Object construction in collectivedecisions

Luigi Marengo

1LEM, Scuola Superiore Sant’Anna, Pisa, l.marengo@sssup.it

based on joint work with Simona Settepanella and CorradoPasquali

“Economics has gained the title of queen of the social sciencesby choosing solved political problems as its domain”

(Abba Lerner, “The economics and politics of consumersovereignty”, American Economic Review, 1972, p. 259).

The general questions

I Decision making=Choice. But choice among what?I Social choice models assume that choice is among

exogenously given, uni-dimensional and “simple” objects.I . . . but where do alternatives come from?I How do (economic) institutions differ in their pre-choice

alternative-generation properties?I How do alternative generation mechanisms frame choice

processes (pattern recognition, identity and logic ofappropriateness)?

I The power of constructing the objects of choice (differentfrom mere issue-raising . . . )

I these are fundamental “political” problems . . .

Applications

I social choiceI organizational designI categorization in individual and collective decision making,

learning and cognition

Outline

I introduction to the problemI application to social choice (aggregation of preferences)

I a computational approachI (hints to) a geometric approach based on hyperplane

arrangements (developed with Simona Settepanella)I application to allocation of decision rights in organizations

A simple example: “What shall we do tonight?”

I C = {movie, theater, restaurant, stay home, . . . }I the object “going to the movies” is defined by:

I with whomI which movieI which theaterI what timeI . . .

I the object “staying at home” is defined by:I with whomI to do whatI e.g. watch TV, or have a drink put on a nice record and see

what happens . . .I which showI which movie

I what we eatI . . .

The big question:

What if C = {movie, theater, restaurant, stay home, . . . } is not“given” but is constructed by an institution or authority byassembling or disassembling in various ways the features “withwhom”, “at what time”, “what we eat”, etc.

Some non-standard properties

1. objects typically do not partition the set of traits/features2. in general there are non-separabilities (interdependencies)

and non-monotonicities among traitsI e.g. I might prefer Françoise to Giovanni as instance of the

“with whom” if associated to “staying at home” and“tête-à-tête dinner”, but Giovanni to Françoise as aninstance of the “with whom” if associated to “going to theAmerican Economic Association meeting”

Our Questions

1. The extent to which object construction can lead to specificsocial outcomes through the selection and categorizationof appropriate traits/features sets

2. The relations between objects and individual preferences3. The extent to which specific objects’ constructions drive

and constrain individual choices.

Sincere majority voting

Some fundamental background:I Condorcet-Arrow paradox: even if everyone has transitive

preferences aggregation may produce social intransitivitiesI Agenda power: the sequence in which alternatives are

voted may determine the outcomeI Median-voter theorem: if there exists a median voter, his or

her most preferred outcome will always beat any otheralternative in any pairwise majority vote

The Model: Definitions I

I Choices are made over bundles of N elements or featuresF = {f1, f2, . . . , fN}, each of which takes on a value out of afinite set possibilities.

I The space of possibilities is given by 2N possible choiceconfigurations: X = {x1, x2, . . . , x2N}.

I There exist h individual agents A = {a1, a2, . . . , ah}, eachcharacterized by complete and transitive preferences) onthe set of choice configurations

I that is each agent has a complete (weak) ranking of choiceconfigurations

Objects

Let I = {1, 2, . . . , N} be the set of indexes.An object Ci ⊆ IThe size of object Ci is its cardinality |Ci |.An objects scheme is a set of objects:

C = {C1, C2, . . . , Ck}

such thatk⋃

i=1

Ci = I

(. . . but not necessarily a partition)

Agendas

An agenda α = Cα1Cα2 . . . Cαk over the object scheme C is apermutation of the set of objects which states the orderaccording to which objects are examined.

Voting procedure

We use the following algorithmic implementation of majorityvoting:

1. repeat for all initial conditions x = x1, x2, . . . , x2N

2. repeat for all objects Cαi = Cα1 , Cα2 , . . . , Cαk until a cycleor a local optimum is found;

3. repeat for j=1 to 2|Cαi |I generate an object instantiation C jαi of object CαiI vote between x and x ′ = C jαi ∨ x(C−αi )I if x ′ º< x then x ′ becomes the new current configuration

