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Simple Search Methods for Finding a Nash Equilibrium Ryan Porter, Eugene Nudelman, & Yoav Shoham Computer Science Department Stanford University

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Finding a Sample Nash Equilibrium Nash equilibrium (NE) Arguably the most important concept in game theory One always exists [N51] Finding a sample NE in a normal form game: Considered hard, but unknown whether it is NP-hard State of the art among existing algorithms: Lemke-Howson [LH64] Simplicial Subdivision [VV87] Govindan-Wilson [GW03] & [BSK03] Our algorithms: simple Artificial Intelligence methods that perform well in practice 2-player games N-player games

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Notation Normal Form Game G = h N,(A i ),(u i ) i : N = {1,…,n} : set of players A i : set of available actions for player i u i : A 1 x …x A n ! < Player i selects a mixed strategy: p i : A i ! [0,1], s.t. a i 2 A i p i (a i ) = 1 Utility function extended to take p=(p 1,…,p n ): u i (p) = a 2 A u i (a) i 2 N p i (a i ) A strategy profile p * is a NE if: 8 i 2 N, a i 2 A i : u i (a i,p * -i ) u i (p * i,p * -i ) 1,-1-1,1 1,-1 1/

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A Harder Game 2,3-1,42,45,21,-1 2,23,04,1-2,41,3 4,67,22,-24,92,1 9,0-2,66,37,00,5 3,26,12,55,31,0 5/11 2/ /11 3/72/7 00

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Searching Over Supports Feasibility Problem: Input: S = (S 1,...,S N ), where 8 i 2 N, S i µ A i Find: p=(p 1,…,p n ) and v=(v 1,…,v n ) Subject to: 8 i 2 N 8 a i 2 S i, p i (a i ) 0 8 a i 2 S i, p i (a i ) = 0 a i 2 A i p i (a i ) = 1 8 a i 2 S i, a -i 2 A -i u i (a i,a -i ) j i p(a j ) = v i 8 a i 2 S i, a -i 2 A -i u i (a i,a -i ) j i p(a j ) v i 2,3-1,42,45,21,-1 2,23,04,1-2,41,3 4,67,22,-24,92,1 9,0-2,66,37,00,5 3,26,12,55,31,0 5/11 2/ /11 3/72/

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Features of Algorithm 1)Prefer balanced supports 2)Prefer small supports Motivated by existing theoretical results for particular distributions (e.g., [MB02]) 3)Separately instantiate supports, and remove conditionally dominated actions: An a i is conditionally dominated, given R -i µ A -i if: 9 a i ' 2 A i, 8 a -i 2 R -i, u i (a i,a -i ) < u i (a i ',a -i ) Especially useful in conjunction with (2)

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Two-Player Algorithm FOR ALL x = (x 1,x 2 ), sorted in increasing order of |x 1 – x 2 | and (x 1 + x 2 ) FOR ALL S 1 µ A 1 s.t. |S 1 | = x 1 A 2 ' {a 2 2 A 2 not conditionally dominated, given S 1 } a 1 2 S 1 conditionally dominated, given A 2 ' FOR ALL S 2 µ A 2 ' s.t. |S 2 | = x 2 a 1 2 S 1 conditionally dominated, given S 2 IF Feasibility Problem satisfied for (S 1,S 2 ) Return found NE p

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N-Player Algorithm Constraint Satisfaction Problem (CSP) for each support size profile x=(x 1,x 2 ) : Variables: S i Domain: all subsets of A i of size x i Constraint: support profile S is consistent with a NE 2-player algorithm: Backtracking, enforcing arc consistency w.r.t. weaker constraints that no conditionally dominated actions in S N-player algorithm: Generalizes the 2-player algorithm Ordering of size and balance reversed

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Experimental Results Most previous empirical tests only on random games: Each payoff drawn independently from uniform distribution GAMUT [NWSL04] Based on extensive literature search Generates games from a wide variety of distributions Available at D1Bertrand OligopolyD2Bidirectional LEG, Complete Graph D3Bidirectional LEG, Random GraphD4Bidirectional LEG, Star Graph D5 Covariance Game: = 0.9 D6 Covariance Game: = 0 D7 Covariance Game: Random 2 [-1/(N-1),1] D8Dispersion Game D9Graphical Game, Random GraphD10Graphical Game, Road Graph D11Graphical Game, Star GraphD12Location Game D13Minimum Effort GameD14Polymatrix Game, Random Graph D15Polymatrix Game, Road GraphD16Polymatrix Game, Small-World Graph D17Random GameD18Travelers Dilemma D19Uniform LEG, Complete GraphD20Uniform LEG, Random Graph D21Uniform LEG, Star GraphD22 War Of Attrition

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2-player Games Tested on player, 300-action games for each of 22 distributions Capped all runs at 1800s

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2-player Games: Scaling

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2-player Games: Covariance Games Covariance Games: For each action profile, payoffs of all players drawn from a multivariate normal distribution, with identical covariance between any two players

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N-player Games Tested on player, 5-action games for each distribution

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N-player Games: Scaling

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N-player Games: Covariance Games

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BFS Lemke-Howson Lemke-Howson algorithm: Pivoting method to solve LCP for a 2-player game First pivot is an arbitrary selection of a 1 2 A 1 Afterwards, a deterministic path to a NE Idea: favor simple solutions Breadth-First Search: FOR ALL a 1 2 A 1 Initialize Lemke-Howson( a 1 ) REPEAT FOR ALL a 1 2 A 1 Pivot Lemke-Howson( a 1 ) IF found a NE, THEN return p

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2-player Random Games

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2-player Games: Covariance Games

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Summary CSP-based algorithms Heuristics: Favor balanced and small supports Eliminate conditionally dominated strategies Perform well in practice BFS Lemke-Howson In preliminary results, performs even better than our 2-player algorithm Commentary on problem: Games researchers care about tend to have at least one simple solution

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Future Work Coming to Gambit Focus on Covariance Games, with low covariance Other techniques from Artificial Intelligence Local Search: State: support profile Operators: add or delete an action Score: based on relaxation of the feasibility problem

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Simple Search Methods for Finding a Nash Equilibrium Ryan Porter, Eugene Nudelman, & Yoav Shoham Computer Science Department Stanford University

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