As the objectives and the entailed bias of the two policy options are clarified now, the
timing of the complete contest game can be summarized in the following way: The
heterogeneity of the contestants (i.e. different marginal cost parameters) is observed.
Based on the ethical perception of this observation a policy option P ∈ {ET, AA}
is selected that determines the weighting parameters (α1P , . . . , αnP ) for the respective
policy. The contestants exert the optimal (with respect to their expected utility)
equilibrium effort level e* (P) for each i ∈ N, taking as given the effort levels of their
rivals and the relevant weights induced by policy P . In the last step the exerted
efforts are observed and the winner of the contest game is determined according to the
announced policy option. After that the total equilibrium effort that is generated by
each policy can be compared which directly answers the question whether a trade-off
between affirmative action and total effort does in fact exist or not.
3 The Two-Player Contest Game
Restricting the number of contestants in the two player case yields the key result of
the comparative policy analysis: in equilibrium both contestants will exert more effort
under AA than under ET. Contrary to the n -player contest game this result holds
without any extra assumption and the derivation of equilibrium in the two-player
contest is based on simple first order conditions.15 Therefore, the two-player contest
game is analyzed separately.
Applying the CSF as specified in eq. (2), the expected utility function for policy
P ∈ {ET, AA} can be expressed as:
ui(ei, ej) =
p αPer p V - βiei for i = 1,2;
,
α1P er1 + α2P er2
where contestant 1 is assumed to be the one with the lowest marginal cost parameter
such that β1 = 1 and β2 > 1. By Definition 1 and 2 the bias for contestant 1 is
normalized to α1P = 1 for P ∈ {ET, AA}. Solving first order conditions for a given
policy parameter P yields the equilibrium effort candidate for i = 1, 2:
α1P α2P βir-1βr
e-(P ) = + P βr )2 rV for i = j,
(6)
that would imply positive expected equilibrium utility by the assumption on r:
Ui(e,∙(P ),e∙(P )) =
(αiPβjr)2 + α1P α2P (β1β2)r (1 - r)
(α1P β2 + α2Pβ1)2
V > 0.
(7)
15Nti (1999) is based on a similar set-up with a non-biased CSF in a rent-seeking framework where
heterogeneity does affect individual valuations instead of marginal costs.