14 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
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
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 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
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
Approximate well-supported Nash equilibria below two-thirds
In an ε-Nash equilibrium, a player can gain at most ε by changing his behaviour. Recent work has addressed the question of how best to compute ε-Nash equilibria, and for what values of ε a polynomial-time algorithm exists. An ε-well-supported Nash equilibrium (ε-WSNE) has the additional requirement that any strategy that is used with non-zero probability by a player must have payoff at most ε less than a 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
Finding Approximate Nash Equilibria of Bimatrix Games via Payoff Queries
We study the deterministic and randomized query complexity of finding approximate equilibria in a k × k bimatrix game. We show that the deterministic query complexity of finding an ϵ-Nash equilibrium when ϵ < ½ is Ω(k2), even in zero-one constant-sum games. In combination with previous results [Fearnley et al. 2013], this provides a complete characterization of the deterministic query complexity of approximate Nash equilibria. We also study randomized querying algorithms. We give a randomized algorithm for finding a (3-√5/2 + ϵ)-Nash equilibrium using O(k.log k/ϵ2) payoff queries, which shows that the ½ barrier for deterministic algorithms can be broken by randomization. For well-supported Nash equilibria (WSNE), we first give a randomized algorithm for finding an ϵ-WSNE of a zero-sum bimatrix game using O(k.log k/ϵ4) payoff queries, and we then use this to obtain a randomized algorithm for finding a (⅔ + ϵ)-WSNE in a general bimatrix game using O(k.log k/ϵ4) payoff queries. Finally, we initiate the study of lower bounds against randomized algorithms in the context of bimatrix games, by showing that randomized algorithms require Ω(k2) payoff queries in order to find an ϵ-Nash equilibrium with ϵ < 1/4k, even in zero-one constant-sum games. In particular, this rules out query-efficient randomized algorithms for finding exact Nash equilibria
Computing Constrained Approximate Equilibria in Polymatrix Games
This paper is about computing constrained approximate Nash equilibria in
polymatrix games, which are succinctly represented many-player games defined by
an interaction graph between the players. In a recent breakthrough, Rubinstein
showed that there exists a small constant , such that it is
PPAD-complete to find an (unconstrained) -Nash equilibrium of a
polymatrix game. In the first part of the paper, we show that is NP-hard to
decide if a polymatrix game has a constrained approximate equilibrium for 9
natural constraints and any non-trivial approximation guarantee. These results
hold even for planar bipartite polymatrix games with degree 3 and at most 7
strategies per player, and all non-trivial approximation guarantees. These
results stand in contrast to similar results for bimatrix games, which
obviously need a non-constant number of actions, and which rely on stronger
complexity-theoretic conjectures such as the exponential time hypothesis. In
the second part, we provide a deterministic QPTAS for interaction graphs with
bounded treewidth and with logarithmically many actions per player that can
compute constrained approximate equilibria for a wide family of constraints
that cover many of the constraints dealt with in the first part