4 Additive noise The definition of an RPF allows for a very general relationship between effort and performance. Indeed, the only substantial requirement is that higher effort implies higher distribution of performance in the sense of first-order stochastic dominance. In this section we add more structure and characterize the case where performance is obtained by adding a random shock to the effort choice. More explicitly, performance is equal to e + X, where X is a random variable representing the noise term. Definition 3. The contest success function W is an Additive RPF (A-RPF) if there is a continuous and strictly increasing cdf F such that W has an RPF representation with Fe(x) = F (x e) for every e E.

4.1 Characterization To characterize Additive RPFs we introduce the following two additional properties. Invariance to Common Shifts: For every e, a E and p k, W (e, p) = W (e + a, p a). Invariance to p Shifts: For every e, e, a E and p, p k, if W (e, p) = W (e, p), then W (e, pa) = W (e, p a). Theorem 2. The contest success function W is an A-RPF if and only if it satisfies e-Continuity,

Monotonicity, Invariance to Common Shifts, and Invariance to p Shifts. 8 Proof. (Necessity) Suppose W is an A-RPF with cdf F . Since W is in particular an RPF, e-Continuity and Monotonicity follow from Theorem 1. Fix some p k and a E. Then (cid:90) [1 F (s(p) + a e)]d(p a)(e) = (cid:90) [1 F (s(p) e)]d(p)(e) = (cid:90) W (e, p)dp(e) = k, where the first equality is a change of variable, the second by the assumption that F represents W , and the last by (1). It follows that s(p a) = s(p) + a. Now, for Invariance to Common Shifts, we have for every e, a E and p k, W (e + a, p a) = 1 F (s(p a) (e + a)) = 1 F (s(p) + a (e + a)) = 1 F (s(p) e) = W (e, p). Similarly, suppose that W (e, p) = W (e, p) holds for some e, e E and p, p k. Then 1 F (s(p) e) = 1 F (s(p) e), which, by the strict monotonicity of F , implies that s(p) e = s(p) e. But then for any a E, s(p a) e = s(p) + a e = s(p) + a e = s(p a) e.

This shows that W (e, p a) = W (e, p a) and hence that W satisfies Invariance to p Shifts. (Sufficiency) Fix some arbitrary (e, p) E k. Define F by F (x) = 1 W (e, p x) 1 W (e x, p) if x 0, if x < 0. We start by showing that F is strictly increasing. If x1 < x2 0 then by Monotonicity W (e x1, p) > W (e x2, p), so F (x1) < F (x2). If 0 x1 < x2 then W (e, p x1) = W (e + (x2 x1), (p x1) (x2 x1))) = W (e + (x2 x1), p x2) > W (e, p x2), where the first equality is by Invariance to Common Shifts, the next is just a simplification, and the inequality is by Monotonicity. Thus, F (x1) < F (x2) and F is strictly increasing. Next, we argue that F is continuous. For x < 0 this follows immediately from e-Continuity. For x 0, fix some e > x. Then by Invariance to Common Shifts, W (e, p x) = W (e + (e x), p (x + (e x))) = W (e + e x, p e). Thus, by e-Continuity we get that F is continuous at any x 0 as well. 9 (3)