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A Polynomial Time Algorithm for 2 -Player Rank 1 Games

A Polynomial Time Algorithm for 2 -Player Rank 1 Games. Ruta Mehta. Based on a joint work with Bharat Adsul , Jugal Garg and Milind Sohoni. m strategies. n strategies. B. A. Mixed Strategy/Randomize. B. A. Goal: Maximize expected payoff. Nash Equilibrium. John Nash (1951):

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A Polynomial Time Algorithm for 2 -Player Rank 1 Games

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  1. A Polynomial Time Algorithm for 2-Player Rank 1 Games Ruta Mehta Based on a joint work with Bharat Adsul, JugalGarg and MilindSohoni

  2. m strategies n strategies B A

  3. Mixed Strategy/Randomize B A Goal: Maximize expected payoff

  4. Nash Equilibrium John Nash (1951): Given a finite game, there exists a tuple of mixed-strategy vectors, one for each player, such that no player gains by deviating unilaterally.

  5. Computation k-Nash: Computing a Nash equilibrium of a finite k player game. 2-Nash:Rational Solutions PPAD-complete (Papadimitriou’92, DGP’06, CD’06) k-Nash, : Algebraic Solutions

  6. Computation k-Nash: Computing a Nash equilibrium of a finite k player game. 2-Nash:Rational Solutions PPAD-complete (Papadimitriou’92, DGP’06, CD’06) k-Nash, : Irrational Solutions (Nash’51) FIXP-complete (Etessami & Yannakakis’07)

  7. 2-Nash • von Neumann (1928):In zero-sum games (), Min-Max strategies are stable. • Linear programming duality In P. (Dantzig’51, Adler’10) • Kannan Theobald (2005): Defined rank of a game as rank(A+B). • Zero-sum rank 0.

  8. 2-Nash • von Neumann (1928):In zero-sum games (), Min-Max strategies are stable. • Linear programming duality In P. (Dantzig’51, Adler’10) • Kannan Theobald (2005): Defined rank of a game as rank(A+B). • Zero-sum rank 0. FPTAS for fixed-rank. • Can we solve rank 1 games in polynomial time?

  9. Difficulty in Rank 1 Games Disconnected set of equilibria(KT’05). Exponentially many disconnected equilibria (von Stengel’12).

  10. 2-Nash n strategies m strategies A B • Mixed strategy • Probability distribution vectors

  11. 2-Nash n strategies m strategies A B • Mixed strategy is a Nash equilibriumiff • fetches the best payoff to Alice, against . • fetches the best payoff to Bob, against

  12. 2-Nash Characterization n strategies m strategies A B If Bob plays then for Alice • Expected payoff from her strategy is A

  13. Bob plays . Then for Alice • Expected payoff from her strategy is A • Best strategies are • achieves max payoffiff Related toComplementarity

  14. For Bob Related toComplementarity Fixing for Alice, Bob’s payoff is from strategy. Best strategies are achieves max payoffiff

  15. Define and Complementarity Complementarity and Nash equilibrium

  16. 2-Nash Quadratic Program

  17. Polyhedra Variable vector Scalar variable Variable vector Scalar variable

  18. Complementarity: Vertex is a NE are the payoffs

  19. At least the sum of max payoff Sum of payoffs

  20. At least the sum of max payoff Sum of payoffs is a NE are the max payoffs

  21. At least the sum of max payoff Sum of payoff 2-Nash max: Complementarity s.t. is a NE are the max payoffs

  22. Rank 1 Game Bilinear 2-Nash max: s.t.

  23. Rank 1 Game Product of two linear terms 2-Nash max: s.t. Rank 1 QP is NP-hard in general

  24. Think Big! Consider a space of rank 1 games S = {

  25. Think Big! Consider a space of rank 1 games

  26. Think Big! 2-Nash max: s.t.

  27. Think Big! Consider game space 2-Nash max: s.t.

  28. Think Big! Consider game space All NE of S Complementarity Idea Captures max: Solutions of LP( s.t. LP()

  29. (,), a NE of S Proof: At any feasible point, cost of LP() is at most zero. is feasible. Complementarity Cost is exactly zero at it. Claim: LP()

  30. LP() Goal: NE of If one of them then done! (m-1)-dimensional space in S Claim:s.t. (,) is a NE of game

  31. Goal: NE of game LP() If = then done!

  32. Goal: NE of game LP() NE of game Fixed points of If = then done!

  33. NE of game Fixed points of LP() Continuous

  34. 1-D Fixed Point Lower Upper 1 Upper Lower 0 1

  35. 1-D Fixed Point Lower Upper 1 Upper Lower 0 1 And so on until the difference becomes small enough

  36. This was in hindsight

  37. Kohlberg and Mertens (1986) Finite player Game space x Homeomorphic Its Nash correspondence g

  38. Kohlberg and Mertens (1986) Finite player Game space x Continuous Bijection Its Nash correspondence g Stability, Index (Shapley’74, KM’86, GW’97) To design homotopy methods. (Govindan & Wilson’03,…) No extension to subspaces known

  39. We Show That Game space S = Its Nash correspondence Homeomorphic

  40. Rank 2 Games Open: A polynomial time algorithm.

  41. PPAD = P?

  42. Simplex method: Practical, although (Dantzig’47) exponential in worst case In P or NP-hard? Khachiyan’79: Polynomial time solvable.

  43. Though exponential in worst case

  44. vs. Stumbling Block

  45. PPAD = P? vs. Stumbling Block?? Our result Not in general

  46. Thanks

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