We specify an oligopoly game, where firms choose quantity in order to maximise profits, that is strategically equivalent to a standard Tullock rent-seeking game. We then show that the Tullock game may be interpreted as an oligopsonistic market for influence.Alternative specifications of the strategic variable give rise to a range of Nash equilibria with varying levels of rent dissipation.Tullock contests, oligopoly

Flavio Menezes

John Quiggin

English

Research Papers in Economics

 
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Markets for Influence 
 
Flavio Menezes  
University of Queensland 
and 
  
John Quiggin 
Australian Research Council Federation Fellow, University of Queensland 
 
 
 
 
 
Risk & Sustainable Management Group 
 
 
Schools of Economics and Political Science 
University of Queensland 
Brisbane, 4072 
rsmg@uq.edu.au 
http://www.uq.edu.au/economics/rsmg 
 
Risk & Uncertainty Program Working Paper: R07#7 Markets for In￿ uence
Flavio M. Menezes
The University of Queensland
John Quiggin
The University of Queensland
July 15 2007Abstract
We specify an oligopoly game, where ￿rms choose quantity in order to maximise
pro￿ts, that is strategically equivalent to a standard Tullock rent-seeking game.
We then show that the Tullock game may be interpreted as an oligopsonistic
market for in￿ uence.Alternative speci￿cations of the strategic variable give rise
to a range of Nash equilibria with varying levels of rent dissipation.1 Introduction
Tullock￿ s (1980) seminal contribution marks the beginning of a large research
e⁄ort aimed at understanding some economic interactions, typically referred to
as contests, where agents spend resources in order to get ahead of their rivals
and win a prize. Economic interactions that have been studied in this literature
include elections, litigation, internal labour market tournaments, sales contests,
R&D races, and rent seeking.1
This note shows that the standard formulation of the Tullock rent-seeking
game, where individuals choose e⁄ort or resources to win a prize, is strategically
equivalent to a Cournot oligopoly game where the elasticity of demand is unitary
and ￿rms choose quantity to maximise their pro￿ts. We derive the associated
isomorphism and show how alternative speci￿cations of the success function
in the Tullock contest correspond to speci￿cations of the cost function for
oligopolists. Conversely, oligopolistic markets may be regarded as a generalized
class of contests.
This result has an important implication; it suggests that the exclusive fo-
cus of the Tullock contest literature on e⁄ort or resources as the strategic vari-
able might be misleading. There is an obvious contrast with oligopoly models,
where both prices and quantities (Bertrand or Cournot models) were considered
as possible strategic variables even before the game theory revolution that has
dominated the ￿eld of industrial organisation over the last three decades. More
recently, a number of papers have proposed alternative strategic variables, such
as supply curves (Klemperer and Meyer, 1989) and markups (Grant and Quig-
gin, 1994). In addition, there have been numerous attempts to motivate the
choice of particular strategic variables, for example as outcomes of a multistage
game (Kreps and Scheinkman, 1983). These issues have received comparatively
less systematic attention in the contests literature.
The contests literature does examine games for which the equilibria resem-
ble Bertrand. For example, when the contest success function is discontinuous,
complete dissipation of rent (equivalent to zero pro￿ts in the oligopoly case) is
an equilibrium outcome. However, our result shows that even con￿ning atten-
tion to the standard success function with probability proportional to relative
e⁄ort, alternative choices of the strategic variable can yield the full gamut of
equilibrium outcomes from Cournot to Bertrand.
This ￿nding leads us to explore the relationship between contests and mar-
kets more carefully. We argue that a natural economic interpretation of contests
may be presented in terms of oligopsonistic markets for in￿ uence. The in￿ uence
variable may be interpreted as electoral support, legal expertise, connections
within labour markets and so on. As markets for in￿ uence become more com-
petitive, the implicit price of in￿ uence increases and the net rent shared by
purchasers of in￿ uence decreases. Once again, this interpretation leads to an
isomorphism with a model of imperfect competition, in this case that of ￿rms
acting as oligopsonists in a factor market.
1For a recent survey of the literature see, for example, Konrad (2007).
1This paper is organised as follows. Section 2 formally de￿nes the strategic
equivalence between the two classes of games. Section 3 then discusses the
implications of such equivalence by recasting contests as markets for in￿ uence.
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 e⁄ort levels e1;e2;:::;en in order to win a prize of ￿xed value V ,
