Economic Theory of Contests
CES Lectures
03/04/2004
Martin Kolmar
University of Mainz
Lecture 1:
Recent developments in institutional economics and
introduction to the field
Overview over the lectures
1. What is Institutional Economics trying to accomplish and how can the Theory of
Contests help to reach this goal?
2. A little taxonomy of contest models.
3. Some basic results.
4. How to use contests in economic modeling.
5. Some recent trends in the literature.
What is a contest and what do contests have to do with institutional economics?
Contest: Game in which players compete over a prize by making irreversible
outlays (Amegashi 2003).
Examples:
•Rent-seeking and lobbying
•Patent races and R&D
•Tournaments
•Arms races
•Political campaigns
•Athletic contests
•Takeover battles
•Lawsuits
•Lotteries
•Academic career incentives
•Appropriation and defense
•Enforcement of property rights
What is a contest and what do contests have to do with institutional economics?
Research Program of New Institutional Economics:
•Why do institutions matter?
•What is the economic role of institutions?
•How can we improve economic performance by designing institutions?
Conjecture:
Differences in institutions are a first-order determinant for differences in
economic success. (North 1990)
Starting Point:
Coase (Irrelevance) Theorem: In the absence of transaction costs there
always exists a first-best efficient decentralized and a first-best efficient
centralized mechanism.
Consequence: Either institutions are irrelevant or one has to identify the
idiosyncratic transaction costs of different institutions.
What is a contest and what do contests have to do with institutional economics?
An institutional structure is efficient if every individual fully internalizes the
marginal effect its actions impose upon itself and others.
(Principle of Marginal Internalization).
•Why is it impossible to completely internalize externalities?
•What are the consequences of non-internalization of externalities?
•Can we discriminate between different institutions?
Three potential implications (Acemoglu 2002):
1. Institutions do not matter (instead: geography,… (Sachs 2000)).
2. Decisive actors differ in their beliefs about good institutions.
(but: strong tendencies to avoid obviously bad institutions)
1. Gains from trade cannot be exhausted because of limited commitment.
(Theories of social conflict)
What are institutions?
North (1993), Kreps (1990): Institutions
define the incentive structure of a society.
Institutions are the rules of the game.
“Outside-Enforcement-Approach”
(Hafer 2002)
Alternatively : Institutions are the stable
behavioral patterns of a society. They are
the equilibria of a game.
“Theory of Expressive Law”
(Cooter 1998)
What are institutions?
Williamson (2000) defines four different institutional levels and two different
directions of causality.
L1:
L2:
L3:
L4:
Embeddedness
Institutional Environment
Governance Structures
Resource Allocation
Potential sources of imperfect internalization:
-
asymmetric information
-
incomplete contracts
-
commitment (renegotiation, time inconsistency, …)
Asymmetric Information
L1:
L2:
L3:
L4:
Embeddedness
Institutional Environment
Governance
Resource Allocation
Outside-Enforcement-Approach
commitment (incomplete contracts, renegotiation, time consistency)
L1:
L2:
L3:
L4:
Embeddedness
Institutional Environment
Governance
Resource Allocation
How do contests enter the picture?
•If one is interested in the ability of L1 and L2 institutions to shape individual
incentives one cannot use the outside-enforcement approach.
•If one is interested in the ability of L3 and L4 institutions to shape individual
incentives it may be more appropriate to include the ability to appropriate,
influence, confiscate, etc. as well as to defend, protect, etc.
“The efforts of men are utilized in two different ways: they are directed
to the production or transformation of economic goods, or else to the
appropriation of goods produced by others.” (V. Pareto)
“Cooperation, with a few obvious exceptions, occurs only in the shadow
of conflict.” (J. Hirshleifer)
•Theory of contests can be used as an analytical tool to better understand
•
the emergence of institutions and their transactions costs and
•
the behavioral patterns of individuals in situations of conflict.
Theory of contests
C
L1:
B
A
L2:
L3:
L4:
pragmatic
inst. design.
inst. as eq.
Embeddedness
Institutional Environment
Governance
Resource Allocation
given
Rent-seeking and lobbying
L2
Patent races and R&D
L2
Political campaigns
L2
Takeover battles
L2
Lawsuits
L2,L3
Academic career incentives
L2,L3
Appropriation and defense
L4
Enforcement of property rights L4
Evolution of morality
L4
D
E
explained
A
A
A
A
L4
L4
B
B
C
Example: Enforcement of Property Rights
Starting Point:
situation with non-existing or non-enforced property rights.
Individuals can invest in production, appropriation, and
defense.
Explanans:
production functions, conflict-success functions, preferences
Explanandum:
emergence of (unchallenged) property rights
(resource-) costs of property rights
centralized versus decentralized enforcement
comparative-static behavior
Introduction
First analysis of a contest (Tullock 1980):
• two risk-neutral individuals compete for
• exogenous rent R
• a1 , a2 : effort spent to win the rent
• p1 , p2 : probability that 1 (2) wins the rent (p1 + p2 = 1) (Contest
Success Function)
• ‘Tullock function:’
a1
a2
p1 =
, p2 =
.
a1 + a2
a1 + a2
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• utility of i = 1, 2:
ai
πi = pi R − ai =
R − ai ,
ai + aj
FOC:
∂pi
aj
R=1⇔
R = 1.
2
∂ai
(ai + aj )
Nash equilibrium: a1
= a2 = R/4
Dissipation rate:
P
D=
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,
R
i
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Taxonomy of Contests
• According to the type of rent: goods or resource contests.
• According to the evaluation of the rent: identical or different
evaluation.
• According to the evaluation of the contest: risk neutrality or risk
aversion.
• According to the relative position of appropriation and defense:
‘common-pool’- and ‘initial-endowment’ contest.
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Taxonomy of Contests (ctd.)
• According to the ability to withdraw from the contest: contests with
or without non-appropriable goods or resources.
• According to the group that wins the rent: private- (one) or public(more then one) goods contests (Katz, Nitzan und Rosenberg
1988, Leininger 2002).
• According to the sequence of moves: simultaneous or sequential
contests.
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Types of Contest-Success Functions
1. Ratio model (Tullock),
fi (ai )
pi (a1 , ..., an ) = Pn
,
j=1 fj (aj )
special cases: (i) fi (ai )
= abi , b ≥ 0, (ii) fi (ai ) = θi ai
axiomatic foundation: Skaperdas (1996)
2. Difference model (Hirshleifer),
1
pi (ai , aj ) =
.
(a
−a
)
i
j
1+e
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Types of Contest-Success Functions (ctd.)
3. Fully discriminating contest


