Markets for Influence



in expenditure by one player raises the market price, and therefore lowers the

equilibrium quantity associated with a given expenditure level.

To verify this we first remind the reader that in the standard analysis of
Tullock games, player i chooses e⅛ to maximise (1). The unique (symmetric)


Nash equilibrium is well-known and given by e*


To see this,
________ej________
(eι + (n-1)e)2
n

up to ∑ e*
i—1


note that n-21 is the solution to °^x I
n2                               oeγ 1


n— 1

= ■

e2 — e3 —.


e* for i = 1,..., n.

_    1


—e„ —e   eι+(n-1)e


1 = 0. This implies that the total resources spent by players add


n— 1

*
n


Second, consider the Cournot-Nash strategic representation where the can-
didates choose quantity θi to maximise:


Ki


pθi

P E3 θ


— pθi.


It is easy to see that this representation has a unique symmetric equilibrium


where


θc _ θc _  _ θc

θ1 = θ2 = ... = θn


√n — 1

nʌ/n + 1 ’


and consequently


= √n — 1

√n + 1


and


ec


(n - 1)
n(n + 1)


n

ΣC C
ei = ne

i—1


(n -1)

(n + 1).


(5)


This impliesless rent dissipation than the standard solution for the Tullock con-
test as (n+1) < n-1 always holds.

Finally, we consider a strategic representation of markets for influence that is
equivalent to a ‘Bertrand’ model of oligopoly. Under this scenario the candidates
compete for voters in the ‘prices’ space. We impose the standard assumptions
in Bertrand competition, where the voters will vote for the candidate who offers
the higher price. In the event that both candidates offer the same price, voters
are equally split among the two candidates. It is not difficult to see that the
Bertrand (auction) logic implies that in equilibrium:

,,b =   =    = 1

p1 = ... = pn = 1.

That is, any price lower than one leads to ‘undercutting’. Under this equilib-
rium, there is zero profit, that is, full rent dissipation, as

θB _  _ θB

θ 1 — ... — θ 2


1= eB
n


... en .


(6)


The discussion suggests that by considering the full range of strategies avail-
able to participants in Tullock contests, it is possible to obtain a wide range of
symmetric equilibrium outcomes, just as in the case of oligopoly.



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