1 < n-ιβ or not. This inequality is satisfied if βB > ((nA + nB)2 — 2nA)∕nB — 1. If
this is the case, all individuals will individually exert higher equilibrium effort under
AA. Otherwise, only group B members will increase their individual effort.
For the second case Proposition 3 is applicable, which provides condition (13) to com-
pare the aggregated equilibrium effort23 under the optimal specified policy parameters
for AA and ET. This condition simplifies for the contest game considered here to the
following expression:
βB <
nA (nA + пв — 1)
nA (nA + пв — 2) — пв
(16)
The intuitive explanation that was given in the last section is that condition 13 is likely
to hold if either the level of dispersion is sufficiently low or the number of contestants
is relatively small. For the case considered here this can be verified explicitly for
the simplified condition given in Eq. (16). In fact, it is satisfied if either βB is low in
comparison to β* (which coincides with low dispersion), or if nA and nB are sufficiently
low (it can be checked that β* is decreasing in nA and nB).
It should also be noticed that condition (16) is not trivial in the sense that, for instance,
satisfying condition (15) automatically implies condition (16) because it can be shown
that β > nn-1. Hence, there are cases in which it is possible that, although not all
contestants are active, the sum of equilibrium effort is higher under AA than under
ET.
5.1 A Partially Informed Contest Organizer
In this section the previous contest game with groups is generalized by relaxing the
assumption on homogeneity within groups and on complete information of the contest
organizer with respect to individual characteristics of the contestants. From now on the
contestants again face different individual marginal costs that are common knowledge
for the contestants. However, the contest designer is only partially informed about
the heterogeneity of the contestants because, by assumption, she can only observe
the group membership of each contestant and is supposed to know an aggregated
measure of heterogeneity given by βA = ɪ ∑j∙∈A βj for all contestants in group A and
вв = ɪ Σj∙∈B βj for all contestants in group B, respectively. Group B is assumed to
23 Proposition 3 does not mention individual equilibrium effort. For the simple contest game analyzed
here, the analytical solutions for individual equilibrium effort can be compared easily to show that
members of group A exert individually less effort under AA than under ET while members of
group B trivially exert more (because they are not active under ET).
18