Stopping rule

We consider two possibilities:1. objects which have already been settled cannot be

re-examined2. objects which have already been settled can be

re-examined and if new social improvements have becomepossible

Reachable configurations

Given an objects’ scheme C = {C1, C2, . . . , Ck}, we say that aconfiguration x i is a preferred neighbor of configuration x j withrespect to object Ch ∈ C if the following three conditions hold:

1. x i º< x j2. x iν = x

jν ∀ν /∈ Ch

3. x i 6= x jWe call Hi(x , Ci) the set of neighbors of a configuration x forobject Ci .A path P(x i , C) from a configuration x i and for an objects’scheme C is a sequence, starting from x i , of preferredneighbors:P(x i , C) = x i , x i+1, x i+2, . . . with x i+m+1 ∈ H(x i+m, C)A configuration x j is reachable from another configuration x i

and for scheme C if there exist a path P(x i , C) such thatx j ∈ P(x i , C).

Social outcomes

I A configuration x is a local optimum for the objects’scheme C if there does not exist a configuration y suchthat y ∈ H(x , C) and y Â< x

I A cycle is a set X0 = {x10 , x20 , . . . , x j0} of configurationssuch that x10 Â< x20 Â< . . . Â< x j0 Â< x10 and that for allx ∈ X0, if x has a preferred neighbor y ∈ H(x , C) thennecessarily y ∈ X0.

The relevance of objects I

I object construction mechanisms forego and constrainchoices.

I Influence of the generative mechanism:1. define the sequence of voting;2. define which subset of alternatives undergoes examination.

The relevance of objects II

I Different sets of objects may generate different socialoutcomes.

I Social optima do – in general – change when objects aredifferent both because:

1. the subset of generated alternatives is different (and somesocial optima may not belong to many of these subsets)

2. the agenda is different (and this may determine differentoutcomes)

I object generation power appears therefore as a moregeneral phenomenon than agenda power

Results in a nutshell

I Under general conditions (notably if preferences are notfully separable) the answer to the previous question isentirely dependent upon the set of objects

I We show algorithmically that, given a set of individualpreferences:

1. by appropriate modifications of the set of objects it ispossible to obtain different social outcomes

2. cycles à la Condorcet-Arrow may also appear anddisappear by appropriately modifying the set of objects

3. the median voter theorem is also dependent upon the set ofobjects (the median voter may be transformed into outrightloser)

Results I

I Social outcomes are, in general, dependent upon theobjects’ scheme

I Consider a very simple example in which 5 agents have acommon most preferred choice.

I By appropriately modifying the objects’ scheme one canobtain different social outcomes or even theappearance/disappearance of intransitive limit cycles.

Results II

Rank Agent1 Agent2 Agent3 Agent4 Agent51st 011 011 011 011 0112nd 111 000 010 101 1113rd 000 001 001 111 0004th 010 110 101 110 0105th 100 010 000 100 0016th 110 111 110 001 1017th 101 101 111 010 1108th 001 100 100 000 100

Results III

I With C = {{f1, f2, f3}} the only local optimum is the globalone 011 whose basin of attraction is the entire set X

I With C = {{f1}, {f2}, {f3}} we have the appearance ofmultiple local optima and agenda-dependence.

I With C = {{f1, f2}, {f3}} multiple local optima butagenda-independence

Different Global Optima

Rank Agent1 Agent2 Agent31st 011 010 0002nd 000 100 1103rd 010 101 1014th 110 011 0115th 100 000 0106th 101 110 1007th 001 001 0018th 111 111 111

Results III

I With C = {{f1, f2}, {f3}} 000 is the (unique) global optimumI With C = {{f1}, {f2, f3}} 011 is the (unique) global optimum

Object-dependent cycles I

Redefining objects can make cyclesI Consider the case of three agents and three objects with

individual preferences expressed by:Order Agent 1 Agent 2 Agent 3

1st x y z2nd y z x3rd z x y

Object-dependent cycles II

I Social preferences expressed through majority rule areintransitive and cycle among the three objects: x Â< y andy Â< z, but z Â< x .

I Imagine that x,y,z are three-features objects which weencode according to the following mapping:

x 7→ 000, y 7→ 100, z 7→ 010

Object-dependent cycles III

I Suppose that individual preferences are given by:

Order Agent 1 Agent 2 Agent 31st 000 100 0102nd 100 010 0003th 010 000 1004th 110 110 1105th 001 001 0016th 101 101 1017th 011 011 0118th 111 111 111

Object-dependent cycles IV

1. With C = {{f1, f2, f3}} the voting process always ends up inthe limit cycle among x,y and z.

2. The same happens is each feature is a separate object:C = {{f1}, {f2}, {f3}}.

3. However, with:C = {{f1}, {f2, f3}}

or with:C = {{f1, f3}, {f2}}

Voting always produces the unique global social optimum010 in both cases.

