The name is absent

inverted to ei = zi /α_i^P for all i ∈ N . The resulting transformed expected utility
function for contestant i, which has the aggregative game property, has then the
following form:

πi(zi, Z) = Zi - δPZi for all i ∈ N and for P ∈ {ET, AA}          (20)

Z

where δ_i^p = _iι∕ⁱV and Z defined as above. This transformed contest game is now
α_i

covered by the model in Cornes and Hartley (2005). The share function can therefore
be constructed in an analogous way by deriving the first order condition:

Zi (Zz2zi — δ^p) = 0 for Zi ≥ 0.                       (21)

The best response z* of player i can be expressed in terms of the aggregated equilibrium
effort:²⁹ zi(Z) = max{Z — δ^pZ², 0}. Finally, define player i’s share function as her
relative contribution

Si(Z) =   J = max{1 — δ^pP Z, 0}.                    (22)

Z

In equilibrium the aggregated effort Zⁱ is implicitly defined by the condition that the
individual share functions sum up to one:

S(Zⁱ) =    si(Zⁱ) = 1                             (23)

i∈N

Theorem 1 in Cornes and Hartley (2005) states that a solution to this equation exists
and is unique by observing that the aggregated share function S(Z) is continuous
and strictly decreasing for positive Z , and that it has a value higher than one for Z
sufficiently small and equal to zero for Z sufficiently large.

Equation (22) already indicates that contestants with a high level of δ might have an
equilibrium share of zero, i.e. they might prefer to stay non-active. Note that due to
the definitions of AA and ET the order of the contestants according to δ_i^p coincides³⁰
for both policies with the one based on marginal costs because δ₁^p ≤ δ₂^p ≤ . . . ≤ δ_n^p .

Now the the set of active contestants M ⊆ N can be characterized, i.e. the m players
with strict positive share in equilibrium. From Eq. (22) it is obvious that in equlibrium
Zⁱ < 1∕δ_i^p for all i ∈ M. Combining Eq. (22) and (23) yields Zⁱ = ^m-ɪP. The last
j∈M δj

two expressions yield the condition that indirectly defines the set M ⊆ N of active

²⁹It should be obvious that the best response and also the share functions depends on the policy
parameter P . But as the finally implemented policy does not affect the proof of equilibrium
existence and uniqueness, it is suppressed in this section for notational convenience.

³⁰in a weak sense for the AA weights because δ_i^AA = δ_j^AA for i = j .

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