incentive compatibility and the bargaining problem by roger b. myerson presented by anshi liang...

Incentive Compatibility and Incentive Compatibility and the Bargaining Problemthe Bargaining Problem

By Roger B. MyersonBy Roger B. Myerson

Presented by Anshi LiangPresented by Anshi Liang

lasnake@eecslasnake@eecs

Outline of this presentationOutline of this presentation

1.1. IntroductionIntroduction

2.2. Bayesian Incentive-CompatibilityBayesian Incentive-Compatibility

3.3. Response-Plan EquilibriaResponse-Plan Equilibria

4.4. Incentive-EfficiencyIncentive-Efficiency

5.5. The Bargaining SolutionThe Bargaining Solution

6.6. ExampleExample

IntroductionIntroduction

Consider the problem of an arbitrator trying Consider the problem of an arbitrator trying to select a collective choice for a group of to select a collective choice for a group of individuals when he does not have complete individuals when he does not have complete information about their preferences and information about their preferences and endowments.endowments.

The goal of this paper is to develop a unique The goal of this paper is to develop a unique solution to this arbitrator’s problem, based solution to this arbitrator’s problem, based on the concept of incentive-compatibility and on the concept of incentive-compatibility and bargaining solution.bargaining solution.


Describe by a Describe by a Bayesian collective choice problemBayesian collective choice problem::

((C, AC, A11, A, A22, …, A, …, Ann, U, U11, U, U22, …, U, …, Unn, P, P))

CC is the set of choices or strategies available to the group;is the set of choices or strategies available to the group;

AAii is the set of possible types for player i;is the set of possible types for player i;

UUii is the utility function for player i such that is the utility function for player i such that UUii(c, a(c, a11, a, a22, …, , …, aann) ) is the payoff which player i would get if cis the payoff which player i would get if cЄЄC were C were chosen and if chosen and if (a(a11, a, a22, …, a, …, ann) ) were the true vector of player were the true vector of player types;types;

PP is the probability distribution such that is the probability distribution such that P(aP(a11, a, a22, …, a, …, ann) ) is the is the probability that probability that (a(a11, a, a22, …, a, …, ann) ) is the true vector of types for is the true vector of types for all players.all players.


Assumptions:Assumptions:

a.a. CC and all the and all the AAii sets are nonempty finite sets are nonempty finite sets;sets;

b.b. The response of each player is The response of each player is communicated to the arbitrator communicated to the arbitrator confidentially and noncooperatively;confidentially and noncooperatively;

c.c. The arbitrator cannot compel a player to The arbitrator cannot compel a player to give the truthful response;give the truthful response;

d.d. The arbitration is binding.The arbitration is binding.


Choice mechanismChoice mechanism is a real-value function is a real-value function ππ with a domain of the form with a domain of the form CX(SCX(S11XSXS22X…X…

XSXSnn)—)—for some collection of response sets for some collection of response sets

SS11, S, S22,…, S,…, Snn——such thatsuch that

∑∑c,c,ЄЄCCππ(c’|s(c’|s11,…,s,…,snn))=1, and =1, and ππ(c|s(c|s11,…,s,…,snn) ) for all for all

c,for every (c,for every (ss11,…,s,…,snn) in ) in SS11XSXS22X…XSX…XSnn..

AAii is the is the standard response setstandard response set..







6.6. ExampleExample

Bayesian Incentive-CompatibilityBayesian Incentive-Compatibility

With a choice mechanism With a choice mechanism ππ, we have , we have ZZii((ππ, ,

bbii|a|aii) ) represents the represents the conditionally-expected conditionally-expected

utility payoffutility payoff for player i, here a for player i, here aii is his true is his true

type, btype, bii is the type he claims. is the type he claims.

A choice mechanism is A choice mechanism is Bayesian incentive-Bayesian incentive-compatiblecompatible if if

ZZii((ππ, a, aii|a|aii)≥ Z)≥ Zii((ππ, b, bii|a|aii) ) for all i, for all i, aaiiЄЄAAii, b, biiЄЄAAii


Define Define VVii((ππ|a|aii)=Z)=Zii((ππ, a, aii|a|aii) ) if choice mechanism if choice mechanism ππ is is

used and if everyone is honest.used and if everyone is honest. Define Define VV((ππ)=(()=((VVii((ππ|a|aii))))a1a1ЄЄA1A1,…,(,…,(VVnn((ππ|a|ann)) )) ananЄЄAnAn).).

