On discrete preferences and coordination

An active line of research has considered games played on networks in which payoffs depend on both a player's individual decision and also the decisions of his or her neighbors. Such games have been used to model issues including the formation of opinions (in which people wish to express views consistent with those of their friends) and technology adoption (in which people or firms seek compatibility with their network neighbors). A basic question that has remained largely open in this area is to consider games where the strategies available to the players come from a fixed, discrete set, and where players may have different intrinsic preferences among the possible strategies. It is natural to model the tension among these different preferences by positing a distance function on the strategy set that determines a notion of "similarity" among strategies; a player's payoff is determined by the distance from her chosen strategy to her preferred strategy and to the strategies chosen by her network neighbors. Even when there are only two strategies available, this framework already leads to natural open questions about a version of the classical Battle of the Sexes problem played on a graph; such questions generalize issues in the study of network coordination games. We develop a set of techniques for analyzing this class of games, which we refer to as discrete preference games. We parametrize the games by the relative extent to which a player takes into account the effect of her preferred strategy and the effect of her neighbors' strategies, allowing us to interpolate between network coordination games and unilateral decision-making. When these two effects are balanced, we show that the price of stability is equal to 1 for any discrete preference game in which the distance function on the strategies is a tree metric; as a special case, this includes the Battle of the Sexes on a graph. We also show that trees essentially form the maximal family of metrics for which the price of stability is 1, and produce a collection of metrics on which the price of stability converges to an asymptotically tight bound of 2.