to one player as the Low-Benefit (LB) player and to the other player as the High-
Benefit (HB) player3. The interest groups engage in a contest that determines the
probabilities of approval and rejection of the proposed policy.4
Player i’s preferred policy is approved in probability Pri . The present
discounted value of this policy to this player is equal to ui and its value to his
opponent player j is equal to vj. By assumption then, for each player, approval of his
preferred policy is associated with a positive payoff, that is, ui > vi . Note that, in
general, the four values uL, vL, uH and vH, viz., the players’ payoffs corresponding to
the approval and rejection of the policy I proposed by the government (a ruling
politician or a bureaucrat) depend on I.
Let xi denote the effort of the risk-neutral player i. The expected net payoff of
i is given by:
(1) E(wi)=Priui(I)+ Prjvj(I)- xi , i≠ j
Given the contestants’ efforts, the probabilities of approval and rejection of the
proposed policy, PrL and PrH , are obtained by the contest success function. As in
Skaperdas (1992), it is assumed that
d PrAχ-- χj )
∂ xi
>0,
∂ Pri (xi , x )
----λ i j, < 0 and
d xj
∂2 Pri (xi, xj )
∂ χi 2
< 0 5 (the latter inequality ensures that the second order conditions are
satisfied). Since Pri(xi, xj )+Prj (xj , xi ) = 1 , i≠ j , it holds that
(2)
∂2Pri(xi, xj) ∂2Prj(xj, xi)
∂ xi ∂ xj ∂ xi ∂ xj
3 See Epstein and Nitzan (2001a).
4 Modeling the contestants as single agents presumes that they have already solved the collective action
problem. The model thus applies to already formed interest groups.
5 The function Pri( xi , x ) is usually referred to as a contest success function (CSF). The functional
ij
forms of the CSF’s commonly assumed in the literature, see Nitzan (1994) and Skaperdas (1996),
satisfy these assumptions.