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Game Theory

Game Theory. Lecture 8. problem set 8. from Osborne’s Introd. To G.T. Ex. (426.1) , 428.1, 429.1, 430.1, 431.1, (431.2). A reminder. Repeated games. The grim (trigger) strategy. Begin by playing C and do not initiate a deviation from C

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Game Theory

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  1. Game Theory Lecture 8

  2. problem set 8 from Osborne’s Introd. To G.T. Ex. (426.1), 428.1, 429.1, 430.1, 431.1, (431.2)

  3. A reminder Repeated games 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. Is the grim strategy a Nash equilibrium? i.e. is the pair (grim , grim) a N.E. ?? Notin a finitely repeated Prisoners’ Dilemma. Punishment does not seem to work in the finitely repeated game.

  4. Every Nash equilibrium of the finitely repeated P.D. generates a path along which the players play only D Proof: Consider the last time that any of the players plays C along the Nash Equilibrium path. (assume it is player 1) After that period they both play D > > > He is better off here If he switches to play D:

  5. A reminder 1 C D 2 2 C C D 1 1 1 1 2 2 2 2 2 2 2 2 1 1 C D C D infinitely An infinitely repeated prisoners’ Dilemma sub-games D

  6. 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.

  7. An infinitely repeated game The payoff of player 1 following a history { a0, a1, ….. at,...… } is a stream { G1(a0), G1(a1), ….. G1(at)...… }

  8. An infinitely repeated game If the payoff stream of a player is a cycle (of length n): w0,w1,w2,……wn-1,w0,w1,w2,……wn-1, w0,w1,w2,……wn-1, ……… his utility is:

  9. An infinitely repeated game

  10. An infinitely repeated game Is the pair (grim , grim) a N.E. ?? Notin a finitely repeated Prisoners’ Dilemma. (grim,grim) is a N.E. in the infinitely repeated P.D. if the discount rate is sufficiently large i.e. if the future is sufficiently important

  11. Assume that player 2 plays ‘grim’: If at some time t player 1 considers deviating from C (for the first time) ? while if he did not deviate:

  12. The payoffs:

  13. Player 1 willnot deviate if: (grim,grim) is a N.E. if the discount rate is sufficiently large i.e. if the future is sufficiently important

  14. However, (grim,grim) is not a Sub-game Perfect equilibrium of the game Assume player 1 follows the grim strategy, and that in the last period C,D was played Player 1’s (grim) reaction will be: If Player 1 follows grim: but he could do better with :

  15. Strategies as Finite Automata • A finite automaton has a finite xxxxnumber of states (+ initial state) • Each state is characterized xxxxby an action • Input changes the state of the xxxxautomaton

  16. C C,D C D D The grim strategy A state and its action Inputs : The actions of the other player { C,D } The transition: How inputs change the state Initial State

  17. C,D D C,D C D C C C C D D D D D D D D D D C,D C,D C,D C,D C,D C,D Some more strategies Modified Grim C,D C C D D D D D C,D C,D C,D 4 1 2 3

  18. Some more strategies Tit for Tat C D D C D C

  19. C D D C D C Robert Axelrod Axelrod’s Tournament Robert Axelrod: The Evolution of Cooperation, 1984 (Nice !!!) Tit for Tat C C D D C D

  20. C D D Can you still bite ??? C C D D C D C D D C D C Modified Tit for Tat Tit for Tat C C C,D ‘simpler’ than D D D C

  21. C D D Can you still bite ??? C C D C D C C D D a modification a strategy that exploits the weakness of C C C,D D D D C

  22. The FEASIBLE payoffs as π2 π1 What payoffs are N.E. payoffs of the infinitely repeated P.D. ??

  23. π2 π1 What payoffs are N.E. payoffs of the infinitely repeated P.D. ?? Clearly, Nash Equilibria payoffs are ≥(1,1) All feasible payoffs above (1,1) can be obtained as Nash Equilibria payoffs ???? The folk theorem (R. Aumann, J. Friedman) (1,1)

  24. All feasible payoffs above (1,1) can be obtained as Nash Equilibria payoffs π2 π1 Proof: choose a point in this region it can be represented as: (1,1)

  25. π2 π1 Proof: The coefficientsαican be approximated by rational numbers (1,1)

  26. π2 π1 Proof: { { { { If the players follow this cycle, their payoff will be approximately the chosen point when the discount rate is close to 1. (1,1)

  27. π2 π1 Proof: { { { { A strategy: Follow the sequence of the cycle as long as the other player does. If not, play D forever. (1,1)

  28. π2 π1 { { { { A strategy: Follow the sequence of the cycle as long as the other player does. If not, play D forever. This pair of strategies is a N.E. (1,1)

  29. One Deviation PropertyandAgent Equilibria One Deviation Property A player cannot increase his payoff in a sub-game in which he is the first to move, by changing his action in that node only. A Theorem A strategy profile is a sub-game perfect equilibrium in an extensive game with perfect information iff both strategies have the one deviation property (no proof)

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