Additionally, an assumption is needed in the n -player contest because for a non-linear
CSF with parameter r < 1 it is not possible to derive closed form solutions.20 As the
existence of closed form solutions is crucial for the comparative analysis of the policy
alternatives, it is assumed from now on that the CSF is linear with r = 1.
The expected utility of the risk-neutral contestant i in the n-player contest can then
be expressed as:
αPei
ui(ei, e-i) =-------p— V — βiei for all i ∈ N and for P ∈ {ET, AA } . (9)
j∈N αjP ej
It is also assumed that the contestants are ordered with respect to their marginal cost
parameter: β1 ≤ β2 ≤ . . . ≤ βn with the normalization β1 = 1.
The equilibrium of this contest game will be derived in appendix A.1, based on the ob-
servation that the contest game can be interpreted as an aggregative game with its con-
venient properties. The following equation provides an expression of the equilibrium
effort for those m contestants of the set M ⊆ N that are active, i.e. that exert a
positive equilibrium effort:
e;(P ) = α1P
i
βi (m — 1) ʌ
αp V βj )
αi l^j∈M αp
j
(m—1)V for all i ∈ M and P ∈ {ET, AA } .
∑j∈M αP
j
(10)
Set M is indirectly defined by the following inequality:
(m — 1) -^P < ^" j for all i ∈ M and P ∈ {ET, AA } . (11)
αiP j∈M αjP
Using the specification of the weights for the AA and ET policy and the charac-
terization of the active set, the following Lemma describes the set of participating
contestants for each policy option.
Lemma 1 Under the policy ET the active set M ⊆ N of contestants is implicitly
defined by the following inequality:
(m — 1)βi < ^ βj∙ for all i ∈ M. (12)
j∈M
Under policy AA all contestants will be active.
that is instead applied is based on the notion of ‘share functions’ as defined in Cornes and Hartley
(2005) which has the advantage that the existence proof of equilibrium is reduced to a simple
fixed point argument in 'R2.
20Cornes and Hartley (2005) give existence results for this class of games by analyzing the properties
of implicit equilibrium conditions. They also show that total equilibrium effort is increasing in r.
12