  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.
