Contest Success Function with the
Possibility of a Draw: Axiomatization
Pavlo R. Blavatskyy
Contest success function with the possibility of a draw:
axiomatization
Pavlo R. Blavatskyy’
Institute for Empirical Research in Economics
University of Zurich
Winterthurerstrasse 30
CH-8006 Zurich
Switzerland
Phone: +41(1)6343586
Fax: +41(1)6344978
e-mail: [email protected]
Abstract:
In imperfectly discriminating contests the contestants contribute effort to win a prize but
the highest contributed effort does not necessarily secure a win. The contest success function
(CSF) is the technology that translates an individual’s effort into his or her probability of
winning. This paper provides an axiomatization of CSF when there is the possibility of a draw
(the sum of winning probabilities across all contestants does not add up to one).
Keywords: contest success function, draw, logit, axiomatization, irrelevant alternatives,
homogeneity, anonymity, imperfectly discriminating
JEL Classification codes: C72, D72, L13
’
I am grateful to Wolfgang Koehler, Patrick Leoni, and Stergios Skaperdas for helpful comments. Ganna Pogrebna
helped a lot with English copy editing.
Contest success function with the possibility of a draw:
axiomatization
1. Introduction
A contest is a competition where the contestants simultaneously contribute effort to win a
prize. All contests can be classified as either perfectly or imperfectly discriminating (e.g. Hillman
and Riley 1989). In perfectly discriminating contests (e.g. Moldovanu and Sela 2001) the highest
contributed effort secures a win (like in an all-pay auction). In imperfectly discriminating
contests (e.g. Dixit 1987) the player who contributed the highest effort has the highest probability
of a win, but the highest effort does not necessarily secure a win. Imperfectly discriminating
contests are extensively used to study political rent-seeking (e.g. Nitzan 1991, 1994), research
and development and patent races (e.g. Nti 1997), labor incentives (e.g. Rosen 1986), sport
competitions (e.g. Szymanski 2003) to name a few.
In imperfectly discriminating contests the technology that translates an individual’s effort
into his or her probability of winning is called the contest success function (CSF). The majority
of studies on imperfectly discriminating contests use the logit CSF introduced by Tullock (1980)
and subsequently axiomatized by Skaperdas (1996) and Clark and Riis (1996, 1998). In some
studies (e.g. Hirshleifer 1989, Baik 2004) the logit CSF is called the ratio-form CSF. Other forms
of CSF include the probit CSF (e.g. Lazear and Rosen 1981, Dixit 1987) and the difference-form
CSF (e.g. Hirshleifer 1989). All above mentioned CSFs share one limitation. The sum of winning
probabilities across all contestants adds up to one by assumption. Thus, only one result is
admitted in the contest—one of the contestants wins the prize unilaterally. On the other hand, the
contests with the possibility of a draw (tie) have been studied in the literature on the labor market
incentives (e.g. Nalebuff and Stiglitz, 1983), where a CSF is derived from the idiosyncratic
shocks distorting individual labor input.
2
In the majority of real life contests it is possible that no one wins the prize outright. The
contest outcome without either side winning is called a draw. A draw is a situation when two or
more contestants obtain equal rights for the prize and they share the price. The possibility of a
draw is apparent in sport tournaments (e.g. chess, soccer), political rent-seeking (e.g. no party
wins the majority and a winning coalition is formed), research and development and patent races
(e.g. a simultaneous independent discovery), labor tournaments (e.g. two individuals are
promoted to share a senior position), military conflicts (e.g. no party can force a win and a ceasefire or peace agreement is reached). The analysis of these imperfectly discriminating contests
requires that the CSF allows for the possibility of a draw. This paper presents the axiomatic
characterization of such a CSF. This axiomatization is a generalization of Skaperdas (1996) and
Clark and Riis (1998) when the sum of winning probabilities across all contestants does not add
up to one.
The remainder of this paper is organized as follows. The logical exposition and the
intuition behind the proposed system of axioms is presented in section two. This section
describes the simplest case when there are only two contestants. In section three, which is more
technical, a general result for any number of contestants is presented. Section four concludes.
2. Contest with two contestants
Consider first the simplest contest with only two contestants. This contest is of special
interest because:
•
The model of a pairwise contest is extensively used in the literature (e.g. Rosen 1986, Baye
and Shin 1999, Gradstein and Konrad 1999, Morgan 2003, Baik 2004),
3
•
The existing axiomatizations of the logit CSF without the possibility of a draw (e.g.
