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.