We consider contests with a large set (continuum) of participants and axiomatize contest success functions that arise when performance is composed of both effort and a random element, and when winners are those whose performance exceeds a cutoff determined by a market clearing condition. A co-monotonicity property is essentially all that is needed for a representation in the general case, but significantly stronger conditions must hold to obtain an additive structure. We illustrate the usefulness of this framework by revisiting some of the classic questions in the contests literature.
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.
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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,
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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.
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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)
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