All-stage Strong Correlated Equilibrium

1. All-stage Strong Correlated Equilibrium Yuval HellerTel-Aviv University (Part of my Ph.D. thesis supervised by Eilon Solan)http://www.tau.ac.il/~helleryu/ Presenting in HUJI Rationality Center 22 June 2008

2. Outline • Introduction & motivation • Examples for strong correlated equilibria • Model • Main Result: • Demonstration • Proof • Discussion: • Comparison with other notions • Coalition-proof notions • Concluding Remarks

3. Introduction & Motivation “What do you think…should we get started on that motivation research or not? ”

4. Reasonable Outcomes With Communication • A non-cooperative game with pre-play communication • Agreements: • Feasible - correlated profiles • Self-enforcing: immune to plausible joint deviations • Nash equilibria are not self-enforcing outcomes • Appropriate notions: • Strong & coalition-proof correlated equilibria Other approaches

5. Strong & Coalition-Proof Correlated Equilibria • Strong correlated equilibrium: • Resistance against all coalitional deviations • Coalition-proof correlated equilibrium: • Resistance against self-enforcing joint deviations • A deviation with no further self-enforcing and improving sub-deviation • A correlated profile is implemented by arevealing protocol: • A mediator: privately recommends each player what to play • A signaling process: payoff-irrelevant private & public signals Other approaches

6. Revealing Protocol • A revealing protocol has to satisfy: • At the end – each player knows his recommended action • No player knows anything about the recommended actions of the other players • Assumption in the literature: all the players receive their recommended actions simultaneously

7. Revealing Protocol • A general signaling process can be more complex. Few examples: • The recommendations are revealed consecutively in a pre-specified order • e.g.: the polite cheap-talk protocol in Heller, 2008 • In a pre-specified order, each player is informed at each stage about a new unrecommended action • The order of the signals depends on a private lottery

8. Ex-ante & Ex-post Stages • When all recommendations are simultaneously reveled, joint deviations can be planned at 2 stages: • Ex-antestage – deviations are only plannedbeforereceiving recommendations • Moreno & Wooders, Milgrom & Roberts, Ray (all 1996) • Ex-poststage – deviations are only planned afterreceiving recommendations • Einy & Peleg (1995), Ray (1998), Bloch & Dutta (2007)

9. Deviating in All Stages • When the players receive several signals, they can communicate, share information & plan joint deviations at all stages • A requirement from the protocol: • Sharing information among deviators would not allow them to know anything non-trivial about other players’ recommendations • Similar to the existing literature

10. Deviating in All Stages • The use of a joint deviation requires the unanimous agreement of all members of the deviating coalition • A player agrees to be a part of a joint deviation if given his own information the deviation is profitable • Thus, if a joint deviation is used, it is common knowledge (among the deviators) that each deviator believes that the deviation is profitable

11. Deviating in All Stages • We assume that deviations are binding: • A deviation is implemented by a new mediator • The deviators truthfully report their information to the new mediator, and they are bound to follow his recommendations • In the spirit of the strong correlated equilibrium notion • We model the information structure of the deviators by an incomplete information model (with a common prior) à la Aumann (1987)

12. All-Stage Strong Correlated Equilibrium • A profile is an all-stage strong correlated equilibrium if for every stage of every revealing protocol that implements it, there is no coalition with a profitable deviation • Aprofile is an ex-ante strong correlated equilibrium if there is no coalition with a profitable deviation at the ex-ante stage • Equivalent to the definition of Moreno & Wooders (1996)

13. Main Result • The two notions (Ex-ante& all-stage) coincide • An ex-ante strong correlatedequilibrium is resistant todeviations at all stages of any signaling process • A robust notion • Inclusion relations:

14. Examples An all-stage strong correlated equilibrium that is the only plausible outcome of a game An ex-post strong correlated equilibrium that is not an ex-ante equilibrium

15. Nash Payoffs: (-1, -1, 2) (-0.5, -0.5, 1) Example 1: 3-PlayerMatching Pennies Game Adapted from Moreno & Wooders (96) • 3 Nash equilibria: • 2 pure equilibria (payoff: (-1,-1, 2)): • A totally mixed equilibrium – Each action is chosen with probability 0.5. Expected payoff: (-0.5, -0.5, 1)

16. Example 1: 3-PlayerMatching Pennies Game • None of the Nash equilibria is a plausible outcome of a game with pre-play communication • Players 1 & 2 can guarantee an expected payoff of (0,0) by playing the correlated profile: Nash Payoffs: (-1, -1, 2) (-0.5, -0.5, 1)

17. Example 1: 3-PlayerMatching Pennies Game • The game has a single strong correlated equilibrium: • Players 1 & 2 play: • Player 3 plays: • Payoff: (0,0,0)

18. Example 1: 3-PlayerMatching Pennies Game • The strong correlated equilibrium is the only “plausible” outcome of the game • Experimental study (Moreno & Wooders, 1998): Players play it (and not any of the Nash equilibria)

19. Examples An all-stage strong correlated equilibrium that is the only reasonable outcome of a game An ex-poststrong correlated equilibrium that is not an ex-anteequilibrium Demonstration

20. Example 2: Chicken Game Adapted from Moreno & Wooders (96) q • q is not an ex-antestrong correlated equilibrium: • Players 1 and 2 have a profitable joint deviation - play the pure action profile (T,L) • Gives a payoff of 6 instead of 5 to each player

