Due to Assumption 2, equilibrium effort increases if a change in the primitives of the model leads to an increase in $\int_{\mathbb{R}} r_{e,i}(x)f_k(x)dx$. As indicated before, this expression has the same structure as a decision maker's expected utility in decision theory, where the function $r_{e,i}$ is replaced by the decision maker's utility function. Since the structure of the problems is the same, we can make extensive use of results from decision theory in our analysis. We obtain the following proposition.
Proposition 3. Consider two contests with skill distributions $(\tilde{F}_k, F_i)$ and $(F_k, F_i)$ where $\text{supp}(\tilde{f}_k)$ and $\text{supp}(f_k)$ both are subsets of $\text{supp}(f_i)$. Let $\tilde{e}^*$ and $e^*$ denote, respectively, the (symmetric) equilibrium efforts associated with these contests. Then, $\tilde{e}^* > e^*$ if either one of the following statements hold:
(i) $r_{e,i}(x)$ is strictly increasing for all $x \in \text{supp}(f_i)$ and all $e \ge 0$, and $\tilde{F}_k$ dominates $F_k$ in the sense of first-order stochastic dominance.
(ii) $r_{e,i}(x)$ is strictly decreasing for all $x \in \text{supp}(f_i)$ and all $e \ge 0$, and $\tilde{F}_k$ is dominated by $F_k$ in the sense of first-order stochastic dominance.
Proof. See Appendix A.6. □
Note that Proposition 3 holds independently of whether $\mu_k \le \mu_i$ or $\mu_k \ge \mu_i$. Combining Definition 1 with Proposition 3, we have the following corollary.¹⁷
Corollary 2. Effort can be higher when contestants are more heterogeneous in a first-order sense.
We illustrate the intuition behind Proposition 3 and Corollary 2 through two examples. In each example, we start from a situation of equal expected skills, and then introduce a first-order stochastic dominance shift. In the first example, which has a somewhat simpler intuition than the second, $r_{e,i}(x)$ is strictly decreasing and effort gets higher as player $k$ becomes weaker, illustrating part (ii) of Proposition 3. In the second example, $r_{e,i}(x)$ is strictly increasing and effort gets higher as player $k$ becomes stronger, illustrating part (i) of Proposition 3.
Example 2. Suppose that $g(\theta, e) = \theta + e$, $\Theta_i \sim \text{Exp}(\frac{4}{3})$, $\Theta_k \sim U[\frac{1}{2}, 1]$, $\tilde{\Theta}_k \sim U[\frac{7}{16}, \frac{15}{16}]$, $c(e) = \frac{e^2}{2}$, $V = 1$. Then $e^* = \frac{2(\exp(\frac{2}{3})-1)}{\exp(\frac{4}{3})} \approx 0.499$ and $\tilde{e}^* = \frac{2(\exp(\frac{2}{3})-1)}{\exp(\frac{5}{4})} \approx 0.543$.
¹⁷There is one small caveat to Corollary 2 that we should mention. If equilibrium effort increases as contestants become more heterogeneous, then a symmetric equilibrium in which both players exert positive effort will fail to exist if the heterogeneity between players becomes too large. The reason is that the weaker player would eventually receive a negative payoff, meaning that this player would prefer to choose zero effort.