Median voter I

Order Ag1 Ag2 Ag3 Ag4 Ag5 Ag6 Ag71st 1 2 3 4 5 6 72nd 2 3 4 5 6 7 63rd 0 1 2 3 4 5 54th 3 4 5 6 7 4 45th 4 0 1 2 3 3 36th 5 5 6 7 2 2 27th 6 6 0 1 1 1 18th 7 7 7 0 0 0 0

Median voter theorem: an example

Median voter II

Order Ag1 Ag2 Ag3 Ag4 Ag5 Ag6 Ag71st 001 010 011 100 101 110 1112nd 010 011 100 101 110 111 1103rd 000 001 010 011 100 101 1014th 011 100 101 110 111 100 1005th 100 000 001 010 011 011 0116th 101 101 110 111 010 010 0107th 110 110 000 001 001 001 0018th 111 111 111 000 000 000 000

If C = {{f1, f2, f3}} there is unique social optimum 100 (medianvoter’s most preferred)If C = {{f1}, {f2}, {f3}} two local optima: 100 and 011 (theopposite of median voter’s most preferred).

Random Preferences

I QUESTION: how likely or plausible are such phenomenaas intransitivities and cycles?

I That is:1. how many local optima may we encounter?2. how different and/or distant from each other are such local

optima?3. how does the number and location of local optima change

with a modification of objects?4. how likely are cycles?

Methodology

1. We ran some simulations of the voting model forpopulations of agents with random (but always completeand transitive) preferences over the elements of the set X

2. We consider a set of 8 binary features and therefore aspace 256 configurations, on which a population of 99random agents vote following the majority rule.

3. Results are averages over 1,000 repetitions of a simulationall with the same parameters but a different randomlygenerated population.

Simulations

I We have tested the following objects’ schemes:I C1 = {{1, 2, 3, 4, 5, 6, 7, 8}}I C2 = {{1, 2, 3, 4}, {5, 6, 7, 8}}I C4 = {{1, 2}, {3, 4}, {5, 6}, {7, 8}}I C8 = {{1}, {2}, {3}, {4}, {5}, {6}, {7}, {8}}

Simulation Results I

I For the objects’ scheme C1, i.e. a single object containingall the features, we have almost always intransitive cyclesand these cycles are rather long (almost 40 differentchoice configuration on average).

I At the other extreme, i.e. the set of finest objects C8, in682 out of 1000 populations we do not observe cycles, butvoting ends in a local optimum. On average there are15.66 local optima

I the number of local optima increases more thanproportionally with N (number of features), with N = 12over 300 local optima

Simulation Results II

I A very clear trade-off between the presence of cycles andthe number of local optima.

I When large objects are employed, cycles almost certainlyoccur.

I The likelihood rapidly drops when finer and finer objectsare employed, but the number of local optima rapidlyincreases.

I This implies that a social outcome becomes well definedbut which social outcome strongly depends upon thespecific objects employed and the sequence in which theyare examined.

A geometric approach (Marengo-Settepanella

I based on hyperplanes arrangementsI also exploits properties of directed graphs

Salvetti’s Complex in social choice

Some theorems

Some theorems:I “object-construction power”: we provide necessary and

sufficient conditions for any social outcome to be a local ora global optimum for a given set of objects

I Condorcet-Arrow intransitive cycles are dependent uponthe set of objects: almost any cycle can be broken byappropriately modification of the set of objects

I we characterize the basin of attraction of every local(global) optimum

Some links with existing literature

I categorization and choice: Lakoff (2004), Mullainathan,Schwartzstein, and Shleifer (2008), Fryer and Jackson(2008)

I multidimensional voting models: Kramer (1972), Shepsle(1979), Denzau and Mackay (1981), Enelow and Hinich(1983)

I context-dependent voting: Callander and Wilson (2006)I logrolling models Tullock (1961), Bernholz (1974), Abrams

(1980)I legislative bundling: Saari and Sieberg (2001)I algebraic and geometric models of voting: Eckmann

(1954), Eckmann, Ganea, and Hilton (1962), Chichilnisky(1980), Chichilnisky (1983)

