but also convex, resolves this trade-off, implying that the overall effect of the shift is to increase the value of the integral expression. Thus, both players have a higher incentive to exert effort as a result of the move from $f_k$ to $\tilde{f}_k$. Intuitively, due to the change in the distribution of player k, situations where the competing players have the same skill become "more likely", implying an increase in equilibrium effort.
Figure 3: Illustration of Example 5
We conclude this section by defining contestant heterogeneity in a second-order sense and we follow the structure of the corresponding definition of heterogeneity in a first-order sense (Definition 1). In Definition 1, we used the ranking of players' mean skills to characterize the initial situation. In the new definition, we do so through the variances of the skill distributions of the competing players (restricting attention to statistical distributions with finite variance). Notice, however, that variance is not always a good measure of uncertainty or risk (see, e.g., Rothschild and Stiglitz 1970). Therefore one should keep in mind, when applying the definition below, that higher variance entails higher uncertainty only for certain skill distributions (e.g., the normal distribution).
Definition 3. Let $Var_k$ and $Var_i$ refer to the variances of the skill distributions ($F_k, F_i$) in an initial contest. Players in a contest with skill distributions ($\tilde{F}_k, \tilde{F}_i$), 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 second-order sense, if either of the following conditions hold:
(i) $Var_k \ge Var_i$ and $F_k$ dominates $\tilde{F}_k$ in the sense of second-order stochastic dominance.