Incentive Compatibility and the Bargaining Problem

1. Incentive Compatibility and the Bargaining Problem By Roger B. Myerson Presented by Anshi Liang lasnake@eecs

2. Outline of this presentation • Introduction • Bayesian Incentive-Compatibility • Response-Plan Equilibria • Incentive-Efficiency • The Bargaining Solution • Example

3. Outline of this presentation • Introduction • Bayesian Incentive-Compatibility • Response-Plan Equilibria • Incentive-Efficiency • The Bargaining Solution • Example

4. Introduction • Consider the problem of an arbitrator trying to select a collective choice for a group of individuals when he does not have complete information about their preferences and endowments. • The goal of this paper is to develop a unique solution to this arbitrator’s problem, based on the concept of incentive-compatibility and bargaining solution.

5. Introduction • Describe by a Bayesian collective choice problem: (C, A1, A2, …, An, U1, U2, …, Un, P) Cis the set of choices or strategies available to the group; Aiis the set of possible types for player i; Uiis the utility function for player i such that Ui(c, a1, a2, …, an) is the payoff which player i would get if cЄC were chosen and if (a1, a2, …, an) were the true vector of player types; P is the probability distribution such that P(a1, a2, …, an) is the probability that (a1, a2, …, an) is the true vector of types for all players.

6. Introduction • Assumptions: a. C and all the Ai sets are nonempty finite sets; • The response of each player is communicated to the arbitrator confidentially and noncooperatively; • The arbitrator cannot compel a player to give the truthful response; • The arbitration is binding.

7. Introduction • Choice mechanism is a real-value function π with a domain of the form CX(S1XS2X…XSn)—for some collection of response sets S1, S2,…, Sn—such that ∑c,ЄCπ(c’|s1,…,sn)=1, and π(c|s1,…,sn) for all c,for every (s1,…,sn) in S1XS2X…XSn. • Ai is the standard response set.

8. Outline of this presentation • Introduction • Bayesian Incentive-Compatibility • Response-Plan Equilibria • Incentive-Efficiency • The Bargaining Solution • Example

9. Bayesian Incentive-Compatibility • With a choice mechanism π, we have Zi(π, bi|ai) represents the conditionally-expected utility payoff for player i, here ai is his true type, bi is the type he claims. • A choice mechanism is Bayesian incentive-compatible if Zi(π, ai|ai)≥ Zi(π, bi|ai) for all i, aiЄAi, biЄAi

10. Bayesian Incentive-Compatibility • Define Vi(π|ai)=Zi(π, ai|ai) if choice mechanism π is used and if everyone is honest. • Define V(π)=((Vi(π|ai))a1ЄA1,…,(Vn(π|an)) anЄAn). • The feasible set of expected allocation vectors: F={V(π): π is a choice mechanism} • The incentive-feasible set of expected allocation vectors: F*={V(π): π is a Bayesian incentive-compatible}

11. Bayesian Incentive-Compatibility • Theorem 1: F* is a nonempty convex and compact subset of F (proof in the paper). • If Vi(π|ai)<Vi(π’|ai), for all i and aiЄAi, we say that π is strictly dominated by π’.

12. Outline of this presentation • Introduction • Bayesian Incentive-Compatibility • Response-Plan Equilibria • Incentive-Efficiency • The Bargaining Solution • Example

13. Response-Plan Equilibria • A response plan for player i is a function σi mapping each type aiЄAi onto a probability distribution over his response set Si. σi(si|ai) is the probability that player i will tell the arbitrator si if his true type is ai • So we have Wi(π, σ1, …, σn|ai) to represent the player i’s expected utility payoff; similarly to before, we have a vector of conditionally-expected payoffs: W(π, σ1, …, σn)=(((Wi(π, σ1, …, σn|ai))aiЄAi)ni=1)

