This paper considers all-pay contests in which the relationship between bids and allocations reflects a small amount of noise. Prior work had focused on one particular equilibrium. However, there may be other equilibria. To address this issue, we introduce a new and intuitive measure for the proximity to the all-pay auction. This allows, in particular, to provide simple conditions under which actually any equilibrium of the contest is both payoff equivalent and revenue equivalent to the unique equilibrium of the corresponding all-pay auction. The results are shown to have powerful implications for monopoly licensing, political lobbying, electoral competition, optimally biased contests, the empirical analysis of rent-seeking, and dynamic contests.