the main proof andprove the lemmas later

Lemma 1 A cardinal allocation rule is incentive compatibleif and only if there exists a closed convex set X sube atilde such thatx4u5 isin arg maxyisinX8u middoty9 forallu isinU We call X the closed convex setthat corresponds to incentive-compatible allocation rule x This isillustrated in Figure C1

We proceed to prove the theorem by induction on S ForS=1 there is nothing to prove as atilde is one point Suppose wehave proven this theorem for all smaller S Let X be the con-vex set that corresponds to allocation rule x Suppose X does

Figure C1 An Incentive-Compatible Allocation Rule with S = 3

(010) (001)

(100)

u

X

x(u)

Note X is an arbitrary closed convex subset of the feasibility simplex atildey = x4u5 is a maximizer of the linear objective u middot y with y isin X

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not intersect the relative interior of atilde int4atilde5=8yisinS2 ygt01sum

sys =19 then some component of x must be restricted tozero so we can set the price for that service to infinity ignorethat service and arrive at a scenario with a smaller numberof services for which the theorem is true by induction Thusit suffices to consider the case Xcapint4atilde5 6=

Let H4u15 denote the 4S minus 15-dimensional hyperplane8y isinS2 u middoty = 9 Let Hminus4u15 denote the half-space 8y2 u middot

y le 9 and let H+4u15 denote 8y2 u middot y ge 9 Let aff4atilde5denote the affine hull atilde aff4atilde5 = 8y isin S2

sum

s ys = 19 Notethat X is a convex subset of aff4atilde5 The following lemmaallows us to express tangents of X in aff4atilde5 in terms of aprice vector a isin 4015S

Lemma 2 If X sube atilde any tangent hyperplane of X in aff4atilde5can be represented as H4a115capaff4atilde5 for some a isin 4015S witha pointing outward from X and not colinear with 1 (a middot y le 1forally isinX and a 6= 1 for any isin)

Recall that D = 8u isin U2 u middot 1 = 09 is the space of relativeutilities For any set A sube D (subset of the relative utilities)let U4A5 = 8u isin U2 ProjD4u5 isin A9 This is the set of utilitiesfor which the projection onto D is in A The average alloca-tion of agents with relative preference in A is

x4U4A55=

int

U4A5x4u5 dF 4u50

Since CEEI without infinite prices is the same as hav-ing X = Hminus4a115 cap atilde it suffices to show that the convexset X has only one tangent in int4atilde5 Intuitively if it has twodifferent tangents H4a115 and H4aprime115 with nonzero andunequal unit projections onto D then we can find a unitvector d isin D st d middot a gt 0 gt d middot aprime Since a and aprime are tangentnormals we can perturb x4u5 in direction d for u near aand perturb x4u5 in direction minusd for u near aprime thus Paretoimproving x4 middot 5 but keeping average x4U5 fixed This is illus-trated in Figure C2 However defining a feasible move withpositive measure in all cases is nontrivial as prior F andclosed convex set X are general To do this we prove thefollowing lemma

Lemma 3 Suppose that H4a0115 is an outward-pointing sup-porting hyperplane of X that intersects X in the relative interior

Figure C2 Exchange Argument to Pareto Improve the Allocation Ruleby Expanding X Along Opposite Directions When ThereIs More Than One Supporting Hyperplane of XIntersecting int4atilde5

(010) (001)

(100)

d

ndashd

X

x(a)

x(aprime)

of the feasibility simplex int4atilde5 Then for any unit vector d isinDsuch that d middot a0 gt 0 there exists 0 gt 0 such that for all isin

40105 there exists allocation rule xprime that strictly dominates xwith xprime4U5= x4U5+ d

Using this we can rigorously carry out the above argu-ment suppose that H4a115 and H4aprime115 are two outward-pointing supporting hyperplanes of X that intersect Xin int4atilde5 with different nonzero unit projections onto U a 6= aprime Take any unit vector d isin int4H+4a105capHminus4aprime1055capD(Such d exists since a 6= aprime) Then d middot a gt 0 gt d middot aprime UsingLemma 3 there exists allocation rule xprime and xprimeprime which bothstrictly dominate x one of which has average allocationx4U5+ d and the other x4U5minus d Taking xprimeprimeprime =

12 4x

prime + xprimeprime5we have that xprimeprimeprime also strictly dominates x but xprimeprimeprime4U5= x4U5contradicting the Pareto efficiency of x4 middot 5 Therefore X hasonly one supporting hyperplane in atilde that intersects it in theinterior int4atilde5

Proof of Lemma 1 Suppose cardinal allocation rule x4u5is incentive compatible Let X be the convex closure of itsrange Since u middot x4u5 ge u middot x4uprime5 for every uprime isin U we haveu middot x4u5ge u middot y for every y isinX So x4u5 isin arg maxyisinX8u middot y9

Conversely if for some closed convex set X for anyu isinU x4u5 isin arg maxyisinX8u middot y9 then foralluprime isin U x4uprime5 isin X sou middot x4u5ge u middot x4uprime5 So x is incentive compatible

Proof of Lemma 2 Any tangent hyperplane Y of X inaff4atilde5 is a 4S minus 25-dimensional affine subset of the 4S minus 15-dimensional affine set aff4atilde5 Take an arbitrary point z isinatilde onthe same side of Y as X Consider the 4S minus 15-dimensionalhyperplane H passing through Y and 41 + 5z For some suf-ficiently small gt 0 by continuity H has all positive inter-cepts so H = 8y2 a middoty = 19 for some a gt 0 and by constructiona is not colinear with 1 Now a middot z = 141 + 5 lt 1 so a pointsoutward from X

In carrying out the exchange argument in Figure C2 weneed to guarantee that a positive measure of agents benefitsfrom the perturbation in the set X If the points x4a5 andx4aprime5 occur at a vertex meaning that a positive measure ofagents obtains each of these allocations then there is noth-ing additional to show The difficulty is if X is ldquosmoothrdquoat a and aprime so we need to do an exchange for agents withutilities in a neighborhood of a and aprime but in that case it isnot clear that we can move in directions d and minusd with-out going past the boundary of the feasibility simplex andit is not clear that we can obtain utility improvements forall these agents For example if the neighborhood is toolarge then d may no longer be a strictly improving direc-tion Lemma 3 is needed to guarantee that we can do thisexchange with a positive measure of agents

The proof of Lemma 3 uses the following rather technicallemma which guarantees that for any gt 0 and any u0 isinDwe can find a small open neighborhood of A of u0 suchthat the average allocation x4A5 is close to x4u05 Notethat for any u isinA8u09 x4u5 itself may not be close to x4u05because u0 could be normal to the convex set X along alinear portion of X in which case x4u5 would veer off fromx4u05 until it reaches the end of the linear portion But thislemma shows that by taking a convex combination of suchu isinA we can have x4A5 arbitrarily close to x4u05

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Figure C3 (Color online) Illustration of the Construction of Cone Aprime inWhich Any u isin Aprime Has x4u5 in the Open Half-Space H+

2 soAny Convex Combination of Such x4u5 Cannot Be in cl4X 5

u0

y0

u2

Aprime

R

cl(X )ndash

on X H2ndashsubOuter bound

H2ndash

H2+

H2

Lemma 4 Given any bounded closed convex set X sube nany nonempty open cone C sube n and any measurable functionx2 C rarrn such that x4u5 isin arg maxyisinX4u middoty5 Let F be an atom-less measure with F 4C5 gt 0 and such that for any nonempty opencone AsubeC we have F 4A5 gt 0 Now define

X =

x4A5=

int

A x4u5 dF 4u5F 4A5

2 F 4A5 gt 01 AsubeC

0

We have that for every u isinC arg maxyisinX8u middot y9sube cl4X5

Proof of Lemma 4 We first show that X is convex fol-lowing the proof of Lemma 33 in Zhou (1992) For anyAsubeC define the 4n+ 15-dimensional measure

m4A5=

(

int

Ax4u5 dF 4u51 F 4A5

)

0

By Lyapunovrsquos convexity theorem since F is atomlessthe range of this measure denoted by M is convex There-fore the cone generated by M cone4M5=8x2 xisinM1gt09is convex and so its intersection with the hyperplane 4middot115is convex (last component restricted to 1) This intersec-tion is nonempty since F 4C5 gt 0 Moreover this intersectionrestricted to the first n components is exactly X so X isconvex

The proof of the lemma proceeds by contradiction Sup-pose on the contrary that there exists u0 isin C and y0 isin

arg maxyisinX8u0 middot y9 but y0 y cl4X5 We will exhibit an opensubset AsubeC such that x4A5y X But F 4A5 gt 0 by the open-ness of A so x4A5 isin X by definition of X This is a contradic-tion The construction is given below Because of its geomet-ric nature we refer readers to Figure C3 for an illustration

Since cl4X5 is closed convex and bounded there exists astrictly separating hyperplane H4u1115 such that for some1 gt 0

u1 middot y0 ge 1 + 1 gt1 minus 1 ge u1 middot y forally isin cl4X50

Since every point of X is a convex combination of pointsin X and since X is closed and convex cl4X5 sube X so byconstruction

u0 middot y0 ge u0 middot y forally isin cl4X50

Let 0 = u0 middot y0 Since C is open by taking 4u2125 =

4u0105+ 4u1115 for some sufficiently small gt 0 we canensure that u2 isinC and by construction H2 =H4u2125 is astrictly separating hyperplane with


where 2 = 1Now let R = supyisinX8y minus y09 R is finite because X is

bounded Let Hminus2 be the closed half-space on the nonposi-

tive side of H2 If y isinX capHminus2 then y isin B4y01R5capHminus

2 whereB4y01R5 is the Euclidean closed ball of radius R centeredat y0 Define A to be the open normal cone to B4y01R5capHminus

2 namely

Aprime=

u isinn2u middot u2

uu2gt

radicR2u2

2 minus 22

Ru2

0

This construction is illustrated in Figure C3 Note thatu2 isin Aprime Moreover forallu isin Aprime x4u5 middot u ge y0 middot u gt y middot u for everyy isin X cap Hminus

2 Thus x4u5 y Hminus2 Let A = Aprime cap C then A is

open since it is the intersection of two open sets and A isnonempty since u2 isin A by construction By the assumptionon F F 4A5 gt 0 but x4A5 y Hminus

2 (since it is a convex combi-nation of points not in this half-space) so x4A5 y X sinceX subeHminus

2 This contradicts the definition of X

Proof of Lemma 3 The goal is to show that we can finda small neighborhood A sube D for which we can perturb theaverage allocation in direction d and yield a strict improve-ment for each u isinA

Let a0 = ProjD a0 the projection of a0 onto D We wishto construct our desired open neighborhood by taking aneighborhood A of a0 in D such that for every u isin U4A5(recall that U4A5 is the subset of U whose projection on Dis in A) the agent prefers the allocation y1 = x4U4A55+ drather than x4u5 Moreover the neighborhood has to be suf-ficiently small so that y1 remains feasible that is y1 isinatilde Todo this we make use of Lemma 4

Let y0 isinXcapH4a0115cap int4atilde5 (atilde is the feasibility simplex)Let be the distance from y0 to the boundary of atilde then gt 0 since y0 is in the interior of atilde Define = 3

4 Definer = 4a0 middot d46a055 Define the following cones

C1 =

a isinD2a

amiddot 4y0 minus x4a55 gtminusr

1

C2 =

a isinD2a

amiddot d gt

a0

2a0middot d = 3

r

0

The first represents relative utilities for which the allocationof y0 is ldquonot too badrdquo and the second represents relativeutilities that are ldquoroughlyrdquo in the direction of d We want toshow that C =C1 capC2 is a nonempty open cone so that wecan apply Lemma 4

