Many players , few participants ∗

We derive equilibrium properties of participation in a large family of economic games that take place when players’ information is incomplete. The fact that participation is almost always costly introduces a crucial distinction between players and participants. The number of players is exogenous but the number, or distribution, of participants is part of equilibrium. We assume a quite general setting in order to conform, as much as possible, to reality. Players receive signals, possibly multi-dimensional, gains may be interdependent, payment rules are arbitrary and participation costs may vary across players. We prove that, independently of the distribution of signals, if there are many players the probability of participation of a single player goes to 0 even if participation costs are very small. The random number of participants has a Poisson distribution with parameter λ ≤ ln(Gmax/c), where Gmax is the largest possible gain and c stands for participation costs. Consequently, the expected number of participants is smaller than ln(Gmax/c), which is a small number given the decimating property of the logarithmic function. Similar, but somewhat weaker, results are obtained when gains have a non-compact support. ∗Most results presented in this paper appeared in preprints co-authored by M. Landsberger and B. Tsirelson. The contribution of Tsirelson to this paper can not be overestimated; I can not imagine this paper could have been written without this inspiring collaboration. I am also grateful to P. Jehiel, S. Hart, D. Levin, L. Ye, J.F. Mertens and R. Radner for useful comments. Remarks made by seminar participants at Caltech, CORE, IHS Vienna, Northwestern, NYU, Ohio State and UCL were helpful as well.