Markov chain entropy games and the geometry of their Nash equilibria

Consider the following two-person mixed strategy game of a probabilist against Nature with respect to the parameters $(f, \mathcal{B},\pi)$, where $f$ is a convex function satisfying certain regularity conditions, $\mathcal{B}$ is either the set $\{L_i\}_{i=1}^n$ or its convex hull with each $L_i$ being a Markov infinitesimal generator on a finite state space $\mathcal{X}$ and $\pi$ is a given positive discrete distribution on $\mathcal{X}$. The probabilist chooses a prior measure $\mu$ within the set of probability measures on $\mathcal{B}$ denoted by $\mathcal{P}(\mathcal{B})$ and picks a $L \in \mathcal{B}$ at random according to $\mu$, whereas Nature follows a pure strategy to select $M \in \mathcal{L}(\pi)$, the set of $\pi$-reversible Markov generators on $\mathcal{X}$. Nature pays an amount $D_f(M||L)$, the $f$-divergence from $L$ to $M$, to the probabilist. We prove that a mixed strategy Nash equilibrium always exists, and establish a minimax result on the expected payoff of the game. This also contrasts with the pure strategy version of the game where we show a Nash equilibrium may not exist. To find approximately a mixed strategy Nash equilibrium, we propose and develop a simple projected subgradient algorithm that provably converges with a rate of $\mathcal{O}(1/\sqrt{t})$, where $t$ is the number of iterations. In addition, we elucidate the relationships of Nash equilibrium with other seemingly disparate notions such as weighted information centroid, Chebyshev center and Bayes risk. This article generalizes the two-person game of a statistician against Nature developed in the literature, and highlights the powerful interplay and synergy between modern Markov chains theory and geometry, information theory, game theory, optimization and mathematical statistics.