and a parameter R > 0. The winner of the contest is the player who spends the
most e⁄ort. Player i￿ s payo⁄ in this class of games is given by:
￿i(e1;e2;:::;en) = V
eR
i
n X
j=1
eR
j
￿ ei: (1)
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:
p =
V
n P
j=1
qj
and assume that i￿ s cost function is given by ci(qi) = q
1
R
i . Thus, the revenue for
player i is
ri =
V qi
n P
j=1
qj
=
c
￿
i P
j c
￿
j
:
Therefore, we can write i￿ s payo⁄ as follows :
￿i(c1;c2;:::;cn) = V
cR
i
n X
j=1
cR
j
￿ ci: (2)
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 speci￿c innovation and patent-race
games.
2Proposition 1 A standard Tullock contest characterised by the payo⁄ function
given in (1) is strategically equivalent to a Cournot oligopoly game with demand
function p = V
n P
j=1
qj
and cost function ci(qi) = q
1
R
i :
2.1 Imperfectly competitive markets as contests
The interpretation of contests as taking places in markets, which is a⁄orded
by the proposition above, may be turned around. Participants in oligopolistic
markets may be considered as taking part in a contest for market share. In the
case where the elasticity of demand is unitary, this interpretation is represented
by the isomorphism given above. More generally, oligopolistic markets may be
considered as analogous to contests where the strategic choices of the players
determine both the value of the prize (total revenue) and the probability of
winning (market share).
One important implication of this literature, which has received only lim-
ited attention in the industrial organization literature, is that, in determining
the rent accruing to participants, the cost function is just as important as the
choice of strategic variable. Depending on the cost function, any outcome in
the range from perfect competition to joint monopoly pricing may be sustained
as a Cournot equilibrium.
3 Tullock Contests as Markets for In￿ uence
The results derived above suggest that individual behavior in Tullock contests
may usefully be related to the behavior of ￿rms in imperfectly competitive mar-
kets. To pursue this idea further, it seems natural to consider more carefully the
idea, familiar from public-choice theoretic discussions of political processes, that
contests represent a particular kind of market, namely a market for in￿ uence.
If this analogy is taken seriously, the participants in contests may be regarded
as buyers in oligopsonistic markets. To formalize the idea, we need to de￿ne
concepts analogous to prices, quantities, and supply schedules.
To address this task, we introduce the idea of a price of in￿ uence which is
given by the inverse demand function
p(￿1;￿2;:::;￿n) =
X
i
￿i
where ￿i is the in￿ uence acquired by player i and p is the unit price of in￿ uence.
In the electoral case, for example, we might adopt the interpretation that p is
the price paid by the candidates for each vote and ￿i the total number of voters
induced to vote for candidate i. Accordingly, the expenditure for player i is
ei = p￿i;i = 1;2:::n:
3As before, we assume that the success probabilities are given by
ei
n X
j=1
ej
=
￿i
n X
j=1
￿j
(3)
where we assume, for simplicity, that R = 1 and the prize is normalized to one
so that i￿ s payo⁄ is given by ￿i￿ei: One can immediately see that such context
is essentially isomorphic to a oligopsony game as described below.
Proposition 2 A standard Tullock contest characterised by payo⁄ function
ei
n X
j=1
ej
￿ ei, i = 1;:::;n; is strategically equivalent to a oligopsony game where:
(i) the strategic variable for ￿rm i is the quantity purchased of an input
xi > 0;
(ii) output is given by the production function f(xi) = xi;
(iii) the (constant) output priceis p = A ￿ 1
(iv) A is su¢ ciently large that the input supply price w = A￿ 1
n P
i=1
xi
is always
positive.
Proof: Each ￿rm i chooses xi to maximise pro￿ts, which can be written as:
￿i = pf(xi) ￿ wxi = (A ￿ 1)xi ￿
0
B
B
@A ￿
1
n P
i=1
xi
1
C
C
Axi = ￿xi +
xi
n P
i=1
xi
: (4)
Then replace xi with ei. ￿
Proposition 2 above suggests that a Tullock contest will deliver outcomes
that are as competitive as those where the strategic variable is the quantity of
in￿ uenced purchased. This result is the oligopsony equivalent of Proposition 1.
This raises the following question: Would outcomes be more competitive if the
strategic variable were total expenditure?
The analogy with oligopoly can help us to answer this question. Grant
and Quiggin (1994) show that the equilibrium outcome with revenue as the
strategic variable is less competitive (higher price, lower aggregate quantity,
higher pro￿t) than the Cournot-Nash equilibrium. This is because (loosely
speaking) if one player chooses to deviate by increasing revenue, this entails an
increase their own output and a reduction in the market price, and the Nash
assumption that other players will hold revenue constant implies that they must
increase quantity. Converse reasoning for the oligopsony case suggests that the
outcome of a standard Tullock contest with expenditure as a strategic variable
