  The Ising model of statistical physics has served as a keystone example of
phase transitions, thermodynamic limits, scaling laws, and many other phenomena
and mathematical methods. We introduce and explore an Ising game, a variant of
the Ising model that features competing agents influencing the behavior of the
spins. With long-range interactions, we consider a mean-field limit resulting
in a nonlocal potential game at the mesoscopic scale. This game exhibits a
phase transition and multiple constant Nash-equilibria in the supercritical
regime.
  Our analysis focuses on a sharp interface limit for which potential
minimizing solutions to the Ising game concentrate on two of the constant
Nash-equilibria. We show that the mesoscopic problem can be recast as a mixed
local/nonlocal space-time Allen-Cahn type minimization problem. We prove, using
a $\Gamma$-convergence argument, that the limiting interface minimizes a
space-time anisotropic perimeter type energy functional. This macroscopic scale
problem could also be viewed as a problem of optimal control of interface
motion. Sharp interface limits of Allen-Cahn type functionals have been well
studied. We build on that literature with new techniques to handle a mixture of
local derivative terms and nonlocal interactions. The boundary conditions
imposed by the game theoretic considerations also appear as novel terms and
require special treatment.
