A Minimax Theorem for Finite Blocklength Joint Source-Channel Coding over an AVC

We pose the finite blocklength communication problem in the presence of a jammer as a zero-sum game between the encoder-decoder team and the jammer, where the communicators, as well as the jammer, are allowed locally randomized strategies. The minimax value of the game corresponds to joint source-channel coding over an Arbitrarily Varying Channel (AVC), which in the channel coding setting is known to admit a strong converse. The communicating team's problem is non-convex and hence, in general, a minimax theorem need not hold for this game. However, we show that an approximate minimax theorem holds in the sense that the minimax and maximin values of the game approach each other asymptotically. In particular, for rates above a critical threshold, both the minimax and maximin values approach unity. This result is stronger than the usual strong converse for channel coding over an AVC, which only says that the minimax value approaches unity for such rates.