Pre-print, Working paper: The paper analyzes how uncertainty on traders' participation affects a competitive security market in which there are some informed traders. We show that discontinuities, or "crashes", can arise at equilibrium, even when no investor posts a priori an increasing demand. Because of uncertain participation, the precision of the information brought by a price is endogenous, affected by the size of the trades. As a result, two prices with different volumes and information revelation may clear the market for the same values of the fundamentals. At one price, insurance motives drive the exchanges, noise is large and little information is revealed. At another price, uninformed trades are small, which makes the clearing price much more informative. This multiplicity of prices with different precision of information generates discontinuities.