The The feasible setfeasible set of expected allocation vectors: of expected allocation vectors:

F={F={VV((ππ): ): ππ is a choice mechanism} is a choice mechanism} The The incentive-feasible setincentive-feasible set of expected allocation of expected allocation

vectors:vectors:

F*={F*={VV((ππ): ): ππ is a Bayesian incentive-compatible} is a Bayesian incentive-compatible}


Theorem 1: Theorem 1: FF** is a nonempty convex and is a nonempty convex and compact subset of Fcompact subset of F (proof in the paper). (proof in the paper).

If If VVii((ππ|a|aii)<)<VVii((ππ’|a’|aii), ), for all i andfor all i and a aiiЄЄAAii, , we say we say

that that ππ is is strictly dominatedstrictly dominated by by ππ’.’.







6.6. ExampleExample

Response-Plan EquilibriaResponse-Plan Equilibria

A A response planresponse plan for player i is a function for player i is a function σσi i

mapping each type amapping each type a iiЄЄAAii onto a probability onto a probability distribution over his response set Sdistribution over his response set S ii. . σσii(s(sii|a|aii) is the ) is the probability that player i will tell the arbitrator sprobability that player i will tell the arbitrator s ii if his if his true type is atrue type is a ii

So we have WSo we have Wii((ππ, , σσ11, …, , …, σσnn|a|aii) to represent the ) to represent the player i’s expected utility payoff; similarly to player i’s expected utility payoff; similarly to before, we have a before, we have a vector of conditionally-expected vector of conditionally-expected payoffspayoffs::

WW((ππ, , σσ11, …, , …, σσnn)=(((W)=(((Wii((ππ, , σσ11, …, , …, σσnn|a|aii))))aiaiЄЄAiAi))nni=1i=1) )

Response-Plan EquilibriaResponse-Plan Equilibria

((σσ11, …, , …, σσnn) is a ) is a response-plan equilibriumresponse-plan equilibrium for the for the choice mechanism choice mechanism ππ if, for any player i and type if, for any player i and type aaiiЄЄAAii, for every possible alternative response plan , for every possible alternative response plan σσ’’

ii for player i: for player i:

WWii((ππ, , σσ11, …, , …, σσnn|a|aii)≥ W)≥ Wii((ππ, , σσ11, …, , …, σσi-1i-1, , σσ’’ii, , σσi+1i+1,…,,…,σσnn|a|aii) ) The The equilibrium-feasible setequilibrium-feasible set of expected allocation of expected allocation

vectors:vectors:

FF****={={WW((ππ, , σσ11, …, , …, σσnn): ): ππ is a choice mechanism, and is a choice mechanism, and ((σσ11, …, , …, σσnn) is a response-plan equilibrium for ) is a response-plan equilibrium for ππ}}

Theorem 2: FTheorem 2: F****=F=F* * (proof in the paper)(proof in the paper)







6.6. ExampleExample

Incentive-EfficiencyIncentive-Efficiency

ππ is is incentive-efficientincentive-efficient if and only if it is a if and only if it is a Bayesian incentive-compatibleBayesian incentive-compatible choice choice mechanism and is mechanism and is not strictly dominatednot strictly dominated by by any other Bayesian incentive-compatible any other Bayesian incentive-compatible mechanism (remind: mechanism (remind: If If VVii((ππ|a|aii)<)<VVii((ππ’|a’|aii), ), for for

all i andall i and a aiiЄЄAAii, , we say that we say that ππ is is strictly strictly

dominateddominated by by ππ’).’).







6.6. ExampleExample

The Bargaining SolutionThe Bargaining Solution

Conflict outcomeConflict outcome: it represents what would happen : it represents what would happen by default if the arbitrator failed to lead the players by default if the arbitrator failed to lead the players to an agreement. Examples:to an agreement. Examples:

MarketMarketPoliticsPoliticsStudentsStudents Conflict payoff vectorConflict payoff vector:: tt=((t=((ta1a1))a1a1ЄЄA1A1, (, (tta2a2))a2a2ЄЄA2A2,…,(,…,(ttanan))ananЄЄAnAn), where each t), where each taiai is is

player i’s conditional expectation, given that aplayer i’s conditional expectation, given that a ii is is his true type, of what his utility payoff would be if his true type, of what his utility payoff would be if the conflict outcome occurred. the conflict outcome occurred.

The Bargaining SolutionThe Bargaining Solution

Given the conflict payoff vector Given the conflict payoff vector tt our collective our collective choice problem becomes a choice problem becomes a bargaining problembargaining problem, , with a feasible set Fwith a feasible set F**, , t t is a reference point in Fis a reference point in F**..

