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Game Dynamics in Extensive Form Dietmar Berwanger LSV, ENS Cachan & CNRS ICLA - Logic & Social Interaction, 2009 Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 1 / 13 Evolution in repeated games 0,1 2,1 1,2

  1. Game Dynamics in Extensive Form Dietmar Berwanger LSV, ENS Cachan & CNRS ICLA - Logic & Social Interaction, 2009 Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 1 / 13

  2. Evolution in repeated games 0,1 2,1 1,2 0,3 1,1 2,0 Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 2 / 13

  3. Evolution in repeated games 0,1 2,1 1,2 0,3 1,1 2,0 Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 2 / 13

  4. Evolution in repeated games 0,1 2,1 1,2 0,3 1,1 2,0 Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 2 / 13

  5. Evolution in repeated games 0,1 2,1 1,2 0,3 1,1 2,0 Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 2 / 13

  6. Evolution in repeated games 0,1 2,1 1,2 0,3 1,1 2,0 Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 2 / 13

  7. Evolution in repeated games 0,1 2,1 1,2 0,3 1,1 2,0 Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 2 / 13

  8. Evolution in repeated games 0,1 2,1 1,2 0,3 1,1 2,0 Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 2 / 13

  9. Evolution in repeated games 0,1 2,1 1,2 0,3 1,1 2,0 Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 2 / 13

  10. Evolution in repeated games 0,1 2,1 1,2 0,3 1,1 2,0 Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 2 / 13

  11. Evolution in repeated games 0,1 2,1 1,2 0,3 1,1 2,0 Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 2 / 13

  12. Evolution in repeated games 0,1 2,1 1,2 0,3 1,1 2,0 Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 2 / 13

  13. Evolution in repeated games 0,1 2,1 1,2 0,3 1,1 2,0 Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 2 / 13

  14. Evolution in repeated games 0,1 2,1 1,2 0,3 1,1 2,0 Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 2 / 13

  15. Evolution in repeated games Justification of Nash Equilibrium: 0,1 2,1 definite state 1,2 0,3 selective pressure 1,1 2,0 ◮ sequential dynamics Project: Reveal extensive structure from the normal-form representation of a game. Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 2 / 13

  16. Game dynamics graph 0,1 2,1 0,1 2,1 Game Γ in normal form: 1,2 0,3 1,2 0,3 n players 1,1 2,0 1,1 2,0 strategy sets S i -- finite utility u i : S → N 0,1 2,1 0,1 2,1 1,2 0,3 1,2 0,3 ◮ Game dynamics graph G ( Γ ) : 1,1 2,0 1,1 2,0 nodes: profiles in S edges: switches s → s ′ 0,1 2,1 0,1 2,1 by any subset of players 1,2 0,3 1,2 0,3 1,1 2,0 1,1 2,0 We consider pure strategies. Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 3 / 13

  17. Best-response dynamics 0,1 2,1 0,1 2,1 1,2 0,3 1,2 0,3 1,1 2,0 1,1 2,0 Greedy walks converge to Nash Equilibrium 0,1 2,1 0,1 2,1 if well-founded 1,2 0,3 1,2 0,3 e.g., in potential games. 1,1 2,0 1,1 2,0 0,1 2,1 0,1 2,1 1,2 0,3 1,2 0,3 1,1 2,0 1,1 2,0 Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 4 / 13

  18. Best-response dynamics 0,1 2,1 0,1 2,1 1,2 0,3 1,2 0,3 1,1 2,0 1,1 2,0 Greedy walks converge to Nash Equilibrium 0,1 2,1 0,1 2,1 if well-founded 1,2 0,3 1,2 0,3 e.g., in potential games. 1,1 2,0 1,1 2,0 What if not? 0,1 2,1 0,1 2,1 1,2 0,3 1,2 0,3 1,1 2,0 1,1 2,0 Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 4 / 13

  19. Sink equilibria [Vetta & al, 2005] [Goemans, Mirokni, Vetta 2005] Sink: terminal connected component of best-response graph. ◮ Price of sinking -- social cost of lack of coordination vs price of anarchy. Σu i ( ♦♣t ) Σu i ( ♦♣t ) Σu i ( ✇♦rst ❙✐♥❦❊q ) � Σu i ( ✇♦rst ◆❛s❤❊q ) Theorem. The price of anarchy can underestimate the price of sinking by a factor of n . Consequences for convergence speed of random best-response walks. Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 5 / 13

  20. Salience of the definite state Example: Tragedy of the commons - variant each player has one responsible and n irresponsible strategies ◮ responsible strategies guarantee 1 Rp ◮ irresponsible strategies pay off 2 Rs for one player ◮ all other irresponsibles -- 0 Rs who wins depends on all chosen strategies. Pareto optimum: n+1 · Nash Eq: n+ ǫ · Sink Eq: 2 ◮ Effect hides when mixing strategies relies on perfect information about the current state Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 6 / 13