To see that C = C1 cap C2 is a nonempty open cone notethat a0 isin C1 and a0 isin C2 so C is nonempty C1 and C2 arecones because the expressions that define them depend onlyon aa so C is a cone (Note also that a middot x4a5 = a middot x4a5for any gt 0 by incentive compatibility) The two conesare open because the left-hand side of the inequalities thatdefine them are continuous functions of a (Note that g4a5=

a middot x4a5 is a continuous function of a as this is the objective

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of the linear maximizer over the bounded convex set X)Therefore C is open

Now we apply Lemma 4 By continuity and full relativesupport of F F 4U4 middot 55 is an atomless measure on C such thatF 4U4C55 gt 0 and for every open cone A sube C F 4U4A55 gt 0Moreover X and x4 middot 5 satisfy the assumptions of Lemma 4so by the lemma existAsubeC F 4U4A55 gt 0 such that x4U4A55minusy0 le r Now define y1 = x4U4A55 + d then y1 isin atilde sincey1 minus y0 le + r le 7

8 Consider the alternative allocationrule

y4u5=

y1 if u isinU4A51

x4u5 otherwise

Then y strictly Pareto improves over x because foralla isinA

a middot y1 minus a middot x4a5

= a middot 4x4U4A55+ d5minus a middot x4a5

= a middot 4x4U4A55minus y05+ a middot 4y0 minus x4a55+ a middot d

ge minusra minus ra + 3ra

gt 00

Now let 0 = F 4U4A55 for any isin 40105 if we setxprime4u5 = 405y4u5 + 41 minus 05x4u5 Then xprime still strictlyPareto improves over x but xprime4U5 = x4U5 + d which iswhat we needed

C2 Characterization for Ordinal Mechanisms

Proof of Theorem 2 The proof is similar to that of The-orem 1 in that we first find an equivalent description ofincentive compatibility and then use an exchange argumentto derive the lottery-plus-cutoffs structure The differenceis that instead of a closed convex set as in the proof ofTheorem 1 we have the base polytope of a polymatroidThe exchange argument is also simpler because the space ofpermutations ccedil is discrete and every member has positiveprobability as a result full relative support As before wefirst apply a series of lemmas and prove them later

Lemma 5 An ordinal allocation rule x45 is incentive com-patible if and only if there exists monotone submodular set func-tion f 2 2S rarr 60117 such that for every permutation isin ccedil andfor every k (1 le k le S)

x4k545= f 484151 0 0 0 14k595minus f 484151 0 0 0 14kminus 15950

We call f the monotone submodular set function that correspondsto x

If X is the range of x then the above lemma says that xis incentive compatible if and only if X is the vertex set ofthe base polytope of polymatroid defined by f

sum

sisinM

xs le f 4M5 forallM sube S1 (C1)

sum

sisinS

xs = 11 (C2)

x ge 00 (C3)

The following lemma embodies the exchange argument

Lemma 6 Let f be the monotone submodular set functionthat corresponds to incentive compatible allocation rule x If x isordinal efficient then for any M11M2 sube S

f 4M1 cupM25= max8f 4M151 f 4M2590

Let as = f 48s95 An easy induction using the above lemmayields forallM sube S f 4M5 = maxsisinM as which together withthe expression in Lemma 5 implies that x is lottery-plus-cutoffs

Proof of Lemma 5 If x45 is an incentive-compatibleordinal allocation rule then for any M sube S define

f 4M5=

M sum

j=1

x4j5451 where 841514251 0 0 0 14M 59=M0

This is well defined because incentive compatibility re-quires each agentrsquos chances of getting a service in M con-ditional on ranking these first in some order (ranking allof M before all of SM) to be fixed regardless of therelative ranking between services in M and between ser-vices in SM If this were not the case then for some largeb gt 0 and small gt 0 consider an agent with utilities us =

4s isinM5b+ s where 4s isinM5 equals 1 if s isinM and 0 other-wise and s rsquos are distinct numbers to be defined later withs le If the agentrsquos chance of getting one of the servicesin M can be altered by changing relative order in M andthe relative order in SM while she ranks M before SM then the agent would for some 8s9rsquos gain b times a positivenumber and lose at most S so for sufficiently large bshe has incentive to misreport

We now show that f is submodular Suppose on the con-trary that f is not submodular then there exists M1 sube M2and s yM2 such that

f 4M1 cup 8s95minus f 4M15 lt f 4M2 cup 8s95minus f 4M250

However let us = 4s isin M1 cup 8s95b with some b gt 0 to bespecified later She prefers all of M1 cup 8s9 before any ofM2M1 Reporting this true ranking gives an expected util-ity of bf 4M1 cup 8s95 However if she instead ranked M1 thenM2M1 then s and she would get b4f 4M15+ f 4M2 cup 8s95minus

f 4M255 gt bf 4M1 cup8s95 So she has incentive to misreport Thiscontradicts incentive compatibility

Now the construction of f implies that f is monotoneand by the definition of f we have that forall isinccedil and 1 le k le

S x4k545 = f 484151 0 0 0 14k595 minus f 484151 0 0 0 14k minus 1595This proves the forward direction

Conversely if f is a monotone submodular set functionwe show that if we define x so that x4k545= f 484151 0 0 0 14k595minus f 484151 0 0 0 14kminus 1595 then x is incentive compat-ible Note that the range of x defined this way is simply thevertex set of the base polytope of the polymatroid definedby f (see Equations (C1)ndash(C3))

Now the agentrsquos utility u middot x is linear in x so using thefact that the greedy algorithm optimizes a linear objectiveover a polymatroid (and also the base polytope) we get thatfor any u if we relabel S so that

u1 ge u2 ge middot middot middot ge uS1

then an optimal point of the base polytope has x1 set tof 48195 x2 set to f 4811295 minus f 48195 and so on which is

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exactly our expression for the allocation rule x Thus x45 isin

arg max primeisinccedil u middot x4 prime5 and x is incentive compatible

Proof of Lemma 6 By monotonicity of f f 4M1 cupM25ge

max8f 4M151 f 4M259 What we need to show is that f 4M1 cup

M25 le max8f 4M151 f 4M259 By monotonicity it suffices toshow this for the case in which M1 capM2 =

Suppose that on the contrary f 4M1 cupM25 gt max8f 4M151f 4M259 M1 capM2 = Consider two preference rankings 1and 2 1 ranks services in M1 first followed by M2 fol-lowed by other services in arbitrary order and 2 ranksservices in M2 first followed by M1 followed by othersBy Lemma 5 since x is incentive compatible

sum

sisinM2x415=

f 4M1 cup M25 minus f 4M15 gt 0 andsum

sisinM1x425 = f 4M1 cup M25 minus

f 4M25 gt 0 Thus agents with preference ranking 1 cantrade probabilities with agents with preference ranking 2and mutually improve in the first-order stochastic domi-nance sense (Agents preferring M1 get additional probabil-ities for services in M1 in place of equal probabilities forM2 whereas agents preferring M2 get additional probabil-ities for M2 in place of M1) By full ordinal support thereexist positive measures of both kinds of agents which con-tradicts the ordinal efficiency of x

C3 Comparing Cardinal and Ordinal MechanismsWe show an example in which the optimal social welfarefrom a cardinal mechanism is arbitrarily many times largerthan the optimal social welfare from an ordinal mechanismThis example uses the intuition that the value of a cardinalmechanism lies mostly in its ability to distinguish betweenan agent who has an extremely large relative preference fora service over another who has only a weak preference

Let M and N be two positive real numbers with M ge

min431N 35 and N ge 1 Suppose that there are three ser-vices Service 1 has capacity 1N 2 whereas services 2 and 3have capacity 1 each Suppose there is only one type ofmass 1 and 1N of the agents have utilities 4M11105 andthe remaining agents have utilities 421110537 The objectiveis to maximize the social welfare An optimal cardinal mech-anism charges price vector 4p11105 where price p gt 2 dif-ferentiates the two types of agents Agents have a virtualbudget 1 The 1N of agents would purchase the bundle41N1011 minus 1N5 whereas the other agents will opt for4011105 Hence the social welfare is MN 2 + 41 minus 1N5 gtMN 2 However with an ordinal mechanism one cannotdistinguish between the two groups of agents and the besta lottery-plus-cutoff mechanism can do is to have cutoffs41N 211115 for the services and every agent gets allocation41N 211 minus 1N 2105 The social welfare is 41N54MN 2 + 1 minus

1N 25 + 41 minus 1N542N 2 + 1 minus 1N 25 le 3MN 3 So the ratiobetween the best cardinal and the best ordinal in this exam-ple is at least N3 which we can make arbitrarily large

C4 Computation Results

Proof of Theorem 3 Define dual variables for (Large-MarketLP) as follows let be the dual variable for the

37 Although this does not satisfy full relative support we can triv-ially modify it to satisfy by having mass of agents with utilities4u11u21u35 where the uj rsquos are standard normals drawn indepen-dently for each agent

cost constraint s be the capacity constraint of school s t

be the constraint of menu probabilities summing to 1 forgeocode t and t be the constraint enforcing the minimumconstraint for geocode t The dual is as follows

(Dual) min

C + m middotEuml+sum

tisinT

t

1

t ge 4wt + t5vt4M5minusnt

sum

s

pt4s1M544t isin Ts5s +Bts5

forall t isin T 1 M sube S1sum

tisinT

t ge 1 minus1

1Euml1Igrave1Iacute ge 00

Label the right-hand side of the first inequality asft41Euml1Iacute1M5 This can be interpreted as follows sup-pose that one unit of expected utility for the student fromgeocode t contributes wt +t ldquocreditsrdquo to the city whereasassigning the student to school s costs the city 4t isin Ts5s +

Bts credits then ft41Euml1Iacute1M5 is the expected number ofcredits a student from geocode t who is given menu M con-tributes to the city taking into account both her expectedutility and the negative externalities of her occupying a slotof a service Maximizing this over menus M is thus anldquooptimal-menurdquo subproblem

Definition 5 Given Euml Iacute the optimal menu subproblemis to find the solution set

arg maxMsubeS

ft41Euml1Iacute1M50

Denote the optimal objective value t41Euml1Iacute5 =

maxMsubeS ft41Euml1Iacute1M5

Lemma 7 The function t41Euml1Iacute5 is convex

Proof of Lemma 7 This follows from ft41Euml1Iacute1M5being linear in Euml and Iacute for fixed M So t is the upperenvelope of a family of linear functions and is thereforeconvex

Therefore the dual can be written as a convex pro-gram with T +S+1 nonnegative variables with objectiveC + m middot Euml+

sum

tisinT t41Euml1Iacute5 and a single linear constraintsum

tisinT t ge 1minus One difficulty is that the optimal menu sub-problem needs to optimize over exponentially many menusM sube S However when preferences are multinomial logitwe can efficiently solve the subproblem Moreover the opti-mum has the nested structure that we need

Lemma 8 Under multinomial-logit utility priors if wt +

t gt 0 then the number of optimal solutions for the optimal menusubproblem is at most S and can all be found in time S log SMoreover the optimal menus will all be nested within oneanother

Proof of Lemma 8 Recall that having multinomial-logitutilities means that uis = uts + btis where the uts rsquos areaverages for this school and geocode and the is rsquos areunobservable idiosyncratic preferences which are standardGumbel distributed Fix a geocode t Let hs = nt44t isin Ts5s

+ Bts544wt + t5bt5 zs = exp4utsbt5 Substituting in theformula for vt4M5 and pt4s1M5 from the multinomial-logit

