1,088 research outputs found
Approximate Well-supported Nash Equilibria below Two-thirds
In an epsilon-Nash equilibrium, a player can gain at most epsilon by changing
his behaviour. Recent work has addressed the question of how best to compute
epsilon-Nash equilibria, and for what values of epsilon a polynomial-time
algorithm exists. An epsilon-well-supported Nash equilibrium (epsilon-WSNE) has
the additional requirement that any strategy that is used with non-zero
probability by a player must have payoff at most epsilon less than the best
response. A recent algorithm of Kontogiannis and Spirakis shows how to compute
a 2/3-WSNE in polynomial time, for bimatrix games. Here we introduce a new
technique that leads to an improvement to the worst-case approximation
guarantee
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
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
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
Distributed Methods for Computing Approximate Equilibria
We present a new, distributed method to compute approximate Nash equilibria
in bimatrix games. In contrast to previous approaches that analyze the two
payoff matrices at the same time (for example, by solving a single LP that
combines the two players payoffs), our algorithm first solves two independent
LPs, each of which is derived from one of the two payoff matrices, and then
compute approximate Nash equilibria using only limited communication between
the players.
Our method has several applications for improved bounds for efficient
computations of approximate Nash equilibria in bimatrix games. First, it yields
a best polynomial-time algorithm for computing \emph{approximate well-supported
Nash equilibria (WSNE)}, which guarantees to find a 0.6528-WSNE in polynomial
time. Furthermore, since our algorithm solves the two LPs separately, it can be
used to improve upon the best known algorithms in the limited communication
setting: the algorithm can be implemented to obtain a randomized
expected-polynomial-time algorithm that uses poly-logarithmic communication and
finds a 0.6528-WSNE. The algorithm can also be carried out to beat the best
known bound in the query complexity setting, requiring payoff
queries to compute a 0.6528-WSNE. Finally, our approach can also be adapted to
provide the best known communication efficient algorithm for computing
\emph{approximate Nash equilibria}: it uses poly-logarithmic communication to
find a 0.382-approximate Nash equilibrium
Computing Approximate Nash Equilibria in Polymatrix Games
In an -Nash equilibrium, a player can gain at most by
unilaterally changing his behaviour. For two-player (bimatrix) games with
payoffs in , the best-known achievable in polynomial time is
0.3393. In general, for -player games an -Nash equilibrium can be
computed in polynomial time for an that is an increasing function of
but does not depend on the number of strategies of the players. For
three-player and four-player games the corresponding values of are
0.6022 and 0.7153, respectively. Polymatrix games are a restriction of general
-player games where a player's payoff is the sum of payoffs from a number of
bimatrix games. There exists a very small but constant such that
computing an -Nash equilibrium of a polymatrix game is \PPAD-hard.
Our main result is that a -Nash equilibrium of an -player
polymatrix game can be computed in time polynomial in the input size and
. Inspired by the algorithm of Tsaknakis and Spirakis, our
algorithm uses gradient descent on the maximum regret of the players. We also
show that this algorithm can be applied to efficiently find a
-Nash equilibrium in a two-player Bayesian game
Approximate Equilibrium and Incentivizing Social Coordination
We study techniques to incentivize self-interested agents to form socially
desirable solutions in scenarios where they benefit from mutual coordination.
Towards this end, we consider coordination games where agents have different
intrinsic preferences but they stand to gain if others choose the same strategy
as them. For non-trivial versions of our game, stable solutions like Nash
Equilibrium may not exist, or may be socially inefficient even when they do
exist. This motivates us to focus on designing efficient algorithms to compute
(almost) stable solutions like Approximate Equilibrium that can be realized if
agents are provided some additional incentives. Our results apply in many
settings like adoption of new products, project selection, and group formation,
where a central authority can direct agents towards a strategy but agents may
defect if they have better alternatives. We show that for any given instance,
we can either compute a high quality approximate equilibrium or a near-optimal
solution that can be stabilized by providing small payments to some players. We
then generalize our model to encompass situations where player relationships
may exhibit complementarities and present an algorithm to compute an
Approximate Equilibrium whose stability factor is linear in the degree of
complementarity. Our results imply that a little influence is necessary in
order to ensure that selfish players coordinate and form socially efficient
solutions.Comment: A preliminary version of this work will appear in AAAI-14:
Twenty-Eighth Conference on Artificial Intelligenc
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