β = 1 and for less able contestants β ∈ {'β, ∞) :
ci (ei) = βiei for all i ∈ N.
(1)
The contestants perceive the outcome of the contest game as probabilistic. However,
they can influence the probability of winning by exerting effort, i.e. the outcome
depends on the vector of effort levels exerted by all individuals. The following Contest
Success Function (CSF), axiomatized in Clark and Riis (1998), that will be applied
in the model allows also an asymmetric treatment of the contestants that can be
interpreted as affirmative action policy:
pi (e) =
αiP er
∑j∈N αPer
(2)
with αiP > 0 for all i ∈ N and r ∈ (0, 1]. This function maps the vector of effort
levels e = (e1,... ,en) into win probabilities for each contestant: pi(e) : 'hv' → [0,1].
The parameter r measures the sensitivity of the outcome of the contest game with
respect to differences in effort.8 Additionally, each individual effort level is weighted
by a positive parameter αiP that depends on the policy P , formally defined in the
next subsection. If no contestant exerts positive effort it is assumed that none of the
individuals receives the prize, i.e. pi(0, . . . , 0) = 0 for all i ∈ N .
The specification of the cost function in eq. (1) and the contest mechanism in eq. (2)
are already the necessary elements to state the following expected (additive separable)
utility function of contestant i:
ui(ei, e-i) = pi(e)V - ci (ei) for all i ∈ N.
(3)
This contest game can therefore be interpreted as a standard non-cooperative game:
Γ[N, P, (Ai)i∈N, (ui)i∈N], where ei ∈ Ai and P is an additional policy parameter.
2.2 The Policy Options
It is assumed that the choice of the policy P is based on the ethical perception of
the heterogeneity of the contestants (i.e. the different marginal cost functions)9 which
directly implies the normative objective of the respective policy option and therefore
8The upper bound r ≤ 1 is imposed because otherwise the existence of pure strategy equilibria
cannot be guaranteed.
9 As the model is formulated under complete information, the individual marginal cost functions are
common knowledge.
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