Game Theory Lecture 7

1. Game Theory Lecture 7

2. problem set 7 from Osborne’s Introd. To G.T. p.442 Ex. 442.1 p.443 Ex. 443.1 from Binmore’s Fun and Games p.342 Ex. 38,40 (p.383 Ex. 8,9 p. 388 Ex. 24,29)

3. payoffs Correlated Equilibria The equilibria of this game are: [B,B] (2 , 1) [X,X] (1 , 2) and the mixed strategy equilibrium: [(2/3 , 1/3),(1/3 , 2/3) ] (2/3 , 2/3)

4. Correlated Equilibria The equilibria of this game are: [B,B] (2 , 1) payoffs [X,X] (1 , 2) and the mixed strategy equilibrium: [(2/3 , 1/3),(1/3 , 2/3) ] (2/3 , 2/3)

5. Correlated Equilibria (2 , 1) (1 , 2) (2/3 , 2/3) π2 π1

6. π2 π1 Correlated Equilibria By varying the roulette one can obtain each point in the convex hull A roulette wheel seen by both players

7. π2 π1 Correlated Equilibria The strategy pair If roulette stops on red play B if on green playX if on grey play (2/3,1/3)or (1/3,2/3) is a Nash equilibrium

8. π2 π1 Correlated Equilibria Can one do more with this coordination device?

9. π2 π1 Correlated Equilibria Can one do more with this coordination device?

10. Correlated Equilibria Hawk Dove game ‘Chicken’ Nash Equilibria: [H,D] [D,H] A mixed strategy equilibrium [ (1/2 , 1/2) , (1/2 , 1/2) ]

11. π2 π1 Correlated Equilibria Hawk Dove game ‘Chicken’ Not a Nash Equilibrium Nash Equilibria: [H,D] [D,H] A mixed strategy equilibrium [ (1/2 , 1/2) , (1/2 , 1/2) ]

12. π2 π1 Correlated Equilibria Hawk Dove game ‘Chicken’ Nash Equilibria: [H,D] [D,H] A mixed strategy equilibrium [ (1/2 , 1/2) , (1/2 , 1/2) ]

13. π2 π1 Correlated Equilibria Hawk Dove game ‘Chicken’

14. Correlated Equilibria When is it an Equilibrium to follow the referees advice?

15. Correlated Equilibria

16. Correlated Equilibria

17. Correlated Equilibria

18. Correlated Equilibria What payoffs can be obtained ???

19. π2 π1 Correlated Equilibria All payoffs in this area can be achieved by choosing α,β,γ.

20. π2 π1 Correlated Equilibria With a simple Roulette

21. Robert Aumann Correlated Equilibria

22. Repeated Games The Prisoners’ Dilemma The unique Nash Equilibrium:

23. Repeated Games The Prisoners’ Dilemma The unique Nash Equilibrium:

24. Playing D in every stage is a Nash equilibrium of the repeated game. In a repeated game,there may be a possibility of coperation by punishing deviations from cooperation.

25. Punishment The grim (trigger) strategy • Begin by playing C and do not initiate a deviation from C • If the other played D, play D for ever after.

26. The grim (trigger) strategy • Begin by playing C and do not initiate a deviation from C • If the other played D, play D for ever after. If both play grim, they never defect Is the grim strategy a Nash equilibrium? i.e. is the pair (grim , grim) a N.E. ?? Notin a finitely repeated Prisoners’ Dilemma.

27. The grim (trigger) strategy • Begin by playing C and do not initiate a deviation from C • If the other played D, play D for ever after. The grim strategy is Nota Nash equilibrium in a finitelyrepeated Prisoners’ Dilemma. Given that the other plays grim it pays to deviate in the last period and play D Indeed, to obtain cooperation in the repeated P.D. it is necessary* to have infinite repetitions. *It will be shown later

28. An infinitely repeated game sub-games history is:{ [C,D], [C,C] } history is:[D,D] 1 C D 2 2 D C C D 1 1 1 1 2 2 2 2 2 2 2 2 1 1 C D C D

29. An infinitely repeated game A history at time t is: { a1, a2, ….. at } where ai is a vector of actions taken at timei ai is [C,C] or [DC] etc. A strategy is a function that assigns an action for each history.

30. An infinitely repeated game The payoff of player 1 following a history { a1, a2, ….. at,...… } is a stream { G1(a1), G1(a2), ….. G1(at)...… }