14. Response-Plan Equilibria • (σ1, …, σn) is a response-plan equilibrium for the choice mechanism π if, for any player i and type aiЄAi, for every possible alternative response plan σ’i for player i: Wi(π, σ1, …, σn|ai)≥ Wi(π, σ1, …, σi-1, σ’i, σi+1,…,σn|ai) • The equilibrium-feasible set of expected allocation vectors: F**={W(π, σ1, …, σn): π is a choice mechanism, and (σ1, …, σn) is a response-plan equilibrium for π} • Theorem 2: F**=F* (proof in the paper)

15. Outline of this presentation • Introduction • Bayesian Incentive-Compatibility • Response-Plan Equilibria • Incentive-Efficiency • The Bargaining Solution • Example

16. Incentive-Efficiency • π is incentive-efficient if and only if it is a Bayesian incentive-compatible choice mechanism and is not strictly dominated by any other Bayesian incentive-compatible mechanism (remind: If Vi(π|ai)<Vi(π’|ai), for all i and aiЄAi, we say that π is strictly dominated by π’).

17. Outline of this presentation • Introduction • Bayesian Incentive-Compatibility • Response-Plan Equilibria • Incentive-Efficiency • The Bargaining Solution • Example

18. The Bargaining Solution • Conflict outcome: it represents what would happen by default if the arbitrator failed to lead the players to an agreement. Examples: Market Politics Students • Conflict payoff vector: t=((ta1)a1ЄA1, (ta2)a2ЄA2,…,(tan)anЄAn), where each tai is player i’s conditional expectation, given that ai is his true type, of what his utility payoff would be if the conflict outcome occurred.

19. The Bargaining Solution • Given the conflict payoff vector t our collective choice problem becomes a bargaining problem, with a feasible set F*, t is a reference point in F*. • Let F*+ be the set of all incentive-feasible payoff vectors which are individually rational: F*+=F*∩{y:yai≥tai for all i and all aiЄAi} • Theorem 3: Suppose that c* is not incentive-efficient, then there exist a unique incentive-feasible bargaining solution.

20. Outline of this presentation • Introduction • Bayesian Incentive-Compatibility • Response-Plan Equilibria • Incentive-Efficiency • The Bargaining Solution • Example

21. Example • Two players share the cost of a project which benefit them both. • The project cost $100, the two players call an arbitrator to divide it. • Project value: Player1: $90 if he is type1.0, $30 if he is type1.1 Player2: $90 4. To the arbitrator and player2, P1(1.0)=.9 and P2(1.1)=.1

22. Example • Some observation points: • No matter what player 1’s type is, the project appears to be worth more than it costs; • The decisions cannot be made separately. • Some intuitive solutions: • 50-50 or 20-80 • 47-53 • 50-50 or 0-0

23. Example • Formal solution: Let C={c0, c1, c2}, A1={1.0, 1.1}, A2={2}. We have P(1.0, 2)=.9 and P(1.1, 2) =.1. c0means “do not undertake the project”; c1 means “undertake the project and make player1 pay for it”; c2means “undertake the project and make player2pay for it”.

24. Example • Strategies can be randomized. • Use the abbreviations π0j=π(cj|1.0, 2) and π1j=π(cj|1.1, 2). • The incentive-compatible choice mechanisms satisfies the following: -10π01+90π02≥-10π11+90π12, -70π11+30π12≥-10π01+90π02, π00+π01+π02=1, π10+π11+π12=1 ,

25. Example • Expected benefits for all players: x1.0=0π00-10π01+π02, x1.1=0π10-10π11+π12, x2=.9(0π00+90π01-10π02)+.1(0π10+90π11-10π12), • Then the incentive-feasible bargaining solution is the solution that maximize ((x1.0).9(x1.1).1x2), x and π satisfy the restrictions above.

26. Example • Result: x1.0=39.5, x1.1=13.2, x2=36 π01=.505, π02=.495, π10=.561 and π12=.439 • Meanings in English

27. Conclusion • A great paper overall • The mathematical derivation is complicated but very clear • This concept can be possibly extended to our networking study. For example, say that the arbitrator is the network designer; the two players are network users, etc.

28. Thank you very much! Anshi Liang