will be more competitive (lower price, higher aggregate quantity, more rent
dissipation) than the Cournot-Nash equilibrium. This is because an increase
4in expenditure by one player raises the market price, and therefore lowers the
equilibrium quantity associated with a given expenditure level.
To verify this we ￿rst remind the reader that in the standard analysis of
Tullock games, player i chooses ei to maximise (1). The unique (symmetric)
Nash equilibrium is well-known and given by e￿
i = n￿1
n2 = e￿ for i = 1;:::;n:
To see this, note that n￿1
n2 is the solution to @￿1
@e1 je2=e3=:::=en=e= 1
e1+(n￿1)e ￿
e1
(e1+(n￿1)e)2 ￿1 = 0. This implies that the total resources spent by players add
up to
n P
i=1
e￿
i = n￿1
n :
Second, consider the Cournot-Nash strategic representation where the can-
didates choose quantity ￿i to maximise:
￿i =
p￿i
p
P
j ￿j
￿ p￿i:
It is easy to see that this representation has a unique symmetric equilibrium
where
￿
C
1 = ￿
C
2 = ::: = ￿
C
n =
p
n ￿ 1
n
p
n + 1
;
and consequently
pC =
p
n ￿ 1
p
n + 1
and
eC =
(n ￿ 1)
n(n + 1)
and
n X
i=1
eC
i = neC =
(n ￿ 1)
(n + 1)
: (5)
This impliesless rent dissipation than the standard solution for the Tullock con-
test as
(n￿1)
(n+1) ￿ n￿1
n always holds.
Finally, we consider a strategic representation of markets for in￿ uence 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 o⁄ers
the higher price. In the event that both candidates o⁄er the same price, voters
are equally split among the two candidates. It is not di¢ cult to see that the
Bertrand (auction) logic implies that in equilibrium:
pB
1 = ::: = pB
n = 1:
That is, any price lower than one leads to ￿ undercutting￿ . Under this equilib-
rium, there is zero pro￿t, that is, full rent dissipation, as
￿
B
1 = ::: = ￿
B
2 =
1
n
= eB
1 = ::: = eB
n: (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.
54 Concluding comments
In this paper, we have shown that the standard Tullock contest game is strate-
gically isomorphic to a Cournot oligopoly game if e⁄ort in the Tullock contest
is mapped to outputs in the oligopoly game. Consideration of this isomorphism
indicates some di⁄erences in the aspects of the problem considered in the two
literatures. Analysis of Tullock contests has focused on di⁄erences in the success
function, while the oligopoly literature has paid more attention to the determi-
nation of the strategic variable. In each case, a range of possible outcomes from
complete rent dissipation to sharing of the maximum rent may be obtained in
appropriate cases.
Understanding of the relationship between contests and imperfectly compet-
itive markets is hampered by the absence of explicit prices and quantities in the
standard contest model. When contests are represented as markets for in￿ u-
ence, we derive a natural strategic equivalence between the standard Tullock
contest and an oligopsonistic market in which expenditure is the strategic vari-
able. Unlike the corresponding case for oligopoly, this outcome turns out to be
less competitive (and hence less dissipative of rent) than the Cournot solution.
Representation of Tullock contests as markets for in￿ uence raises a wide
range of possible future developments. Most obviously, the literature on indus-
trial organization focuses on the extent to which the choice of strategic variable
determines whether market outcomes will yield competitive (Bertrand) out-
comes, less competitive (Cournot outcomes) or joint monopoly outcomes, not
to mention a wide range of intermediate possibilities. Analogies with Tullock
contests, including elections, litigation and so on may be fruitful. Beyond this,
it would be natural to consider the implications of the literature on mergers to
determine conditions under which participants in a Tullock contests, such as
political parties, might bene￿t from the formation of a coalition.
References
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seeking innovation, and patent-race games,￿Games and Economic Behavior
44, 217 -226.
[2] Baye, M. R., Kovenock, D., and de Vries, C. G. (1994), ￿ The solution to
the Tullock rent-seeking game when R > 2: Mixed-strategy equilibria and
dissipation rates,￿Public Choice 81, 362-380.
[3] Grant, S. and Quiggin, J. (1994), ￿ Nash equilibrium with markup-pricing
oligopolists￿ , Economics Letters 45, 245￿ 51.
[4] Klemperer, P. and M. Meyer, 1989, ￿ Supply function equilibria in oligopoly
under uncertainty,￿Econometrica 57, 1243-1277.
6[5] Konrad, K. A. (2007), ￿ Strategy in contests ￿ an introduction￿ . WZB-
Markets and Politics Working Paper No. SP II 2007-01. Available at SSRN:
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[6] Kreps, D. and J. Scheinkman, 1983, ￿ Quantity precommitment and Bertrand
competition yield Cournot outcomes￿ , Bell Journal of Economics 14(2), 326-
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G. Tullock, (eds.), Towards a Theory of the Rent-Seeking Society, College
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