The Majority and Minority Models on Regular and Random Graphs Chris Cannings Abstract— 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. I. I NTRODUCTION Games on networks are of increasing interest in many disciplines. In economics, sociology and biology it is clear that individuals often interact at a local level, and hence with a restricted set of other individuals. The nature of the interactions and the structure of the network will impact on the dynamics of the pattern of behaviors, both locally and globally. Perhaps the most famous result in the theory of epidemics, that there is a critical level of interaction and infectivity, above which a substantial proportion of the population will be affected, and below which the infection will die out fairly quickly, is a long standing and important insight derived from graph theory. This result is usually derived in the context of the Erdos-Renyi graph [4], where the links between individuals are present, or absent, with constant probabilities, independently of each other. Recent interest has tended to be motivated by the complex networks such as the WWW (World Wide Web), or email traffic, and there the focus is often on networks with small-world structure [13]. Here the focus is somewhat different. We examine networks with substantial symmetries, and often of fairly small size. These are likely to be relevant to social groups, perhaps of primates which tend to be of limited size [6], though no claim is made here that they are likely to be associated together in any of the specific structures considered here. Similarly small groups occur as families or small businesses. Imposition of certain symmetries will allow us to extract features of the behavior which we can This work was partially supported by the U.K Engineering and Physical Sciences Research Council C.Cannings is at the Department of Probability and Statistics, School of Mathematics and Statistics, University of Sheffield, The United King- dom.c.cannings@shef.ac.uk investigate mathematically, hopefully then being able to derive some results which prove to be robust across less structured networks, or alternatively allow us to deduce the way in which outcomes depend on presence or absence of symmetries. More immediate relevance will be to algorithms or to computer devices where we wish some element of self-assembly to occur, possibly in response to external signals. We in fact confine our attention to complete graphs, hypercubes and certain cubic graphs ( a cubic graph is one with each vertex having three neighbors). The class of graphs we examine are what we shall call cylindrical graphs, they are constructed by taking two polygons of the same type (here only triangles, rectangles and pentagons), and making cross-links between the vertices of the two, using each vertex only once. The reason for choosing cubic graphs is that it avoids the necessity of introducing tie-breaking in the models considered, as any graph with any vertices of even degree will require. We similarly only consider complete graphs of even size when we restrict our attention to synchronized regimes. Our focus will be entirely on a game of great simplicity but rich behavior. For ease we discuss it in the case where individuals are able to choose only between a pair of alternatives. The so-called majority game has been discussed by various authors, and its origin has been attributed to Anderlini and Ianni [1]. Essentially (the mathematical set-up is given in section II), an individual receives information from some set of neighbors regarding their current choice of some set of alternatives. The individual then adjusts their own choice to match that being currently played by the majority of the neighbors. This is thus a conformist behavior, and should, one would expect, lead to a consensus resulting throughout the population. Indeed this is the case providing that there are no errors in play, except in the cases, as we shall see below when there is some specific structure for the network. An alternative game is the minority game [2] where the individuals are faced with the same set-up but adjust their choice to match that used by the minority of their neighbours. Such a rebellious behavior may again lead to stable outcome, though such outcomes may have any number of the two available choices being used, the population always being repelled from a consensus. The above games are both examples of what has been