Part II - Application to Organizational Design

I let us now suppose that each feature fi ∈ {f1, f2, . . . , fN} isa decision to be taken (a policy)

I decisions interact and generate externalitiesI the “problem” is to allocate decisions to individual decision

makersI compare to the “Coase theorem”I we begin with the problem of a principal who has to

allocate decisions to agents

The general questions

I agency models assume that conflict in organizations arisesbecause individuals have diverging objectives

I but conflict arises also from individuals having differentviews of how things should be done, objectives being(roughly) equal

I the two sources of conflict are related and intertwinedI but conflicting views may be harder to reconcile (if people

feel strongly about their views)I and one may not want to fully reconcile them if there is

uncertainty of what should be doneI mis-aligned views may be a fundamental driver of learning

Agency models

I what is to be done for the organization is not problematicI the agent has no interest in and no opinion on what should

be done for the organization (but only on what should bedone for himself, i.e. applying the lowest possible level ofeffort)

I the problem is mis-alignment of objectives, not of “visions”

In our model

I principal and agents have diverging opinions/preferenceson alternative courses of action, not only on the policiesunder their control but on policies under the control of theother members

I this may indeed also reflect differences in objectivesI especially likely and relevant when facing new and

uncertain situationsI the principal faces a trade off between:

I having her views implemented as closely as possibleI use the agents’ different views to learn and discover better

policies

Decision rights and incentives

I if views are different decisions cause a form of externalities(negative and positive) on other agents

I the allocation of decision rights may mitigate or increasesuch externalities

I organizational design as a device to control externalitiesI incentives as a device to compensate for externalities

The Model: policy landscape

I a set of of n (binary) policies P = {p1, p2, . . . , pn}I X is the set of 2n policy vectors and xi = [pi1, p

i2, . . . , p

in]

one generic elementI policy vectors have an exogenously determined

performance level: F : X 7→ [0, 1]I F defines the policy landscape whose ruggedness reflects

the extent of interdependencies among policies and thecomplexity of the problem

The Model: principal, agents and organization

I a principal Π and h agents A = {a1, a2, . . . , ah} with1 ≤ h ≤ n

I all defined by a complete and transitive preferenceordering over policy vectors: ºΠ and ºai

I a decomposition of decision rights D = {d1, d2, . . . , dk}

such that:h⋃

i=1

di = P and di⋂

dj = ∅ ∀i 6= j

(for simplicity the principal does not take directly anydecision)

I the organizational structure is a mapping of the set Donto the set A of agents, plus an agenda (a permutation ofthe set of agents) giving the sequence of decision (if any)

Examples of organizational structure

assuming four policy items:I {a1 ← {p1, p2, p3, p4}}, i.e. one agent has control on all

four policiesI {a1 ← {p1}, a2 ← {p2}, a3 ← {p3}, a4 ← {p4}}, i.e. four

agents have each control on one policyI {a1 ← {p1, p2}, a2 ← {p3, p4}}, i.e. two agents have each

control on two policiesI {a1 ← {p1}, a2 ← {p2, p3, p4}}, i.e. two agents with

“asymmetric” responsibilities: one has control on the firstpolicy item and the other on the remaining three

Agents’ decisions

I when asked to choose between xi and xj an agent selectsthe vector which ranks higher in his preference ordering

I unless the principal provides extra-incentives for reversingthe decision

I simple linearity assumption, if xi ºak xj then in order toinduce agent ak to choose xj the principal must payc|rankk (xi)− rankk (xj)|

Organizational decisions

I an initial status quo policy is (randomly) givenI following the agenda, agents are called to generate

alternative policy sub-vector for the policy items assignedto them by the organizational structure

I between status quo and alternative they choose accordingto own preference combined with incentives

I the process is repeated until the agent has tried allalternatives

I the process is repeated for all agents (according toagenda) until an organizational equilibrium or a cycle arereached

Getting what you want when you know what you want

Order Agent1 Agent2 Agent31st 011 011 0112nd 111 000 0103rd 000 001 1004th 010 110 1015th 100 010 0006th 110 111 1107th 101 101 1118th 001 100 001

Example I: a relatively homogeneous set of agents

Getting what you want when you know what you want

Order Agent1 Agent2 Agent3 Principal1st 011 011 011 0002nd 111 000 010 1013rd 000 001 100 1114th 010 110 101 1105th 100 010 000 1006th 110 111 110 0017th 101 101 111 0108th 001 100 001 011