Skaperdas 1996, Clark and Riis 1998) provide axiomatic characterization only when there are
three or more contestants (see example in the Appendix).
•
In the special case when there are only two contestants it is possible to generalize the existing
axiomatization by Skaperdas (1996) and Clark and Riis (1998) by relaxing their axioms of
additivity and homogeneity.
In what follows, the notation of Skaperdas (1996) and Clark and Riis (1998) is used
wherever possible. Let p i ( y1 , y 2 ) ∈ [0,1] be the probability that the i-th player wins the contest,
i ∈ {1,2}, when the first (second) contestant contributes effort
y1 ≥ 0 ( y 2 ≥ 0 ).1 Let
p 12 ( y1 , y 2 ) ∈ [0,1] be the probability that the first and the second player end up in a draw.
Axiom 1 (Additivity) p 1 ( y1 , y 2 ) + p 2 ( y1 , y 2 ) + p 12 ( y1 , y 2 ) = 1 , ∀y1 , y 2 ≥ 0
Axiom 2 (Nontriviality) Probability p i ( y1 , y 2 ) is strictly increasing in y i , i ∈ {1,2}
Axioms 1 and 2 guarantee that the winning probabilities of two contestants together with their
probability of a draw indeed constitute a non-trivial probability distribution. Axioms 1 and 2 are
straightforward extensions of axioms (A1) and (A2) in Skaperdas (1996) and Clark and Riis
(1998).2 Axiom 2 implies that p i ( y1 , y 2 ) < 1 for every y i ≥ 0 , i ∈ {1,2}. In other words, there
exists no combination of (finite) efforts such that one of the contestants wins the contest with
certainty. Even when the first contestant contributes zero effort, the second contestant does not
automatically win the contest if his or her effort is positive. Both contestants can end up in a draw
1
Since the event of a draw is explicitly allowed, there is no problem in considering a zero effort of (all) contestants.
Clark and Riis (1998) rule this case out because without the possibility of a draw, the Tullock CSF exhibits a
discontinuous jump when all contestants contribute zero effort.
2
Skaperdas (1996) and Clark and Riis (1998) assumed in addition that for every contestant the probability of
winning is nonincreasing in his or her opponent’s effort. This property follows from the conjunction of axioms 1-3
and there is no need to assume it explicitly.
4
or the first contestant may even win, for example, when the second contestant contributes very
small positive effort. Similar as in Nalebuff and Stiglitz (1983), the contestant contributing zero
effort has a chance to end up in a draw or even to win because the contest is imperfectly
discriminating (“noisy”). In contrast, Clark and Riis (1998) interpret zero effort as nonparticipation, which implies that a contestant who contributes zero effort cannot win3.
Axiom 3 (Independence of irrelevant alternatives) The probability that the i-th
contestant wins given that the j-th contestant does not win depends only on the effort level of the
i-th contestant (and it does not depend on the effort level of the j-th contestant), i, j ∈ {1,2}, i ≠ j .
The intuition behind axiom 3 is the following. Suppose that it is already known that the
first contestant is successful in winning the contest prize either unilaterally or by sharing it
together with the second contestant. In this case, the probability that the first contestant wins the
contest unilaterally depends only on the effort level contributed by the first contestant. In other
words, the probability that the first contestant wins, given that he or she either wins or draws,
depends only on the effort level of the first contestant. Axiom 3 is a straightforward extension of
the axiom (A4’) in Clark and Riis (1998) to the case when the contest can end up in a draw. For
the contest with only two participants, axiom (A4’) in Clark and Riis (1998) states that
p 1 ( y1 ,0 ) =
p 1 ( y1 , y 2 )
and a symmetric expression for the second contestant. When there is no
1 − p 2 ( y1 , y 2 )
possibility of a draw, the right hand side of axiom (A4’) just equals to unity. When the contest
can end up in a draw, the right hand side of axiom (A4’) equals to the odds ratio “the first
3
The problem with such interpretation is that when all but one contestant contribute zero effort i.e. withdraw from
participation, and there is no possibility of a draw, the remaining contestant wins with certainty. The probability of
winning for this remaining contestant is then not strictly increasing in his or her effort as required by axiom (A2) in
Clark and Riis (1998). Therefore, the CSF axiomatized by Clark and Riis (1998) actually does not satisfy their axiom
(A2).
