A tight characterization of strategic games with a unique equlibrium

Media access protocols in wireless networks require each contending node to wait for a backoff time chosen randomly from a fixed range, before attempting to transmit on a shared channel. However, nodes acting in their own selfish interest may not follow the protocol. In this paper, we use a game-theoretic approach to study how nodes might be induced to adhere to the protocol. In particular, a static version of the problem is modeled as a strategic game played by non-cooperating, rational players (the nodes). A strategy for a player corresponds to a backoff value in the medium access protocol. We are interested in designing a game which exhibits a unique Nash equilibrium corresponding to a pre-specified full-support distribution profile. In the context of the media access problem, the equilibrium of the game would correspond to nodes following the protocol, viz. choosing backoff times randomly from a given range of values according to the prespecified distribution. Building on results described in earlier work, we identify the exact relationship that must hold between the cardinalities of the players´ action sets that would make it possible to design such a game.