2013 | OriginalPaper | Buchkapitel
Polylogarithmic Supports Are Required for Approximate Well-Supported Nash Equilibria below 2/3
verfasst von : Yogesh Anbalagan, Sergey Norin, Rahul Savani, Adrian Vetta
Erschienen in: Web and Internet Economics
Verlag: Springer Berlin Heidelberg
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In an
ε
-approximate Nash equilibrium, a player can gain at most
ε
in expectation
by unilateral deviation. An
ε
-
well-supported
approximate Nash equilibrium has the stronger requirement that every pure strategy used with positive probability must have payoff within
ε
of the best response payoff. Daskalakis, Mehta and Papadimitriou [8] conjectured that every win-lose bimatrix game has a
$\frac{2}{3}$
-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
$\frac23$
payoff guarantee cannot be broken with constant size supports; we construct win-lose games that require supports of cardinality at least
$\Omega(\sqrt[3]{\log n})$
in any
ε
-well supported equilibrium with
$\epsilon < \frac23$
. The key tool in showing the validity of the construction is a proof of a bipartite digraph variant of the well-known Caccetta-Häggkvist conjecture [4]. A probabilistic argument [13] shows that there exist
ε
-well-supported equilibria with supports of cardinality
$O(\frac{1}{\epsilon^2}\cdot \log n)$
, for any
ε
> 0; thus, the polylogarithmic cardinality bound presented cannot be greatly improved. We also show that for any
δ
> 0, there exist win-lose games for which no pair of strategies with support sizes at most two is a (1 −
δ
)-well-supported Nash equilibrium. In contrast, every bimatrix game with payoffs in [0,1] has a
$\frac{1}{2}$
-approximate Nash equilibrium where the supports of the players have cardinality at most two [8].