This paper studies a large class of imperfectly discriminating contests, referred to as elastic contests, that induce players to either overbid a standing bid or to abstain from bidding altogether. Many common forms of contest are elastic. In any equilibrium of an elastic contest, there is complete rent dissipation for all but at most one player. This result is used to show that in any sufficiently decisive anonymous standard contest, any equilibrium is an all-pay auction equilibrium. Thus, the analysis offers strong support for the robustness of the all-pay auction. The approach also delivers definite answers regarding the extent of rent dissipation in Tullock contests with intermediate values of the decisiveness parameter.