In the upcoming example, it can be verified that $r_{e,i}(x) = a_e(x)f_i(x)$ is neither always increasing nor always decreasing, by virtue of the multiplicative production technology combined with the bell-shaped normal distribution. Nonetheless, equilibrium effort increases as players become more heterogeneous in the sense of increasing the distance $|\mu_i - \mu_k|$.
Example 4. Consider Proposition 4 and assume that $(\sigma_i, \sigma_k) = (1,1)$, $(\mu_i, \mu_k) = (\frac{1}{2}, \frac{1}{2})$, $V=1$, and $c(e) = \frac{e^2}{2}$. Then equilibrium effort is $e^* = (2\pi^{\frac{1}{4}})^{-1} \approx 0.38$. If we increase $\mu_i$ from $\frac{1}{2}$ to $\frac{3}{2}$, keeping $\mu_k$ constant, equilibrium effort increases to $\tilde{e}^* = (\sqrt{2}\exp(\frac{1}{8})\pi^{\frac{1}{4}})^{-1} \approx 0.47$.
6.2 Second-Order Stochastic Dominance
The studies by Hvide (2002), Kräkel and Sliwka (2004), Kräkel (2008), Gilpatric (2009), and DeVaro and Kauhanen (2016) investigate how “risk” or “uncertainty” affects players’ incentive to exert effort in contests. One result that is common to all of these analyses is that in contests between equally able players, higher risk (as measured by a higher variance of the random variables capturing the uncertainty of the contest outcome) leads to lower efforts. We revisit this result in the context of our model, and show that effort may increase as the skill distribution of one of the players becomes more uncertain.
The economic literature has identified different ways to conceptualize risk or uncertainty. We follow Rothschild and Stiglitz 1970 by using second-order stochastic dominance to measure the uncertainty regarding players’ skill distributions.¹⁸
Definition 2. The skill distribution $\tilde{F}_i$ is said to be more uncertain than the distribution $F_i$ if $\tilde{F}_i$ is a mean-preserving spread of $F_i$. This is equivalent to $\tilde{F}_i$ being dominated by $F_i$ in the sense of second-order stochastic dominance.
Equipped with this definition, we can use well-known results from decision theory to obtain our next proposition:
Proposition 5. 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. Suppose that $\tilde{F}_k$ is more uncertain than $F_k$. Then, the following results hold:
¹⁸Gerchak and He (2003) analyze how effort in two-player contests is determined by the Rényi entropy, and Drugov and Ryvkin (2020) generalize their insights to the case of more than two players. Their results, however, rely on the assumptions of an additive production function and homogeneous players (i.e., players with the same skill distributions). It is not obvious how these results transfer to a model with general production technologies and asymmetric skill distributions, which is the primary focus here.