On Sparse Discretization for Graphical Games

TL;DR

This paper presents a new discretization scheme for graphical games that efficiently approximates Nash equilibria, significantly reducing the complexity compared to previous methods, and highlights the role of AI constraint networks in this context.

Contribution

The main contribution is a representation theorem showing that a uniform discretization size linear in inverse approximation quality suffices, with a logarithmic dependency for graphical games, improving over prior linear dependencies.

Findings

Discretization size is linear in inverse approximation quality.

Graphical games allow logarithmic discretization complexity.

Simplified proof techniques and open problems are discussed.

Abstract

This short paper concerns discretization schemes for representing and computing approximate Nash equilibria, with emphasis on graphical games, but briefly touching on normal-form and poly-matrix games. The main technical contribution is a representation theorem that informally states that to account for every exact Nash equilibrium using a nearby approximate Nash equilibrium on a grid over mixed strategies, a uniform discretization size linear on the inverse of the approximation quality and natural game-representation parameters suffices. For graphical games, under natural conditions, the discretization is logarithmic in the game-representation size, a substantial improvement over the linear dependency previously required. The paper has five other objectives: (1) given the venue, to highlight the important, but often ignored, role that work on constraint networks in AI has in simplifying the derivation and analysis of algorithms for computing approximate Nash equilibria; (2) to summarize the state-of-the-art on computing approximate Nash equilibria, with emphasis on relevance to graphical games; (3) to help clarify the distinction between sparse-discretization and sparse-support techniques; (4) to illustrate and advocate for the deliberate mathematical simplicity of the formal proof of the representation theorem; and (5) to list and discuss important open problems, emphasizing graphical-game generalizations, which the AI community is most suitable to solve.