Saddle-Point Properties and Nash Equilibria for Channel Games

Abstract

In this paper, transmission over a wireless channel is interpreted as a two-person zero-sum game, where the transmitter gambles against an unpredictable channel, controlled by nature. Mutual information is used as payoff function. Both discrete and continuous output channels are investigated. We use the fact that mutual information is a convex function of the channel matrix or noise distribution densities, respectively, and a concave function of the input distribution to deduce the existence of equilibrium points for certain channel strategies. The case that nature makes the channel useless with zero capacity is discussed in detail. For each, the discrete, continuous, and mixed discrete-continuous output channel, the capacity-achieving distribution is characterized by help of the Karush-Kuhn-Tucker conditions. The results cover a number of interesting examples like the binary asymmetric channel, the Z-channel, the binary asymmetric erasure channel, and the n-ary symmetric channel. In each case, explicit forms of the optimum input distribution and the worst channel behavior are achieved. In the mixed discrete-continuous case, all convex combinations of some noise-free and maximum-noise distributions are considered as channel strategies. Equilibrium strategies are determined by extending the concept of entropy and mutual information to general absolutely continuous measures.

Encoding of cellular positional information and maximum capacity of parallel coupled channels

“…For computation of the channel capacity, the following characterization of the optimal input distribution from [26,Proposition 5], which is based on the KKT optimality conditions, is useful. Here w k is the k-th row of the channel matrix W. …”

Threshold optimization for capacity-achieving discrete input one-bit output quantization

“…From [9], see also [10], Th. 4.5.1, we conclude that p * is capacity-achieving if and only if D(v i p * V ) = ζ for some ζ > 0 and all i with p * i > 0, and furthermore, D(v j p * V ) ≤ ζ for all j with p * j = 0.…”

“…The capacityachieving distribution of the asymmetric binary channel is obtained as a special case, cp. [4], [9].…”