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ved


structure and simplifying the optimal menu subproblem isequivalent to finding all solutions to

maxMsubeS

log(

sum

sisinM

zs

)

minus

sum

sisinM hszssum

sisinM zs0

Consider the continuous relaxation of this in which ys isa continuous variable constrained to be in 601 zs7 and thereare S such variables

maxysisin601 zs 7 forall s

log(

sum

sisinS

ys

)

minus

sum

sisinS hsyssum

sisinS ys0

Now if hs lt hsprime and ysprime gt 0 but ys lt zs then by decreas-ing ysprime by and by increasing ys by for small gt 0 wecan decrease

sum

s hsys while keepingsum

s ys the same so thiscannot occur at an optimum Relabel services so that

h1 le h2 le middot middot middot le hS0

We first consider the case in which the hs values are alldistinct In this case by the above an optimal solution ofthe continuous relaxation must be of the following form forsome 1 le k le S

ys = zs forall s lt k1 yk isin 601 zk71 ys = 0 forall s gt k0

We show that at an optimal solution it must be that yk isin

801 zk9 Suppose on the contrary that yk isin 401 zk5 Let d1 =sum

sltk zs d2 = hkd1 minussum

sltk hszs As a function of yk the objec-tive is g4yk5 = log4d1 + yk5+ d24d1 + yk5minus hk The first andsecond derivatives are gprime4yk5= 414d1 +yk5541 minusd24d1 +yk55and gprimeprime4yk5= 414d1 + yk5

2542d24d1 + yk5minus 15 respectivelySince yk is an interior optimum d24d1 + yk5 = 1 and

so the second derivative is gprimeprime4yk5 = 14d1 + yk52 gt 0 which

implies that yk is a strict local minimum which contradictsour assumption Therefore the objective is maximized whenyk isin 801 zk9

This implies that all optimal solutions are restricted tobe of the form Mk = 811 0 0 0 1 k9 (the services are sorted inincreasing order of hs) and so we only need to searchthrough 1 le k le S This can be done in S log S time asit is a linear search after sorting services in nondecreasingorder of hs This also implies that the number of optimalsolutions is at most S

Now if some of the 8hs9 are equal then we can col-lapse them into one service in the continuous relaxationand the above argument implies that an optimal menu Meither contains all of them or none of them Thus arbitrarilybreaking ties when sorting hs in nondecreasing order andsearching through the Mkrsquos for k isin 811 0 0 0 1 S9 still yields alloptimal solutions Moreover because the Mkrsquos are nested bydefinition all the optimal menus will be nested within oneanother

Since the subproblems are efficiently solvable we canefficiently solve the dual to arbitrary precision If Mt is thesolution set to the optimal menu subproblem for geocode tusing the optimal dual variables then an optimal primalfeasible solution can be recovered using complementary

slackness38 by finding a feasible solution to the polynomial-sized LPsum

MisinMt

vt4M5zt4M5gey foralltisinT with equality if tgt01

sum

MisinMt

zt4M5=1 foralltisinT 1

sum

tisinTs

sum

MisinMt

ntpt4s1M5zt4M5lems forallsisinS with equality if sgt01

sum

s1t

sum

MisinMt

ntpt4s1M5Btszt4M5leC with equality if gt01

zt4M5ge0 foralltisinT 1M isinMt 0

ReferencesAbdulkadiroglu A Soumlnmez T (2003) School choice A mechanism

design approach Amer Econom Rev 93(3)729ndash747Abdulkadiroglu A Soumlnmez T (2013) Matching markets Theory

and practice Acemoglu D Arellano M Dekel E eds Advancesin Economics and Econometrics 10th World Congress Volume 1Economic Theory (Cambridge University Press New York)3ndash47

Abdulkadiroglu A Che Y-K Yasuda Y (2015) Expanding ldquochoicerdquoin school choice Amer Econom J Microeconomics 7(1)1ndash42

Abdulkadiroglu A Pathak PA Roth AE (2009) Strategy-proofnessversus efficiency in matching with indifferences Redesign-ing the NYC high school match Amer Econom Rev 99(5)1954ndash1978

Abdulkadiroglu A Pathak PA Roth AE Soumlnmez T (2006) Changingthe Boston school choice mechanism Boston College WorkingPapers in Economics 639 Boston College Chestnut Hill MA

Ashlagi I Shi P (2014) Improving community cohesion in schoolchoice via correlated-lottery implementation Oper Res 62(6)1247ndash1264

Aumann RJ (1964) Markets with a continuum of traders Economet-rica 32(1)39ndash50

Azevedo E Leshno J (2015) A supply and demand framework fortwo-sided matching markets J Political Econom Forthcoming

Bogomolnaia A Moulin H (2001) A new solution to the randomassignment problem J Econom Theory 100(2)295ndash328

Budish E (2012) Matching versus mechanism design ACM SIGecomExchanges 11(2)4ndash15

Budish E Cantillon E (2012) The multi-unit assignment problemTheory and evidence from course allocation at Harvard AmerEconom Rev 102(5)2237ndash2271

Caro F Shirabe T Guignard M Weintraub A (2004) School redis-tricting Embedding GIS tools with integer programmingJ Oper Res Soc 55(8)836ndash849

Chakravarty S Kaplan TR (2013) Optimal allocation without trans-fer payments Games Econom Behav 77(1)1ndash20

Che Y Kojima F (2011) Asymptotic equivalence of probabilis-tic serial and random priority mechanisms Econometrica78(5)1625ndash1672

Clarke S Surkis J (1968) An operations research approach to racialdesegregation of school systems Socio-Econom Planning Sci1(3)259ndash272

Condorelli D (2012) What money canrsquot buy Efficient mechanismdesign with costly signals Games Econom Behav 75(2)613ndash624

38 Because the convex primal problem is induced by an LP it suf-fices to solve to a finite level of precision to guarantee that theright dual variables are positive when we stop the optimization Inpractice we may stop the optimization earlier and look for primalsolutions that approximately satisfy complementary slackness

Dow

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2015

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righ

ts r

eser

ved


Echenique F Yenmez MB (2015) How to control controlled schoolchoice Amer Econom Rev 105(8)2679ndash2694

Ehlers L Hafalir IE Yenmez MB Yildirim MA (2014) School choicewith controlled choice constraints Hard bounds versus softbounds J Econom Theory 153(September)648ndash683

Erdil A Ergin H (2008) Whatrsquos the matter with tie-breakingImproving efficiency in school choice Amer Econom Rev98(3)669ndash689

Gale D Shapley LS (1962) College admissions and the stability ofmarriage Amer Math Monthly 69(1)9ndash15

Hartline JD Roughgarden T (2008) Optimal mechanism design andmoney burning Proc 40th Annual ACM Sympos Theory Com-put 4STOC rsquo085 (ACM New York) 75ndash84

Hoppe HC Moldovanu B Sela A (2009) The theory of assorta-tive matching based on costly signals Rev Econom Stud 76(1)253ndash281

Hylland A Zeckhauser R (1979) The efficient allocation of individ-uals to positions J Political Econom 87(2)293ndash314

Kominers SD Soumlnmez T (2015) Matching with slot-specific priori-ties Theory Theoret Econom Forthcoming

Liu Q Pycia M (2013) Ordinal efficiency fairness and incentivesin large markets Working paper University of California LosAngeles Los Angeles

Maglaras C Zeevi A (2005) Pricing and design of differentiatedservices Approximate analysis and structural insights OperRes 53(2)242ndash262

Miralles A (2012) Cardinal Bayesian allocation mechanisms withouttransfers J Econom Theory 147(1)179ndash206

Miralles Asensio A Pycia M (2014) Prices and efficient assign-ments without transfers Working paper Boston UniversityBoston httppapersssrncomsol3paperscfmabstract_id=2505241

Myerson RB (1981) Optimal auction design Math Oper Res6(1)58ndash73

Pathak PA Sethuraman J (2011) Lotteries in student assignmentAn equivalence result Theor Econom 6(1)1ndash17

Pathak PA Shi P (2015) Demand modeling forecasting and coun-terfactuals part I Working paper Massachusetts Institute ofTechnology Cambridge httparxivorgabs14017359

Perry O Whitt W (2009) Responding to unexpected overloads inlarge-scale service systems Management Sci 55(8)1353ndash1367

Pycia M (2014) The cost of ordinality Working paper University ofCalifornia Los Angeles Los Angeles httppapersssrncomsol3paperscfmabstract_id=2460511

Roth AE Sotomayor M (1990) Two-Sided Matching A Study in Game-Theoretic Modeling and Analysis (Cambridge University PressCambridge UK)

Russell J Ebbert S (2011) The high price of school assignment BostonGlobe (June 12) httpwwwbostoncomnewseducationk_12articles20110612the_high_price_of_school_assignment

Schummer J Vohra R (2007) Mechanism design without moneyNisan N Roughgarden T Tardos E Vazirani VV eds Algo-rithmic Game Theory (Cambridge University Press CambridgeUK) 243ndash266

Shi P (2013) Closest types A simple non-zone-based frameworkfor school choice Memo Massachusetts Institute of Tech-nology Cambridge httpwwwmitedu~pengshipapersclosest-typespdf

Soumlnmez T Uumlnver MU (2010) Course bidding at business schoolsInternat Econom Rev 51(1)99ndash123

Sutcliffe C Board J Cheshire P (1984) Goal programming and allo-cating children to secondary schools in reading J Oper ResSoc 35(8)719ndash730

Thomson W Zhou L (1993) Consistent allocation rules in atomlesseconomies Econometrica 61(3)575ndash587

Tirole J (1988) The Theory of Industrial Organization (MIT Press Cam-bridge MA)

Zhou L (1992) Strictly fair allocations in large exchange economiesJ Econom Theory 57(1)160ndash175

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MANAGEMENT SCIENCEArticles in Advance pp 1ndash20ISSN 0025-1909 (print) ISSN 1526-5501 (online) httpdxdoiorg101287mnsc20152162

copy 2015 INFORMS

Optimal Allocation Without MoneyAn Engineering Approach

Itai Ashlagi Peng ShiMassachusetts Institute of Technology Cambridge Massachusetts 02139 iashlagimitedu pengshimitedu

We study the allocation of heterogeneous services to agents with incomplete information and withoutmonetary transfers Agents have private multidimensional utilities over services drawn from commonly

known priors The social plannerrsquos goal is to maximize a potentially complex public objective For tractabilitywe take an ldquoengineeringrdquo approach in which we solve a large-market approximation and convert the solutioninto a feasible finite-market mechanism that still yields good results We apply this framework to real data fromBoston to design a mechanism that assigns students to public schools in order to maximize a linear combinationof utilitarian and max-min welfare subject to capacity and transportation constraints We show how to opti-mally solve a large-market formulation with more than 868 types of students and 77 schools and we translatethe solution into a finite-market mechanism that significantly outperforms the baseline plan chosen by the cityin terms of efficiency equity and predictability

Data as supplemental material are available at httpdxdoiorg101287mnsc20152162

Keywords market design priors assignment school choice optimal design large-market approximationHistory Received May 22 2014 accepted November 30 2014 by Martin Lariviere operations management

Published online in Articles in Advance

1 IntroductionIn many settings goods or services are allocated toagents without monetary transfers Examples includethe allocation of seats in public schools spaces in col-lege dorms or courses and positions in medical res-idency programs Social planners have multifacetedconcerns such as social welfare equity and systemcosts

One example of such a problem was faced by thecity of Boston in the 2012ndash2013 school assignmentreform Prior to 2013 seats in Boston Public Schools(BPS) had been allocated as follows The city wasdivided into three zones and each family rankedschools within a menu of choices that consisted ofschools within the familyrsquos zone and schools withinone mile of the familyrsquos home A centralized algo-rithm then allocated seats based on familiesrsquo rankinglists priorities and lotteries These large menus ofschools led to unsustainably high busing costs repre-senting approximately 10 of the total school boardbudget (Russell and Ebbert 2011) In 2012 the cityused simulation to choose a new plan from a shortlist of proposals to decrease the choice menus whilemaintaining equity among neighborhoods Althoughthe proposals in the short list were ad hoc the tech-niques in this paper can be used to design the optimalallocation in a systematic way

