1 / 23

E. Lehrer , E. Kalai , A. Kalai , D. Samet

How to commit to cooperation. Lehrer. Ehud. E. Lehrer , E. Kalai , A. Kalai , D. Samet. Kalai. Ehud. www.tau.ac.il/~dsamet. Dov Samet. Kalai. Adam. Game Theory. Non-cooperative game G’. Non-Cooperative. Non-cooperative game G. Cooperative. Strategic considerations

cira
Download Presentation

E. Lehrer , E. Kalai , A. Kalai , D. Samet

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. How to commit to cooperation Lehrer Ehud E. Lehrer, E. Kalai,A. Kalai, D. Samet Kalai Ehud www.tau.ac.il/~dsamet Dov Samet Kalai Adam

  2. Game Theory Non-cooperative game G’ Non-Cooperative Non-cooperative game G Cooperative • Strategic considerations • Nash equilibrium • Possible outcomes • Enforcing commitment equ. non-equ. equ. How to cooperate non-cooperatively?How can non-equilibrium outcomes be achieved non-cooperatively? Each player’s strategy is a best response to the other players’ strategies.

  3. The Prisoner’s Dilemma Clyde 1 yrs 1 yrs Bonnie 10 yrs 10 yrs Bonnieand Clydeare apprehended after robbing a bank. The police have little incriminating evidence. free 20 yrs 20 yrs Each of the suspects can choose to confess or to deny. free

  4. Clyde reasons… Clyde 1 yrs 1 yrs Bonnie 10 yrs 10 yrs If Bonnie denies … … I’d betterconfess. > free If Bonnie confesses … … I’d better confess. 20 yrs > 20 yrs No matter whatBonnie does,I am better off confessing. free

  5. Bonnie thinks too… Clyde 1 yrs 1 yrs Bonnie 10 yrs 10 yrs … and she reasons exactlythe same way. free 20 yrs No matter whatClyde does,I am better off confessing. 20 yrs free

  6. The outcome… Clyde deny confess deny 1 yrs free 1 yrs 20 yrs Bonnie confess 20 yrs 10 yrs free 10 yrs Both Bonnie and Clyde confess.

  7. PD: cooperative perspective Clyde deny confess deny 1 yrs free 1 yrs 20 yrs Bonnie confess 20 yrs 10 yrs free 10 yrs 10 12 10 0 2 0 12 2

  8. PD: cooperative perspective dc dd Clyde cc cd Bonnie Feasible outcomes 10 12 10 0 2 0 12 2

  9. Repeated PD dc dd PD Clyde cc dc dd cd cc cd PD PD PD PD Bonnie Feasible outcomes cc dc dd cd An equilibrium strategy that guarantees dd Keep denying as long as your opponent does. Else, keep confessing for ever.

  10. Repeated PD ½ dc + ½ dd An equilibrium strategy that guarantees ½ dc + ½ dd Bonnie’s role: Keep denyingas long as Clyde sticks to his role. Else, keep confessing for ever. Clyde’s role: Keep denying on odd days and confessing on even days as long as Bonnie sticks to her role. Else, keep confessing for ever. dc dd PD Clyde cc dc dd cd cc cd PD PD PD PD Bonnie Feasible outcomes cc dc dd cd

  11. Time in service of cooperation: • Commitments are long term plans, • Enforcement by punishment, • Enables generation of any frequency of pure outcomes. The Folk Theorem: Any cooperative outcome, is attainable as an equilibrium in the repeated game. in which each player gets at least her individually rational payoff, • How can commitments be made without repetition? • What outcomes can be achieved?

  12. Commitments to act Choosing which commitment to make is a voluntary non-cooperative action. Suppose Bonnie and Clyde can submit irrevocable commitments. Clyde Bonnie I hereby commit to deny our involvement in the robbery. I hereby commit to confess our involvement in the robbery. I hereby commit to deny our involvement in the robbery. An unconditional commitment to act does not help.

  13. Commitments to act Conditional commitments The commitment is incomplete.What if Clyde commits to confess? Hmmmm...... These commitments fail to determine players’ actions. Suppose Bonnie and Clyde can submit irrevocable commitments. Clyde Bonnie I hereby commit to deny ifClyde denies, to confess ifClyde confesses. I hereby commit to deny ifBonnie denies, to confess ifBonnie confesses. I hereby commit to confess our involvement in the robbery. I hereby commit to deny ifClyde denies. I hereby commit to deny our involvement in the robbery.

  14. Conditional commitments May even be incompatible... I commit to undersell my competitor. I commit to undersell my competitor. grocer I grocer II Conditional commitments may be incomplete, undefined or incompatible... The problem is that the action is conditioned on the opponent’s action.

  15. commitment condition on commitments = Bonnie confesses, Clyde denies. Clyde Bonnie C1 C2 C3 B1 B2 B3 ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... B2 C3 If C1, confess; If C2, deny; If C3, confess. If B1, deny; If B2, deny; If B3, confess.

  16. Nigel Howard (1971) Paradoxes of Rationality: Theory of Metagames and Political Behaviour John Harsanyi (1967-8) Games with incomplete information played by Bayesian players A hierarchy of responses(commitments?) Player I: actions Player II: Responses to I’s action. Player I: Responses to II’s responses to I’s actions. and so on… A hierarchy of beliefs Player I: Beliefs about G Player II: Beliefs about I’s beliefs about G. Player I: Beliefs about II’s beliefs about I’s beliefs about G. and so on… Types A player’s type is her beliefs about her opponents’ types

  17. commitment condition on commitments = Clyde Bonnie C1 C2 C3 B1 B2 B3 ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... B2 C3 If C1, confess; If C2, deny; If C3, confess. If B1, deny; If B2, deny; If B3, confess.

  18. They get rid of me... Good-bye!

  19. Program Equilibria (Tennenholtz) Both Bonnie and Clyde deny Text of program Program = = Both Bonnie and Clyde deny Each player’s program is the best response to the opponent’s program Name of commitment Commitment to act conditioned on opponent’s commitment Both Bonnie and Clyde deny Clyde Bonnie function Act (opp_prog) { if (opp_prog = this_prog) thenreturn “deny”; elsereturn “confess”; } function Act (opp_prog) { if (opp_prog = this_prog) thenreturn “deny”; elsereturn “confess”; }

  20. How to mix pure outcome Agent II desired agreement q 1-q OI 1 0 p 10 1 Agent 2 Agent I 10 0 1-p II 0 0 IO OO Agent I Entry game Choose mixed action Agent I’s commitment: If II’s commitment is A, play I with probability p and O with prob. 1-p.If II’s commitment is B, .....

  21. How to mix pure outcome desired agreement OI Jointly controlled lotteries Agent 2 I Agent IIComm-s II IO 1/2 1/2 OO Agent I b II g II I O g I 1/2 I Agent IComm-s O O I b I 1/2 I O

  22. A “Folk Theorem” for the commitment submission game There exist (infinite) commitment sets for player I and II, such that every feasible outcome (above the individual rational level) is attained as a mixed Nash equilibrium of the commitment submission game.

  23. Some morals… • Base your commitment on the whole scheme of your opponent’s commitment, not on her action. • You can make the choices of your actions unequivocal (deterministic), but… • You should allow for ambiguity concerning the choice of your scheme of commitment.

More Related