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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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Presentation on theme: "Simple Search Methods for Finding a Nash Equilibrium Ryan Porter, Eugene Nudelman, & Yoav Shoham Computer Science Department Stanford University."— Presentation transcript:

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

2 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

3 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/

4 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

5 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/

6 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)

7 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

8 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

9 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

10 2-player Games Tested on player, 300-action games for each of 22 distributions Capped all runs at 1800s

11 2-player Games: Scaling

12 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

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

14 N-player Games: Scaling

15 N-player Games: Covariance Games

16 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

17 2-player Random Games

18 2-player Games: Covariance Games

19 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

20 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

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


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