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Cooperative Games, Mechanism Cooperative Games, Mechanism Design, and AuctionsDesign, and Auctions

Onn Shehory Onn Shehory

March 9-13 2009March 9-13 2009Politecnico di MilanoPolitecnico di Milano

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The Glove GameThe Glove Game Players set: {1,2,3}Players set: {1,2,3} 1,2 have right-had gloves1,2 have right-had gloves 3 has left-had gloves3 has left-had gloves Players may stay alone and profit 0Players may stay alone and profit 0 Or join together: {1,3}, {2,3}, {1,2,3} to gain 1Or join together: {1,3}, {2,3}, {1,2,3} to gain 1

What mechanism should they follow?What mechanism should they follow? How do they behave given such mechanism?How do they behave given such mechanism? How is profit divided?How is profit divided?

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The English AuctionThe English Auction

An item is placed for saleAn item is placed for sale Players free to bidPlayers free to bid New bid must be higher than currentNew bid must be higher than current Winner: highest bidWinner: highest bid

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OutlineOutline Protocols and strategiesProtocols and strategies Attitudes and rationalityAttitudes and rationality Stability and equilibriumStability and equilibrium

– Nash revisitedNash revisited Cooperative gamesCooperative games Solution conceptsSolution concepts Coalition FormationCoalition Formation RFP coalitionsRFP coalitions Mechanism designMechanism design AuctionsAuctions Auctions – field resultsAuctions – field results

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What is a Mechanism/Protocol?What is a Mechanism/Protocol? A protocol (aka mechanism):A protocol (aka mechanism):

– provides a set of rules and behaviors to be provides a set of rules and behaviors to be followed by its participantsfollowed by its participants

– following the rules of a protocol is to a following the rules of a protocol is to a player’s discretion, though deviation may player’s discretion, though deviation may leave her “out of the game” leave her “out of the game”

– examples: auctions, negotiation, votingexamples: auctions, negotiation, voting

Desired properties:Desired properties:– maximize payoffsmaximize payoffs– not manipulable/enforceablenot manipulable/enforceable– simple to implement and executesimple to implement and execute

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What are Strategies?What are Strategies? A strategy:A strategy:

– is one of the possible actions a player can select given is one of the possible actions a player can select given the protocolthe protocol

– is not dictated (or provided) by the protocol is not dictated (or provided) by the protocol – is usually the result of the player’s reasoning and is usually the result of the player’s reasoning and

decisions, based on local algorithms and informationdecisions, based on local algorithms and information– examples: examples:

in an auction – bid as low as possiblein an auction – bid as low as possible in elections – vote for your factionin elections – vote for your faction

A good strategy:A good strategy:– should maximize the player’s payoff given the protocol should maximize the player’s payoff given the protocol

and the behavior of other playersand the behavior of other players– should be difficult or impossible to manipulateshould be difficult or impossible to manipulate– should be computationally feasibleshould be computationally feasible– may depend on the strategies of other playersmay depend on the strategies of other players

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Players AttitudesPlayers Attitudes Self-interestSelf-interest: a self-interested player is : a self-interested player is

attempting to maximize its own personal attempting to maximize its own personal payoffpayoff

Benevolence/altruismBenevolence/altruism: a benevolent player : a benevolent player is attempting to increase others’ payoffs is attempting to increase others’ payoffs and the cumulative payoff of the societyand the cumulative payoff of the society

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Risk AttitudesRisk Attitudes

Risk prone:Risk prone: a player that has a preference for risk a player that has a preference for risk Risk averse:Risk averse: a player that has a preference for a player that has a preference for

avoiding riskavoiding risk Risk neutral:Risk neutral: a player that has no risk preference a player that has no risk preference Players do not need to be strictly prone, neutral or Players do not need to be strictly prone, neutral or

averse – they may mix theseaverse – they may mix these Human players tend to have alternating and Human players tend to have alternating and

context dependent risk attitudescontext dependent risk attitudes

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Example Risk AttitudesExample Risk Attitudes

You are offered two options:You are offered two options:1.1. Get 1000 Euro, cashGet 1000 Euro, cash2.2. Get a lottery certificate, with a prize value of Get a lottery certificate, with a prize value of

10,000 Euro, and a 10% chance of winning10,000 Euro, and a 10% chance of winning What should you choose?What should you choose?

– Select 2, for a chance for getting 10,000?Select 2, for a chance for getting 10,000?– Select 1 to avoid risk?Select 1 to avoid risk?

What should you choose if chance is 20%?What should you choose if chance is 20%?– Select 2, to maximize expected utility (2000)?Select 2, to maximize expected utility (2000)?

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RationalityRationality A rational behavior is such that prefers a A rational behavior is such that prefers a

greater payoff over a smaller onegreater payoff over a smaller one A rational player should always behave A rational player should always behave

rationally. That is, from among several rationally. That is, from among several options available, he should select the one options available, he should select the one that results in maximum payoffthat results in maximum payoff

The problem:The problem:– in may cases the number of options is in may cases the number of options is

overwhelmingoverwhelming– there may be no algorithm for finding the bestthere may be no algorithm for finding the best

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Bounded RationalityBounded Rationality To overcome problems of rationality, To overcome problems of rationality,

bounded rationality:bounded rationality:– limits the time/computation for option limits the time/computation for option

considerationconsideration– prunes the search spaceprunes the search space– imposes restrictions on the types of optionsimposes restrictions on the types of options

Results in fewer possibilities, henceResults in fewer possibilities, hence– computationally feasiblecomputationally feasible– may be too restrictive, far from optimalmay be too restrictive, far from optimal– strategically inferior to rationalstrategically inferior to rational

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““Good Enough” BehaviorGood Enough” Behavior Make the bounded rationality rational:Make the bounded rationality rational:

– modify linear payoff functions to incorporate modify linear payoff functions to incorporate computational costscomputational costs

– put a cap on payoffput a cap on payoff– add a small-amounts’ indifference add a small-amounts’ indifference

The payoff of an option is good enough ifThe payoff of an option is good enough if– too much additional computation to find other good too much additional computation to find other good

options, or options, or – other options do not provide a significant payoff increase, other options do not provide a significant payoff increase,

oror– the player is indifferent w.r.t. the increasethe player is indifferent w.r.t. the increase

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Stability and EquilibriaStability and Equilibria

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Protocol EvaluationProtocol Evaluation Payoff maximizationPayoff maximization: can refer to : can refer to

individual payoffs, group payoffs, or individual payoffs, group payoffs, or social welfare - the sum of individual social welfare - the sum of individual payoffspayoffs

Pareto-optimalityPareto-optimality: a payoff vector : a payoff vector p(xp(x11,x,x22,…,x,…,xnn) is Pareto-optimal if there is ) is Pareto-optimal if there is no other feasible payoff vector p' such no other feasible payoff vector p' such that at least one payoff is better in p' that at least one payoff is better in p' and no payoff is worse in pand no payoff is worse in p

StabilityStability: a protocol is stable if once the : a protocol is stable if once the players arrived at a solution they do not players arrived at a solution they do not deviate from itdeviate from it

