When coordination is required, we often attend closely to categorical distinctions while overlooking continuous variation. For instance, the norm against chemical weapons is based on the type of weapon used and is applied irrespective of the number of civilians killed. Building off the game theory literature on Global Games, and using a standard Nash Equilibrium analysis, we show that it is harder to coordinate on the basis of a continuous variable than a discrete variable whenever the variable is observed with idiosyncratic noise. We demonstrate our result using several stylized signal structures, and also present a general theorem specifying which signal structures permit coordination on the basis of one's signal. We also present dynamic models that do not rely on any rationality assumptions. We end by mapping the model to possible applications, including territoriality, human rights, inefficient altruism, institutionalized racism, territorial disputes, revolutions, and collusion.
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