When agentsrsquo preferences are publicly known theallocation is merely an optimization problem This

study addresses how to allocate services when agentsrsquopreferences are only privately known In our modelthere are multiple kinds of services and agentshave private multidimensional utilities over the ser-vices drawn from commonly known priors whichmay depend on agentsrsquo observable information Sincemechanism design with multidimensional valuationshas traditionally been difficult especially for generalobjectives we adopt an engineering approach wefirst find the optimal mechanism for a large-marketformulation then we convert it into a feasible finite-market mechanism and we evaluate it using data-driven simulation

In the large-market formulation there are finitelymany types of agents and a continuum of agents ofeach type Types in this paper represent the publicinformation of agents For example in school choicea type may represent students from a certain neigh-borhood of a certain race or of a certain socioeco-nomic status We require mechanisms to be incentivecompatible and Pareto optimal among agents of thesame type and we refer to such mechanisms as validmechanisms Whereas agents of the same type aretreated symmetrically agents of different types maybe differentiated The goal is to find a valid mech-anism that maximizes the social plannerrsquos objectivewhich can be fairly general

We first characterize all valid mechanisms Undermild assumptions on the utility priors we show that

1

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all

righ

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eser

ved









atilde=

p isinS2 p ge 01sum

s

ps = 1

0



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all

righ

ts r

eser

ved








u middot xt4uprime5

0



int

Uxprime

t4u5 dFt =int

Uxt4u5 dFt1


u middot xprime








vt =1nt

int




t

int










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ved



Budget line

(010) (001)

(100)

u

ax(u)















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eser

ved









u middot xt4prime5

0



int

ccedilxprime

t dFt =int

ccedilxt dFt1



k le Sksum

j=1

xprime

t4j545ge

ksum

j=1

xt4j5451







1

a1

a2

x3(1) = a3 ndash a1

a3

a4

2 3

0

12

34

1

1 3 2 2 1 3 4




x4k545=k

maxj=1

a4j5 minuskminus1

maxj=1

a4j50





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ved















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eser

ved















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all

righ

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eser

ved










W = sum

t

(


vtprime

)

0




Bts =


0 otherwise


sum

t

wtBtspts leC1








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eser

ved


















uis = uts + btis1





vt4M5=1nt

int

UmaxsisinM

us dFt4u50


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ved



pt4s1M5=


usprime

∣

∣

∣

∣

u sim Ft

0


sum


exp4utsbt5sum



(LargeMarketLP)

max W = sum

t1M


st yminussum

M


sum

M


sum

tisinTs1M


sum

s1 t1M








= 1 = 005 = 0




sum






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ved














Q1

Q2

Q3

Q4

0 10 20



Qua

rtile

s by

infe

rred

qua

lity

(Q1

is b

est)

30 40 50


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all

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eser

ved




Lyon k-8

Edison Baldwin

WinshipJacksonMann

Gardner

Lyon k-8

Edison Baldwin

Winship

JacksonMann

Gardner

Bradley

Kennedy Patrick

Guild

BradleyGuild

East boston EEC

AdamsEliot

QuincyBlackstone

Hurley Condon

HarvardKent Otis



Perkins PerryTynan



MendellTrotter

KingEverett


Lyndon

Kilmer

Beethoven

Conley

MozartBates

Sumner

BTULee

UP Acad Dor

HendersonMurphy

Kenny

Taylor

Young Achievers


ChittickE Greenwood

Grew

Channing

Roosevelt

Dever

S GreenwoodHolmes

EllisHernandez

PhilbrickHaley

Eliot Adams

East Boston EEC

Otis

WarrenPrescott


MarioUmana

Roosevelt

Channing

Grew

Beethoven

Kilmer

LyndonMozart

Bates

Sumner



E GreenwoodChittick

BTU

ManningMission Hill

Curley

Tobin

HurleyBlackstone

Condon

Quincy

Tynan


ClapDudleyHale



EverettDever

MatherKingEllis

Hernandez

Holland

UP Acad Dor

Lee


Henderson

Kenny

WinthropHaynes

PhilbrickHaley











(010) (001)

(100)

u

X

x(u)


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ded

from

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all

righ

ts r

eser

ved






sum




x4U4A55=

int

U4A5x4u5 dF 4u50




(010) (001)

(100)

d

ndashd

X

x(a)

x(aprime)




12 4x







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all

righ

ts r

eser

ved




u0

y0

u2

Aprime

R

cl(X )ndash


H2ndash

H2+

H2


X =

x4A5=

int

A x4u5 dF 4u5F 4A5


0



m4A5=

(

int

Ax4u5 dF 4u51 F 4A5

)

0















2 namely

Aprime=

u isinn2u middot u2

uu2gt

radicR2u2

2 minus 22

Ru2

0










C1 =

a isinD2a


1

C2 =

a isinD2a

amiddot d gt

a0

2a0middot d = 3

r

0




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all

righ

ts r

eser

ved





y4u5=

y1 if u isinU4A51

x4u5 otherwise






gt 00








sum

sisinM


sum

sisinS

xs = 11 (C2)

x ge 00 (C3)



f 4M1 cupM25= max8f 4M151 f 4M2590



f 4M5=

M sum

j=1

x4j5451 where 841514251 0 0 0 14M 59=M0













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se o

nly

all

righ

ts r

eser

ved








sum

sisinM2x415=













(Dual) min


tisinT

t

1


sum

s



tisinT

t ge 1 minus1





arg maxMsubeS

ft41Euml1Iacute1M50






sum







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all

righ

ts r

eser

ved



maxMsubeS

log(

sum

sisinM

zs

)

minus

sum

sisinM hszssum

sisinM zs0



log(

sum

sisinS

ys

)

minus

sum

sisinS hsyssum

sisinS ys0


sum

















MisinMt


sum

MisinMt


sum

tisinTs

sum

MisinMt


sum

s1t

sum

MisinMt





















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r pe

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all

righ

ts r

eser

ved




























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righ

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eser

ved













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at 1

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Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved









atilde=

p isinS2 p ge 01sum

s

ps = 1

0



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eser

ved








u middot xt4uprime5

0



int

Uxprime

t4u5 dFt =int

Uxt4u5 dFt1


u middot xprime








vt =1nt

int




t

int










Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved



Budget line

(010) (001)

(100)

u

ax(u)















Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved









u middot xt4prime5

0



int

ccedilxprime

t dFt =int

ccedilxt dFt1



k le Sksum

j=1

xprime

t4j545ge

ksum

j=1

xt4j5451







1

a1

a2

x3(1) = a3 ndash a1

a3

a4

2 3

0

12

34

1

1 3 2 2 1 3 4




x4k545=k

maxj=1

a4j5 minuskminus1

maxj=1

a4j50





Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved















Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved















Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved










W = sum

t

(


vtprime

)

0




Bts =


0 otherwise


sum

t

wtBtspts leC1








Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved


















uis = uts + btis1





vt4M5=1nt

int

UmaxsisinM

us dFt4u50


Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved



pt4s1M5=


usprime

∣

∣

∣

∣

u sim Ft

0


sum


exp4utsbt5sum



(LargeMarketLP)

max W = sum

t1M


st yminussum

M


sum

M


sum

tisinTs1M


sum

s1 t1M








= 1 = 005 = 0




sum






Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved














Q1

Q2

Q3

Q4

0 10 20



Qua

rtile

s by

infe

rred

qua

lity

(Q1

is b

est)

30 40 50


Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved



















Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved




Lyon k-8

Edison Baldwin

WinshipJacksonMann

Gardner

Lyon k-8

Edison Baldwin

Winship

JacksonMann

Gardner

Bradley

Kennedy Patrick

Guild

BradleyGuild

East boston EEC

AdamsEliot

QuincyBlackstone

Hurley Condon

HarvardKent Otis



Perkins PerryTynan



MendellTrotter

KingEverett


Lyndon

Kilmer

Beethoven

Conley

MozartBates

Sumner

BTULee

UP Acad Dor

HendersonMurphy

Kenny

Taylor

Young Achievers


ChittickE Greenwood

Grew

Channing

Roosevelt

Dever

S GreenwoodHolmes

EllisHernandez

PhilbrickHaley

Eliot Adams

East Boston EEC

Otis

WarrenPrescott


MarioUmana

Roosevelt

Channing

Grew

Beethoven

Kilmer

LyndonMozart

Bates

Sumner



E GreenwoodChittick

BTU

ManningMission Hill

Curley

Tobin

HurleyBlackstone

Condon

Quincy

Tynan


ClapDudleyHale



EverettDever

MatherKingEllis

Hernandez

Holland

UP Acad Dor

Lee


Henderson

Kenny

WinthropHaynes

PhilbrickHaley











(010) (001)

(100)

u

X

x(u)


Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved






sum




x4U4A55=

int

U4A5x4u5 dF 4u50




(010) (001)

(100)

d

ndashd

X

x(a)

x(aprime)




12 4x







Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved




u0

y0

u2

Aprime

R

cl(X )ndash


H2ndash

H2+

H2


X =

x4A5=

int

A x4u5 dF 4u5F 4A5


0



m4A5=

(

int

Ax4u5 dF 4u51 F 4A5

)

0















2 namely

Aprime=

u isinn2u middot u2

uu2gt

radicR2u2

2 minus 22

Ru2

0










C1 =

a isinD2a


1

C2 =

a isinD2a

amiddot d gt

a0

2a0middot d = 3

r

0




Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved





y4u5=

y1 if u isinU4A51

x4u5 otherwise






gt 00








sum

sisinM


sum

sisinS

xs = 11 (C2)

x ge 00 (C3)



f 4M1 cupM25= max8f 4M151 f 4M2590



f 4M5=

M sum

j=1

x4j5451 where 841514251 0 0 0 14M 59=M0













Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved








sum

sisinM2x415=













(Dual) min


tisinT

t

1


sum

s



tisinT

t ge 1 minus1





arg maxMsubeS

ft41Euml1Iacute1M50






sum







Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved



maxMsubeS

log(

sum

sisinM

zs

)

minus

sum

sisinM hszssum

sisinM zs0



log(

sum

sisinS

ys

)

minus

sum

sisinS hsyssum

sisinS ys0


sum

















MisinMt


sum

MisinMt


sum

tisinTs

sum

MisinMt


sum

s1t

sum

MisinMt





















Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved




























Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved









atilde=

p isinS2 p ge 01sum

s

ps = 1

0



Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved








u middot xt4uprime5

0



int

Uxprime

t4u5 dFt =int

Uxt4u5 dFt1


u middot xprime








vt =1nt

int




t

int










Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved



Budget line

(010) (001)

(100)

u

ax(u)















Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved









u middot xt4prime5

0



int

ccedilxprime

t dFt =int

ccedilxt dFt1



k le Sksum

j=1

xprime

t4j545ge

ksum

j=1

xt4j5451







1

a1

a2

x3(1) = a3 ndash a1

a3

a4

2 3

0

12

34

1

1 3 2 2 1 3 4




x4k545=k

maxj=1

a4j5 minuskminus1

maxj=1

a4j50





Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved















Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved















Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved










W = sum

t

(


vtprime

)

0




Bts =


0 otherwise


sum

t

wtBtspts leC1








Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved


















uis = uts + btis1





vt4M5=1nt

int

UmaxsisinM

us dFt4u50


Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved



pt4s1M5=


usprime

∣

∣

∣

∣

u sim Ft

0


sum


exp4utsbt5sum



(LargeMarketLP)

max W = sum

t1M


st yminussum

M


sum

M


sum

tisinTs1M


sum

s1 t1M








= 1 = 005 = 0




sum






Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved














Q1

Q2

Q3

Q4

0 10 20



Qua

rtile

s by

infe

rred

qua

lity

(Q1

is b

est)