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Stability and EquilibriaStability and Equilibria There are multiple stability concepts. In game There are multiple stability concepts. In game

theory, the notion of equilibrium is used:theory, the notion of equilibrium is used:– dominant strategiesdominant strategies: the agents have some strategies : the agents have some strategies

that, regardless of what others do, maximize payoffthat, regardless of what others do, maximize payoff– Nash equilibriumNash equilibrium: the agents have strategies that, as : the agents have strategies that, as

long as other stick to theirs, maximize payofflong as other stick to theirs, maximize payoff– Mixed NashMixed Nash: the agents each have a set of strategies : the agents each have a set of strategies

from among which they select one with some probabilityfrom among which they select one with some probability– Bayes-NashBayes-Nash: adds types (e.g. history) to the previous : adds types (e.g. history) to the previous

oneone

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The Prisoner’s DilemmaThe Prisoner’s Dilemma

Player 2’s strategiesPayoff table for a 2-player, no-repetitionPrisoner’s Dilemmagame

Cooperate Defect

Cooperate2, 2 -2, 4

Player 1’sstrategies

Defect4,-2 -1,-1

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No Pure Nash EquilibriumNo Pure Nash Equilibrium

Player 2’s strategiesPayoff table for a 2-player game with nopure Nash equilibrium Cooperate Defect

Cooperate4, 0 0, 2

Player 1’sstrategies

Defect2, 0 3,-2

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No Pure Nash EquilibriumNo Pure Nash Equilibrium

Player 2’s strategiesPayoff table for a 2-player game with nopure Nash equilibrium Cooperate Defect

Cooperate6, 2 0, 3

Player 1’sstrategies

Defect0, 1 5, 0

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Mixed NashMixed Nash Player 1 will cooperate with probability pPlayer 1 will cooperate with probability pcc and and

defect with probability pdefect with probability pdd

Player 2 will cooperate with probability qPlayer 2 will cooperate with probability qcc and and

defect with probability qdefect with probability qdd

Expected utility of an agent is the utility from a Expected utility of an agent is the utility from a strategy times the probability of this strategy strategy times the probability of this strategy being selectedbeing selected

When there are multiple possibilities, the When there are multiple possibilities, the expected utility is a sum over these possibilities expected utility is a sum over these possibilities

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Computing the ProbabilitiesComputing the Probabilities The expected utility of an player x when the other The expected utility of an player x when the other

player y follows strategy s is denoted by Uplayer y follows strategy s is denoted by Uxx(s)(s)

In the case of equilibrium (mixed Nash), the In the case of equilibrium (mixed Nash), the expected utility of x should be the same for all of expected utility of x should be the same for all of the possible strategies of ythe possible strategies of y

In our case we have players In our case we have players 11,,22 and strategies and strategies c,dc,d

We require that UWe require that Uxx(c) = U(c) = Uxx(d), which means that (d), which means that

for each of the two players, the expected utility for each of the two players, the expected utility from the other cooperating should be equal to the from the other cooperating should be equal to the expected utility from the other defectingexpected utility from the other defecting

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Computation DetailsComputation Details

For For player 1player 1 we have: we have: UU11((cc)=)= 6 p6 pcc+ 0 p+ 0 pdd, U, U11((dd)=)= 0 p0 pcc+ 5 p+ 5 pdd

For For player 2player 2 we have: we have: UU22((cc)=)= 2 q2 qcc+ 3 q+ 3 qdd, U, U22((dd)=)= 1 q1 qcc+ 0 q+ 0 qdd

The requirement that UThe requirement that Uxx(c) = U(c) = Uxx(d) results in:(d) results in:

qqcc= 0.642, q= 0.642, qd d = 0.358= 0.358

ppcc= 0.317, p= 0.317, pd d = 0.683= 0.683

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Tragedy of Common GoodsTragedy of Common Goods Information on the web is (mostly) freeInformation on the web is (mostly) free Web agents that seek up to date information may Web agents that seek up to date information may

query web site as frequently as desiredquery web site as frequently as desired If all agents will do so, the network will be overly If all agents will do so, the network will be overly

congested, and some servers will crashcongested, and some servers will crash So, is it undesirable to behave this way?So, is it undesirable to behave this way? If all (or most) of the agents prevent congestion, If all (or most) of the agents prevent congestion,

it is in the best interest of each individual agent it is in the best interest of each individual agent to increase network use ... to increase network use ...

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Cooperative GamesCooperative Games

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Cooperative GamesCooperative Games

A cooperative game (aka coalitional game):A cooperative game (aka coalitional game):– Cooperation within groups is enforceableCooperation within groups is enforceable– Groups compete, and not individualsGroups compete, and not individuals

– Each group (=coalition) has a value Each group (=coalition) has a value vv– Characteristic function: Characteristic function:

vv : 2 : 2NN, from coalitions to payments , from coalitions to payments – Players decide which coalitions to form (to Players decide which coalitions to form (to

maximize payoff)maximize payoff)

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Coalitional GamesCoalitional Games Coalitional game (Coalitional game (N,vN,v))

– A set of players A set of players NN

– A A coalitioncoalition SS is a group of players, subset of is a group of players, subset of NN, which cooperate, which cooperate

– Value (or utility) of a coalition Value (or utility) of a coalition vv v(S)v(S) is a real, represents the gain of coalition is a real, represents the gain of coalition SS in the game ( in the game (NN,,vv)) v(N)v(N) is the value of forming the is the value of forming the grand coalition,grand coalition, coalition of all players coalition of all players

– Player payoff Player payoff xxii

The portion of The portion of v(S)v(S) received by a player received by a player ii in coalition in coalition SS

Characteristic function form implies:Characteristic function form implies:

– vv depends only on the depends only on the internal structureinternal structure of the coalition of the coalition Transferable utilityTransferable utility

– The value of a coalition can be distributed arbitrarily among its The value of a coalition can be distributed arbitrarily among its playersplayers

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Coalitional Games: ExampleCoalitional Games: Example Example: Majority VoteExample: Majority Vote

– Prime minister is elected by majority votePrime minister is elected by majority vote

– A coalition consisting of a majority of players has a A coalition consisting of a majority of players has a worth of 1 since it is a decision makerworth of 1 since it is a decision maker

– Value of a coalition does not depend on the Value of a coalition does not depend on the external strategies of the users => characteristic external strategies of the users => characteristic function form function form

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Super-additive GamesSuper-additive Games

Super-additive gameSuper-additive game– (N,v)(N,v) is is super-additivesuper-additive if if

– Here, cooperation is always beneficialHere, cooperation is always beneficial

– Unification of two coalitions increases overall payoffUnification of two coalitions increases overall payoff

Monotonicity: larger coalitions gain more Monotonicity: larger coalitions gain more Not all games are super-additive! Not all games are super-additive!

