Georgia Kastidou David R. Cheriton School of Computer Science University of Waterloo

1. Simple search methods for finding a Nash equilibriumRyan Porter, Eugene Nudelman, and Yoav ShohamGames and Economic Behavior, Vol. 63, Issue 2. pp. 642-661, 2004 Georgia Kastidou David R. Cheriton School of Computer Science University of Waterloo

2. Outline • Problem • Contribution • Algorithm for 2-player game • Experimental Results • Algorithm for n-player game • Experimental Results • Conclusions • Future Work • Take Home Message

3. Problem we will tackle… • Consider a n-player normal-form game G=N, (Ai), (ui) where: • N=[1,..n] is the set of players • Ai=[ai1,…,aimi] is the set of actions available to player i • ui:A1x…xAnR is the utility function for each player • Each player selects a mixed strategy from the set of available strategies: • “Support” of a mixed strategy pi is the set of all actions ai in Ai such that pi(ai)>0. • x=(x1,..,xn): xi is the size of agent’s i support • The expected utility for player i for a strategy profile p=(p1,...,pn) is: where: • Problem: Design an algorithm which can find a Nash Equilibrium for a normal form game • A strategy profile p* in P is a Nash Equilibrium (NE) if:

4. Development of Algorithms • Two extremes: • 1st: Aims to design a low complexity algorithm • Gain deep insight in the structure of the problem • Design highly specialized algorithms • 2nd: Aims to design more simple algorithms with an “acceptable” complexity • Identify shallow heuristics • Hoping that given the increasing computing power they will be sufficient. • focus on more “common” problems • Which one is better? • Although the first is more interesting, in practice in a number of cases the second is preferred either because is simpler or because it outperforms the first. • Note: There are cases of optimal algorithms that have never been implemented because they are too complicate.

5. Background/Related Work • Nash equilibrium is an important concept in game theory • “little is known about the problem of computing a sample NE in a normal-form game” • Normal form game is guaranteed to have at least one NE • It does not fall into a standard complexity class (Papadimitriou, 2001) • it cannot be cast as a decision problem.

6. Background/Related Work • 2-player game: • Lemke-Howson, (1964) • Dickhaut and Kaplan (1991) • Enumerates all possible pairs of supports for 2-play game • For each pair it solves a feasibility program • n-player game: • Simplicial Subdivision (van der Laan et al., 1987) • Approximates a fixed point of a function which is defined on a simplotope • Govindan and Wilson (2003) • first perturbs a game to one that has a known equilibrium, and • then traces the solution back to the original game as the magnitude of the perturbation approaches zero.

7. Background/Related Work • GAMUT: a “recently” (2004) introduced computational testbed for game theory “Run the GAMUT:A Comprehensive Approach to Evaluating Game-Theoretic Algorithms” by E. Nudelman et al.

8. What the author propose and what’s their contribution? • Propose: • heuristic-based algorithms for 2-player games and for n-player games • explore the space of support profiles using a backtracking procedure to instantiate the support for each player separately. • test using a variety of different distributions • Use of GAMUT (computational testbed for game theory) • Contribution: • in a big number of cases the proposed algorithms outperform the algorithm of Lemke-Howson on 2-player games and the Siplicial Subdivision on n-player games.

9. The Proposed Algorithms • explore “support” profiles: • pure strategies played with nonzero probability • use backtracking procedures • are biased towards simple solutions • preference for small supports • based on the observation that a number of games in the past proved to have at least one simple solution. • e.g. for n = 2, the probability that there exists a NE consistent with a particular support profile varies inversely with the size of the supports, and is zero for unbalanced support profiles.

10. Proposed Algorithm for 2-players game

11. Proposed Algorithm for 2-players game υi : expected utility of agent i in an equilibrium

12. Proposed Algorithm for 2-players game • The first two classes of constraints require that: • each player must be indifferent between all actions within his/her support, and • must not strictly prefer an action outside of his/her support.

