On a Game Between a Delay-Constrained Communication System and a Finite State Jammer

This paper considers a zero-sum game between a team of delay-constrained encoder and decoder, and a finite state jammer, with average probability of error as the payoff. The team attempts to communicate a discrete source using a finite blocklength over a finite family of discrete channels whose state is controlled by the jammer. For each strategy of the jammer, the team's problem has nonclassical information structure and hence is nonconvex, whereby a saddle point solution may not exist for the game. Our main contributions consist of a novel lower bound on the min-max and max-min value of the game via a linear programming (LP) relaxation and a new upper bound on the min-max value. These bounds imply that for rates of transmission $R$ satisfying $R < \underline{C}$ and $R > \overline{C}$} where $\underline{C}, \overline{C}$ are precomputable thresholds depending on the channel kernels, the min-max and max-min values of the game approach each other as the blocklength $n\rightarrow\infty$. In the intermediate range, we give a fundamental lower bound on the limiting max-min value, which for a two-state jammer is shown to be $\frac{1}{2}$.