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Coalitional StabilityCoalitional Stability

Stability of a coalitionStability of a coalition– Depends on how the value Depends on how the value vv isis distributed distributed

among the playersamong the players

– How to divide How to divide vv ?? Improper payoff division => players may leave Improper payoff division => players may leave

the coalition, unhappy with their sharethe coalition, unhappy with their share

– Multiple solution concepts address this Multiple solution concepts address this pointpoint

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Coordination GameCoordination Game

Strategy table

Player 2’s strategies

A, a B, c Player 1’s strategies

C, b D,d

• For 1, A>C, D>B

• For 2, a>c, d>b

• Red circles are pure Nash

• For driving side, all benefit if all adopt the same side, but two equilibria points exist

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Solution ConceptsSolution Concepts

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Solution ConceptsSolution Concepts Assumption: the Assumption: the grand coalitiongrand coalition will form will form

– Even when the solution includes multiple coalitionsEven when the solution includes multiple coalitions Solve for sub-gamesSolve for sub-games

– Each players gets a payoff Each players gets a payoff xxii

Major issue: payoff distributionMajor issue: payoff distribution A solution concept is a payoff allocation vector A solution concept is a payoff allocation vector xx NN

Efficiency: Efficiency: ∑ ∑ xxii = = vv(N)(N) Individual rationality: Individual rationality: xxii ≥ ≥ vv((ii)) Group rationality: Group rationality: vv((SS)) ≤ ∑ ≤ ∑ xxii SS ImputationImputation: a payoff allocation vector which is efficient : a payoff allocation vector which is efficient

and individually rational (and group rational for N)and individually rational (and group rational for N) Many solution concepts are imputationsMany solution concepts are imputations Players prefer coalitions based on their respective Players prefer coalitions based on their respective

payoffspayoffs

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ExampleExample

Socks sellingSocks selling– Sold in pairs for 3euro a pairSold in pairs for 3euro a pair– 2 sellers, each holds 5 socks2 sellers, each holds 5 socks– Each can get 6euro, but together they get Each can get 6euro, but together they get

15euro.15euro.– Imputations: (6, 9), (7, 8), (7.5, 7.5)Imputations: (6, 9), (7, 8), (7.5, 7.5)

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Some PropertiesSome Properties

Null playerNull player: when added to a coalition, contributes : when added to a coalition, contributes nothing to its valuenothing to its value

Existence Existence (of a solution concept): determines (of a solution concept): determines whether the solution concept exists for every whether the solution concept exists for every game game – Examples: Kernel exists, Core does notExamples: Kernel exists, Core does not

SymmetrySymmetry: symmetric players receive equal : symmetric players receive equal payoffspayoffs

UniquenessUniqueness: the solution concept is unique: the solution concept is unique

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The Shapley Value (1953)The Shapley Value (1953)

Unique, efficient, symmetric, allocates zero to Unique, efficient, symmetric, allocates zero to null playersnull players

Individually rational for super-additive gamesIndividually rational for super-additive games The payoff allocated to player The payoff allocated to player ii is is

φφii((vv) = 1/n! Σ) = 1/n! ΣSSN\N\ii (s!-(n-s-1)!) ( (s!-(n-s-1)!) (vv((SS∪∪ii)-)-vv((SS))))

s = |s = |SS|| This is considered a fair allocationThis is considered a fair allocation

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The Glove Game ExampleThe Glove Game Example

N={1,2,3}N={1,2,3} 1,2 have right-had gloves1,2 have right-had gloves 3 has left-had gloves3 has left-had gloves vv((SS) = 1 for {1,3}, {2,3}, {1,2,3}, otherwise 0) = 1 for {1,3}, {2,3}, {1,2,3}, otherwise 0 φφ11((vv) = 1/6) = 1/6

φφ22((vv) = 1/6) = 1/6

φφ33((vv) = 4/6) = 4/6

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The Stable SetThe Stable Setvon Neumann & Morgenstern (1944)von Neumann & Morgenstern (1944)

Given a game Given a game vv and imputations and imputations xx,,yy xx dominates dominates yy if if

– there is a nonempty coalition there is a nonempty coalition S, S, such thatsuch that– the members of the members of SS prefer payoff from prefer payoff from xx over over

payoff from payoff from yy

– vv((SS) ) ≥ ∑ ≥ ∑ xxii SS

SS players can threaten to quit the grand players can threaten to quit the grand coalition if coalition if xx not implemented not implemented

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Stable Set ExampleStable Set Example

xx=(1,2,3), =(1,2,3), yy=(3,2,1)=(3,2,1) For players 1,2, For players 1,2, yy is better than is better than xx Does Does yy dominate dominate xx? ?

– Depends on Depends on vv({1,2})({1,2})

If If vv({1,2}) ({1,2}) ≥ 5, ≥ 5, yy dominates dominates x x via {1,2}via {1,2}

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Stable Set Example 2Stable Set Example 2

xx=(50,50,0), =(50,50,0), yy=(0,60,40), =(0,60,40), zz=(15,0,85)=(15,0,85) vv({1,2}) ({1,2}) = = vv({1,3}) = ({1,3}) = vv({2,3}) =100({2,3}) =100 yy dominates dominates x x via {2,3}via {2,3} zz dominates dominates y y via {1,3}via {1,3} xx dominates dominates z z via {1,2}via {1,2}

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Some PropertiesSome Properties

Existence: the stable set may or may not existExistence: the stable set may or may not exist Uniqueness: when exists, it is usually not uniqueUniqueness: when exists, it is usually not unique Internal stability: no imputation within the stable Internal stability: no imputation within the stable

set dominate one anotherset dominate one another External stability: all payoff vectors outside the External stability: all payoff vectors outside the

stable set are dominated by at least on member of stable set are dominated by at least on member of the setthe set

Interpretation: the stable set represents conflicts, Interpretation: the stable set represents conflicts, but excludes inferior behaviorsbut excludes inferior behaviors

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The CoreThe Core The coreThe core is a set of imputationsis a set of imputations (x(x11, . . .,, . . ., xxNN)) satisfying satisfying

two conditionstwo conditions

No coalition has a value greater than the sum of its No coalition has a value greater than the sum of its members’ payoffs (coalition rationality)members’ payoffs (coalition rationality)

The core can be emptyThe core can be empty A non-empty core in a super-additive game => stable A non-empty core in a super-additive game => stable

grand coalitiongrand coalition– No coalition has an incentive to leave (and receive a greater No coalition has an incentive to leave (and receive a greater

payoff)payoff)

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Properties of the CoreProperties of the Core

May be empty (rather common)May be empty (rather common) Not uniqueNot unique Subset of the stable setSubset of the stable set Has several variantsHas several variants

– E.g., epsilon-coreE.g., epsilon-core– Least-coreLeast-core

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Shoes ExampleShoes Example A pair: a left and a right shoe. Sold for €50A pair: a left and a right shoe. Sold for €50 21 players: 10 have 1 left shoe, 11 have 1 right shoe21 players: 10 have 1 left shoe, 11 have 1 right shoe The core: a single imputation, gives 10 to left shoe owners, 0 to right The core: a single imputation, gives 10 to left shoe owners, 0 to right

shoe ownersshoe owners Any left-right pair can form a coalition and sell for €50Any left-right pair can form a coalition and sell for €50 Any such pair getting less than that will block the imputationAny such pair getting less than that will block the imputation For imputation in the core, any of these pairs gets exactly 50 (we can For imputation in the core, any of these pairs gets exactly 50 (we can

only sell 10 pairs, totaling to 500)only sell 10 pairs, totaling to 500) One right-shoe owner gets 0 paymentOne right-shoe owner gets 0 payment Examine the pairs: if any left-shoe owner gets less than 50, say 40, it Examine the pairs: if any left-shoe owner gets less than 50, say 40, it

can join this player, sell their shoes, give her 5, and keep 45 to herself. can join this player, sell their shoes, give her 5, and keep 45 to herself. Both are better offBoth are better off

But such a left-shoe owner cannot exist: all left shoe owners get But such a left-shoe owner cannot exist: all left shoe owners get already 50.already 50.

