inverted to ei = zi /αiP 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 δip = iι∕iV and Z defined as above. This transformed contest game is now
αi
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 — δp) = 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 — δpZ2, 0}. Finally, define player i’s share function as her
relative contribution
Si(Z) = J = max{1 — δpP 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 δip coincides30
for both policies with the one based on marginal costs because δ1p ≤ δ2p ≤ . . . ≤ δnp .
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∕δip 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 δiAA = δjAA for i = j .
24