Tournaments of Weakly Heterogeneous Players

We consider three prominent tournament formats—contests, binary elimination tournaments, and round‐robin tournaments—in the case when players are heterogeneous in their abilities but the heterogeneity is, in a well‐defined sense, weak. Using linear approximation, we characterize equilibrium strategies and payoffs in the three tournament games and compare them to the benchmark symmetric case of identical players. We describe small deviations from the symmetric equilibria by elasticities of a player's equilibrium effort with respect to her own ability and the abilities of her rivals. Our results only require general symmetry and smoothness assumptions but not specific functional forms for the probabilities of winning in tournaments. We show that, in equilibrium: (i) a player's effort and payoff depend on her rivals' abilities in a model‐independent way, either through the average ability of the field (for static games), or through the properly discounted average ability of the field (for dynamic elimination tournaments); (ii) players respond stronger to changes in their own relative abilities than to changes in their rivals' relative abilities; (iii) aggregate effort (dissipated rent) does not change compared to the benchmark case; (iv) it is not possible to manipulate aggregate effort through seeding in binary elimination tournaments, although optimal seeding schemes for other purposes can be identified; and (v) balanced seeding and a uniform distribution of relative abilities cancel out the dependence of a player's effort on her rivals' abilities in binary elimination tournaments.