Markets for Influence

This paper is organised as follows. Section 2 formally defines the strategic
equivalence between the two classes of games. Section 3 then discusses the
implications of such equivalence by recasting contests as markets for influence.
Section 4 concludes.

2 Tullock Contests and Cournot Competition

Our starting point is the most well-known model of contests, namely, the Tullock
rent-seeking game. This class of games can be represented by a set of n players,
who choose effort levels e₁,e₂,...,e_n in order to win a prize of fixed value V,
and a parameter R> O. The winner of the contest is the player who spends the
most effort. Player Ps payoff in this class of games is given by:

ɛ^

π_i(eι,e₂ ,...,e_n) = V —--e_i.                       (1)

∑ef
J=ι

The equilibria for this family of games (both symmetric and asymmetric, pure
and mixed-strategy) are well-known.²

To develop the isomorphism with oligopoly games, consider now a unit elas-
ticity demand curve with normalisation such that:

V

P = ——

Σ⅞,∙

J=ι

Therefore, we can write Ps payoff as follows :

Thus, incentives in the oligopoly game, with Ci as the strategic variable, are
strategically equivalent to those in a standard Tullock contest with ei substituted
for Ci. That is, we have established the following result:

²See, for example, Baye, Kovenock and de Vries (1994). Importantly, Baye and Hoppe
(2003) show that this family of games is isomorphic to certain innovation and patent-race
games. It follows then that our main result also applies to these other classes of games. That
is, there are isomorphisms between oligopoly games and specific innovation and patent-race
games.

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