The ability of a contestant j to convert effort into probability of winning the contest
can be represented by the marginal effect of a change in his effort on his winning
probability. By assumption, this marginal effect is declining with his own effort. A
change in his effort also affects, however, the marginal winning probability of his
opponent i. The opponent i has an advantage in terms of ability if a change in j’s
effort positively affects his marginal winning probability. In other words, a positive
(negative) sign of the cross second-order partial derivative of Pri (xi,xj)
implies that , has an advantage (disadvantage) when j’s effort changes. At some
given combination of efforts (x,,xj), the ratio between the effect of a change in j’s
∂2 Pr,-
∂xj∂xi
effort on the marginal winning probability of , and the effect of a change in j’s effort
between the abilities of , and j. This asymmetry together with two types of stakes-
asymmetry that are presented below play a crucial rule in determining the
comparative statics effects on which this study focuses.
∂2 Pr,
on his own ability, ,
∂x,∂xj
P
/l * ,
is therefore a local measure of the asymmetry
Denote by n, = (u, -v,) the stake of player i (his real benefit from winning the
contest), (see Baik, 1999, Epstein and Nitzan, 2001b and Nti, 1999). A player’s stake
is secured when he wins the contest, that is, when his preferred policy is the outcome
of the contest. Recall that for one player the desirable outcome is associated with the
approval of the proposed policy while for the other player the desirable outcome is
realized when the proposed policy is rejected. The expected net payoff (surplus) of
interest group , can be rewritten as follows:
(3) E(w,)=v,(I)+ Pr,n,(I)- x,
In general, the stakes of the contestants are different, that is, one of them has an
advantage over the other in terms of his benefit from winning the contest. With no
loss of generality, we assume that nL ≤ nH . The ratio nL/nH is a measure of the
asymmetry between the stakes of the contestants.