  21. Strategising equilibrium selection 1,1 0,0 1,1 0,0 0,0 2,1 0,0 2,1 1,1 0,0 1,1 0,0 0,0 2,1 0,0 2,1 Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 7 / 13

  22. Strategising equilibrium selection 1,1 0,0 1,1 0,0 0,0 2,1 0,0 2,1 1,1 0,0 1,1 0,0 0,0 2,1 0,0 2,1 Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 7 / 13

  23. Efficient orbit supported by a non-greedy switch 1,1,1 0,0,0 1,1,0 2,0,1 0,0,0 2,2,2 1,2,1 1,3,3 Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 8 / 13

  24. Efficient orbit supported by a non-greedy switch 1,1,1 0,0,0 1,1,0 2,0,1 0,0,0 2,2,2 1,2,1 1,3,3 Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 8 / 13

  25. Efficient orbit supported by a non-greedy switch 1,1,1 0,0,0 1,1,0 2,0,1 0,0,0 2,2,2 1,2,1 1,3,3 Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 8 / 13

  26. Efficient orbit supported by a non-greedy switch 1,1,1 0,0,0 1,1,0 2,0,1 0,0,0 2,2,2 1,2,1 1,3,3 Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 8 / 13

  27. Efficient orbit supported by a non-greedy switch 1,1,1 0,0,0 1,1,0 2,0,1 0,0,0 2,2,2 1,2,1 1,3,3 Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 8 / 13

  28. Efficient orbit supported by a non-greedy switch 1,1,1 0,0,0 1,1,0 2,0,1 0,0,0 2,2,2 1,2,1 1,3,3 Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 8 / 13

  29. Efficient orbit supported by a non-greedy switch 1,1,1 0,0,0 1,1,0 2,0,1 0,0,0 2,2,2 1,2,1 1,3,3 Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 8 / 13

  30. Efficient orbit supported by a non-greedy switch 1,1,1 0,0,0 1,1,0 2,0,1 0,0,0 2,2,2 1,2,1 1,3,3 Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 8 / 13

  31. Efficient orbit supported by a non-greedy switch 1,1,1 0,0,0 1,1,0 2,0,1 0,0,0 2,2,2 1,2,1 1,3,3 Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 8 / 13

  32. Efficient orbit supported by a non-greedy switch 1,1,1 0,0,0 1,1,0 2,0,1 0,0,0 2,2,2 1,2,1 1,3,3 Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 8 / 13

  33. Efficient orbit supported by a non-greedy switch 1,1,1 0,0,0 1,1,0 2,0,1 0,0,0 2,2,2 1,2,1 1,3,3 Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 8 / 13

  34. Efficient orbit supported by a non-greedy switch 1,1,1 0,0,0 1,1,0 2,0,1 0,0,0 2,2,2 1,2,1 1,3,3 Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 8 / 13

  35. Efficient orbit supported by a non-greedy switch 1,1,1 0,0,0 1,1,0 2,0,1 0,0,0 2,2,2 1,2,1 1,3,3 Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 8 / 13

  36. Efficient orbit supported by a non-greedy switch 1,1,1 0,0,0 1,1,0 2,0,1 0,0,0 2,2,2 1,2,1 1,3,3 Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 8 / 13

  37. Dynamics metagame Game graph: G ( Γ ) ; 0,0 0,0 utility: cumulative ◮ mean payoff, discounted 0,1 0,0 perfect information ◮ no procedural rules ◮ fair tie breaking 0,0 1,0 Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 9 / 13

  38. Dynamics metagame Game graph: G ( Γ ) ; 0,0 0,0 utility: cumulative ◮ mean payoff, discounted 0,1 0,0 perfect information ◮ no procedural rules ◮ fair tie breaking 0,0 1,0 Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 9 / 13

  39. Basic case: two players 0,0 0,0 0,1 0,1 0,0 0,0 0,0 0,0 0,0 0,0 1,0 1,0 Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 10 / 13

  40. Basic case: two players 0,0 0,0 0,1 0,1 0,0 0,0 0,0 0,0 0,0 0,0 1,0 1,0 Theorem. [Ehrenfeucht, Mycielski] Mean-payoff zero-sum games are determined with memoryless strategies. feasible outcomes ◮ ◭ Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 10 / 13

  41. Basic case: two players 0,0 0,0 0,1 0,1 0,0 0,0 0,0 0,0 0,0 0,0 1,0 1,0 Theorem. [Ehrenfeucht, Mycielski] Mean-payoff zero-sum games are determined with memoryless strategies. feasible outcomes ◮ ◭ Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 10 / 13

  42. Basic case: two players 0,0 0,0 0,1 0,1 0,0 0,0 0,0 0,0 0,0 0,0 1,0 1,0 Theorem. [Ehrenfeucht, Mycielski] Mean-payoff zero-sum games are determined with memoryless strategies. feasible outcomes ◮ ◭ Dietmar Berwanger (CNRS) Game Dynamics Logic & Social Interaction 10 / 13

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