pares individual equilibrium effort under each policy. However, a comparable result
for the n-player contest is not possible because the set of active agents depends on the
underlying distribution of the marginal cost parameter. Hence, the additional assump-
tion of full participation by all contestants under both policies shall be considered to
get some further insights into the individual equilibrium behavior. This assumption
would imply that the dispersion of cost parameters is sufficiently low such that also
under policy ET all contestant would be active.
The following Proposition mirrors Proposition 1 for this class of restricted distributions
of the marginal cost parameter. Although the sum of equilibrium effort in this special
case is higher under the optimal AA policy versus the optimal ET policy (without
any further conditions) as in the two-player case, the result with respect to individual
equilibrium effort is different: In the n-player contest game the set of contestants
that individually exert higher equilibrium effort under policy AA than under ET is
restricted to contestants with either very low marginal cost or higher than average
marginal cost.
Proposition 4 If all contestants in the n-person contest game are active under policy
ET, then (i) the sum of equilibrium effort levels is higher under policy AA, and (ii)
the individual equilibrium effort of all contestants with marginal cost parameter β ∈
[1, n-ιβ} ∪ (β, n--ιβ} is higher under policy AA, while it is lower for contestants with
β ∈ (n⅛ιβ,β} . For contestants with β ∈ {n⅛ιβ,β} the individual equilibrium effort
is the same under both policies.
Proof: If all contestants are active set M and N coincide, and condition (13) simplifies
to β > βN. This inequality is always satisfied which proves the first part of the
proposition.
For the second part the following inequality has to be analyzed: e*(AA) > e*(ET).
Simplifying this expression yields after some algebra the following inequality:
^ βj^ - n2βi f ^ βj - (n - 1)βi)^ > 0.
(14)
j∈N j∈N
This inequality is satisfied if βi ∈ [1, ɪβyj, where the lower bound stems from the
assumption that βi ≥ 1 for all i ∈ N ,or if βi ∈ ^β, ɪ βyj, where the upper bound
comes from the assumption of full participation under the optimal ET policy. The left
hand side of Eq. (14) is equal to zero for β ∈ ∣n⅛ι∕β,∕β}. Continuity of the left hand
side of Eq. (14) in βi implies the condition for the reversed inequality. Note also that
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