  In competitive multi-player interactions, simultaneous optimality is a key
requirement for establishing strategic equilibria. This property is explicit
when the game-theoretic equilibrium is the simultaneously optimal solution of
coupled optimization problems. However, no such optimization problems exist for
the correlated equilibrium, a strategic equilibrium where the players can
correlate their actions. We address the lack of a coupled optimization
framework for the correlated equilibrium by introducing an {unnormalized game}
-- an extension of normal-form games in which the player strategies are lifted
to unnormalized measures over the joint actions. We show that the set of fully
mixed generalized Nash equilibria of this unnormalized game is a subset of the
correlated equilibrium of the normal-form game. Furthermore, we introduce an
entropy regularization to the unnormalized game and prove that the
entropy-regularized generalized Nash equilibrium is a sub-optimal correlated
equilibrium of the normal form game where the degree of sub-optimality depends
on the magnitude of regularization. We prove that the entropy-regularized
unnormalized game has a closed-form solution, and empirically verify its
computational efficacy at approximating the correlated equilibrium of
normal-form games.
