Mutual Information Jammer-Relay Games

Abstract

We consider a two-person zero-sum mutual information game between one jammer ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">J</i> ) and one relay (Rfr) in both nonfading and fading scenarios. Assuming that the source ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</i> ) and the destination ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">D</i> ) are unaware of the game, we derive optimal pure or mixed strategies for <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">J</i> and Rfr depending on the link qualities and whether the players are active during the <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</i> rarr <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">D</i> channel training. In nonfading scenarios, when both <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">J</i> and Rfr have full knowledge of the source signal, linear jamming (LJ) and linear relaying (LR) are shown optimal in the sense of achieving Nash equilibrium. When the <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</i> rarr <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">J</i> and <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</i> rarrRfr links are noisy, LJ strategies (pure or mixed) are still optimal under LR. In this case, instead of always transmitting with full power as when the <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</i> rarrRfr link is perfect, Rfr should adjust the transmit power according to its power constraint and the reliability of the source signal it receives. Furthermore, in fading scenarios, it is optimal for <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">J</i> to jam only with Gaussian noise if it cannot determine the phase difference between its signal and the source signal. When LR is considered with fading, Rfr should forward with full power when the <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</i> rarrRfr link is better than the jammed <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</i> rarr <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">D</i> link, and defer forwarding otherwise. Optimal parameters are derived based on exact Nash equilibrium solutions or upper and lower bounds when a closed-form solution cannot be found.