The name is absent

As equilibrium effort is given by eq. (10), the two policies can now be compared
with respect to the aggregated equilibrium effort Ep = ∑_i∈M e*(P) that they induce.
However, Lemma 1 already reveals that the comparison between the two policy options
will not be as straight forward as in the two-player contest game because the total
equilibrium effort depends on the distribution of the cost parameter that determines
the active set.

The following notation will simplify the characterization of the relevant distribution for
a subset J ⊆ N of contestants: the arithmetic mean of the cost parameters of agents
of set J will be denoted as βJ = j ∑_i∈Jβ_i (where β = βN to facilitate notation), and
the harmonic mean respectively as: βH = [¹ ∑_ii∈J ¹ ]-1.

The subsequent proposition states the condition under which policy AA generates
higher aggregated effort.

Proposition 3 In the n-player contest game the sum of equilibrium effort levels is
higher under policy AA than under policy ET if:

m-1
m
n-1 .

n

Proof : Calculation of the sum of equilibrium effort for each policy under consideration

of lemma 1 yields EAAA = n^-1 V Σ∈n j: and EET = ^m^-1¾ V.
inequality EAA > E*_eT leads to condition (13). □

The following intuitive explanation is provided for the condition in Proposition 3 which
is afterwards clarified by a numerical example. As already observed in the two-player
contest game, AA in general induces higher competitive pressure because contestants
are more similar than under ET. Increasing the number of active contestants therefore
yields higher total effort for both policies because this implies more intense compe-
tition. However, inducing heavily discriminated contestants to participate comes at
a non-negligible cost, especially for the AA policy because by lemma 1 all partici-
pants will be active under AA. This effect is less profound for ET because highly
discriminated contestant will not participate under ET.

Numerical Example: Consider the following contest game with three contestants that
have marginal costs of (β₁, β₂, β₃) = (1, 2, 2). The underlying dispersion is measured
by the coefficient of variation (defined as CV = σ(β)//3) which is in this case CV ≈
0.2828. For these parameters AA will generate E_A^A _A ≈ 0.4444 that is higher than the
aggregated effort under ET which is E_E^A _T = 0.4. If a fourth contestant with β₄ = 2.43

13

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