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Abstract In this paper we study the typical dilemma of social coordination between a risk-dominant convention and a payoff-dominant convention. In particular, we consider a model where a population of agents play a coordination game over time, choosing both the action and the network of agents with whom to interact. The main modeling novelties with respect to the existing literature are: (1) Agents come in two distinct types, (2) the interaction with a different type is costly, and (3) an agent’s type is unobservable prior to interaction. We show that when the cost of interacting with a different type is small with respect to the payoff of coordination, the payoff-dominant convention is the only stochastically stable convention; instead, when the cost of interacting with a different type is large, the only stochastically stable conventions are those where all agents of one type play the payoff-dominant action and all agents of the other type play the risk-dominant action.