in this framework), i.e. there might be distributions where for each policy option
different sets of contestants are active. A condition that guarantees that the sum
of equilibrium effort is higher under AA’ than under ET (parallel to Proposition 3)
would depend on the number of active contestants under each policy options and on
the underlying distribution of marginal cost parameters in both groups.26
To reduce the complexity of the policy comparison, the same special case as in the last
section shall be considered, i.e. it is assumed that all contestants are active under both
policy options. This implies the analysis is restricted to distributions with a sufficiently
low degree of dispersion such that condition (11) is satisfied for all contestants under
ET and AA’. As each contestant takes the effort level of the competitors as given, the
aggregated equilibrium effort can be calculated as usually, i.e. Ep = ∑i∈N ei(P) for
P = {ET,AAt}. The following result about the consequences of optimal affirmative
action AA’ is possible:
Proposition 5 If all contestants are active under policy ET and AA’ in a contest
game with a partially informed contest organizer, then (i) the sum of equilibrium effort
levels is higher under AA’ than under ET , and (ii) the individual equilibrium effort
is higher under AA’ than under ET for all contestants i ∈ A with discrimination level
βi∈A <
n ∕a∕
(n - 1)(y0 + ∕a)
and for all contestants i ∈ B with discrimination level
∕i∈B <
n ∕b β
(n - 1)(/3 + ∕b)
Proof : Forthefirstpartthefollowinginequalityhastobeanalyzed: EAA > E*et. Ifall
contestants are active, this inequality is reduced to: nA∕βA + nB/βB > n2/(∑i∈Nβi).
This inequality can be further reduced to nAnB (∕3A - ∕3B)2 > 0 which is always satisfied
by assumption.
For the second part the individual equilibrium effort has to be compared. Starting
with a member of group A, the inequality e*∈A (AAz) > e*∈A(ET) can be reduced to
βi∈A < (ni n Γ B 3 β i) with the analogous derivation for members of group B.□
26The condition for EAA > EET is in fact:
∣3maa, > mET - 1 mAA'
βMET mET mAA' - 1
where MP denotes the active set under policy P ∈ {ET, AA'}. As the characterization of the
active set will now depend on the distribution of each group and also its interrelation, an intuitive
interpretation of this condition seems to be overly complex and is therefore omitted.
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