Saddle-Point Properties and Nash Equilibria for Channel Games
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Research Article Saddle-Point Properties and Nash Equilibria for Channel Games Rudolf Mathar1 and Anke Schmeink2 1 Institute 2 UMIC
for Theoretical Information Technology, RWTH Aachen University, 52056 Aachen, Germany Research Center, RWTH Aachen University, 52056 Aachen, Germany
Correspondence should be addressed to Rudolf Mathar, [email protected] Received 15 September 2008; Accepted 4 March 2009 Recommended by Holger Boche 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. Copyright © 2009 R. Mathar and A. Schmeink. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction Transmission over a band-limited wireless channel is often considered as a game where players compete for a scarce medium, the channel capacity. Nash bargaining solutions are determined for interference games with Gaussian additive noise. In the works [1, 2], different fairness and allocation criteria arise from this paradigm leading to useful access control policies for wireless networks. The engineering problem of transmitting messages over a channel with varying states may also be gainfully considered from a game-theoretic point of view, particularly if the channel state is unpredictable. Here, two players are entering the scene, the transmitter and the channel state selector. The transmitter gambles against the channel state, chosen by a malicious nature, for example. Mutual information I(X; Y ) is considered as payoff function, the transmitter aims at maximizing, nature at minimizing I(X; Y ). A simple motivating example is the additive scalar
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