Learning in Games

1. Learning in Games

2. Fictitious Play

3. Notation! For n Players we have: • n Finite Player’s Strategies Spaces S1, S2, …, Sn • n Opponent’s Strategies Spaces S-1, S-2, …, S-n • n Payoff Functions u1, u2,…, un • For each i and each s-i in S-i a set of Best Responses BRi (s-i)

4. What is Fictitious Play? Each player creates an assessment about the opponent’s strategies in form of a weight function:

5. Prediction Probability of player i assigning to player –i playing s-i at time t:

6. Fictious Play is … • … any rule that assigns NOT UNIQUE!

7. Further Definitions In 2 Player games: Marginal empirical distributions of j’s play (j=-i)

8. Asymptotic Behavior • Propositions: • Strict Nash equilibria are absorbing for the process of fictitious play. • Any pure-strategy steady state of fictitous play must be a Nash equilibrium

9. Example “matching pennies” H T H T

10. Example “matching pennies” H T H T Weights: Row Player Col Player H T H T T T

11. Example “matching pennies” H T H T Weights: Row Player Col Player H H T T H H T T T T H T H T H T H T H T H T

12. Example “matching pennies” H T H T Weights: Row Player Col Player H T H T T T H T H T T H H T H T H T H T

13. Example “matching pennies” H T H T Weights: Row Player Col Player H T H T T T H T H T T H H T H T H T H T

14. Example “matching pennies” H T H T Weights: Row Player Col Player H T H T T T H T H T T H H T H T T H H T H T

15. Example “matching pennies” H T H T Weights: Row Player Col Player H T H T T T H T H T T H H T H T T H H T H T

16. Example “matching pennies” H T H T Weights: Row Player Col Player H T H T T T H T H T T H H T H T T H H T H T H H

17. Example “matching pennies” H T H T Weights: Row Player Col Player H T H T T T H T H T T H H T H T T H H T H T H H

18. Example “matching pennies” H T H T Weights: Row Player Col Player H T H T T T H T H T T H H T H T T T H H T H T H H

19. Weights: Row Player Col Player T T T H T H H H H H H H H T H T

20. Convergence? Strategies cycle and do not converge … …but the marginal empirical distributions?

21. MATLAB Simulation - Pennies Game Play Weight / Time Payoff

22. Proposition Under fictitious play, if the empirical distributions over each player’s choices converge, the strategy profile corresponding to the product of these distributions is a Nash equilibrium.

23. A B C 1,0 0,1 A 0,1 1,0 B 1,0 0,1 C Rock-Paper-Scissors Game Play Weight / Time Payoff

24. 1,0 0,1 0,1 1,0 1,0 0,1 Rock-Paper-Scissors Game Play Weight / Time Payoff

25. 0,0 1,0 0,1 0,1 0,0 1,0 1,0 0,1 0,0 Shapley Game Game Play Weight / Time Payoff Does not converge!

26. A B A 0,0 1,1 1,1 0,0 B Initial weights: 1 1.4 1 1.4 Persistent miscoordination Game Play Weight / Time Payoff Nash: (1,0) (0,1) (0.5,0.5)

27. A B A 0,0 1,1 1,1 0,0 B Initial weights: 2 1.4 2 1.4 Persistent miscoordination Game Play Weight / Time Payoff Nash: (1,0) (0,1) (0.5,0.5)

28. A B A 0,0 1,1 1,1 0,0 B Initial weights: 2 2.4 2 2.4 Persistent Miscoordination Game Play Weight / Time Payoff Nash: (1,0) (0,1) (0.5,0.5)

29. Summary on fictitious play • In case of convergence, the time average of strategies forms a Nash Equilibrium • The average payoff does not need to be the one of a Nash (e.g. Miscoordination) • Time average may not converge at all (e.g. Shapley Game)

30. References • Fudenberg D., Levine D. K. (1998) The Theory of Learning in Games MIT Press

31. Nash Convergence of Gradient Dynamics in General-Sum Games

32. Notation • 2 Players: • Strategies and • Payoff matricies R= C=

33. Objective Functions • Payoff Functions: • Vr(,)=r11()+r22((1-)(1-)) +r12((1-))+r21((1-)) • Vc(,)=c11()+c22((1-)(1-)) +c12((1-))+c21((1-))

34. Hillclimbing Idea

35. Gradient Ascent for Iterated Games • With u=(r11+r22)-(r21+r12) • u’=(c11+c22)-(c21+c12)

36. can be arbitrary strategies Stepsize Update Rule

37. Problem • Gradient can lead the players to an infeasible point outside the unit square. 1 0 1

38. Solution: • Redefine the gradient to the projection of the true gradient onto the boundary. Let this denote the constrained dynamics! 1 0 1

39. U Infinitesimal Gradient Ascent (IGA) Become functions of time!

40. 1. Case: U is invertible The two possible qualitative forms of the unconstrained strategy pair:

41. 2. Case: U is not invertible Some examples of qualitative forms of the unconstrained strategy pair:

42. Convergence If both players follow the IGA rule, then both player’s average payoffs will converge to the expected payoff of some Nash equilibrium If the strategy pair trajectory converges at all, then it converges to a Nash pair.

43. Proposition Both previous propositions also hold with finite decreasing step size

44. References • Singh S., Kearns M., Yishay M. (2000) Nash Convergence of Gradient Dynamics in General-Sum Games Proceedings of the Sixteenth Conference on Uncertainty in Artificial Intelligence, Morgan Kaufmann, pages 541-548

45. Dynamic computation of Nash equilibria in Two-Player general-sum games.

46. Notation • 2 Players: • Strategies and • Payoff matricies R= C=

47. Objective Functions • Payoff Functions: • Row Player: • Col Player:

48. Observation! is linear in each pi and qj Let xi denote the pure strategy for action i. This means: If then the value of pi the payoff. increasing increases

49. Hill climbing (again) Multiplicative Update Rules

50. Hill climbing (again) System of Differential Equations (i=1..n)