21. Example 2: Chicken Game q • q is an ex-poststrong correlated equilibrium: • No player has a unilateral profitable deviation (because q is a correlated equilibrium) • Assume to the contrary: there is a profitable joint deviation • When the players deviate, it is common knowledge that both earn from it

22. Example 2: Chicken Game q • If player 1 received a recommendation B then he can not earn from any deviation (his payoff is maximal) • The same for player 2 and R • Thus: it is common knowledge that the action profile is (T,L), • No deviation can make both players earn more than 6

23. Model & Definitions Demonstration

24. Notation • A finite game in strategic form: • A Coalition: • A (correlated) profile: • A (correlated) S-profile:

25. State Space (Aumann, 1987) • A probability space - • Ω - space of possible states of the world • - -algebra of all measurable events • μ- The common prior the players share (assumption) • Notation: Given an event E and denote the posterior distribution of x conditioned on E

26. A Recommendation Profile • - a random variable with a prior distribution equal to the agreement: • Interpretation: The mediator’s chosen recommendation profile

27. A Deviation (of a coalition S) • A random variable satisfies: • dS and a-S are conditionally independent given aS • Interpretation: • The new mediator who implements dS only knows the recommendations of the deviators • His output can’t depend on the recommendations of the non-deviators

28. Information Structure of S • When considering the use of a deviation, it is a situation of incomplete information • Private information each player may have: • His recommended action (or partial information about it) • Information acquired while communicating with the others • We model it by partitions on Ω

29. Information Structure of S • - partitions of Ω • - the information partition of player i • In ω, player i is informed of the element Fi(ω) • A technical assumption: join consists of non-null events • Consistency requirement: The deviators have no direct information about the recommendations of the non-deviating players:

30. Conditional Expected Payoffs • When everyone follows q: • When S members deviate (and –S follow q):

31. Common Knowledge (Aumann, 1976) • An event E is common knowledge in a state ω, if E includes that member of the meet that contains ω

32. A Profitable Deviation • A deviation dS is profitable (w.r.t. ) if: • There is a consistent information structure • There is a state in which it is common knowledge that deviating is profitable to all S members

33. All-stage Strong Correlated Equilibrium • A profile q is an all-stage strong correlated equilibrium, if no coalition has a profitable deviation

34. Ex-Ante Strong Correlated Equilibrium • A profile q is an ex-antestrong correlated equilibrium, if no coalition has a profitable deviation w.r.t. the ex-anteinformation structure that satisfies:

35. Main Result • A profile is an ex-antestrong correlated equilibrium  it is an all-stage strong correlated equilibrium • ex-ante  all-stage:Straightforward • Main Theorem: ex-ante all-stage

36. Main Result - Demonstration

37. Example 3 • An ex-anteStrong correlated Equilibrium - q: • An ex-ante symmetric payoff - 10

38. Example 3 • Why it is an ex-ante Strong correlated equilibrium: • No Unilateral deviations – q is a correlated equilibrium • No 2 players can earn together more than 20 by deviating • The grand coalition can’t earn together more than 30

39. Example 3 • An intermediate Stage : • Player 1 (2) received a recommendation a1 (b1) (expected payoff ) • Player 3 hasn’t received a recommendation yet (expected payoff 10) • Each player doesn’t know whether the others have been recommended

40. Example 3 • A deviation that may look profitable to all players: • Playing (a3, b3, c3) with probability 1 • Player 1 earns 7 instead of • Player 2 earns 11 instead of • Player 3 earns 12 instead of 10

41. Example 3 • Further analysis shows that deviating is unprofitable to player 3 • Player 1 agrees  his recommended action is a1 (common knowledge) • Given only a1  Player 2 & 3 expects to get • Player 2 agrees  his recommended action is b1(common knowledge) • Player 3 expects to get 15  Deviating is unprofitable for him

42. Main Result – Proof Coalition-proof

43. Main Result • A profile is ex-ante strong correlated equilibrium  It is an all-stagestrong correlatedequilibrium • In other words: a profitable deviation exists  a profitable ex-ante deviation exists Coalition-proof

44. Main Result – Proof (1) • Let be an agreement that is not an all-stage strong correlated equilibrium • There is a coalition S and a profitable deviation dS w.r.t. • Thus, there is a state such that it is common knowledge that the deviation is profitable:

45. Main Result – Proof (2) • The ex-anteprofitable deviation is: • Why is it ex-ante profitable?

46. Comparison with Other Notions Ex-post & Ex-ante SCE

47. Milgrom-Roberts ex-ante SCE Rayex-anteSCE Moreno-Woodersex-ante SCE = Our all-stage SCE Comparing to Other Ex-Ante Strong Correlated Equilibria • Ray (96) – Deviating coalitions are not allowed to construct new correlating devices, but only use uncorrelated deviations • Milgrom & Roberts (96) – Only some of the coalitions can communicate and plan deviations • Our set is included in the other ex-ante sets

48. Comparing to Other Ex-post Strong Correlated Equilibria • Our ex-post notion: The information structure must satisfy that each player knows his recommend action • Einy & Peleg (95) – A coalition can only use deviations that improve all conditional utilities for all possible recommendations • In our notation:

49. Comparing to Other Ex-post Strong Correlated Equilibria • Ray (98) –Using only pure deviations • Bloch & Dutta (07): A coalition can only use deviations that satisfy for some • Each player earns from the deviation • There is a player that looses from the deviation • Implies the existence of a profitable deviation

50. Comparing to Other Ex-post Strong Correlated Equilibria Bloch-Dutta ex-postSCE Ray ex-post SCE Einy-Peleg ex-postSCE Ourex-postSCE