13. Experimental results • The authors consider games from a number of different distributions • D18: most common one • D5, D6, and D7 are also important distributions

14. Experimental ResultsAlgorithms for 2-player games • Experiment-Setup: • 2-player, 300-action games drawn from 24 of GAMUT’s 2- player distributions. • executed on 100 games drawn from each distribution. • First diagram: • compares the unconditional median running times of the algorithms, • might reflect the fact that there is a greater than 50% chance that the distribution will generate a game with a pure • Second diagram: • Compares the percentage of instances solved • Third diagram: • the average running time conditional on solving an instance

15. Experimental ResultsAlgorithms for 2-player games Compares the unconditional median running times of the algorithms. (“Might reflect the fact that there is a greater than 50% chance that the distribution will generate a game with a pure”)

16. Experimental ResultsAlgorithms for 2-player games Compares the percentage of instances solved

17. Experimental ResultsAlgorithms for 2-player games Compares the average running time conditional on solving an instance (unconditional average running time)

18. Experimental ResultAlgorithms for 2-player games Compare the scaling behavior as the number of actions increases (unconditional average running time)

19. Experimental ResultAlgorithms for 2-player games • Covariance Games • neither of the algorithms solved any of the games in another “Covariance Game” distribution in which ρ =−0.9,

20. Proposed Algorithm for n-players games • Uses a general backtracking algorithm to solve a constraint satisfaction problem (CSP) for each support size profile • The variables in each CSP are: • the supports Si, and • the domain of each Siis the set of supports of size xi. • Constraints: • no agent plays a conditionally dominated action.

21. Proposed Algorithm for n-players games • IRSDS: • Input a domain for each player’s support. • For each agent whose support has been instantiated the domain contains only that instantiated support, • For each other agent i it contains all supports of size xithat were not eliminated in a previous call to this procedure. • On each pass of the repeat-until loop, • every action found in at least one support in a player’s domain is checked for conditional domination. • If a domain becomes empty after the removal of a conditionally dominated action, • the current instantiations of the Recursive-Backtracking are inconsistent, and IRSDS returns failure. • IRSDS repeats until it either returns failure or iterates through all actions of all players without finding a dominated action.

22. Proposed Algorithm for n-players games

23. Algorithm 2 For all x=(x1,..xn) sorted in increasing order first by: and then by: R-B … IRSDS IRSDS IRSDS R-B R-B R-B … IRSDS IRSDS IRSDS R-B R-B R-B Failed IRSDS R-B: Recursive Backtracking IRSDS:Iterated Removal of Strictly Dominated Strategies RBT Failed

24. Experimental Results: n-player games • Experiment-Setup: • 6-player, 5-action games drawn from 22 of GAMUT’s n-player distributions. • 15,625 outcomes and 93,750 payoffs • executed on 100 games drawn from each distribution. • First diagram: • compares the unconditional median running times of the algorithms, • might reflect the fact that there is a greater than 50% chance that the distribution will generate a game with a pure • Second diagram: • Compares the percentage of instances solved • Third diagram: • the average running time conditional on solving an instance

25. Experimental Results: n-player games • Compares the unconditional median running times of the algorithms Compares the percentage of instances solved

26. Experimental Results: n-player games Compares the average running time conditional on solving an instance

27. Experimental Results: n-player games • Compare the scaling behavior: number of players constant at 6 number of actions varies. • (unconditional average running time) • Compare the scaling behavior: number of players varies, number of actions constant 5. • (unconditional average running time)

28. Experimental ResultsPercentage of Pure Strategy NE • (2-player game) • n-player game

29. Experimentals ResultsAverage measure of support balanced • 6-player, 5-action games • 2-player, 300-action games

30. Conclusions • Propose algorithms that use backtracking approaches to search the space of support profiles, favoring supports that are small and balanced. • Both algorithms outperform the current state of the art. • The most difficult games • “Covariance Game” model, as the covariance approaches its minimal value • hard because authors found that: • as the covariance decreases, the number of equilibria decreases, and • the equilibria that do exist are more likely to have support sizes near one half of the number of actions

31. Future Work • Employ more sophisticated CSP techniques • Explore local search, in which the state space is the set of all possible supports, and the available moves are to add or delete an action from the support of a player • Study the games that are generated by the Covariance Game distribution

32. Take home message • Studying the results of complicated problems can lead to observations that although might not provide ideas to find optimal solutions can provide insights on how to improve current approaches. • The selection of the tests and the parameters that will be examined very important. • Not only because they can show that your algorithm is working… • E.g. “Covariance Game” model might proved a good starting point for game theoretic algorithms