The Majority and Minority Models on Regular and Random Graphs

In the two strategy majority/minority game on a network, at time t, an individual observes some subset of its neighbors strategies, and then adopts at time t+1 that strategy which was more/less frequently played by its neighbors at time t. We shall examine in this paper a variety of distinct models which vary the subsets observed, the synchrony of actions, the regularity, or otherwise, of the networks, the mix of majority and minority players. Important measures of the dynamics of such systems such as the nature of the fixed points, and limit cycles are discussed. The expected payoffs under certain games are computed and compared. In particular we shall examine complete networks, hypercubes and a certain class of cubic networks. These graphs have rather different numbers of symmetries which impact on the properties of the dynamics. The work is of an exploratory nature and hopefully will suggest many potential lines of enquiry.