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

2. Outline of this presentation 1.Introduction 2.Bayesian Incentive-Compatibility 3.Response-Plan Equilibria 4.Incentive-Efficiency 5.The Bargaining Solution 6.Example

3. Outline of this presentation 1.Introduction 2.Bayesian Incentive-Compatibility 3.Response-Plan Equilibria 4.Incentive-Efficiency 5.The Bargaining Solution 6.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, A 1, A 2, …, A n, U 1, U 2, …, U n, P) C is the set of choices or strategies available to the group; A i is the set of possible types for player i; U i is the utility function for player i such that U i (c, a 1, a 2, …, a n ) is the payoff which player i would get if cЄC were chosen and if (a 1, a 2, …, a n ) were the true vector of player types; P is the probability distribution such that P(a 1, a 2, …, a n ) is the probability that (a 1, a 2, …, a n ) is the true vector of types for all players.

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

7. Introduction  Choice mechanism is a real-value function π with a domain of the form CX(S 1 XS 2 X…XS n )—for some collection of response sets S 1, S 2,…, S n —such that ∑ c,ЄC π(c’|s 1,…,s n )=1, and π(c|s 1,…,s n ) for all c,for every (s 1,…,s n ) in S 1 XS 2 X…XS n.  A i is the standard response set.

8. Outline of this presentation 1.Introduction 2.Bayesian Incentive-Compatibility 3.Response-Plan Equilibria 4.Incentive-Efficiency 5.The Bargaining Solution 6.Example

9. Bayesian Incentive-Compatibility  With a choice mechanism π, we have Z i (π, b i |a i ) represents the conditionally-expected utility payoff for player i, here a i is his true type, b i is the type he claims.  A choice mechanism is Bayesian incentive- compatible if Z i (π, a i |a i )≥ Z i (π, b i |a i ) for all i, a i ЄA i, b i ЄA i

10. Bayesian Incentive-Compatibility  Define V i (π|a i )=Z i (π, a i |a i ) if choice mechanism π is used and if everyone is honest.  Define V(π)=((V i (π|a i )) a1ЄA1,…,(V n (π|a n )) 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 V i (π|a i )<V i (π’|a i ), for all i and a i ЄA i, we say that π is strictly dominated by π’.

12. Outline of this presentation 1.Introduction 2.Bayesian Incentive-Compatibility 3.Response-Plan Equilibria 4.Incentive-Efficiency 5.The Bargaining Solution 6.Example

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

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

15. Outline of this presentation 1.Introduction 2.Bayesian Incentive-Compatibility 3.Response-Plan Equilibria 4.Incentive-Efficiency 5.The Bargaining Solution 6.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 V i (π|a i )<V i (π’|a i ), for all i and a i ЄA i, we say that π is strictly dominated by π’).

17. Outline of this presentation 1.Introduction 2.Bayesian Incentive-Compatibility 3.Response-Plan Equilibria 4.Incentive-Efficiency 5.The Bargaining Solution 6.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: MarketPoliticsStudents  Conflict payoff vector: t=((t a1 ) a1ЄA1, (t a2 ) a2ЄA2,…,(t an ) anЄAn ), where each t ai is player i’s conditional expectation, given that a i 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:y ai ≥t ai for all i and all a i ЄA i }  Theorem 3: Suppose that c * is not incentive- efficient, then there exist a unique incentive- feasible bargaining solution.

20. Outline of this presentation 1.Introduction 2.Bayesian Incentive-Compatibility 3.Response-Plan Equilibria 4.Incentive-Efficiency 5.The Bargaining Solution 6.Example

21. Example 1.Two players share the cost of a project which benefit them both. 2.The project cost $100, the two players call an arbitrator to divide it. 3.Project value: Player1: $90 if he is type1.0, $30 if he is type1.1 Player2: $90 4.To the arbitrator and player2, P 1 (1.0)=.9 and P 2 (1.1)=.1

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

23. Example  Formal solution: Let C={c 0, c 1, c 2 }, A 1 ={1.0, 1.1}, A 2 ={2}. We have P(1.0, 2)=.9 and P(1.1, 2) =.1. c 0 means “do not undertake the project”; c 1 means “undertake the project and make player1 pay for it”; c 2 means “undertake the project and make player2 pay for it”. (u 1, u 2 ) c0c0c0c0 c1c1c1c1 c2c2c2c2 a 1 =1.0 (0, 0) (-10, 90) (90, -10) a 2 =1.1 (0, 0) (-70, 90) (30, -10)

24. Example  Strategies can be randomized.  Use the abbreviations π 0 j =π(c j |1.0, 2) and π 1 j =π(c j |1.1, 2).  The incentive-compatible choice mechanisms satisfies the following: -10π 0 1 +90π 0 2 ≥-10π 1 1 +90π 1 2, -70π 1 1 +30π 1 2 ≥-10π 0 1 +90π 0 2, π 0 0 +π 0 1 +π 0 2 =1, π 1 0 +π 1 1 +π 1 2 =1,

25. Example  Expected benefits for all players: x 1.0 =0π 0 0 -10π 0 1 +π 0 2, x 1.1 =0π 1 0 -10π 1 1 +π 1 2, x 2 =.9(0π 0 0 +90π 0 1 -10π 0 2 )+.1(0π 1 0 +90π 1 1 - 10π 1 2 ),  Then the incentive-feasible bargaining solution is the solution that maximize ((x 1.0 ).9 (x 1.1 ).1 x 2 ), x and π satisfy the restrictions above.

26. Example  Result: x 1.0 =39.5, x 1.1 =13.2, x 2 =36 π 0 1 =.505, π 0 2 =.495, π 1 0 =.561 and π 1 2 =.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