30 40 50


Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved



















Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved




Lyon k-8

Edison Baldwin

WinshipJacksonMann

Gardner

Lyon k-8

Edison Baldwin

Winship

JacksonMann

Gardner

Bradley

Kennedy Patrick

Guild

BradleyGuild

East boston EEC

AdamsEliot

QuincyBlackstone

Hurley Condon

HarvardKent Otis



Perkins PerryTynan



MendellTrotter

KingEverett


Lyndon

Kilmer

Beethoven

Conley

MozartBates

Sumner

BTULee

UP Acad Dor

HendersonMurphy

Kenny

Taylor

Young Achievers


ChittickE Greenwood

Grew

Channing

Roosevelt

Dever

S GreenwoodHolmes

EllisHernandez

PhilbrickHaley

Eliot Adams

East Boston EEC

Otis

WarrenPrescott


MarioUmana

Roosevelt

Channing

Grew

Beethoven

Kilmer

LyndonMozart

Bates

Sumner



E GreenwoodChittick

BTU

ManningMission Hill

Curley

Tobin

HurleyBlackstone

Condon

Quincy

Tynan


ClapDudleyHale



EverettDever

MatherKingEllis

Hernandez

Holland

UP Acad Dor

Lee


Henderson

Kenny

WinthropHaynes

PhilbrickHaley











(010) (001)

(100)

u

X

x(u)


Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved






sum




x4U4A55=

int

U4A5x4u5 dF 4u50




(010) (001)

(100)

d

ndashd

X

x(a)

x(aprime)




12 4x







Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved




u0

y0

u2

Aprime

R

cl(X )ndash


H2ndash

H2+

H2


X =

x4A5=

int

A x4u5 dF 4u5F 4A5


0



m4A5=

(

int

Ax4u5 dF 4u51 F 4A5

)

0















2 namely

Aprime=

u isinn2u middot u2

uu2gt

radicR2u2

2 minus 22

Ru2

0










C1 =

a isinD2a


1

C2 =

a isinD2a

amiddot d gt

a0

2a0middot d = 3

r

0




Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved





y4u5=

y1 if u isinU4A51

x4u5 otherwise






gt 00








sum

sisinM


sum

sisinS

xs = 11 (C2)

x ge 00 (C3)



f 4M1 cupM25= max8f 4M151 f 4M2590



f 4M5=

M sum

j=1

x4j5451 where 841514251 0 0 0 14M 59=M0













Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved








sum

sisinM2x415=













(Dual) min


tisinT

t

1


sum

s



tisinT

t ge 1 minus1





arg maxMsubeS

ft41Euml1Iacute1M50






sum







Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved



maxMsubeS

log(

sum

sisinM

zs

)

minus

sum

sisinM hszssum

sisinM zs0



log(

sum

sisinS

ys

)

minus

sum

sisinS hsyssum

sisinS ys0


sum

















MisinMt


sum

MisinMt


sum

tisinTs

sum

MisinMt


sum

s1t

sum

MisinMt





















Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved




























Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved








u middot xt4uprime5

0



int

Uxprime

t4u5 dFt =int

Uxt4u5 dFt1


u middot xprime








vt =1nt

int




t

int










Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved



Budget line

(010) (001)

(100)

u

ax(u)















Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved









u middot xt4prime5

0



int

ccedilxprime

t dFt =int

ccedilxt dFt1



k le Sksum

j=1

xprime

t4j545ge

ksum

j=1

xt4j5451







1

a1

a2

x3(1) = a3 ndash a1

a3

a4

2 3

0

12

34

1

1 3 2 2 1 3 4




x4k545=k

maxj=1

a4j5 minuskminus1

maxj=1

a4j50





Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved















Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved















Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved










W = sum

t

(


vtprime

)

0




Bts =


0 otherwise


sum

t

wtBtspts leC1








Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved


















uis = uts + btis1





vt4M5=1nt

int

UmaxsisinM

us dFt4u50


Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved



pt4s1M5=


usprime

∣

∣

∣

∣

u sim Ft

0


sum


exp4utsbt5sum



(LargeMarketLP)

max W = sum

t1M


st yminussum

M


sum

M


sum

tisinTs1M


sum

s1 t1M








= 1 = 005 = 0




sum






Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved














Q1

Q2

Q3

Q4

0 10 20



Qua

rtile

s by

infe

rred

qua

lity

(Q1

is b

est)

30 40 50


Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved



















Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved




Lyon k-8

Edison Baldwin

WinshipJacksonMann

Gardner

Lyon k-8

Edison Baldwin

Winship

JacksonMann

Gardner

Bradley

Kennedy Patrick

Guild

BradleyGuild

East boston EEC

AdamsEliot

QuincyBlackstone

Hurley Condon

HarvardKent Otis



Perkins PerryTynan



MendellTrotter

KingEverett


Lyndon

Kilmer

Beethoven

Conley

MozartBates

Sumner

BTULee

UP Acad Dor

HendersonMurphy

Kenny

Taylor

Young Achievers


ChittickE Greenwood

Grew

Channing

Roosevelt

Dever

S GreenwoodHolmes

EllisHernandez

PhilbrickHaley

Eliot Adams

East Boston EEC

Otis

WarrenPrescott


MarioUmana

Roosevelt

Channing

Grew

Beethoven

Kilmer

LyndonMozart

Bates

Sumner



E GreenwoodChittick

BTU

ManningMission Hill

Curley

Tobin

HurleyBlackstone

Condon

Quincy

Tynan


ClapDudleyHale



EverettDever

MatherKingEllis

Hernandez

Holland

UP Acad Dor

Lee


Henderson

Kenny

WinthropHaynes

PhilbrickHaley











(010) (001)

(100)

u

X

x(u)


Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved






sum




x4U4A55=

int

U4A5x4u5 dF 4u50




(010) (001)

(100)

d

ndashd

X

x(a)

x(aprime)




12 4x







Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved




u0

y0

u2

Aprime

R

cl(X )ndash


H2ndash

H2+

H2


X =

x4A5=

int

A x4u5 dF 4u5F 4A5


0



m4A5=

(

int

Ax4u5 dF 4u51 F 4A5

)

0















2 namely

Aprime=

u isinn2u middot u2

uu2gt

radicR2u2

2 minus 22

Ru2

0










C1 =

a isinD2a


1

C2 =

a isinD2a

amiddot d gt

a0

2a0middot d = 3

r

0




Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved





y4u5=

y1 if u isinU4A51

x4u5 otherwise






gt 00








sum

sisinM


sum

sisinS

xs = 11 (C2)

x ge 00 (C3)



f 4M1 cupM25= max8f 4M151 f 4M2590



f 4M5=

M sum

j=1

x4j5451 where 841514251 0 0 0 14M 59=M0













Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved








sum

sisinM2x415=













(Dual) min


tisinT

t

1


sum

s



tisinT

t ge 1 minus1





arg maxMsubeS

ft41Euml1Iacute1M50






sum







Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved



maxMsubeS

log(

sum

sisinM

zs

)

minus

sum

sisinM hszssum

sisinM zs0



log(

sum

sisinS

ys

)

minus

sum

sisinS hsyssum

sisinS ys0


sum

















MisinMt


sum

MisinMt


sum

tisinTs

sum

MisinMt


sum

s1t

sum

MisinMt





















Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved




























Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved



Budget line

(010) (001)

(100)

u

ax(u)















Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved









u middot xt4prime5

0



int

ccedilxprime

t dFt =int

ccedilxt dFt1



k le Sksum

j=1

xprime

t4j545ge

ksum

j=1

xt4j5451







1

a1

a2

x3(1) = a3 ndash a1

a3

a4

2 3

0

12

34

1

1 3 2 2 1 3 4




x4k545=k

maxj=1

a4j5 minuskminus1

maxj=1

a4j50





Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved















Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved















Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved










W = sum

t

(


vtprime

)

0




Bts =


0 otherwise


sum

t

wtBtspts leC1








Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved


















uis = uts + btis1





vt4M5=1nt

int

UmaxsisinM

us dFt4u50


Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved



pt4s1M5=


usprime

∣

∣

∣

∣

u sim Ft

0


sum


exp4utsbt5sum



(LargeMarketLP)

max W = sum

t1M


st yminussum

M


sum

M


sum

tisinTs1M


sum

s1 t1M








= 1 = 005 = 0




sum






Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved














Q1

Q2

Q3

Q4

0 10 20



Qua

rtile

s by

infe

rred

qua

lity

(Q1

is b

est)

30 40 50


Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved



















Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved




Lyon k-8

Edison Baldwin

WinshipJacksonMann

Gardner

Lyon k-8

Edison Baldwin

Winship

JacksonMann

Gardner

Bradley

Kennedy Patrick

Guild

BradleyGuild

East boston EEC

AdamsEliot

QuincyBlackstone

Hurley Condon

HarvardKent Otis



Perkins PerryTynan



MendellTrotter

KingEverett


Lyndon

Kilmer

Beethoven

Conley

MozartBates

Sumner

BTULee

UP Acad Dor

HendersonMurphy

Kenny

Taylor

Young Achievers


ChittickE Greenwood

Grew

Channing

Roosevelt

Dever

S GreenwoodHolmes

EllisHernandez

PhilbrickHaley

Eliot Adams

East Boston EEC

Otis

WarrenPrescott


MarioUmana

Roosevelt

Channing

Grew

Beethoven

Kilmer

LyndonMozart

Bates

Sumner



E GreenwoodChittick

BTU

ManningMission Hill

Curley

Tobin

HurleyBlackstone

Condon

Quincy

Tynan


ClapDudleyHale



EverettDever

MatherKingEllis

Hernandez

Holland

UP Acad Dor

Lee


Henderson

Kenny

WinthropHaynes

PhilbrickHaley











(010) (001)

(100)

u

X

x(u)


Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved






sum




x4U4A55=

int

U4A5x4u5 dF 4u50




(010) (001)

(100)

d

ndashd

X

x(a)

x(aprime)




12 4x







Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved




u0

y0

u2

Aprime

R

cl(X )ndash


H2ndash

H2+

H2


X =

x4A5=

int

A x4u5 dF 4u5F 4A5


0



m4A5=

(

int

Ax4u5 dF 4u51 F 4A5

)

0















2 namely

Aprime=

u isinn2u middot u2

uu2gt

radicR2u2

2 minus 22

Ru2

0










C1 =

a isinD2a


1

C2 =

a isinD2a

amiddot d gt

a0

2a0middot d = 3

r

0




Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved





y4u5=

y1 if u isinU4A51

x4u5 otherwise






gt 00








sum

sisinM


sum

sisinS

xs = 11 (C2)

x ge 00 (C3)



f 4M1 cupM25= max8f 4M151 f 4M2590



f 4M5=

M sum

j=1

x4j5451 where 841514251 0 0 0 14M 59=M0













Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved








sum

sisinM2x415=













(Dual) min


tisinT

t

1


sum

s



tisinT

t ge 1 minus1





arg maxMsubeS

ft41Euml1Iacute1M50






sum







Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved



maxMsubeS

log(

sum

sisinM

zs

)

minus

sum

sisinM hszssum

sisinM zs0



log(

sum

sisinS

ys

)

minus

sum

sisinS hsyssum

sisinS ys0


sum

















MisinMt


sum

MisinMt


sum

tisinTs

sum

MisinMt


sum

s1t

sum

MisinMt





















Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved




























Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved









u middot xt4prime5

0



int

ccedilxprime

t dFt =int

ccedilxt dFt1



k le Sksum

j=1

xprime

t4j545ge

ksum

j=1

xt4j5451







1

a1

a2

x3(1) = a3 ndash a1

a3

a4

2 3

0

12

34

1

1 3 2 2 1 3 4




x4k545=k

maxj=1

a4j5 minuskminus1

maxj=1

a4j50





Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved















Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved















Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved










W = sum

t

(


vtprime

)

