winning probability of a contestant in equilibrium is positively (negatively) related to
his rival’s effort, his strategic substitution effect is positive (negative). Following
Bulow, Geanakoplos and Klemprer (1985), in such a case we say that a contestant’s
effort is a strategic complement (substitute) to his rival’s effort. When the cross-
partial derivative of the contest success function is equal to zero the contestants’
efforts are independent. Note that, by (2), in our setting the strategic substitution
effects are asymmetric; if a player’s effort is a strategic complement to his opponent’s
effort, then his opponent’s effort is a strategic substitute to his effort.
In the symmetric case where, ∀xH and xL, PrH (xL,xH) = 1 - PrH (xH ,xL), there
exists a pure strategy Nash equilibrium, such that x*H > x *L and, in equilibrium,
Sign (
∂ 2PrH
∂xH ∂xL
) = Sign (
x*H -x *L ) > 0 , which implies that
∂ 2PrL
∂xL∂xH
< 0 . Hence,
by Corollary 1.1,
Corollary 1.2: If, ∀xH and xL, PrH (xL,xH) = 1 - PrH (xH ,xL), then
∂**
xH ∂xL
------> 0, -----
∂nH ∂nL
∂ x * H
∂nL
> 0 and
d-± < 0 .
∂nH
This second corollary generalizes the result obtained by Nti (1999) where Pri is
assumed to take the particular symmetric logit form, as in Tullock (1980),
r
xi
Pri(xi, xj)=—r------, I = L, H. In this special case of symmetric lobbying abilities
xi r + x r
ij
of the contestants, the HB player can be referred to as the favored player and the LB
player can be referred to as the underdog, see Dixit (1987). Corollary 1.2 establishes
that effort of the favored player increases with both his own stake (valuation of the
contested prize) and with the stake (prize) valuation) of the underdog. Effort of the
underdog increases with his stake (prize valuation), but decreases with the stake (prize
valuation) of the favored player.
Another more general asymmetric form of the logit contest success function is:
PrH =----σh(xH )----, where σ > 0, h(0) ≥ 0 and h(xi) is increasing in x i 10, see
σh(xH)+h(xL)
10
In this special case we keep the assumption that a contestant’s marginal winning probability is
13
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