Players receive a return to investment that is increasing in the proportion of others who invest and the state, and incur a small cost for acquiring information about the state. Their information is reflected in a stochastic choice rule, specifying the probability of a signal leading to investment. If discontinuous stochastic choice rules are in finitely costly, there is a unique equilibrium as costs become small, in which actions are a best response to a uniform (Laplacian) belief over the proportion of others investing. Infeasibility of discontinuous stochastic choice rules captures the idea that it is impossible to perfectly distinguish states that are arbitrarily close together and is both empirically documented and satisfied by many natural micro-founded cost functionals on information. Our results generalize global game selection results (Carlsson and van Damme (1993) and Morris and Shin (2003)), and establish that they do not depend on the specifi c additive noise information structure.