46 research outputs found
Large Supports are required for Well-Supported Nash Equilibria
We prove that for any constant and any , there exist bimatrix
win-lose games for which every -WSNE requires supports of cardinality
greater than . To do this, we provide a graph-theoretic characterization of
win-lose games that possess -WSNE with constant cardinality supports.
We then apply a result in additive number theory of Haight to construct
win-lose games that do not satisfy the requirements of the characterization.
These constructions disprove graph theoretic conjectures of Daskalakis, Mehta
and Papadimitriou, and Myers
Polylogarithmic Supports are required for Approximate Well-Supported Nash Equilibria below 2/3
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.Comment: Added details on related work (footnote 7 expanded
Approximate well-supported Nash equilibria in symmetric bimatrix games
The -well-supported Nash equilibrium is a strong notion of
approximation of a Nash equilibrium, where no player has an incentive greater
than to deviate from any of the pure strategies that she uses in
her mixed strategy. The smallest constant currently known for
which there is a polynomial-time algorithm that computes an
-well-supported Nash equilibrium in bimatrix games is slightly
below . In this paper we study this problem for symmetric bimatrix games
and we provide a polynomial-time algorithm that gives a
-well-supported Nash equilibrium, for an arbitrarily small
positive constant
Parameterized Two-Player Nash Equilibrium
We study the computation of Nash equilibria in a two-player normal form game
from the perspective of parameterized complexity. Recent results proved
hardness for a number of variants, when parameterized by the support size. We
complement those results, by identifying three cases in which the problem
becomes fixed-parameter tractable. These cases occur in the previously studied
settings of sparse games and unbalanced games as well as in the newly
considered case of locally bounded treewidth games that generalizes both these
two cases
On the Approximation Performance of Fictitious Play in Finite Games
We study the performance of Fictitious Play, when used as a heuristic for
finding an approximate Nash equilibrium of a 2-player game. We exhibit a class
of 2-player games having payoffs in the range [0,1] that show that Fictitious
Play fails to find a solution having an additive approximation guarantee
significantly better than 1/2. Our construction shows that for n times n games,
in the worst case both players may perpetually have mixed strategies whose
payoffs fall short of the best response by an additive quantity 1/2 -
O(1/n^(1-delta)) for arbitrarily small delta. We also show an essentially
matching upper bound of 1/2 - O(1/n)
An Empirical Study of Finding Approximate Equilibria in Bimatrix Games
While there have been a number of studies about the efficacy of methods to
find exact Nash equilibria in bimatrix games, there has been little empirical
work on finding approximate Nash equilibria. Here we provide such a study that
compares a number of approximation methods and exact methods. In particular, we
explore the trade-off between the quality of approximate equilibrium and the
required running time to find one. We found that the existing library GAMUT,
which has been the de facto standard that has been used to test exact methods,
is insufficient as a test bed for approximation methods since many of its games
have pure equilibria or other easy-to-find good approximate equilibria. We
extend the breadth and depth of our study by including new interesting families
of bimatrix games, and studying bimatrix games upto size .
Finally, we provide new close-to-worst-case examples for the best-performing
algorithms for finding approximate Nash equilibria
Computing Equilibria in Anonymous Games
We present efficient approximation algorithms for finding Nash equilibria in
anonymous games, that is, games in which the players utilities, though
different, do not differentiate between other players. Our results pertain to
such games with many players but few strategies. We show that any such game has
an approximate pure Nash equilibrium, computable in polynomial time, with
approximation O(s^2 L), where s is the number of strategies and L is the
Lipschitz constant of the utilities. Finally, we show that there is a PTAS for
finding an epsilo
Approximate Equilibria in Games with Few Players
We study the problem of computing approximate Nash equilibria (epsilon-Nash
equilibria) in normal form games, where the number of players is a small
constant. We consider the approach of looking for solutions with constant
support size. It is known from recent work that in the 2-player case, a
1/2-Nash equilibrium can be easily found, but in general one cannot achieve a
smaller value of epsilon than 1/2. In this paper we extend those results to the
k-player case, and find that epsilon = 1-1/k is feasible, but cannot be
improved upon. We show how stronger results for the 2-player case may be used
in order to slightly improve upon the epsilon = 1-1/k obtained in the k-player
case