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  • Research Article
  • Open Access

Saddle-Point Properties and Nash Equilibria for Channel Games

EURASIP Journal on Advances in Signal Processing20092009:823513

  • Received: 15 September 2008
  • Accepted: 4 March 2009
  • Published:


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 -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.


  • Information Technology
  • Nash Equilibrium
  • Nash
  • Quantum Information
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Authors’ Affiliations

Institute for Theoretical Information Technology, RWTH Aachen University, 52056 Aachen, Germany
UMIC Research Center, RWTH Aachen University, 52056 Aachen, Germany


© R. Mathar and A. Schmeink. 2009

This article is published under license to BioMed Central Ltd. 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.