Exploiting Concavity in Bimatrix Games: New Polynomially Tractable Subclasses

@inproceedings{Kontogiannis2010ExploitingCI,
  title={Exploiting Concavity in Bimatrix Games: New Polynomially Tractable Subclasses},
  author={Spyros C. Kontogiannis and Paul G. Spirakis},
  booktitle={APPROX-RANDOM},
  year={2010}
}
We study the fundamental problem of computing an arbitrary Nash equilibrium in bimatrix games. We start by proposing a novel characterization of the set of Nash equilibria, via a bijective map to the solution set of a (parameterized) quadratic program, whose feasible space is the (highly structured) set of correlated equilibria. We then proceed by proposing new subclasses of bimatrix games for which either an exact polynomial-time construction, or at least a FPTAS, is possible. In particular… 

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References

SHOWING 1-10 OF 32 REFERENCES

Well Supported Approximate Equilibria in Bimatrix Games

TLDR
This paper provides the first polynomial time algorithms constructing ε-SuppNE for normalized or win lose bimatrix games, for any nontrivial constant 0≤ε<1, bounded away from 1.

Well Supported Approximate Equilibria in Bimatrix Games: A Graph Theoretic Approach

We study the existence and tractability of a notion of approximate equilibria in bimatrix games, called well supported approximate Nash Equilibria (SuppNE in short).We prove existence of e-SuppNE for

Games of fixed rank: a hierarchy of bimatrix games

TLDR
It is shown that even for k = 1 the set of Nash equilibria of these games can consist of an arbitrarily large number of connected components and the question of exact polynomial time algorithms to find a Nash equilibrium remains open for games of fixed rank.

New algorithms for approximate Nash equilibria in bimatrix games

Exponentially many steps for finding a Nash equilibrium in a bimatrix game

TLDR
A class of bimatrix games for which the Lemke-Howson algorithm takes, even in the best case, exponential time in the dimension d of the game, requiring /spl Omega/((/spl theta//sup 3/4/)/sup d/) many steps, where /spl theTA/ is the golden ratio.

An Optimization Approach for Approximate Nash Equilibria

TLDR
An efficient algorithm is provided that computes 0.3393-approximate Nash equilibria, the best approximation to date, based on the formulation of an appropriate function of pairs of mixed strategies reflecting the maximum deviation of the players' payoffs from the best payoff each player could achieve given the strategy chosen by the other.

Efficient Computation of Nash Equilibria for Very Sparse Win-Lose Bimatrix Games

TLDR
A linear time algorithm is described which computes a Nash equilibrium for win-lose bimatrix games where the number of winning positions per strategy of each of the players is at most two.

A Graph Spectral Approach for Computing Approximate Nash Equilibria

TLDR
The general two-person problem is reduced to an indefinite quadratic programming problem of special structure involving the adjacency matrix of an induced simple graph specified by the input data of the game, where $n$ is the number of players' strategies.

Playing large games using simple strategies

TLDR
The existence of ε-Nash equilibrium strategies with support logarithmic in the number of pure strategies is proved and it is proved that if the payoff matrices of a two person game have low rank then the game has an exact Nash equilibrium with small support.