Example I: how to get a different local optimum

With the organizational structure{a1 ← {p1}, a2 ← {p2}, a3 ← {p3}, with agenda (a1, a2, a3) andthe initial status quo [1, 1, 0], [0, 0, 0] is a local optimum

Different Global Optima

Order Agent1 Agent2 Agent31st 001 000 0012nd 110 111 1103rd 000 001 0004th 010 010 0105th 100 100 1006th 011 011 0117th 111 101 1118th 101 110 101

Example II: cycles or different global optima

I Structure {a1 ← {p1, p2}, a2 ← {p3}} always generates thecycle [001] → [000] → [110] → [111] → [001]. It istherefore a structure in which intra-organizational conflictdoes never settle into an equilibrium

I Structure {a1 ← {p1}, a2 ← {p2}, a3 ← {p3}} has theunique global optimum [001] that is reached from everyinitial condition

I Structure {a1 ← {p1}, a2 ← {p2, p3} also produces aunique global optimum but a different one, i.e. vector [000]

The role of organizational structure

We simulate random problems with 8 policies, randomprincipals and agents and the following organizationalstructures:

I O1: a1 ← {1, 2, 3, 4, 5, 6, 7, 8}I O2: a1 ← {1, 2, 3, 4}, a2 ← {5, 6, 7, 8}I O4: a1 ← {1, 2}, a2 ← {3, 4}, a3 ← {5, 6}, a4 ← {7, 8}I O8: a1 ← {1}, a2 ← {2}, a3 ← {3}, a4 ← {4}, a5 ←{5}, a6 ← {6}, a7 ← {7}, a8 ← {8}

The role of organizational structureAverage number of local optima over 1000 randomly generatedproblems:

Org. Structure N. of local optima

O1 1(0.0)

O2 10.3(1.2)

O4 27.7(2.4)

O8 41.9(3.1)

Number of local optima for different organizations(n=8, 1000 repetitions)

Divide and conquer!!

Organizational structure and control without incentivesMinimum attainable distance between principal’s preferredoutcome and actually implemented policy (1000 randomlygenerated problems):

Org. Structure Best attainable control

O1 127.6(76.0)

O2 22.1(21.9)

O4 7.7(8.0)

O8 4.9(5.1)

Best attainable control at zero cost(n=8, 1000 repetitions)

Organizational structure and control with incentivesCost incurred in order to ontain full control (i.e. zero distancebetween principal’s preferred outcome and actuallyimplemented policy) (1000 randomly generated problems):

Org. Structure Average cost of full control

O1 472.9(219.9)

O2 461.4(64.2)

O4 400.6(32.4)

O8 342.7(24.9)

The cost of full control(n=8, 1000 repetitions)

Organizational structure and performance

Minimum attainable distance between best policy and actuallyimplemented policy (1000 randomly generated problems):

Org. Structure Best attainable control

O1 128.5(75.1)

O2 22.1(21.1)

O4 8.2(8.5)

O8 4.8(5.4)

Best attainable performance(n=8, 1000 repetitions)

Learning principal

I trade-off between aligning the agents’ decisions to theprincipal’s preferences or letting agents more free tochoose policies

I if, by means of appropriate incentive and/or organizationalstructures, the principal optimizes such alignment she willhave her preferred policies efficiently implemented(exploitation)

I but she may lose opportunity to learn that some agents’preferred policies (or combinations thereof) are moreeffective

I Assumption: the principal learns through experimentation,agents do not learn

Learning principal and no extra incentives

0

0.2

0.4

0.6

0.8

1

0 200 400 600 800 1000

Per

form

ance

Iterations

O2

O4

O8

O1

Figure: Learning and organizational structures

Learning principal and no extra incentives

in environments with low complexity:I fine decompositions of decision rights and medium or low

powered powered incentives are more efficient, as theorganization quickly climbs one of the few local optimawhose location is easily learnt by the principal (because oflow complexity);

Learning principal and no extra incentives

in environments with high complexity:I we must distinguish cases in which the competitive

pressure is low or high:I with low competitive pressure, learning is fostered if

decision rights are kept together under the control of one orvery few agents and incentives are medium powered.However learning tends to be slow

I as competitive pressure increases and it becomesnecessary to fast increase the performance, the division ofdecision rights must increase and incentives must becomestronger