This holds for any unequal partition This holds for any unequal partition

The core is very sensitive to oversupply.The core is very sensitive to oversupply.

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The KernelThe KernelDavis & Maschler (1965)Davis & Maschler (1965)

We do not assume imputationsWe do not assume imputations That’s it – no group rationalityThat’s it – no group rationality Efficiency still requiredEfficiency still required Result: Result: we are interested no only in payoff we are interested no only in payoff

distribution, but in the coalitions that formdistribution, but in the coalitions that form Implicitly assumes bargaining (but in practice Implicitly assumes bargaining (but in practice

computes its result)computes its result)

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SurplusSurplus

Maximum surplusMaximum surplus– Given an efficient payoff vector Given an efficient payoff vector xx and players and players ii,,j, j, the the

maximum surplus of maximum surplus of ii over over jj is: is:

ssijij(x)= max((x)= max(vv((SS)- )- ∑∑kkSS xxk k : : iiS, S, jjSS))

– the maximal amount the maximal amount ii can gain without the cooperation can gain without the cooperation of of jj by withdrawing from the grand coalition by withdrawing from the grand coalition NN (where (where x x applies), other players in applies), other players in ii's withdrawing coalition are 's withdrawing coalition are satisfied with their satisfied with their xx payoffs payoffs

The maximum surplus measures a player's The maximum surplus measures a player's bargaining power over anotherbargaining power over another

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The Kernel DefinedThe Kernel Defined

The The kernelkernel is the set of payoff vectors is the set of payoff vectors xx that that satisfy, for all pairs satisfy, for all pairs ii,,jj::(s(sijij((xx)-s)-sjiji((xx))))··((xxj j - - vv((jj)) )) ≤≤ 0 0andand(s(sjiji((xx)-s)-sijij((xx))))··((xxi i - - vv((ii)) )) ≤≤ 0 0

If sIf sijij((xx) > s) > sjiji((xx) then ) then ii outweighsoutweighs j – j – it has more it has more bargaining powerbargaining power

But ifBut if xxj j = = vv((jj), ), jj can obtain can obtain xxjj on his own, thus on his own, thus ii’s ’s threat is invalidatedthreat is invalidated

For vectors in the kernel, no player can outweigh For vectors in the kernel, no player can outweigh another – no valid threat another – no valid threat stability stability

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Kernel PropertiesKernel Properties Equilibrium of agents‘ surpluses:Equilibrium of agents‘ surpluses:

In each coalition no player can outweigh another, In each coalition no player can outweigh another, thus getting a better payoff (surplus) in an alternative thus getting a better payoff (surplus) in an alternative coalition excluding the opponentcoalition excluding the opponent„„I can get more without you, than you can without I can get more without you, than you can without me.“me.“

Exists, Pareto-optimal, not uniqueExists, Pareto-optimal, not unique A subset of the bargaining setA subset of the bargaining set Exponentially hard to computeExponentially hard to compute Computational solution: Stearns (1968) Computational solution: Stearns (1968)

– May converge very slowlyMay converge very slowly– A single point out of manyA single point out of many

Polynomial variants (Shehory/Kraus 1996; Polynomial variants (Shehory/Kraus 1996; Klusch/Shehory 1996)Klusch/Shehory 1996)

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ExampleExample

Three playersThree players vv((ii)=0, )=0, vv(12)=2, (12)=2, vv(13)=3, (13)=3, vv(23)=4, (23)=4, vv(123)=8(123)=8

– Kernel: 2, 2.5, 3.5Kernel: 2, 2.5, 3.5– Shapley: 13/6, 16/6, 19/6Shapley: 13/6, 16/6, 19/6

vv((ii)=0, )=0, vv(12)=2, (12)=2, vv(13)=3, (13)=3, vv(23)=4, (23)=4, vv(123)=5(123)=5– Kernel: 2/3, 5/3, 8/3Kernel: 2/3, 5/3, 8/3– Shapley: 7/6, 10/6, 13/6Shapley: 7/6, 10/6, 13/6

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More ConceptsMore Concepts

The Bargaining setThe Bargaining set The NucleolusThe Nucleolus Both based on Both based on excess: excess:

vv((SS)- )- ∑∑kkSS xxk k

Possible gain of players when quitting a Possible gain of players when quitting a coalitioncoalition

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Coalition FormationCoalition Formation

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Coalitions and ComplexityCoalitions and Complexity Given N players, there are 2Given N players, there are 2NN-1 different -1 different

possible coalitionspossible coalitions– If there are k tasks, may need to multiply by If there are k tasks, may need to multiply by

exp(k)exp(k) The number of configurations is O(NThe number of configurations is O(N(N/2)(N/2))) Hence, exhaustive search is infeasible Hence, exhaustive search is infeasible Additionally, players may have conflicting Additionally, players may have conflicting

preferences over the possible preferences over the possible configurationsconfigurations

Nevertheless, coalitions are beneficialNevertheless, coalitions are beneficial

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Issues in Coalition FormationIssues in Coalition Formation

Given a set of tasks and a set of players, which Given a set of tasks and a set of players, which coalitions should a player attempt to form?coalitions should a player attempt to form?

What What mechanismmechanism can agents use for coalition can agents use for coalition formation?formation?

What guarantees regarding efficiency and quality What guarantees regarding efficiency and quality can the mechanism provide?can the mechanism provide?

Once a coalition has formed, how should its Once a coalition has formed, how should its members go about distribution of work/payoff?members go about distribution of work/payoff?

When, and how, does a coalition dissolve?When, and how, does a coalition dissolve?

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Agent2

Agent11

Agent3

Agent10

Agent8

Agent13Agent12

Agent9

Agent6Agent5

Agent1

Agent4

Agent7T1 T2

T4

T3

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Factors Affecting SolutionFactors Affecting Solution Self-interest vs. benevolence: Self-interest vs. benevolence:

– Simpler mechanisms for benevolent playersSimpler mechanisms for benevolent players– Do not need means for individual payoff maximizationDo not need means for individual payoff maximization

Centralization vs. distribution: Centralization vs. distribution: – Central design of coalitions is usually much simpler to Central design of coalitions is usually much simpler to

execute and enforceexecute and enforce

Super-additivity: Super-additivity: – In super-additive environments any unification of two In super-additive environments any unification of two

coalitions increases overall payoffcoalitions increases overall payoff– Strongly influences the mechanism - simplifiesStrongly influences the mechanism - simplifies

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Example Application Domain: Example Application Domain: E-CommerceE-Commerce

B2C: Wholesale marketsB2C: Wholesale markets– Purchasing more of the same product reduces price per Purchasing more of the same product reduces price per

unitunit– Sellers benefit from selling more and spending less on Sellers benefit from selling more and spending less on

marketing/distributionmarketing/distribution– Usually, buyers do not need large quantitiesUsually, buyers do not need large quantities– Can form coalitions of buyersCan form coalitions of buyers

B2B: Request for Proposal (RFP) marketsB2B: Request for Proposal (RFP) markets– A requested product/service can be provided only by A requested product/service can be provided only by

groups of suppliers, hence coalitions are a mustgroups of suppliers, hence coalitions are a must– Difficulties:Difficulties:

Valuations of tasks vary across suppliers, and are private Valuations of tasks vary across suppliers, and are private informationinformation

Time for submitting proposals is limited, and value may be Time for submitting proposals is limited, and value may be discounteddiscounted

5555

Game Theoretic Solutions?Game Theoretic Solutions?