5
contestant wins” and “the first contestant either wins or draws”. The left hand side of axiom
(A4’) is just a function of the effort contributed by the first contestant. Thus, axiom 3 follows
from axiom (A4’) in Clark and Riis (1998) when there are two only contestants who can draw.
Theorem 1 Axioms 1-3 hold if and only if there exist strictly increasing functions
H 1 ( y1 )
H : R + → R + , i ∈ {1,2}, such that p ( y1 , y 2 ) =
,
1 + H 1 ( y1 ) + H 2 ( y 2 )
i
1
H 2 ( y2 )
1
and p 12 ( y1 , y 2 ) =
p ( y1 , y 2 ) =
1
2
1
1 + H ( y1 ) + H ( y 2 )
1 + H ( y1 ) + H 2 ( y 2 )
2
All proofs are delegated to the Appendix.
If axioms 1-3 hold then for every contestant the probability of winning is increasing in his
or her own effort and decreasing in the opponent’s effort. For every contestant the probability of
a draw is decreasing both in his or her own effort and in the opponent’s effort. Thus, the higher
the effort levels of two contestants, the less likely is a draw. This property reflects an intuitive
observation that in real-life contests a draw is more (less) probable when the contestants
contribute low (high) effort on average. For example, in football it is more common to observe
draws with low scores (e.g. “0:0”, “1:1”) than those with high scores (e.g. “2:2”, “3:3”). This can
be interpreted that when both teams put a lot of effort into the game they are less likely to end up
in a draw.
Theorem 1 also implies that, for example, when the first player contributes zero effort, the
second player does not automatically win. In such a case the most probable outcome of a contest
is either a draw (when the second contestant contributes low effort) or a win for the second player
(if he or she contributes high effort). More generally, under any CSF that satisfies axioms 1-3 a
contestant cannot secure a win with certainty. The logit CSF without the possibility of a draw
6
(e.g. Tullock 1980) allows a contestant to secure a win with certainty if all opponents contribute
zero effort.
Skaperdas (1996) and Clark and Riis (1998) also proposed the following homogeneity
axiom: p i ( y1 , y2 ) = p i (λ ⋅ y1 , λ ⋅ y2 ) for any λ > 0 , i ∈ {1,2}. This homogeneity axiom is
formulated in absolute terms. Intuitively, this property of homogeneity means that the winning
probabilities do not change if we change the units in which effort is measured. A similar property
is postulated in axiom 4 below.
p 1 ( y1 , y 2 ) p 1 (λ ⋅ y1 , λ ⋅ y 2 )
Axiom 4 (relative homogeneity) 2
, ∀λ > 0 , ∀y1 , y 2 > 0
=
p ( y1 , y 2 ) p 2 (λ ⋅ y1 , λ ⋅ y 2 )
According to axiom 4 the ratio of winning probabilities remains unchanged if both
contestants multiply their effort by the same amount. When a draw is possible and CSF satisfies
axioms 1-3, as shown above, the probability of a draw decreases when both contestants increase
their efforts. In absolute terms, the sum of winning probabilities increases when both contestants
increase their efforts. Therefore, the homogeneity axiom does not hold in absolute terms but only
in relative terms.
Theorem 2 Axioms 1-4 hold if and only if there exist constant r , α i > 0 such that
α i y ir
1
, i ∈ {1,2}, and p 12 ( y1 , y 2 ) =
.
p ( y1 , y 2 ) =
r
r
r
1 + α 1 y1 + α 2 y 2
1 + α 1 y1 + α 2 y 2r
i
Axiom 4 states that the relative winning probabilities are homogeneous of degree zero.
This implies that the strictly increasing function H i ( y i ) introduced in theorem 1 is uniquely
specified. In particular, H i ( y i ) is a power function. The constant r is the same for all contestants.
It captures the marginal increase of the possibility to win if effort is increased. Contests with low
7
r can be regarded as poorly discriminating or “noisy” contests (e.g. Blavatskyy 2004). When r
converges to zero the contest outcome converges to a random lottery with no dependence on the
contributed efforts of the contestants.
Contests with high r can be regarded as highly
discriminating. When r converges to infinity, the contest becomes an all-pay auction (e.g.
Moldovanu and Sela 2001).
