[Paper Review] Games of fixed rank: a hierarchy of bimatrix games

Ravi Kannan, Thorsten Theobald|arXiv (Cornell University)|Jan 7, 2007

Game Theory and Applications31 references30 citations

TL;DR

This paper introduces and studies bimatrix games of fixed rank k, where the sum of the two players' payoff matrices has rank k, generalizing zero-sum games. It shows that even for k=1, Nash equilibria can form arbitrarily many connected components, and presents both a deterministic and a randomized polynomial-time approximation algorithm for finding approximate equilibria, with the latter potentially yielding exact solutions if a conjecture holds.

ABSTRACT

We propose and investigate bimatrix games, whose (entry-wise) sum of the pay-off matrices of the two players is of rank k, where k is a constant. We will say the rank of such a game is k. For every fixed k, the class of rank k-games strictly generalizes the class of zero-sum games, but is a very special case of general bimatrix games. We show that even for k = 1 the set of Nash equilibria of these games can consist of an arbitrarily large number of connected components. While the question of exact polynomial time algorithms to find a Nash equilibrium remains open for games of fixed rank, we can provide a deterministic polynomial time algorithm for finding an e-approximation (whose running time is polynomial in 1\e) as well as a randomized polynomial time approximation algorithm (whose running time is similar), but which offers the possibility of finding an exact solution in polynomial time if a conjecture is valid. The latter algorithm is based on a new application of random sampling methods to quadratic optimization problems of fixed rank.

Motivation & Objective

• To investigate the structure and complexity of Nash equilibria in bimatrix games of fixed rank k.
• To generalize zero-sum games by studying games where the sum of payoff matrices has rank k.
• To develop efficient algorithms for approximating Nash equilibria in fixed-rank games.
• To explore the potential for exact polynomial-time solutions under a conjecture.

Proposed method

• Define the rank of a bimatrix game as the rank of the sum of the two players' payoff matrices.
• Analyze the topological structure of Nash equilibrium sets, showing they can consist of arbitrarily many connected components even for rank 1.
• Develop a deterministic polynomial-time algorithm that computes an ε-approximate Nash equilibrium, with runtime polynomial in 1/ε.
• Propose a randomized polynomial-time approximation algorithm based on random sampling for fixed-rank quadratic optimization problems.
• Use random sampling techniques to reduce the dimensionality and complexity of fixed-rank quadratic programs.
• Leverage a conjecture to suggest that the randomized algorithm may yield an exact solution in polynomial time under certain conditions.

Experimental results

Research questions

• RQ1 How does the number of connected components in the set of Nash equilibria grow with the rank k of the game?
• RQ2 Can polynomial-time algorithms be designed to approximate or compute Nash equilibria in fixed-rank games?
• RQ3 What is the computational complexity of finding exact Nash equilibria in fixed-rank games, especially for k=1?
• RQ4 Can random sampling methods be effectively applied to fixed-rank quadratic optimization problems arising in game theory?
• RQ5 Under what conditions might a randomized algorithm for fixed-rank games yield an exact solution in polynomial time?

Key findings

• For every fixed k, including k=1, the set of Nash equilibria in rank k-games can consist of an arbitrarily large number of connected components.
• A deterministic polynomial-time algorithm exists for computing an ε-approximate Nash equilibrium, with runtime polynomial in 1/ε.
• A randomized polynomial-time approximation algorithm is proposed, which shares similar runtime efficiency.
• The randomized algorithm may compute an exact Nash equilibrium in polynomial time if a certain conjecture about fixed-rank quadratic programs is true.
• The application of random sampling to fixed-rank quadratic optimization problems is shown to be a viable and efficient method for approximating solutions.