Assumption 2. The primitives of the model are such that $q:E \to \mathbb{R}$, defined by
is strictly decreasing.
As $c$ is strictly convex, Assumption 2 is not very strong and is always satisfied if $\int_{\mathbb{R}} r_{e,i}(x)f_k(x)dx$ is non-increasing in $e$. To give a specific example, consider the CES production function $g(\theta_i, e_i) = (\alpha\theta_i^\rho + \beta e_i^\rho)^{\frac{1}{\rho}}$, with $\alpha, \beta > 0$ and $\rho \le 1$. Here $a_e(x) = \frac{\beta}{\alpha}(\frac{x}{e})^{1-\rho}$, implying that $\int_{\mathbb{R}} a_e(x)f_1(x)f_2(x)dx = e^{\rho-1}\int_{\mathbb{R}} \frac{\beta}{\alpha}x^{1-\rho}f_1(x)f_2(x)dx$. For this specification, Assumption 2 is satisfied in all cases where players have an incentive to exert positive effort (i.e., $\int_{\mathbb{R}} \frac{\beta}{\alpha}x^{1-\rho}f_1(x)f_2(x)dx > 0$). Furthermore, the assumption ensures that effort is always increasing in the prize and that the considered equilibrium is unique in the class of symmetric equilibria (the latter result follows from the assumption ensuring that there is a unique $e$ solving equation (5)).
6.1 First-Order Stochastic Dominance
A standard result in contest theory is that heterogeneity among players with respect to their skills reduces the incentive to exert effort (see, e.g., Schotter and Weigelt 1992, or Observation 1 in the survey by Chowdhury, Esteve-Gonzalez, and Mukherjee 2019). In our model, this standard result is potentially reversed, as we will now show.
Consider a contest with two players with skills drawn from two distributions with expected values $\mu_k$ and $\mu_i$, respectively. If, from the outset, $\mu_k \ge \mu_i$ and the difference $\mu_k - \mu_i$ is increased, then the two players become more heterogeneous in terms of their expected skill. Based on this idea, we proceed by investigating the consequences of making players more heterogeneous in the sense of first-order stochastic dominance, as captured by the following definition.
Definition 1. Let $\mu_k$ and $\mu_i$ refer to the expected values of the skill distributions ($F_k, F_i$) in an initial contest. Players in a contest with skill distributions ($\tilde{F}_k, F_i$) are said to be more heterogeneous (with respect to their skills) relative to players in the initial contest with skill distributions ($F_k, F_i$), in a first-order sense, if either of the following conditions hold:
(i) $\mu_k \ge \mu_i$ and $\tilde{F}_k$ dominates $F_k$ in the sense of first-order stochastic dominance.
(ii) $\mu_k \le \mu_i$ and $\tilde{F}_k$ is dominated by $F_k$ in the sense of first-order stochastic dominance.