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**Simple Search Methods for Finding a Nash Equilibrium**

3/25/2017 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**

3/25/2017 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 “...most fundamental computational problem whose complexity is wide open” [P01] N-player games

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**Notation Normal Form Game G = h N,(Ai),(ui) i:**

3/25/2017 Notation Normal Form Game G = h N,(Ai),(ui) i: N = {1,…,n}: set of players Ai: set of available actions for player i ui: A1 x …x An ! < Player i selects a mixed strategy: pi: Ai ! [0,1], s.t. ai 2 Ai pi(ai) = 1 Utility function extended to take p=(p1,…,pn): ui(p) = a 2 A ui(a) i 2 N pi(ai) A strategy profile p* is a NE if: 8 i 2 N, ai 2 Ai: ui(ai,p*-i) ≤ ui(p*i,p*-i) 1/2 1/2 1,-1 -1,1 1/2 1/2

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3/25/2017 A Harder Game 2/7 3/7 2/7 2,3 -1,4 2,4 5,2 1,-1 2,2 3,0 4,1 -2,4 1,3 4,6 7,2 2,-2 4,9 2,1 9,0 -2,6 6,3 7,0 0,5 3,2 6,1 2,5 5,3 1,0 2/11 4/11 The problem of finding a NE can be formulated as a complementarity problem, which, even in the 2-player case, is hard 5/11

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**Searching Over Supports**

3/25/2017 Searching Over Supports Feasibility Problem: Input: S = (S1,...,SN), where 8 i 2 N, Si µ Ai Find: p=(p1,…,pn) and v=(v1,…,vn) Subject to: 8 i 2 N 8 ai 2 Si, pi(ai) ≥ 0 8 ai 2 Si, pi(ai) = 0 ai 2 Ai pi(ai) = 1 8 ai 2 Si, a-i 2 A-i ui(ai,a-i) ji p(aj) = vi 8 ai 2 Si, a-i 2 A-i ui(ai,a-i) ji p(aj) ≤ vi 2/7 3/7 2/7 2,3 -1,4 2,4 5,2 1,-1 2,2 3,0 4,1 -2,4 1,3 4,6 7,2 2,-2 4,9 2,1 9,0 -2,6 6,3 7,0 0,5 3,2 6,1 2,5 5,3 1,0 2 2 2/11 4 4 4/11 5/11 4 2 3 3 3 2

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**Features of Algorithm Prefer balanced supports Prefer small supports**

3/25/2017 Features of Algorithm Prefer balanced supports Prefer small supports Motivated by existing theoretical results for particular distributions (e.g., [MB02]) Separately instantiate supports, and remove conditionally dominated actions: An ai is conditionally dominated, given R-i µ A-i if: 9 ai' 2 Ai, 8 a-i 2 R-i, ui(ai,a-i) < ui(ai',a-i) Especially useful in conjunction with (2) The distribution of [MB02] is equivalent to one in which each payoff is independently drawn from a normal distribution In 2-player games under this model, there is zero probability that a NE is consistent with any particular unbalanced support profile Probability of existence of NE consistent with support profile varies inversely with the size of the support

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3/25/2017 Two-Player Algorithm FOR ALL x = (x1,x2), sorted in increasing order of |x1 – x2| and (x1 + x2) FOR ALL S1 µ A1 s.t. |S1| = x1 A2' ← {a2 2 A2 not conditionally dominated, given S1} a1 2 S1 conditionally dominated, given A2' FOR ALL S2 µ A2' s.t. |S2| = x2 a1 2 S1 conditionally dominated, given S2 IF Feasibility Problem satisfied for (S1,S2) Return found NE p

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3/25/2017 N-Player Algorithm Constraint Satisfaction Problem (CSP) for each support size profile x=(x1,x2): Variables: Si Domain: all subsets of Ai of size xi 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 Second set of constraints is weaker, but easier to check

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3/25/2017 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 D1 Bertrand Oligopoly D2 Bidirectional LEG, Complete Graph D3 Bidirectional LEG, Random Graph D4 Bidirectional LEG, Star Graph D5 Covariance Game: = 0.9 D6 Covariance Game: = 0 D7 Covariance Game: Random 2 [-1/(N-1),1] D8 Dispersion Game D9 Graphical Game, Random Graph D10 Graphical Game, Road Graph D11 Graphical Game, Star Graph D12 Location Game D13 Minimum Effort Game D14 Polymatrix Game, Random Graph D15 Polymatrix Game, Road Graph D16 Polymatrix Game, Small-World Graph D17 Random Game D18 Traveler’s Dilemma D19 Uniform LEG, Complete Graph D20 Uniform LEG, Random Graph D21 Uniform LEG, Star Graph D22 War Of Attrition based on an extensive literature search of economics, game theory, computer science

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3/25/2017 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**

3/25/2017 2-player Games: Scaling

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**2-player Games: Covariance Games**

3/25/2017 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 1000 points

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3/25/2017 N-player Games Tested on player, 5-action games for each distribution

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**N-player Games: Scaling**

3/25/2017 N-player Games: Scaling

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**N-player Games: Covariance Games**

3/25/2017 N-player Games: Covariance Games

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**BFS Lemke-Howson Lemke-Howson algorithm:**

3/25/2017 BFS Lemke-Howson Lemke-Howson algorithm: Pivoting method to solve LCP for a 2-player game First pivot is an arbitrary selection of a1 2 A1 Afterwards, a deterministic path to a NE Idea: favor “simple” solutions Breadth-First Search: FOR ALL a1 2 A1 Initialize Lemke-Howson(a1) REPEAT Pivot Lemke-Howson(a1) IF found a NE, THEN return p

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**2-player “Random” Games**

3/25/2017 2-player “Random” Games

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**2-player Games: Covariance Games**

3/25/2017 2-player Games: Covariance Games

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**Summary CSP-based algorithms BFS Lemke-Howson Commentary on problem:**

3/25/2017 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**

3/25/2017 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**

3/25/2017 Simple Search Methods for Finding a Nash Equilibrium Ryan Porter, Eugene Nudelman, & Yoav Shoham Computer Science Department Stanford University

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