Let FLet F**+ + be the set of all incentive-feasible payoff be the set of all incentive-feasible payoff

vectors which are individually rational:vectors which are individually rational:

FF**++=F=F**∩{∩{yy:y:yaiai≥t≥taiai for all i and all a for all i and all a iiЄЄAAii}}

Theorem 3: Suppose that cTheorem 3: Suppose that c** is not incentive- is not incentive-efficient, then there exist a unique incentive-efficient, then there exist a unique incentive-feasible bargaining solution.feasible bargaining solution.







6.6. ExampleExample

ExampleExample

1.1. Two players share the cost of a project which Two players share the cost of a project which benefit them both.benefit them both.

2.2. The project cost $100, the two players call an The project cost $100, the two players call an arbitrator to divide it.arbitrator to divide it.

3.3. Project value: Project value:

Player1: $90 if he is type1.0, $30 if he is type1.1Player1: $90 if he is type1.0, $30 if he is type1.1

Player2: $90Player2: $90

4.4. To the arbitrator and player2, PTo the arbitrator and player2, P11(1.0)=.9 and (1.0)=.9 and

PP22(1.1)=.1 (1.1)=.1

ExampleExample

Some observation points:Some observation points:

a.a. No matter what player 1’s type is, the project No matter what player 1’s type is, the project appears to be worth more than it costs;appears to be worth more than it costs;

b.b. The decisions cannot be made separately.The decisions cannot be made separately. Some intuitive solutions:Some intuitive solutions:

a.a. 50-50 or 20-8050-50 or 20-80

b.b. 47-5347-53

c.c. 50-50 or 0-0 50-50 or 0-0

ExampleExample Formal solution:Formal solution:

Let Let C={cC={c00, c, c11, c, c22}, A}, A11={1.0, 1.1}, A={1.0, 1.1}, A22={2}. ={2}. We have We have

P(1.0, 2)=.9 P(1.0, 2)=.9 and and P(1.1, 2) =.1. P(1.1, 2) =.1.

cc00 means means ““do not undertake the projectdo not undertake the project”; c”; c1 1 means means

“undertake the project and make player1 pay for “undertake the project and make player1 pay for it”; it”; cc22 means “undertake the project and make means “undertake the project and make

player2player2 pay for it”.pay for it”.(u(u1, 1, uu22)) cc00 cc11 cc22

aa11=1.0=1.0 (0, 0)(0, 0) (-10, 90)(-10, 90) (90, -10)(90, -10)

aa22=1.1=1.1 (0, 0)(0, 0) (-70, 90)(-70, 90) (30, -10)(30, -10)

ExampleExample

Strategies can be randomized.Strategies can be randomized. Use the abbreviations Use the abbreviations ππ00

jj==ππ(c(cjj|1.0, 2) and |1.0, 2) and ππ11jj==ππ(c(cjj||

1.1, 2).1.1, 2). The incentive-compatible choice mechanisms The incentive-compatible choice mechanisms

satisfies the following:satisfies the following:

-10-10ππ0011+90+90ππ00

22≥-10≥-10ππ1111+90+90ππ11

2,2,

-70-70ππ1111+30+30ππ11

22≥-10≥-10ππ0011+90+90ππ00

2,2,

ππ0000++ππ00

11++ππ0022=1, =1, ππ11

00++ππ1111++ππ11

22=1=1

,,

ExampleExample

Expected benefits for all players:Expected benefits for all players:

xx1.01.0=0=0ππ0000-10-10ππ00

11++ππ0022,,

xx1.11.1=0=0ππ1100-10-10ππ11

11++ππ1122,,

xx22=.9(0=.9(0ππ0000+90+90ππ00

11-10-10ππ0022)+.1()+.1(00ππ11

00+90+90ππ1111-10-10ππ11

22),), Then the incentive-feasible bargaining Then the incentive-feasible bargaining

solution is the solution that maximizesolution is the solution that maximize

((x((x1.01.0)).9.9(x(x1.11.1)).1.1xx22), x and ), x and ππ satisfy the restrictions satisfy the restrictions above. above.

ExampleExample

Result:Result:

xx1.01.0=39.5, x=39.5, x1.11.1=13.2, x=13.2, x22=36=36

ππ0011=.505, =.505, ππ00

22=.495, =.495, ππ1100=.561 and =.561 and ππ11

22=.439=.439

Meanings in English Meanings in English

ConclusionConclusion

A great paper overallA great paper overall The mathematical derivation is complicated The mathematical derivation is complicated

but very clearbut very clear This concept can be possibly extended to This concept can be possibly extended to

our networking study. For example, say that our networking study. For example, say that the arbitrator is the network designer; the the arbitrator is the network designer; the two players are network users, etc.two players are network users, etc.

Thank you Thank you very much!very much!

Anshi LiangAnshi Liang

incentive compatibility and the bargaining problem by roger b. myerson presented by anshi liang...

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