Computation usually hyper-exponentialComputation usually hyper-exponential Centralized Centralized Stable, butStable, but

– Have multiple equilibria points in a solutionHave multiple equilibria points in a solution– Are sensitive to small changesAre sensitive to small changes

Not in strategic form. Transformation is Not in strategic form. Transformation is complexcomplex

No formation mechanism providedNo formation mechanism provided No dynamics: agents cannot join or leave No dynamics: agents cannot join or leave

existing coalitionsexisting coalitions

5656

Classical Solutions: Kernel RevisitedClassical Solutions: Kernel Revisited

Attaches payoff vector(s) to a configurationAttaches payoff vector(s) to a configuration The solution is Pareto-optimal, non-emptyThe solution is Pareto-optimal, non-empty Based on bargaining: proposals/objectionsBased on bargaining: proposals/objections Bargaining/proposals are bilateralBargaining/proposals are bilateral Convergence to an equilibrium is guaranteedConvergence to an equilibrium is guaranteed 22nn coalitions, O(n coalitions, O(nn/2n/2) configurations) configurations No distribution, no algorithms, etc.No distribution, no algorithms, etc.

5757

Practical SolutionsPractical Solutions Prune the solution search spacePrune the solution search space Design a mechanism to motivate players to followDesign a mechanism to motivate players to follow Provide strategies, (and algorithms) for a player to Provide strategies, (and algorithms) for a player to

maximize utility given the mechanism maximize utility given the mechanism Make sure that the mechanism + strategies arrive at Make sure that the mechanism + strategies arrive at

stability and near optimumstability and near optimum Outstanding issues for most of the solutions:Outstanding issues for most of the solutions:

– Complexity/scalability: near-optimal solutions are only good Complexity/scalability: near-optimal solutions are only good for dozens of agentsfor dozens of agents

– Dynamics: joining/leaving coalitions usually not resolvedDynamics: joining/leaving coalitions usually not resolved– Unrealistic, restrictive assumptions:Unrealistic, restrictive assumptions:

Complete informationComplete information No uncertainty and subjectivity with regards to informationNo uncertainty and subjectivity with regards to information

5858

Example: the Modified KernelExample: the Modified Kernel

Use a small subset of coalitions, Use a small subset of coalitions, configurationsconfigurations

Method to compute the Kernel (even with Method to compute the Kernel (even with pruned space) is exponential (Stearns 68): pruned space) is exponential (Stearns 68): allow Kernel+allow Kernel+

Modification violates equilibrium: correct Modification violates equilibrium: correct by applying a computation cost functionby applying a computation cost function

Dynamism, subjective valuations? Dynamism, subjective valuations?

5959

Coalition Formation Coalition Formation Mechanism Example Mechanism Example

6060

Example MechanismExample Mechanism

RFP coalitionsRFP coalitions Protocol and strategiesProtocol and strategies Use of heuristicsUse of heuristics Experimental equilibriumExperimental equilibrium

6161

RFP CoalitionsRFP Coalitions

Problem properties:Problem properties:– Tasks can only be performed by groupsTasks can only be performed by groups– A task is comprised from subtasksA task is comprised from subtasks– A task has a deadline and a value (discounted over time)A task has a deadline and a value (discounted over time)– Agents have Agents have private, subjective valuationsprivate, subjective valuations of subtasks of subtasks– Agents are self-interested utility maximizersAgents are self-interested utility maximizers

Solution approach:Solution approach:– Agents negotiate under time pressure to form coalitionsAgents negotiate under time pressure to form coalitions– Decisions during negotiation are derived via strategiesDecisions during negotiation are derived via strategies– Complete search of the problem space is infeasibleComplete search of the problem space is infeasible– Consequently – a simulation-based solutionConsequently – a simulation-based solution

6262

The ProtocolThe Protocol

General structure: General structure: – Auction-like mechanism for task allocation to coalitionsAuction-like mechanism for task allocation to coalitions– Negotiation for coalition formationNegotiation for coalition formation

Participants: Participants: – Businesses in pressing need for complex products/services Businesses in pressing need for complex products/services

issue RFPs (with a price and a discount rate)issue RFPs (with a price and a discount rate)– Suppliers that can address parts of the RFPs negotiate and Suppliers that can address parts of the RFPs negotiate and

join potential coalitions, then submit joint proposalsjoin potential coalitions, then submit joint proposals– A neutral third party manager serves two roles:A neutral third party manager serves two roles:

An auctioneerAn auctioneer A coalition negotiation managerA coalition negotiation manager

Suppliers are represented by autonomous agents, Suppliers are represented by autonomous agents, these negotiate and submit proposals on their behalfthese negotiate and submit proposals on their behalf

6363

The Manager RoleThe Manager Role

Auctioneer:Auctioneer:– Publishes available RFPsPublishes available RFPs– Collects proposals and determines winning coalitionCollects proposals and determines winning coalition– Monitors completion of task and distributes paymentMonitors completion of task and distributes payment

Negotiation manager:Negotiation manager:– Brokers proposals among agentsBrokers proposals among agents– Verifies joint capabilities and net costsVerifies joint capabilities and net costs– Sets payments according to costs and a profit Sets payments according to costs and a profit

distribution scheme (equal)distribution scheme (equal)– Determines time ordering among proposals Determines time ordering among proposals – Monitors adherence with protocol Monitors adherence with protocol

6464

Coalition NegotiationCoalition Negotiation

Iterative – one proposal per agent at each iterationIterative – one proposal per agent at each iteration Agents either propose or wait, committedAgents either propose or wait, committed More beneficial proposals are preferredMore beneficial proposals are preferred Time is an issue because of discountTime is an issue because of discount Agents must follow protocol, can use any strategy Agents must follow protocol, can use any strategy

for proposal preparation/decisionfor proposal preparation/decision Strategy space is intractable – we propose some Strategy space is intractable – we propose some

strategies based on heuristicsstrategies based on heuristics

6565

StrategiesStrategies Goal: decide which coalitions to propose to which agents and Goal: decide which coalitions to propose to which agents and

accept/reject proposalsaccept/reject proposals Strategies are based on heuristics for ranking coalitions Strategies are based on heuristics for ranking coalitions

according to desirabilityaccording to desirability General guidelines: General guidelines:

– Inspect RFP tasks and subtasksInspect RFP tasks and subtasks– Inspect capabilities and capacities of other agentsInspect capabilities and capacities of other agents– Compute candidate coalitions, then rank themCompute candidate coalitions, then rank them

Ranking heuristics:Ranking heuristics:– MarginalMarginal: prefer coalitions where the estimated marginal profit of : prefer coalitions where the estimated marginal profit of

the coalition is maximalthe coalition is maximal– ExpertExpert: prefer coalitions where only a few others have the right : prefer coalitions where only a few others have the right

capabilities. Consequence - a better chance of winning (being an capabilities. Consequence - a better chance of winning (being an expert)expert)

– Mixture of theseMixture of these These were examined experimentally, shown beneficial with These were examined experimentally, shown beneficial with

respect to a centralized solutionrespect to a centralized solution

6666

Experimental DesignExperimental Design Settings: Settings:

– The number of agentsThe number of agents– The number of tasks The number of tasks – The number of subtasks per task (bounded)The number of subtasks per task (bounded)

Parameters: Parameters: – The value of each taskThe value of each task– The capabilities of each agent The capabilities of each agent – The cost of a given agent to perform tasks it is capable ofThe cost of a given agent to perform tasks it is capable of

For each settings we randomly generated between 100 For each settings we randomly generated between 100 and 1,500 configurations making sure there are “experts”and 1,500 configurations making sure there are “experts”

6767

Experimental Design (contd)Experimental Design (contd)

Each subtask was randomly assigned a mean cost with Each subtask was randomly assigned a mean cost with a uniform distribution. a uniform distribution.

The actual cost of a given subtask for a specific agent The actual cost of a given subtask for a specific agent was determined using a normal distribution, with the was determined using a normal distribution, with the mean of the subtask, and a certain deviation (2 in the mean of the subtask, and a certain deviation (2 in the basic settings)basic settings)

We consider:We consider:– ““Complete information” case, each agent knows the costs of the Complete information” case, each agent knows the costs of the

other agentsother agents

– ““Incomplete information” case, they know only the mean values Incomplete information” case, they know only the mean values of the costsof the costs

– In both cases each agent knows the capabilities of all the agentsIn both cases each agent knows the capabilities of all the agents

6868

Simulation profits / Optimal ProfitsIncomplete Information

0.3

0.35

0.4

0.45

0.5

0.55

Simulation profits / Optimal ProfitsComplete Information

0.76

0.78

0.8

0.82

0.84

0.86

0.88

Expert vs. Marginal

Incomplete Information Complete Information

Social welfare

6969

Experimental EquilibriumExperimental Equilibrium Given a set of agents A and a set strategies Given a set of agents A and a set strategies ΣΣ;;

a profile F of strategies is an experimental a profile F of strategies is an experimental equilibrium iff any agent i in A, by using another equilibrium iff any agent i in A, by using another strategy in strategy in ΣΣ does not increase its estimated does not increase its estimated expected utility, given that the other agents expected utility, given that the other agents follow their F’s strategies.follow their F’s strategies.

Two main differences from Bayesian equilibrium:Two main differences from Bayesian equilibrium:– We limit the deviation to strategies in We limit the deviation to strategies in ΣΣ– We use experimental estimation of the expected utilityWe use experimental estimation of the expected utility

7070

Simulation profits / Optimal Profits

0.30.40.50.60.70.80.9

1

One MarginalIncomplete

All but oneExpert

Incomplete

One Expert All but oneMarginal

Expert vs. Marginal : Deviation

7171

Strategies for Payment Strategies for Payment DistributionDistribution

•Equal Distribution: estimated net divided “equally” •Proportional: proportional to the estimated cost. •Kernel: based on the game-theory concept of stability (adjustments for negotiation)

•Compromise: propose/agree to an offer that is α<1 times the “deserved” share”.

7272

Deviation to Compromising StrategyDeviation to Compromising Strategy

utility ratio for compromise - heterogenous

0

0.5

1

1.5

2

00.30.40.450.50.70.80.850.911.02

alpha

rati

o

Incomplete Information Complete Information

7373

It is not beneficial to deviate from “equal, 0.8”It is not beneficial to deviate from “equal, 0.8”

Compromise - Homogeneous

0

0.2

0.4

0.6

0.8

1

Incomplete informationComplete information

rati

o t

o o

pti

mal

0.8 no compromise

Deviation from 0.8 to 0.5 - no adaptation

0.74

0.76

0.78

0.8

0.82

0.84

0.86

Incomplete InformationComplete Information

rati

o t

o o

pti

mal

Major 0.8 One 0.5

Deviation from equal to kernel - compromise 0.8

0

0.2

0.4

0.6

0.8

1

Incomplete informationComplete information

rati

o t

o o

pti

mal

major equal one kernel

Equal vs Proportional - homogenouswith adaptation

0

0.2

0.4

0.6

0.8

1

Incomplete InfoComplete Info

ratio

to o

ptim

al

Equal Proportional

7474

It is beneficial to deviate from It is beneficial to deviate from “Kernel”“Kernel”

Deviation from kernel to equal - no compromise

00.10.20.30.40.50.60.70.8

Incomplete informationComplete information

major kernel one equal

The protocol does not allow to leave a coalition that has been formed.

7575

Restrictive vs. Non-Restrictive Restrictive vs. Non-Restrictive ProtocolsProtocols

Simulation Contracts / Optimal Contracts

0.00

0.20

0.40

0.60

0.80

1.00

Incomplete Information Complete Information

Non Restrictive Restrictive

Simulation Value / Optimal Value

0.00

0.20

0.40

0.60

0.80

1.00

Incomplete Information Complete Information

Non Restrictive Restrictive

7676

RFP Coalitions SummaryRFP Coalitions Summary

Subjective valuations of tasks and subtasks Subjective valuations of tasks and subtasks impose difficulty to use traditional mechanismsimpose difficulty to use traditional mechanisms

Surprisingly, simple heuristic strategies result in Surprisingly, simple heuristic strategies result in beneficial coalitionsbeneficial coalitions

AAdaptation and compromise further improves daptation and compromise further improves resultsresults

Stability is also arrived at, and complexity is low – Stability is also arrived at, and complexity is low – hundreds of agents and tasks can be handledhundreds of agents and tasks can be handled

7777

Recent StudiesRecent Studies

Fuzzy utilities Fuzzy utilities Ordinal utilitiesOrdinal utilities Effects of search costsEffects of search costs Coalitions with risk attitudesCoalitions with risk attitudes

7878

ConclusionConclusion

Coalitions are an important means for agent Coalitions are an important means for agent collaborationcollaboration

Mechanisms are usually Mechanisms are usually – too complex, ortoo complex, or– too rigid, ortoo rigid, or– do not scale, ordo not scale, or– hold too restrictive, unrealistic assumptionshold too restrictive, unrealistic assumptions

We have seen a few improvements on these We have seen a few improvements on these shortcomingsshortcomings

Yet, all solutions require further improvements to Yet, all solutions require further improvements to become applicable for practical usebecome applicable for practical use

7979

Mechanism DesignMechanism Design

8080

What is Mechanism Design?What is Mechanism Design?

