1 / 75

All-stage Strong Correlated Equilibrium

All-stage Strong Correlated Equilibrium. Yuval Heller Tel-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. Outline. Introduction & motivation Examples for strong correlated equilibria

obert
Download Presentation

All-stage Strong Correlated Equilibrium

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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

More Related