(i) If $r_{e,i}(x)$ is strictly convex on $\text{supp}(f_i)$ for all $e \ge 0$, then $\tilde{e}^* > e^*$.
(ii) If $r_{e,i}(x)$ is linear on $\text{supp}(f_i)$ for all $e \ge 0$, then $\tilde{e}^* = e^*$.
(iii) If $r_{e,i}(x)$ is strictly concave on $\text{supp}(f_i)$ for all $e \ge 0$, then $\tilde{e}^* < e^*$.
Proof. See Appendix A.8. □
The key insight needed to understand Proposition 5 is that applying a mean-preserving spread to the distribution $F_k$ shifts probability mass from the center to the tails of the distribution, and the impact of this change on the incentive to exert effort depends on the curvature of $r_{e,i}(x)$. Notice that Proposition 5 also holds if players have the same expected skill, namely $\mu_i = \mu_k$. This means that, in a contest with two players who are expected to be equally able, higher uncertainty regarding players’ skills may increase the incentive to exert effort.
Next, we illustrate and provide intuition for Proposition 5 by presenting an example set in the context of the Lazear-Rosen model with an additive production technology. The example demonstrates that increasing the uncertainty of the contest while keeping the expected skill of both players unchanged, can increase equilibrium effort.
Example 5. Consider a contest with the additive production function $g(\theta, e) = \theta + e$, the parameter $V = 1$, and the cost function $c(e) = \frac{e^2}{2}$. Suppose $\Theta_i \sim \text{Exp}(1)$ and $\Theta_k \sim U[\frac{1}{2}, \frac{3}{2}]$ (implying $\mu_i = \mu_k = 1$). Equilibrium effort is then $e^* = \frac{\exp(1)-1}{\exp(\frac{3}{2})} \approx 0.38$. Now, consider a mean-preserving spread of the skill distribution of player k, enlarging the support of the uniform distribution, such that $\tilde{\Theta}_k \sim U[0,2]$. Then effort increases to $\tilde{e}^* = \frac{\exp(2)-1}{2\exp(2)} \approx 0.43$.
In Example 5, we have imposed the additive production technology which implies $a_e(x) = 1$. Thus, the convexity of $r_{e,i}(x)$ referred to in part (i) of Proposition 5 is de- termined by the convexity of $f_i(x)$. To understand how the shift from $f_k$ to $\tilde{f}_k$ affects the incentive to exert effort, we need to study how the integral in (4) is affected. Sim- ilar to Example 2, given that $a_e(x) = 1$, it is sufficient to compare $\int f_i(x)f_k(x)dx$ with $\int f_i(x)\tilde{f}_k(x)dx$. The shift from $f_k$ to $\tilde{f}_k$ entails an enlargement of the support of the uni- form distribution. This implies that the density decreases for intermediate values of x, but increases for low and high values of x (see Figure 3 for an illustration). Given that $f_i(x)$ is strictly decreasing, the part of the skill distribution of player k that is stretched out to the left will collide with relatively large values of $f_i$, whereas the part of the skill distribution of player k that is stretched out to the right will collide with relatively small values of $f_i$, creating a trade-off. The fact that $f_i$ is not only strictly decreasing,