Clark and Riis (1998) interpret an individual specific constant α i as a degree of
unfairness inside the contest. The higher the constant α i the more probable is the i-th contestant
to win given that he or she submits the same effort as the opponents. According to the CSF
described in theorem 2, the probability of a draw converges to zero when y1 , y 2 → +∞ . Thus, for
high effort levels the CSF described in theorem 2 converges to the logit CSF without the
possibility of a draw (e.g. Clark and Riis 1998).
Skaperdas (1996) proposed an anonymity axiom that individual identities are irrelevant
for the CSF (only individual efforts matter), which could be formally expressed as
p 1 ( y1 , y 2 ) = p 2 ( y 2 , y1 ) . The anonymity axiom in conjunction with axiom 1 immediately implies
that the probability of a draw is symmetric i.e. p 12 ( y1 , y 2 ) = p 12 ( y 2 , y1 ) . When the anonymity
axiom is added to axioms 1-3 (axioms 1-4) we obtain theorem 1 (theorem 2) with H 1 (.) = H 2 (.)
( α 1 = α 2 = α ). Notice that when α converges to infinity the probability of a draw converges to
zero and we obtain the logit CSF without the possibility of a draw p i ( y1 , y 2 ) = y ir
(y
r
1
)
+ y 2r ,
i ∈ {1,2}. When α converges to zero the contest ends up in a draw with certainty. Thus, we can
interpret parameter α as a relative likelihood of a draw in the contest.
8
3. Contest with n>2 contestants
Let N = {1,..., n} denote the set of all contestants and Ω be the set of all subsets in N. For
any group of contestants S ∈ Ω let S denote the number of contestants in this group. The i-th
{
}
contestant contributes effort y i ≥ 0 , i ∈ N . For any group of contestants S = i1 ,..., i S ∈ Ω ,
(
y S = yi1 ,..., yi S
) denotes the vector of efforts contributed by the contestants in S. Note that
y{i} = y i , ∀i ∈ N and y N = ( y1 ,..., y n ) . Finally, let p S ( y N ) ∈ [0,1] denote the probability that a
group S ∈ Ω ends up in a draw and all remaining contestants (i.e. N \ S ) lose the contest. The
probability that the i-th contestant wins the contest unilaterally is denoted by p {i} ( y N ) , i ∈ N .
Axioms 1a-3a are the generalizations of axioms 1-3 for the contest with n > 2 contestants.
Axiom 1a
∑ p ( y ) = 1 , ∀y
S
S∈Ω
N
N
∈ R n+
When there are more than two contestants many outcomes are possible. All n contestants
can end up in a draw. Additionally any group of n − 1 , n − 2 etc contestants can end up in a
draw. Finally, any one of n contestants can win the contest unilaterally. These outcomes are
mutually exclusive: only one of them is true ex post. Therefore, the probabilities of these
outcomes add up to one.
Axiom 2a Probability p {i} ( y N ) is strictly increasing in y i , i ∈ N
Axiom 3a For any S ∈ Ω there exists a function
p S ( yN )
= f S ( yS )
1 − ∑ pT ( y N )
T ∈Ω
S ⊆/ T
9
f S : R + → [0,1] such that
S
The members of a group S are successful if all of them end up in a draw either with each
other or together with contestants that are not in S. According to axiom 3a the probability that a
group S ends up in a draw, conditional on the success of all its members, depends only on the
efforts of its members and it does not depend on the efforts of other contestants that are not in S.
Theorem 3 Axioms 1a-3a hold if and only if for any group S ∈ Ω , S ≠ N , there exist a
function H S ( y S ) , which is strictly increasing in all its arguments, such that
⎡
⎡
⎤⎤
⎢
A
B
N
⎢
H ( y S ) ⋅ ⎢ ∑ H ( y A ) ∑ H ( y B )...H ( y N )⎥⎥⎥
⎢B∈Ω
⎥
A∈Ω
⎢
⎢
⎥⎦⎥⎦
S
⊂
A
A
⊂
B
⎣
⎣
p S (yN ) =
, where H N ( y N ) = 1 .