0




Bts =


0 otherwise


sum

t

wtBtspts leC1








Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved


















uis = uts + btis1





vt4M5=1nt

int

UmaxsisinM

us dFt4u50


Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved



pt4s1M5=


usprime

∣

∣

∣

∣

u sim Ft

0


sum


exp4utsbt5sum



(LargeMarketLP)

max W = sum

t1M


st yminussum

M


sum

M


sum

tisinTs1M


sum

s1 t1M








= 1 = 005 = 0




sum






Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved














Q1

Q2

Q3

Q4

0 10 20



Qua

rtile

s by

infe

rred

qua

lity

(Q1

is b

est)

30 40 50


Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved



















Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved




Lyon k-8

Edison Baldwin

WinshipJacksonMann

Gardner

Lyon k-8

Edison Baldwin

Winship

JacksonMann

Gardner

Bradley

Kennedy Patrick

Guild

BradleyGuild

East boston EEC

AdamsEliot

QuincyBlackstone

Hurley Condon

HarvardKent Otis



Perkins PerryTynan



MendellTrotter

KingEverett


Lyndon

Kilmer

Beethoven

Conley

MozartBates

Sumner

BTULee

UP Acad Dor

HendersonMurphy

Kenny

Taylor

Young Achievers


ChittickE Greenwood

Grew

Channing

Roosevelt

Dever

S GreenwoodHolmes

EllisHernandez

PhilbrickHaley

Eliot Adams

East Boston EEC

Otis

WarrenPrescott


MarioUmana

Roosevelt

Channing

Grew

Beethoven

Kilmer

LyndonMozart

Bates

Sumner



E GreenwoodChittick

BTU

ManningMission Hill

Curley

Tobin

HurleyBlackstone

Condon

Quincy

Tynan


ClapDudleyHale



EverettDever

MatherKingEllis

Hernandez

Holland

UP Acad Dor

Lee


Henderson

Kenny

WinthropHaynes

PhilbrickHaley











(010) (001)

(100)

u

X

x(u)


Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved






sum




x4U4A55=

int

U4A5x4u5 dF 4u50




(010) (001)

(100)

d

ndashd

X

x(a)

x(aprime)




12 4x







Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved




u0

y0

u2

Aprime

R

cl(X )ndash


H2ndash

H2+

H2


X =

x4A5=

int

A x4u5 dF 4u5F 4A5


0



m4A5=

(

int

Ax4u5 dF 4u51 F 4A5

)

0















2 namely

Aprime=

u isinn2u middot u2

uu2gt

radicR2u2

2 minus 22

Ru2

0










C1 =

a isinD2a


1

C2 =

a isinD2a

amiddot d gt

a0

2a0middot d = 3

r

0




Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved





y4u5=

y1 if u isinU4A51

x4u5 otherwise






gt 00








sum

sisinM


sum

sisinS

xs = 11 (C2)

x ge 00 (C3)



f 4M1 cupM25= max8f 4M151 f 4M2590



f 4M5=

M sum

j=1

x4j5451 where 841514251 0 0 0 14M 59=M0













Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved








sum

sisinM2x415=













(Dual) min


tisinT

t

1


sum

s



tisinT

t ge 1 minus1





arg maxMsubeS

ft41Euml1Iacute1M50






sum







Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved



maxMsubeS

log(

sum

sisinM

zs

)

minus

sum

sisinM hszssum

sisinM zs0



log(

sum

sisinS

ys

)

minus

sum

sisinS hsyssum

sisinS ys0


sum

















MisinMt


sum

MisinMt


sum

tisinTs

sum

MisinMt


sum

s1t

sum

MisinMt





















Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved




























Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved















Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved















Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved










W = sum

t

(


vtprime

)

0




Bts =


0 otherwise


sum

t

wtBtspts leC1








Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved


















uis = uts + btis1





vt4M5=1nt

int

UmaxsisinM

us dFt4u50


Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved



pt4s1M5=


usprime

∣

∣

∣

∣

u sim Ft

0


sum


exp4utsbt5sum



(LargeMarketLP)

max W = sum

t1M


st yminussum

M


sum

M


sum

tisinTs1M


sum

s1 t1M








= 1 = 005 = 0




sum






Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved














Q1

Q2

Q3

Q4

0 10 20



Qua

rtile

s by

infe

rred

qua

lity

(Q1

is b

est)

30 40 50


Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved



















Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved




Lyon k-8

Edison Baldwin

WinshipJacksonMann

Gardner

Lyon k-8

Edison Baldwin

Winship

JacksonMann

Gardner

Bradley

Kennedy Patrick

Guild

BradleyGuild

East boston EEC

AdamsEliot

QuincyBlackstone

Hurley Condon

HarvardKent Otis



Perkins PerryTynan



MendellTrotter

KingEverett


Lyndon

Kilmer

Beethoven

Conley

MozartBates

Sumner

BTULee

UP Acad Dor

HendersonMurphy

Kenny

Taylor

Young Achievers


ChittickE Greenwood

Grew

Channing

Roosevelt

Dever

S GreenwoodHolmes

EllisHernandez

PhilbrickHaley

Eliot Adams

East Boston EEC

Otis

WarrenPrescott


MarioUmana

Roosevelt

Channing

Grew

Beethoven

Kilmer

LyndonMozart

Bates

Sumner



E GreenwoodChittick

BTU

ManningMission Hill

Curley

Tobin

HurleyBlackstone

Condon

Quincy

Tynan


ClapDudleyHale



EverettDever

MatherKingEllis

Hernandez

Holland

UP Acad Dor

Lee


Henderson

Kenny

WinthropHaynes

PhilbrickHaley











(010) (001)

(100)

u

X

x(u)


Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved






sum




x4U4A55=

int

U4A5x4u5 dF 4u50




(010) (001)

(100)

d

ndashd

X

x(a)

x(aprime)




12 4x







Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved




u0

y0

u2

Aprime

R

cl(X )ndash


H2ndash

H2+

H2


X =

x4A5=

int

A x4u5 dF 4u5F 4A5


0



m4A5=

(

int

Ax4u5 dF 4u51 F 4A5

)

0















2 namely

Aprime=

u isinn2u middot u2

uu2gt

radicR2u2

2 minus 22

Ru2

0










C1 =

a isinD2a


1

C2 =

a isinD2a

amiddot d gt

a0

2a0middot d = 3

r

0




Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved





y4u5=

y1 if u isinU4A51

x4u5 otherwise






gt 00








sum

sisinM


sum

sisinS

xs = 11 (C2)

x ge 00 (C3)



f 4M1 cupM25= max8f 4M151 f 4M2590



f 4M5=

M sum

j=1

x4j5451 where 841514251 0 0 0 14M 59=M0













Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved








sum

sisinM2x415=













(Dual) min


tisinT

t

1


sum

s



tisinT

t ge 1 minus1





arg maxMsubeS

ft41Euml1Iacute1M50






sum







Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved



maxMsubeS

log(

sum

sisinM

zs

)

minus

sum

sisinM hszssum

sisinM zs0



log(

sum

sisinS

ys

)

minus

sum

sisinS hsyssum

sisinS ys0


sum

















MisinMt


sum

MisinMt


sum

tisinTs

sum

MisinMt


sum

s1t

sum

MisinMt





















Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved




























Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved















Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved










W = sum

t

(


vtprime

)

0




Bts =


0 otherwise


sum

t

wtBtspts leC1








Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved


















uis = uts + btis1





vt4M5=1nt

int

UmaxsisinM

us dFt4u50


Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved



pt4s1M5=


usprime

∣

∣

∣

∣

u sim Ft

0


sum


exp4utsbt5sum



(LargeMarketLP)

max W = sum

t1M


st yminussum

M


sum

M


sum

tisinTs1M


sum

s1 t1M








= 1 = 005 = 0




sum






Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved














Q1

Q2

Q3

Q4

0 10 20



Qua

rtile

s by

infe

rred

qua

lity

(Q1

is b

est)

30 40 50


Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved



















Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved




Lyon k-8

Edison Baldwin

WinshipJacksonMann

Gardner

Lyon k-8

Edison Baldwin

Winship

JacksonMann

Gardner

Bradley

Kennedy Patrick

Guild

BradleyGuild

East boston EEC

AdamsEliot

QuincyBlackstone

Hurley Condon

HarvardKent Otis



Perkins PerryTynan



MendellTrotter

KingEverett


Lyndon

Kilmer

Beethoven

Conley

MozartBates

Sumner

BTULee

UP Acad Dor

HendersonMurphy

Kenny

Taylor

Young Achievers


ChittickE Greenwood

Grew

Channing

Roosevelt

Dever

S GreenwoodHolmes

EllisHernandez

PhilbrickHaley

Eliot Adams

East Boston EEC

Otis

WarrenPrescott


MarioUmana

Roosevelt

Channing

Grew

Beethoven

Kilmer

LyndonMozart

Bates

Sumner



E GreenwoodChittick

BTU

ManningMission Hill

Curley

Tobin

HurleyBlackstone

Condon

Quincy

Tynan


ClapDudleyHale



EverettDever

MatherKingEllis

Hernandez

Holland

UP Acad Dor

Lee


Henderson

Kenny

WinthropHaynes

PhilbrickHaley











(010) (001)

(100)

u

X

x(u)


Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved






sum




x4U4A55=

int

U4A5x4u5 dF 4u50




(010) (001)

(100)

d

ndashd

X

x(a)

x(aprime)




12 4x







Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved




u0

y0

u2

Aprime

R

cl(X )ndash


H2ndash

H2+

H2


X =

x4A5=

int

A x4u5 dF 4u5F 4A5


0



m4A5=

(

int

Ax4u5 dF 4u51 F 4A5

)

0















2 namely

Aprime=

u isinn2u middot u2

uu2gt

radicR2u2

2 minus 22

Ru2

0










C1 =

a isinD2a


1

C2 =

a isinD2a

amiddot d gt

a0

2a0middot d = 3

r

0




Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved





y4u5=

y1 if u isinU4A51

x4u5 otherwise






gt 00








sum

sisinM


sum

sisinS

xs = 11 (C2)

x ge 00 (C3)



f 4M1 cupM25= max8f 4M151 f 4M2590



f 4M5=

M sum

j=1

x4j5451 where 841514251 0 0 0 14M 59=M0













Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved








sum

sisinM2x415=













(Dual) min


tisinT

t

1


sum

s



tisinT

t ge 1 minus1





arg maxMsubeS

ft41Euml1Iacute1M50






sum







Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved



maxMsubeS

log(

sum

sisinM

zs

)

minus

sum

sisinM hszssum

sisinM zs0



log(

sum

sisinS

ys

)

minus

sum

sisinS hsyssum

sisinS ys0


sum

















MisinMt


sum

MisinMt


sum

tisinTs

sum

MisinMt


sum

s1t

sum

MisinMt





















Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved




























Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved










W = sum

t

(


vtprime

)

0




Bts =


0 otherwise


sum

t

wtBtspts leC1








Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved


















uis = uts + btis1





vt4M5=1nt

int

UmaxsisinM

us dFt4u50


Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved



pt4s1M5=


usprime

∣

∣

∣

∣

u sim Ft

0


sum


exp4utsbt5sum



(LargeMarketLP)

max W = sum

t1M


st yminussum

M


sum

M


sum

tisinTs1M


sum

s1 t1M








= 1 = 005 = 0




sum






Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved














Q1

Q2

Q3

Q4

0 10 20



Qua

rtile

s by

infe

rred

qua

lity

(Q1

is b

est)

