Contagion and uninvadability in local interaction games: The bilingual game and general supermodular games
In a setting where an infinite population of players interact locally and repeatedly, we study the impacts of payoff structures and network structures on contagion of a convention beyond 2×2 coordination games. First, we consider the “bilingual game”, where each player chooses one of two conventions or adopts both (i.e., chooses the “bilingual option”) at an additional cost. For this game, we completely characterize when a convention spreads contagiously from a finite subset of players to the entire population in some network, and conversely, when a convention is never invaded by the other convention in any network. We show that the Pareto-dominant (risk-dominant, resp.) convention is contagious if the cost of bilingual option is low (high, resp.). Furthermore, if the cost is in a medium range, both conventions are each contagious in respective networks, and in particular, the Pareto-dominant convention is contagious only in some non-linear networks. Second, we consider general supermodular games, and compare networks in terms of their power of inducing contagion. We show that if there is a weight-preserving node identification from one network to another, then the latter is more contagion-inducing than the former in all supermodular games.