Designing rules of a game, a protocolDesigning rules of a game, a protocol– Achieve a specific outcomeAchieve a specific outcome– Account for agent self-interested behaviorAccount for agent self-interested behavior

IncentivizeIncentivize– Set rules which players prefer to followSet rules which players prefer to follow

8181

Preferred PropertiesPreferred Properties

Individual rationalityIndividual rationality Budget balanceBudget balance Social welfareSocial welfare TruthfulnessTruthfulness Resistance to attacks/collusionsResistance to attacks/collusions Fair distributionsFair distributions Computational efficiency Computational efficiency

Unfortunately, not all can be met simultaneouslyUnfortunately, not all can be met simultaneously Trade-offs are studied widelyTrade-offs are studied widely

8282

Mechanism DefinedMechanism Defined

GivenGiven N N players with types players with types ttii T Tii

E.g. in an auction, the type of a player may be her E.g. in an auction, the type of a player may be her valuation of the goodvaluation of the good

Depending Depending ttii, player , player ii will select an action will select an action aaii((ttii) ) AAii, , AAii the the set of possible actions of set of possible actions of ii offered by the mechanism offered by the mechanism

E.g. an action in an auction may be a bid of a certain valueE.g. an action in an auction may be a bid of a certain value A player has utility A player has utility uuii, based on the outcome generated by , based on the outcome generated by

the mechanismthe mechanism E.g., in an auction the outcome may be the final allocation E.g., in an auction the outcome may be the final allocation

of goodsof goods

A mechanism A mechanism MM is a pair ( is a pair (AA,,gg), where ), where AA= = ΠΠi i AAii the set of the set of actions and g a mapping function from actions to outcomesactions and g a mapping function from actions to outcomes

8383

Algorithmic Mechanism DesignAlgorithmic Mechanism Design

Considers computational issuesConsiders computational issues– Complexity: poly-time is soughtComplexity: poly-time is sought– Optimality and worst-case analysis employedOptimality and worst-case analysis employed– An algorithm is developedAn algorithm is developed

– If exponential – out of scope:If exponential – out of scope: E.g., VCG (Vickrey-Clarke-Groves) acutionE.g., VCG (Vickrey-Clarke-Groves) acution

8484

ApplicationApplication

In recent years, mostly applied toIn recent years, mostly applied to– AuctionsAuctions– MarketsMarkets– NegotiationNegotiation

8585

AuctionsAuctions

8686

AuctionsAuctions

A centralized protocol, includes one auctioneer A centralized protocol, includes one auctioneer and multiple biddersand multiple bidders

The auctioneer puts a good for sale. In some The auctioneer puts a good for sale. In some cases, the good may be a combination of other cases, the good may be a combination of other goods, or a good with multiple attributesgoods, or a good with multiple attributes

The bidders make offers. This may be repeated for The bidders make offers. This may be repeated for several times, depending on the auction typeseveral times, depending on the auction type

The auctioneer determines the winnerThe auctioneer determines the winner

8787

Auctions: pros and consAuctions: pros and cons

Usually easier to prevent bidder lyingUsually easier to prevent bidder lying Simple protocolsSimple protocols Centralized: a single point of failureCentralized: a single point of failure Multi-attribute exponentially complexMulti-attribute exponentially complex Allows collusion “behind the scenes”Allows collusion “behind the scenes” May favor the auctioneer May favor the auctioneer

8888

Auction TypesAuction Types

Private valuePrivate value: the value of a good to a : the value of a good to a bidder agent depends only on its private bidder agent depends only on its private preferences. Assumed to be known exactlypreferences. Assumed to be known exactly

Common valueCommon value: the good’s value depends : the good’s value depends entirely on other agents’ valuationentirely on other agents’ valuation

Correlated valueCorrelated value: the good’s value depends : the good’s value depends on internal and external valuationson internal and external valuations

8989

Auction ProtocolsAuction Protocols English auction (aka first-price open-cry):English auction (aka first-price open-cry):

– bidders free to raise their bidbidders free to raise their bid– end: no more raises, winner: highest bidder at bid end: no more raises, winner: highest bidder at bid – agent strategy: a series of bids, based on private value, agent strategy: a series of bids, based on private value,

estimates of others’ valuations, their past bidsestimates of others’ valuations, their past bids– dominant strategy: bid a small amount more than dominant strategy: bid a small amount more than

current highest bid, stop when private value reachedcurrent highest bid, stop when private value reached

For correlated value:For correlated value:– auctioneer increases price by constant or other rateauctioneer increases price by constant or other rate– open-exit allows to quit without re-entry open-exit allows to quit without re-entry

9090

More ProtocolsMore Protocols

First-price sealed-bid auction:First-price sealed-bid auction:– each bidder submits one bid, not knowing others’each bidder submits one bid, not knowing others’– highest wins, pays his bidhighest wins, pays his bid– agent strategy: function of private value and beliefs about agent strategy: function of private value and beliefs about

others’ valuationsothers’ valuations– no dominant strategy. Best: bid less than true valueno dominant strategy. Best: bid less than true value– how much less? Nash is computable if probability distribution how much less? Nash is computable if probability distribution

of agents’ values is knownof agents’ values is known

Example: n agents, uniform value distribution, agent i Example: n agents, uniform value distribution, agent i has value vhas value vii, there is Nash if each agent i bids v, there is Nash if each agent i bids vii(n-1)/n(n-1)/n

9191

Yet More AuctionsYet More Auctions

Dutch auction (decending):Dutch auction (decending):– the seller lower the price until a bidder takes itthe seller lower the price until a bidder takes it– strategically, equivalent to first-price sealed-bidstrategically, equivalent to first-price sealed-bid– advantage: auctioneer can accelerate auctionadvantage: auctioneer can accelerate auction

9292

… … and Moreand More

All-pay auction:All-pay auction:– each bidder pays its bid to the auctioneereach bidder pays its bid to the auctioneer– several types of such auctions are used for several types of such auctions are used for

resource (re-)allocationresource (re-)allocation– E.g., olympic games, political lobbying, R&D races E.g., olympic games, political lobbying, R&D races

Equilibrium bidding strategy must be a mixed Equilibrium bidding strategy must be a mixed strategystrategy– Consider a common-value all-pay auction with Consider a common-value all-pay auction with

prize worth 1prize worth 1

9393

Vickrey (second-price sealed-bid)Vickrey (second-price sealed-bid) Each bidder submits one bid, not knowing others’Each bidder submits one bid, not knowing others’ The highest bid wins, but bidder pays second-highest bidThe highest bid wins, but bidder pays second-highest bid Agent strategy: base bid on private value and beliefs about Agent strategy: base bid on private value and beliefs about

others’ valuesothers’ values Dominant strategy: bid true valuationDominant strategy: bid true valuation

– if it bids more and this increment made him win, the agent ends up if it bids more and this increment made him win, the agent ends up with a loss, since it may pay more that its true valuewith a loss, since it may pay more that its true value

– if it bids less, there is a smaller chance of winning (but winning if it bids less, there is a smaller chance of winning (but winning price is not affected)price is not affected)

Meaning: bid true value regardless of othersMeaning: bid true value regardless of others

9494

So, Which Auction is Better?So, Which Auction is Better? Computation: auctions with dominant strategies Computation: auctions with dominant strategies

(Vickrey and English) are more efficient - no need (Vickrey and English) are more efficient - no need to speculate regarding other biddersto speculate regarding other bidders

Auctioneer’s revenue: Auctioneer’s revenue: – second-price is less than the true price, however first-second-price is less than the true price, however first-

price bidders under-bid. Which effect is stronger?price bidders under-bid. Which effect is stronger?– for risk-neutral bidders with private independent values, for risk-neutral bidders with private independent values,

the effects are equivalentthe effects are equivalent– for risk-averse bidders, Dutch and first-price sealed-bid for risk-averse bidders, Dutch and first-price sealed-bid

auctions maximize auctioneer’s revenueauctions maximize auctioneer’s revenue

So, are revenues equivalent?So, are revenues equivalent?