⎡
⎡
⎤⎤
⎢
A
B
C
N
⎢
H ( y A )⎢ ∑ H ( y B ) ∑ H ( yC )...H ( y N )⎥⎥⎥
∑
⎢C∈Ω
⎥
A∈Ω
⎢⎣BA∈⊂ΩB
⎢⎣B⊂C
⎥⎦⎥⎦
S
According to theorem 3 axioms 1a-3a hold if and only if the CSF has the following
structure. Consider the situation when all n contestants end up in a draw. The probability of this
outcome is p N ( y N ) =
1
⎡
⎤⎤
⎡
⎢
N
B
C
⎢
H ( y A )⎢ ∑ H ( y B ) ∑ H ( yC )...H ( y N )⎥⎥⎥
∑
⎥
⎢C∈Ω
A∈Ω
B∈Ω
⎢⎣ A⊂ B
⎥⎦⎥⎦
⎢⎣B⊂C
. Probability p N ( y N ) strictly
A
decreases in every contestant’s effort. Now consider the situation when a group N \ {}
i ends up in
a draw, i ∈ N . The probability of this outcome p N \{i} ( y N ) , when compared with the probability
p N ( y N ) , is a strictly increasing function of the efforts of every member of the group N \ {}
i i.e.
p N \{i} ( y N ) = H N \{i} ( y N \{i} ) ⋅ p N ( y N ) . Thus, when all members of group N \ {}
i contribute higher
effort the outcome when the group N \ {}
i shares the prize becomes relatively more probable than
the outcome when all contestants share the prize.
10
Now consider a group N \ {i,j} of all contestants except for the i-th and the j-th contestant,
i, j ∈ N , i ≠ j . When compared with the probability p N ( y N ) + p N \{i} ( y N ) + p N \{ j} ( y N )
the
probability that the members of the group N \ {i,j} end up in a draw is a strictly increasing
function of the efforts of every member of this group. Intuitively, every member of the group
N \ {i,j} is also a member of the larger groups N \ {}
i and N \ { j} . Thus, higher efforts of the
members of the group N \ {i,j} automatically increase the relative likelihood that the groups
N \ {}
i and N \ { j} share the prize. According to theorem 3 when the members of the group
N \ {i,j} contribute higher effort the outcome when the group N \ {i,j} shares the prize becomes
relatively more probable than the (also more probable) outcome when the groups N \ {}
i , N \ { j}
and
N
share
the
prize.
Formally,
(
this
corresponds
to
)
p N \{i , j} ( y N ) = H N \{i , j} ( y N \{i , j} ) ⋅ 1 + H N \{i} ( y N \{i} ) + H N \{ j} ( y N \{ j} ) ⋅ p N ( y N ) . The same logical structure
of the CSF is preserved for all other groups of contestants.
When there are more than two contestants the homogeneity axiom of Skaperdas (1996)
and Clark and Riis (1998) cannot be generalized without imposing a restriction on the structure
of contest. Specifically, the homogeneity of relative winning probabilities (axiom 4a below) is
compatible with theorem 3 only when two outcomes are possible. Either one contestant wins or a
group of m contestants draws, m ∈ [2, n ] .4 Thus, n = 2 is the only case where the homogeneity
axiom does not restrict the structure of the contest. Intuitively, when axioms 1a-3a hold and all
contestants multiply their efforts by the same amount, the probability of bilateral draws
decreases. At the same time, the probability of trilateral draws decreases even to a greater extent
and so forth. Thus, the relative probabilities of winning do not remain constant.
4
An example when only all contestants together can end up in a draw ( m = n ) is analyzed in section 3.2 below.
11
3.1.
Example of a CSF when the contest outcome is unrestricted
Keeping the assumption that any group S ∈ Ω of contestants can draw, the present
axiomatization allows to characterize a CSF only in general monotone functions H S ( y S ) as in
theorem 3. Figures 1 and 2 show a CSF that satisfies axioms 1a-3a when there are three
2
⎛
⎞
contestants and H ( y S ) = ⎜ ∑ y i ⎟ . Figure 1 (2) demonstrates the probability of a win and a
⎝ i∈S ⎠
S
draw for the first contestant when his or her opponents contribute effort y 2 = y 3 = 0
( y 2 = 0.5, y 3 = 0.4 ). When the first contestant contributes low effort ( y1 → 0 ) the most likely
outcome for the first contestant (given that he or she is successful) is a draw together with two
opponents. If the first contestant does not end up in a draw with both opponents, he or she is
more likely to end up in a draw with one opponent who contributed positive effort rather than to
win the contest unilaterally (e.g. figure 2).
