Polylogarithmic Supports are required for Approximate Well-Supported Nash Equilibria below 2/3



Anbalagan, Yogesh, Norin, Sergey, Savani, Rahul ORCID: 0000-0003-1262-7831 and Vetta, Adrian
Polylogarithmic Supports are required for Approximate Well-Supported Nash Equilibria below 2/3. .

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Abstract

In an epsilon-approximate Nash equilibrium, a player can gain at most epsilon in expectation by unilateral deviation. An epsilon well-supported approximate Nash equilibrium has the stronger requirement that every pure strategy used with positive probability must have payoff within epsilon of the best response payoff. Daskalakis, Mehta and Papadimitriou conjectured that every win-lose bimatrix game has a 2/3-well-supported Nash equilibrium that uses supports of cardinality at most three. Indeed, they showed that such an equilibrium will exist subject to the correctness of a graph-theoretic conjecture. Regardless of the correctness of this conjecture, we show that the barrier of a 2/3 payoff guarantee cannot be broken with constant size supports; we construct win-lose games that require supports of cardinality at least Omega((log n)^(1/3)) in any epsilon-well supported equilibrium with epsilon < 2/3. The key tool in showing the validity of the construction is a proof of a bipartite digraph variant of the well-known Caccetta-Haggkvist conjecture. A probabilistic argument shows that there exist epsilon-well-supported equilibria with supports of cardinality O(log n/(epsilon^2)), for any epsilon> 0; thus, the polylogarithmic cardinality bound presented cannot be greatly improved. We also show that for any delta > 0, there exist win-lose games for which no pair of strategies with support sizes at most two is a (1-delta)-well-supported Nash equilibrium. In contrast, every bimatrix game with payoffs in [0,1] has a 1/2-approximate Nash equilibrium where the supports of the players have cardinality at most two.

Item Type: Conference or Workshop Item (Paper)
Additional Information: Added details on related work (footnote 7 expanded)
Uncontrolled Keywords: cs.GT, cs.GT
Depositing User: Symplectic Admin
Date Deposited: 28 Jan 2015 10:00
Last Modified: 12 Nov 2019 14:17
Related URLs:
URI: http://livrepository.liverpool.ac.uk/id/eprint/2005760

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