A non-interior equilibrium in which a contestant exerts zero effort cannot exist because
there always exists a profitable deviation for one of the contestants.16 The second order
conditions can be expressed in the following way:
which proves concavity by the assumption on r. Hence, the equilibrium is interior and
unique. From eq. (6) it can also be noted that the relative equilibrium effort levels
are independent of the implemented policy because:
∂ 2ui(ei,ej )
∂e2
αpαPrVe2r 2err
(αP er + αp er)3
αP (r - 1) f ej) - αP (r + 1)
ei
< 0,
P ' = β for P ∈ {ET, AA} . (8)
e2(P )
The two policy alternatives ET and AA can now be evaluated with respect to the
sum of equilibrium effort Ep = ∑i=1,2 e*(P) that each policy generates. The following
Proposition states the result for the two-player contest game: The affirmative action
policy as specified in Definition 2 will induce higher individual and also higher aggregated
effort than the equal treatment policy. This result refutes the above mentioned critique
of affirmative action policy because in the contest game as specified here a trade-off
between affirmative action and aggregate effort does not exist.
Proposition 1 In the two-player contest game (i) the sum of equilibrium effort, and
(ii) each individual equilibrium effort level is higher under policy AA than under the
policy ET.
Proof : Using eq. (6) and Definition 1 and 2, the inequality EA A > E*e t can be
reduced to p p+1 > (ɪɪʃr^ ββ+1, which is always satisfied because it can be simplified
to (1 - β2r )2 > 0. This establishes part (i) of the Proposition.
Using the fact that the relation between the equilibrium effort levels remains constant,
as stated in eq. (8), proves part (ii).□
The reason for this at first sight surprising result lies in the fact that the implemen-
tation of the AA policy yields a contest game that is more balanced with respect to
the characteristics of the contestants (the heterogeneity of the contestants is reduced
16 If both contestants would exert zero effort a deviating player i will always win the contest with
certainty by exerting a slightly positive effort level e: ui(e, 0) > ui(0, 0) = 0. If only one contestant
j would exert zero effort player i can deviate profitably by decreasing his chosen effort level by a
small amount e because then he still wins the contest game with certainty: ui(ei — e, 0) > ui(ei, 0)
as long as ei — e > 0.
10