a i > aj

 1,
pi (ai , aj ) =
0.5, ai = aj



0,
a i < aj
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Typical Contests
• Rent-seeking contest (Tullock 1980, already analyzed), tradeoff
between appropriation/defense and alternative activity (leisure,
investment in capital market with save return).
• Contest with endogenous rent (Skaperdas 1992), tradeoff
between productive and appropriative/defensive investment.
ui = pi (ai , aj )x(li , lj ), ai + li = ¯l, i = 1, 2, j 6= i.
FOC for interior solution:
∂x
∂pi
x = pi .
∂ai
∂li
study conditions for perfect/partial cooperation etc.
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Drawback: there is no meaningful distinction between defense
and appropriation. In order to get this distinction, production has
to be additive x
= x1 + x2 (Grossman 2001):
ui = pi (di , aj )xi (li ) + (1 − pj (dj , ai )xj (lj ),
ai + di + li = ¯l, i = 1, 2, j 6= i.
FOC:
∂pj
∂xi
= −xj
,
pi
∂li
∂ai
∂xi
∂pi
pi
= xi
,
∂li
∂di
∂pi
∂pj
xi
= −xj
.
∂di
∂ai
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Risk Aversion
Skaperdas and Gan (1995), Konrad and Schlesinger (1993)
What is the influence of risk aversion on the behavior in contests?
So far there has been no need to distinguish between probability and
share contests
Probability contest: pi is individual i’s probability of winning the
prize R
Share contest: pi is individual i’s share of the total prize R
Observation:
risk aversion matters only for probability contests.
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Research agenda:
• does risk aversion increase or decrease investments in the
contest?
• who spends more resources in the contest, the more or the less
risk averse individual?
• do individuals spend more effort in share or probability contests?
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What do we expect?
Two competing intuitions:
1. A less risk averse individual is more willing to gamble, exerts
higher efforts, and has a higher probability of winning.
2. A more risk averse individual is more fearful of losing. Putting in
effort is a means to self insure.
Which one is true?
It depends
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Expected utility of individual i:
E[ui ] = pi ui (R + Y − ai ) + (1 − pi )ui (Y − ai )
FOC:
p0i (ūi − ui ) = pi ū0i + (1 − pi )u0i
risk neutrality:
p0i (ūi − ui ) = 1
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