30 40 50


Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved



















Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved




Lyon k-8

Edison Baldwin

WinshipJacksonMann

Gardner

Lyon k-8

Edison Baldwin

Winship

JacksonMann

Gardner

Bradley

Kennedy Patrick

Guild

BradleyGuild

East boston EEC

AdamsEliot

QuincyBlackstone

Hurley Condon

HarvardKent Otis



Perkins PerryTynan



MendellTrotter

KingEverett


Lyndon

Kilmer

Beethoven

Conley

MozartBates

Sumner

BTULee

UP Acad Dor

HendersonMurphy

Kenny

Taylor

Young Achievers


ChittickE Greenwood

Grew

Channing

Roosevelt

Dever

S GreenwoodHolmes

EllisHernandez

PhilbrickHaley

Eliot Adams

East Boston EEC

Otis

WarrenPrescott


MarioUmana

Roosevelt

Channing

Grew

Beethoven

Kilmer

LyndonMozart

Bates

Sumner



E GreenwoodChittick

BTU

ManningMission Hill

Curley

Tobin

HurleyBlackstone

Condon

Quincy

Tynan


ClapDudleyHale



EverettDever

MatherKingEllis

Hernandez

Holland

UP Acad Dor

Lee


Henderson

Kenny

WinthropHaynes

PhilbrickHaley











(010) (001)

(100)

u

X

x(u)


Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved






sum




x4U4A55=

int

U4A5x4u5 dF 4u50




(010) (001)

(100)

d

ndashd

X

x(a)

x(aprime)




12 4x







Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved




u0

y0

u2

Aprime

R

cl(X )ndash


H2ndash

H2+

H2


X =

x4A5=

int

A x4u5 dF 4u5F 4A5


0



m4A5=

(

int

Ax4u5 dF 4u51 F 4A5

)

0















2 namely

Aprime=

u isinn2u middot u2

uu2gt

radicR2u2

2 minus 22

Ru2

0










C1 =

a isinD2a


1

C2 =

a isinD2a

amiddot d gt

a0

2a0middot d = 3

r

0




Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved





y4u5=

y1 if u isinU4A51

x4u5 otherwise






gt 00








sum

sisinM


sum

sisinS

xs = 11 (C2)

x ge 00 (C3)



f 4M1 cupM25= max8f 4M151 f 4M2590



f 4M5=

M sum

j=1

x4j5451 where 841514251 0 0 0 14M 59=M0













Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved








sum

sisinM2x415=













(Dual) min


tisinT

t

1


sum

s



tisinT

t ge 1 minus1





arg maxMsubeS

ft41Euml1Iacute1M50






sum







Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved



maxMsubeS

log(

sum

sisinM

zs

)

minus

sum

sisinM hszssum

sisinM zs0



log(

sum

sisinS

ys

)

minus

sum

sisinS hsyssum

sisinS ys0


sum

















MisinMt


sum

MisinMt


sum

tisinTs

sum

MisinMt


sum

s1t

sum

MisinMt





















Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved




























Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved


















uis = uts + btis1





vt4M5=1nt

int

UmaxsisinM

us dFt4u50


Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved



pt4s1M5=


usprime

∣

∣

∣

∣

u sim Ft

0


sum


exp4utsbt5sum



(LargeMarketLP)

max W = sum

t1M


st yminussum

M


sum

M


sum

tisinTs1M


sum

s1 t1M








= 1 = 005 = 0




sum






Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved














Q1

Q2

Q3

Q4

0 10 20



Qua

rtile

s by

infe

rred

qua

lity

(Q1

is b

est)

30 40 50


Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved



















Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved




Lyon k-8

Edison Baldwin

WinshipJacksonMann

Gardner

Lyon k-8

Edison Baldwin

Winship

JacksonMann

Gardner

Bradley

Kennedy Patrick

Guild

BradleyGuild

East boston EEC

AdamsEliot

QuincyBlackstone

Hurley Condon

HarvardKent Otis



Perkins PerryTynan



MendellTrotter

KingEverett


Lyndon

Kilmer

Beethoven

Conley

MozartBates

Sumner

BTULee

UP Acad Dor

HendersonMurphy

Kenny

Taylor

Young Achievers


ChittickE Greenwood

Grew

Channing

Roosevelt

Dever

S GreenwoodHolmes

EllisHernandez

PhilbrickHaley

Eliot Adams

East Boston EEC

Otis

WarrenPrescott


MarioUmana

Roosevelt

Channing

Grew

Beethoven

Kilmer

LyndonMozart

Bates

Sumner



E GreenwoodChittick

BTU

ManningMission Hill

Curley

Tobin

HurleyBlackstone

Condon

Quincy

Tynan


ClapDudleyHale



EverettDever

MatherKingEllis

Hernandez

Holland

UP Acad Dor

Lee


Henderson

Kenny

WinthropHaynes

PhilbrickHaley











(010) (001)

(100)

u

X

x(u)


Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved






sum




x4U4A55=

int

U4A5x4u5 dF 4u50




(010) (001)

(100)

d

ndashd

X

x(a)

x(aprime)




12 4x







Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved




u0

y0

u2

Aprime

R

cl(X )ndash


H2ndash

H2+

H2


X =

x4A5=

int

A x4u5 dF 4u5F 4A5


0



m4A5=

(

int

Ax4u5 dF 4u51 F 4A5

)

0















2 namely

Aprime=

u isinn2u middot u2

uu2gt

radicR2u2

2 minus 22

Ru2

0










C1 =

a isinD2a


1

C2 =

a isinD2a

amiddot d gt

a0

2a0middot d = 3

r

0




Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved





y4u5=

y1 if u isinU4A51

x4u5 otherwise






gt 00








sum

sisinM


sum

sisinS

xs = 11 (C2)

x ge 00 (C3)



f 4M1 cupM25= max8f 4M151 f 4M2590



f 4M5=

M sum

j=1

x4j5451 where 841514251 0 0 0 14M 59=M0













Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved








sum

sisinM2x415=













(Dual) min


tisinT

t

1


sum

s



tisinT

t ge 1 minus1





arg maxMsubeS

ft41Euml1Iacute1M50






sum







Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved



maxMsubeS

log(

sum

sisinM

zs

)

minus

sum

sisinM hszssum

sisinM zs0



log(

sum

sisinS

ys

)

minus

sum

sisinS hsyssum

sisinS ys0


sum

















MisinMt


sum

MisinMt


sum

tisinTs

sum

MisinMt


sum

s1t

sum

MisinMt





















Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved




























Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved



pt4s1M5=


usprime

∣

∣

∣

∣

u sim Ft

0


sum


exp4utsbt5sum



(LargeMarketLP)

max W = sum

t1M


st yminussum

M


sum

M


sum

tisinTs1M


sum

s1 t1M








= 1 = 005 = 0




sum






Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved














Q1

Q2

Q3

Q4

0 10 20



Qua

rtile

s by

infe

rred

qua

lity

(Q1

is b

est)

30 40 50


Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved



















Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved




Lyon k-8

Edison Baldwin

WinshipJacksonMann

Gardner

Lyon k-8

Edison Baldwin

Winship

JacksonMann

Gardner

Bradley

Kennedy Patrick

Guild

BradleyGuild

East boston EEC

AdamsEliot

QuincyBlackstone

Hurley Condon

HarvardKent Otis



Perkins PerryTynan



MendellTrotter

KingEverett


Lyndon

Kilmer

Beethoven

Conley

MozartBates

Sumner

BTULee

UP Acad Dor

HendersonMurphy

Kenny

Taylor

Young Achievers


ChittickE Greenwood

Grew

Channing

Roosevelt

Dever

S GreenwoodHolmes

EllisHernandez

PhilbrickHaley

Eliot Adams

East Boston EEC

Otis

WarrenPrescott


MarioUmana

Roosevelt

Channing

Grew

Beethoven

Kilmer

LyndonMozart

Bates

Sumner



E GreenwoodChittick

BTU

ManningMission Hill

Curley

Tobin

HurleyBlackstone

Condon

Quincy

Tynan


ClapDudleyHale



EverettDever

MatherKingEllis

Hernandez

Holland

UP Acad Dor

Lee


Henderson

Kenny

WinthropHaynes

PhilbrickHaley











(010) (001)

(100)

u

X

x(u)


Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved






sum




x4U4A55=

int

U4A5x4u5 dF 4u50




(010) (001)

(100)

d

ndashd

X

x(a)

x(aprime)




12 4x







Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved




u0

y0

u2

Aprime

R

cl(X )ndash


H2ndash

H2+

H2


X =

x4A5=

int

A x4u5 dF 4u5F 4A5


0



m4A5=

(

int

Ax4u5 dF 4u51 F 4A5

)

0















2 namely

Aprime=

u isinn2u middot u2

uu2gt

radicR2u2

2 minus 22

Ru2

0










C1 =

a isinD2a


1

C2 =

a isinD2a

amiddot d gt

a0

2a0middot d = 3

r

0




Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved





y4u5=

y1 if u isinU4A51

x4u5 otherwise






gt 00








sum

sisinM


sum

sisinS

xs = 11 (C2)

x ge 00 (C3)



f 4M1 cupM25= max8f 4M151 f 4M2590



f 4M5=

M sum

j=1

x4j5451 where 841514251 0 0 0 14M 59=M0













Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved








sum

sisinM2x415=













(Dual) min


tisinT

t

1


sum

s



tisinT

t ge 1 minus1





arg maxMsubeS

ft41Euml1Iacute1M50






sum







Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved



maxMsubeS

log(

sum

sisinM

zs

)

minus

sum

sisinM hszssum

sisinM zs0



log(

sum

sisinS

ys

)

minus

sum

sisinS hsyssum

sisinS ys0


sum

















MisinMt


sum

MisinMt


sum

tisinTs

sum

MisinMt


sum

s1t

sum

MisinMt





















Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved




























Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved














Q1

Q2

Q3

Q4

0 10 20



Qua

rtile

s by

infe

rred

qua

lity

(Q1

is b

est)

30 40 50


Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved



















Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved




Lyon k-8

Edison Baldwin

WinshipJacksonMann

Gardner

Lyon k-8

Edison Baldwin

Winship

JacksonMann

Gardner

Bradley

Kennedy Patrick

Guild

BradleyGuild

East boston EEC

AdamsEliot

QuincyBlackstone

Hurley Condon

HarvardKent Otis



Perkins PerryTynan



MendellTrotter

KingEverett


Lyndon

Kilmer

Beethoven

Conley

MozartBates

Sumner

BTULee

UP Acad Dor

HendersonMurphy

Kenny

Taylor

Young Achievers


ChittickE Greenwood

Grew

Channing

Roosevelt

Dever

S GreenwoodHolmes

EllisHernandez

PhilbrickHaley

Eliot Adams

East Boston EEC

Otis

WarrenPrescott


MarioUmana

Roosevelt

Channing

Grew

Beethoven

Kilmer

LyndonMozart

Bates

Sumner



E GreenwoodChittick

BTU

ManningMission Hill

Curley

Tobin

HurleyBlackstone

Condon

Quincy

Tynan


ClapDudleyHale



EverettDever

MatherKingEllis

Hernandez

Holland

UP Acad Dor

Lee


Henderson

Kenny

WinthropHaynes

PhilbrickHaley











(010) (001)

(100)

u

X

x(u)


Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved






sum




x4U4A55=

int

U4A5x4u5 dF 4u50




(010) (001)

(100)

d

ndashd

X

x(a)

x(aprime)




12 4x







Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved




u0

y0

u2

Aprime

R

cl(X )ndash


H2ndash

H2+

H2


X =

x4A5=

int

A x4u5 dF 4u5F 4A5


0



m4A5=

(

int

Ax4u5 dF 4u51 F 4A5

)