9595

Real AuctionsReal Auctions In real auctions, values are not privateIn real auctions, values are not private As a result, for 3 or more bidders, English auctions As a result, for 3 or more bidders, English auctions

provides auctioneer revenue higher than Vickrey provides auctioneer revenue higher than Vickrey doesdoes

Explanation: when it observes other bidders Explanation: when it observes other bidders increasing their bid, a bidder increases its own increasing their bid, a bidder increases its own valuation of the goodvaluation of the good

Both English and Vickrey are better for the Both English and Vickrey are better for the auctioneer than Dutch and first-price sealed-bidauctioneer than Dutch and first-price sealed-bid

9696

CollusionCollusion Bidders can coordinate their bids to lower themBidders can coordinate their bids to lower them In English and Vickrey auctions, collusion is a In English and Vickrey auctions, collusion is a

dominant Strategy!dominant Strategy! Example: Example:

– agents a,b,c values of the good are 10,10,12, agents a,b,c values of the good are 10,10,12, respectively respectively

– they can agree to bid 5,5,6 respectivelythey can agree to bid 5,5,6 respectively– if one defects, all observe that, and can increase to real if one defects, all observe that, and can increase to real

value, so there is no benefit from defection value, so there is no benefit from defection

9797

Avoiding CollusionAvoiding Collusion In the first-price sealed-bid and Dutch auctions, In the first-price sealed-bid and Dutch auctions,

bidder collusion is not dominant, but possible:bidder collusion is not dominant, but possible:– in the previous example, after a,b,c decided on bidding in the previous example, after a,b,c decided on bidding

5,5,6, it is beneficial for a,b to bid more than 5. For any 5,5,6, it is beneficial for a,b to bid more than 5. For any bid of c below 10 they can bid and winbid of c below 10 they can bid and win

In first-price sealed-bid, Vickrey and Dutch In first-price sealed-bid, Vickrey and Dutch auctions, all bidders must identify each other and auctions, all bidders must identify each other and collude jointly. External bidder can wincollude jointly. External bidder can win

In the English auction identifying is through In the English auction identifying is through bidding. Computerized anonymization can prevent bidding. Computerized anonymization can prevent identification and collusionidentification and collusion

9898

Insincere AuctioneerInsincere Auctioneer Private value auctions:Private value auctions:

– Vickrey: auctioneer can overstate the Vickrey: auctioneer can overstate the second highest bid to the winnersecond highest bid to the winner

– Solution: electronic signatureSolution: electronic signature– Other auctions do not motivate auctioneer Other auctions do not motivate auctioneer

lying, since the winner pays its bidlying, since the winner pays its bid

Non-private value:Non-private value:– English: auctioneer can use shills that bid in English: auctioneer can use shills that bid in

the auction to increase real bidders the auction to increase real bidders valuationvaluation

– Any auction: auctioneer may bid, to Any auction: auctioneer may bid, to guarantee a minimum priceguarantee a minimum price

9999

Example: Auctioneer Bid Example: Auctioneer Bid

In the Vickrey auction, auctioneer is motivated In the Vickrey auction, auctioneer is motivated to bid over its true reservation priceto bid over its true reservation price

In case his bid is second, it determines the In case his bid is second, it determines the good’s price higher than the reservation pricegood’s price higher than the reservation price

On the other hand, auctioneer may win On the other hand, auctioneer may win although others value the good at more than although others value the good at more than reservation pricereservation price

100100

Insincere BiddersInsincere Bidders Non-private value:Non-private value:

– winner’s curse: an agent that bids its true value and wins winner’s curse: an agent that bids its true value and wins knows that it was too highknows that it was too high

– this means that a win is a loss (of money)this means that a win is a loss (of money)– hence, agents should bid less than true valuehence, agents should bid less than true value– this is the best strategy even in Vickrey (unlike private this is the best strategy even in Vickrey (unlike private

value Vickrey)value Vickrey) Private value, Vickrey:Private value, Vickrey:

– dominant truthful bidding reveals true valuationsdominant truthful bidding reveals true valuations– this may be disadvantageous:this may be disadvantageous:

when subcontracting, subcontractors may re-negotiatewhen subcontracting, subcontractors may re-negotiate

101101

Auctions of Interrelated GoodsAuctions of Interrelated Goods Multiple homogeneous goods: truth revelation Multiple homogeneous goods: truth revelation

of Vickrey holdsof Vickrey holds Heterogeneous goods, one at a time, Heterogeneous goods, one at a time,

interdependent values:interdependent values:– for optimal bidding, agents need full lookaheadfor optimal bidding, agents need full lookahead– but then agents don’t bid true values per goodbut then agents don’t bid true values per good

Protocol modifications to overcome that:Protocol modifications to overcome that:– pool of goods at a single auctionpool of goods at a single auction– allow decommitemt, with penalties allow decommitemt, with penalties

Note: lookahead requires speculationNote: lookahead requires speculation

102102

Continuous double auctions Continuous double auctions (CDA)(CDA)

At any time during the trading period of At any time during the trading period of a good a good – buyers may submit bids buyers may submit bids – sellers may submit askssellers may submit asks

If open buy and sell orders are If open buy and sell orders are compatible, a trade is executed compatible, a trade is executed immediatelyimmediately

Typically, an announcement is made to Typically, an announcement is made to all participantsall participants

103103

CDA propertiesCDA properties

Widely used for securities, derivative, commoditiesWidely used for securities, derivative, commodities Highly efficient: can respond rapidly to changing Highly efficient: can respond rapidly to changing

market conditions, despite limited info. Available to market conditions, despite limited info. Available to participants:participants:– Private utility functionPrivate utility function– Stream of bids, asks, tradesStream of bids, asks, trades

Prices converge fast (close) to theoretical Prices converge fast (close) to theoretical competitive equilibrium competitive equilibrium (for human subjects)(for human subjects)

104104

LimitationsLimitations

CentralizedCentralized Requires a neutral brokerRequires a neutral broker Vulnerable to manipulationsVulnerable to manipulations Inefficient for small volumesInefficient for small volumes Requires a matching mechanismRequires a matching mechanism

105105

Auctions –Field ResultsAuctions –Field Results

106106

Thank You!Thank You!

107107

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