(ii) $\mathrm{Var}_k \le \mathrm{Var}_i$ and $F_k$ is dominated by $\tilde{F}_k$ in the sense of second-order stochastic dominance.
Combining Proposition 5 with Definition 3, we have the following corollary.
Corollary 3. Effort can be higher when contestants are more heterogeneous in a second-order sense.
6.3 Implications for Optimal Team Composition
The results in the preceding two subsections have implications for optimal team composition and organizational design.¹⁹ In particular, our results suggest that employers could find it desirable to employ a more heterogeneous workforce as an instrument to induce higher effort. In Section 6.1, we analyzed the effects of increasing the heterogeneity in players’ expected skills, and showed how this can increase equilibrium effort. This means that a firm could benefit (from the perspective of inducing higher effort) by hiring some workers with a high expected skill and some with a low expected skill, based on, for example, signals such as the quality of the institution where a college graduate received his or her degree. In Section 6.2, we showed how increased uncertainty regarding skills of some players can increase equilibrium effort. Thus, a firm could benefit from hiring a mix of experienced workers (for whom the uncertainty regarding skills is relatively small) and inexperienced workers (for whom the uncertainty regarding skills is relatively large).
To see this more formally, suppose a firm already employs a worker with skill distribution $F_1$ and considers to hire another worker with skill distribution $F_2$. Moreover, assume that $r_{e,1}(x)$ is strictly decreasing and strictly convex (for example, by assuming that the production function is given by $g(\theta, e) = \theta + e$ and skills are Exponentially distributed with parameter $\lambda$).²⁰ Then the firm may gain from hiring another worker with a lower expected skill ($\mu_2 < \mu_1$), but where $F_2$ is more uncertain (meaning that worker 2’s skill is drawn from a more uncertain distribution). This finding can be understood from the perspective of Proposition 3, that tells us that effort will be higher due to the lower expected skill of worker 2, combined with Proposition 5, which tells us that effort will
¹⁹See, e.g., Gershkov, Li, and Schweinzer (2009, 2016) and Fu, Lu, and Pan (2015).
²⁰An alternative skill distribution that would also be decreasing and convex would be a normal distribution that is truncated to the left at a point to the right of the second inflection point. Such a distribution could be motivated by the observation that skills are often normally distributed and that, when employing worker 1, the firm tried to hire the most able applicant, meaning that skills in the higher end of the distribution are most relevant (see, e.g., Aguinis and O’Boyle Jr. 2014).