41 research outputs found
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
Constant Rank Bimatrix Games are PPAD-hard
The rank of a bimatrix game (A,B) is defined as rank(A+B). Computing a Nash
equilibrium (NE) of a rank-, i.e., zero-sum game is equivalent to linear
programming (von Neumann'28, Dantzig'51). In 2005, Kannan and Theobald gave an
FPTAS for constant rank games, and asked if there exists a polynomial time
algorithm to compute an exact NE. Adsul et al. (2011) answered this question
affirmatively for rank- games, leaving rank-2 and beyond unresolved.
In this paper we show that NE computation in games with rank , is
PPAD-hard, settling a decade long open problem. Interestingly, this is the
first instance that a problem with an FPTAS turns out to be PPAD-hard. Our
reduction bypasses graphical games and game gadgets, and provides a simpler
proof of PPAD-hardness for NE computation in bimatrix games. In addition, we
get:
* An equivalence between 2D-Linear-FIXP and PPAD, improving a result by
Etessami and Yannakakis (2007) on equivalence between Linear-FIXP and PPAD.
* NE computation in a bimatrix game with convex set of Nash equilibria is as
hard as solving a simple stochastic game.
* Computing a symmetric NE of a symmetric bimatrix game with rank is
PPAD-hard.
* Computing a (1/poly(n))-approximate fixed-point of a (Linear-FIXP)
piecewise-linear function is PPAD-hard.
The status of rank- games remains unresolved
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
A Polynomial-Time Algorithm for 1/2-Well-Supported Nash Equilibria in Bimatrix Games
Since the seminal PPAD-completeness result for computing a Nash equilibrium
even in two-player games, an important line of research has focused on
relaxations achievable in polynomial time. In this paper, we consider the
notion of -well-supported Nash equilibrium, where corresponds to the approximation guarantee. Put simply, in an
-well-supported equilibrium, every player chooses with positive
probability actions that are within of the maximum achievable
payoff, against the other player's strategy. Ever since the initial
approximation guarantee of 2/3 for well-supported equilibria, which was
established more than a decade ago, the progress on this problem has been
extremely slow and incremental. Notably, the small improvements to 0.6608, and
finally to 0.6528, were achieved by algorithms of growing complexity. Our main
result is a simple and intuitive algorithm, that improves the approximation
guarantee to 1/2. Our algorithm is based on linear programming and in
particular on exploiting suitably defined zero-sum games that arise from the
payoff matrices of the two players. As a byproduct, we show how to achieve the
same approximation guarantee in a query-efficient way
A Polynomial-Time Algorithm for 1/3-Approximate Nash Equilibria in Bimatrix Games
Since the celebrated PPAD-completeness result for Nash equilibria in bimatrix games, a long line of research has focused on polynomial-time algorithms that compute ?-approximate Nash equilibria. Finding the best possible approximation guarantee that we can have in polynomial time has been a fundamental and non-trivial pursuit on settling the complexity of approximate equilibria. Despite a significant amount of effort, the algorithm of Tsaknakis and Spirakis [Tsaknakis and Spirakis, 2008], with an approximation guarantee of (0.3393+?), remains the state of the art over the last 15 years. In this paper, we propose a new refinement of the Tsaknakis-Spirakis algorithm, resulting in a polynomial-time algorithm that computes a (1/3+?)-Nash equilibrium, for any constant ? > 0. The main idea of our approach is to go beyond the use of convex combinations of primal and dual strategies, as defined in the optimization framework of [Tsaknakis and Spirakis, 2008], and enrich the pool of strategies from which we build the strategy profiles that we output in certain bottleneck cases of the algorithm