By our assumptions, both players participate in the contest (xL and xH are
positive). We therefore focus on interior Nash equilibria of the contest. Solving the
c . j ʃɪ- Γ∂ E(wL ) л ∂E E(wH ) ɔ
first order conditions I---- l = 0 and ---- = 0 I we obtain:
I d xL d xH )
∂ Pri (xi, xj )
(4) ∆i =-----------nni(I)-1 = 0 ∀ i ≠ j and i, j = L,H
∂ xi
Thus, the first order conditions require6 that:
(5)
∂ Pri
∂ xi
1
ni(I )
∀ i = L, H
By the expressions in (5) that determine the equilibrium efforts of the players and
their probabilities of winning the contest and by the assumed properties of the CSF,
we directly obtain that under a symmetric contest success function7
(∀xi, xj, Pri (xi, xj) = Prj(xj, xi)), the player with the higher stake makes a larger
effort and has a higher probability of winning the contest. The probability of the
socially more efficient outcome of the contest is thus higher than the probability of the
less efficient outcome. For a similar result see Baik (1994) and Nti (1999). This type
of efficiency criterion has been used by Ellingsen (1991), Fabella (1995) and, more
recently, by Hurley (1998).
III. Public Policy and the Prize System (The Contestants’ Stakes)
A change in the policy instrument I has an effect on the stakes of the players and thus
on their efforts and on their probability of winning the contest8. In this section we
examine how a change in the proposed policy affects the prize system, that is, the
6 It can be easily verified that the second order conditions hold.
7
Such symmetry implies that the two players share an equal ability to convert effort into probability of
winning the contest.
8 Note that the domain of the policy instrument I , the closed interval I ∈[I, I ] , may reflect
economic feasibility or political feasibility.