0















2 namely

Aprime=

u isinn2u middot u2

uu2gt

radicR2u2

2 minus 22

Ru2

0










C1 =

a isinD2a


1

C2 =

a isinD2a

amiddot d gt

a0

2a0middot d = 3

r

0




Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved





y4u5=

y1 if u isinU4A51

x4u5 otherwise






gt 00








sum

sisinM


sum

sisinS

xs = 11 (C2)

x ge 00 (C3)



f 4M1 cupM25= max8f 4M151 f 4M2590



f 4M5=

M sum

j=1

x4j5451 where 841514251 0 0 0 14M 59=M0













Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved








sum

sisinM2x415=













(Dual) min


tisinT

t

1


sum

s



tisinT

t ge 1 minus1





arg maxMsubeS

ft41Euml1Iacute1M50






sum







Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved



maxMsubeS

log(

sum

sisinM

zs

)

minus

sum

sisinM hszssum

sisinM zs0



log(

sum

sisinS

ys

)

minus

sum

sisinS hsyssum

sisinS ys0


sum

















MisinMt


sum

MisinMt


sum

tisinTs

sum

MisinMt


sum

s1t

sum

MisinMt





















Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved




























Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved



















Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved




Lyon k-8

Edison Baldwin

WinshipJacksonMann

Gardner

Lyon k-8

Edison Baldwin

Winship

JacksonMann

Gardner

Bradley

Kennedy Patrick

Guild

BradleyGuild

East boston EEC

AdamsEliot

QuincyBlackstone

Hurley Condon

HarvardKent Otis



Perkins PerryTynan



MendellTrotter

KingEverett


Lyndon

Kilmer

Beethoven

Conley

MozartBates

Sumner

BTULee

UP Acad Dor

HendersonMurphy

Kenny

Taylor

Young Achievers


ChittickE Greenwood

Grew

Channing

Roosevelt

Dever

S GreenwoodHolmes

EllisHernandez

PhilbrickHaley

Eliot Adams

East Boston EEC

Otis

WarrenPrescott


MarioUmana

Roosevelt

Channing

Grew

Beethoven

Kilmer

LyndonMozart

Bates

Sumner



E GreenwoodChittick

BTU

ManningMission Hill

Curley

Tobin

HurleyBlackstone

Condon

Quincy

Tynan


ClapDudleyHale



EverettDever

MatherKingEllis

Hernandez

Holland

UP Acad Dor

Lee


Henderson

Kenny

WinthropHaynes

PhilbrickHaley











(010) (001)

(100)

u

X

x(u)


Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved






sum




x4U4A55=

int

U4A5x4u5 dF 4u50




(010) (001)

(100)

d

ndashd

X

x(a)

x(aprime)




12 4x







Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved




u0

y0

u2

Aprime

R

cl(X )ndash


H2ndash

H2+

H2


X =

x4A5=

int

A x4u5 dF 4u5F 4A5


0



m4A5=

(

int

Ax4u5 dF 4u51 F 4A5

)

0















2 namely

Aprime=

u isinn2u middot u2

uu2gt

radicR2u2

2 minus 22

Ru2

0










C1 =

a isinD2a


1

C2 =

a isinD2a

amiddot d gt

a0

2a0middot d = 3

r

0




Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved





y4u5=

y1 if u isinU4A51

x4u5 otherwise






gt 00








sum

sisinM


sum

sisinS

xs = 11 (C2)

x ge 00 (C3)



f 4M1 cupM25= max8f 4M151 f 4M2590



f 4M5=

M sum

j=1

x4j5451 where 841514251 0 0 0 14M 59=M0













Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved








sum

sisinM2x415=













(Dual) min


tisinT

t

1


sum

s



tisinT

t ge 1 minus1





arg maxMsubeS

ft41Euml1Iacute1M50






sum







Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved



maxMsubeS

log(

sum

sisinM

zs

)

minus

sum

sisinM hszssum

sisinM zs0



log(

sum

sisinS

ys

)

minus

sum

sisinS hsyssum

sisinS ys0


sum

















MisinMt


sum

MisinMt


sum

tisinTs

sum

MisinMt


sum

s1t

sum

MisinMt





















Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved




























Dow

nloa

ded

from

info

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org

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171

662

081

45]

on 1

0 Se

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ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved




Lyon k-8

Edison Baldwin

WinshipJacksonMann

Gardner

Lyon k-8

Edison Baldwin

Winship

JacksonMann

Gardner

Bradley

Kennedy Patrick

Guild

BradleyGuild

East boston EEC

AdamsEliot

QuincyBlackstone

Hurley Condon

HarvardKent Otis



Perkins PerryTynan



MendellTrotter

KingEverett


Lyndon

Kilmer

Beethoven

Conley

MozartBates

Sumner

BTULee

UP Acad Dor

HendersonMurphy

Kenny

Taylor

Young Achievers


ChittickE Greenwood

Grew

Channing

Roosevelt

Dever

S GreenwoodHolmes

EllisHernandez

PhilbrickHaley

Eliot Adams

East Boston EEC

Otis

WarrenPrescott


MarioUmana

Roosevelt

Channing

Grew

Beethoven

Kilmer

LyndonMozart

Bates

Sumner



E GreenwoodChittick

BTU

ManningMission Hill

Curley

Tobin

HurleyBlackstone

Condon

Quincy

Tynan


ClapDudleyHale



EverettDever

MatherKingEllis

Hernandez

Holland

UP Acad Dor

Lee


Henderson

Kenny

WinthropHaynes

PhilbrickHaley











(010) (001)

(100)

u

X

x(u)


Dow

nloa

ded

from

info

rms

org

by [

171

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081

45]

on 1

0 Se

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ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved






sum




x4U4A55=

int

U4A5x4u5 dF 4u50




(010) (001)

(100)

d

ndashd

X

x(a)

x(aprime)




12 4x







Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved




u0

y0

u2

Aprime

R

cl(X )ndash


H2ndash

H2+

H2


X =

x4A5=

int

A x4u5 dF 4u5F 4A5


0



m4A5=

(

int

Ax4u5 dF 4u51 F 4A5

)

0















2 namely

Aprime=

u isinn2u middot u2

uu2gt

radicR2u2

2 minus 22

Ru2

0










C1 =

a isinD2a


1

C2 =

a isinD2a

amiddot d gt

a0

2a0middot d = 3

r

0




Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved





y4u5=

y1 if u isinU4A51

x4u5 otherwise






gt 00








sum

sisinM


sum

sisinS

xs = 11 (C2)

x ge 00 (C3)



f 4M1 cupM25= max8f 4M151 f 4M2590



f 4M5=

M sum

j=1

x4j5451 where 841514251 0 0 0 14M 59=M0













Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved








sum

sisinM2x415=













(Dual) min


tisinT

t

1


sum

s



tisinT

t ge 1 minus1





arg maxMsubeS

ft41Euml1Iacute1M50






sum







Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved



maxMsubeS

log(

sum

sisinM

zs

)

minus

sum

sisinM hszssum

sisinM zs0



log(

sum

sisinS

ys

)

minus

sum

sisinS hsyssum

sisinS ys0


sum

















MisinMt


sum

MisinMt


sum

tisinTs

sum

MisinMt


sum

s1t

sum

MisinMt





















Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved




























Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved






sum




x4U4A55=

int

U4A5x4u5 dF 4u50




(010) (001)

(100)

d

ndashd

X

x(a)

x(aprime)




12 4x







Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved




u0

y0

u2

Aprime

R

cl(X )ndash


H2ndash

H2+

H2


X =

x4A5=

int

A x4u5 dF 4u5F 4A5


0



m4A5=

(

int

Ax4u5 dF 4u51 F 4A5

)

0















2 namely

Aprime=

u isinn2u middot u2

uu2gt

radicR2u2

2 minus 22

Ru2

0










C1 =

a isinD2a


1

C2 =

a isinD2a

amiddot d gt

a0

2a0middot d = 3

r

0




Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved





y4u5=

y1 if u isinU4A51

x4u5 otherwise






gt 00








sum

sisinM


sum

sisinS

xs = 11 (C2)

x ge 00 (C3)



f 4M1 cupM25= max8f 4M151 f 4M2590



f 4M5=

M sum

j=1

x4j5451 where 841514251 0 0 0 14M 59=M0













Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved








sum

sisinM2x415=













(Dual) min


tisinT

t

1


sum

s



tisinT

t ge 1 minus1





arg maxMsubeS

ft41Euml1Iacute1M50






sum







Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved



maxMsubeS

log(

sum

sisinM

zs

)

minus

sum

sisinM hszssum

sisinM zs0



log(

sum

sisinS

ys

)

minus

sum

sisinS hsyssum

sisinS ys0


sum

















MisinMt


sum

MisinMt


sum

tisinTs

sum

MisinMt


sum

s1t

sum

MisinMt





















Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved




























Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved




u0

y0

u2

Aprime

R

cl(X )ndash


H2ndash

H2+

H2


X =

x4A5=

int

A x4u5 dF 4u5F 4A5


0



m4A5=

(

int

Ax4u5 dF 4u51 F 4A5

)

0















2 namely

Aprime=

u isinn2u middot u2

uu2gt

radicR2u2

2 minus 22

Ru2

0










C1 =

a isinD2a


1

C2 =

a isinD2a

amiddot d gt

a0

2a0middot d = 3

r

0




Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved





y4u5=

y1 if u isinU4A51

x4u5 otherwise






gt 00








sum

sisinM


sum

sisinS

xs = 11 (C2)

x ge 00 (C3)



f 4M1 cupM25= max8f 4M151 f 4M2590



f 4M5=

M sum

j=1

x4j5451 where 841514251 0 0 0 14M 59=M0













Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved








sum

sisinM2x415=













(Dual) min


tisinT

t

1


sum

s



tisinT

t ge 1 minus1





arg maxMsubeS

ft41Euml1Iacute1M50






sum







Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved



maxMsubeS

log(

sum

sisinM

zs

)

minus

sum

sisinM hszssum

sisinM zs0



log(

sum

sisinS

ys

)

minus

sum

sisinS hsyssum

sisinS ys0


sum

















MisinMt


sum

MisinMt


sum

tisinTs

sum

MisinMt


sum

s1t

sum

MisinMt





















Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved




























Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved





y4u5=

y1 if u isinU4A51

x4u5 otherwise






gt 00








sum

sisinM


sum

sisinS

xs = 11 (C2)

x ge 00 (C3)



f 4M1 cupM25= max8f 4M151 f 4M2590



f 4M5=

M sum

j=1

x4j5451 where 841514251 0 0 0 14M 59=M0













Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved








sum

sisinM2x415=













(Dual) min


tisinT

t

1


sum

s



tisinT

t ge 1 minus1





arg maxMsubeS

ft41Euml1Iacute1M50






sum







Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved



maxMsubeS

log(

sum

sisinM

zs

)

minus

sum

sisinM hszssum

sisinM zs0



log(

sum

sisinS

ys

)

minus

sum

sisinS hsyssum

sisinS ys0


sum

















MisinMt


sum

MisinMt


sum

tisinTs

sum

MisinMt


sum

s1t

sum

MisinMt





















Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved




























Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved








sum

sisinM2x415=













(Dual) min


tisinT

t

1


sum

s



tisinT

t ge 1 minus1





arg maxMsubeS

ft41Euml1Iacute1M50






sum







Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved



maxMsubeS

log(

sum

sisinM

zs

)

minus

sum

sisinM hszssum

sisinM zs0



log(

sum

sisinS

ys

)

minus

sum

sisinS hsyssum

sisinS ys0


sum

















MisinMt


sum

MisinMt


sum

tisinTs

sum

MisinMt


sum

s1t

sum

MisinMt





















Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved




























Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved



maxMsubeS

log(

sum

sisinM

zs

)

minus

sum

sisinM hszssum

sisinM zs0



log(

sum

sisinS

ys

)

minus

sum

sisinS hsyssum

sisinS ys0


sum

















MisinMt


sum

MisinMt


sum

tisinTs

sum

MisinMt


sum

s1t

sum

MisinMt





















Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved




























Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved




























Dow

nloa

ded

from

info

rms

org

by [

171

662

081

45]

on 1

0 Se

ptem

ber

2015

at 1

131

Fo

r pe

rson

al u

se o

nly

all

righ

ts r

eser

ved

optimal allocation without money: an engineering...

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