When the first contestant contributes high effort ( y1 → +∞ ) the most likely outcome is
that the first contestant wins unilaterally. If the first contestant does not win, he or she is more
likely to draw with one opponent rather than with both opponents. A draw with an opponent who
contributed higher effort is more probable that a draw with an opponent who contributed lower
effort. When one of the opponents contributes high effort, a draw with this opponent can be the
most likely successful outcome for the contestant who contributes low or moderate effort (e.g.
figure 2 when y1 = 0.6 ). Figures 1 and 2 also demonstrate that:
•
the probability that one contestant wins the contest is strictly increasing in his or her effort,
•
the probability that one contestant ends up in a draw with one of the opponents is first
increasing and then decreasing in his or her effort, and
12
•
the probability that one contestant ends up in a draw with all opponents is strictly decreasing
in his or her effort.
Probability
1
0.9
0.8
0.7
0.6
P{1}
P{1,2}
P{1,3}
P{1,2,3}
0.5
0.4
0.3
0.2
0.1
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Effort y1
Figure 1 Probability p {1} ( p {1, 2} , p {1,3} , p {1, 2,3} ) that the first contestant wins in a contest with
three contestants (ends up in a draw with the second, the third, the second and the third
contestant) while contributing effort y1 ( y 2 = y 3 = 0 )
13
Probability
1
0.9
0.8
0.7
0.6
P{1}
P{1,2}
P{1,3}
P{1,2,3}
0.5
0.4
0.3
0.2
0.1
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Effort y1
Figure 2 Probability p {1} ( p {1, 2} , p {1,3} , p {1, 2,3} ) that the first contestant wins in a contest with
three contestants (ends up in a draw with the second, the third, the second and the third
contestant) while contributing effort y1 ( y 2 = 0.5, y 3 = 0.4 )
14
3.2.
Example of a CSF when the contest outcome is restricted
Consider a special case when a contest outcome is restricted so that either one of n
contestants wins the contest unilaterally or all contestants end up in a draw. This kind of a draw
can be interpreted as a stalemate when competition ends up in a deadlock with none of the
contestants being able to force a unilateral win. Formally, a restriction p S ( y N ) = 0 is imposed
for every S ∈ Ω , S ≠ {}
i , i ∈ N , and S ≠ N . This restriction allows to simplify theorem 3 into
corollary 1 below and to impose homogeneity on the relative winning probabilities as done in
axiom 4a below. Proof of corollary 1 follows immediately from theorem 3
Corollary 1 If p S ( y N ) = 0 for any S ∈ Ω , S ≠ {}
i and S ≠ N , then axioms 1a-3a hold if
and only if there exist strictly increasing functions H i ( y i ) such that
p {i} ( y N ) =
H i ( yi )
1 + ∑ H (y j )
n
j
j =1
Axiom 4a
, i ∈ N , and p N ( y N ) =
1
1 + ∑ H (y j )
n
.
j
j =1
p i ( y N ) p i (λ ⋅ y N )
, ∀λ > 0 , ∀i, j ∈ N , y i , y j ≠ 0
=
p j ( y N ) p j (λ ⋅ y N )
Axiom 4a is sufficiently restrictive to specify uniquely the monotone functions H i ( y i ) from
corollary 1. Specifically, the only function H i ( y i ) compatible with axiom 4a is a power function
(corollary 2)
15
Corollary 2 If p S ( y N ) = 0 for any S ∈ Ω , S ≠ {}
i and S ≠ N , then axioms 1a-4a hold if
and only if there exist constant r , α i > 0 such that p {i} ( y N ) =
α i y ir
1 + ∑α j y
j =1
p N (yN ) =
, i ∈ N , and
n
r
j
1
n
1 + ∑ α j y rj
j =1
Proof of corollary 2 is analogous to the proof of theorem 2 with axioms 1a-4a and
corollary 1 being used instead of axioms 1-4 and theorem 1 correspondingly.
The CSF from corollary 2 generically applies to “unfair” contests (e.g. Clark and Riis,
1998). In “unfair” contests, the contestants are heterogeneous: for different contestants the same
effort yields a different probability of winning. The anonymity axiom as in Skaperdas (1996)
imposes a restriction that a contest is fair, i.e. two different contestants have the same probability
of winning when they contribute the same effort. Formally, this corresponds to
p {i} ( y N ) = p { j } ( y N ) when y i = y j , ∀i, j ∈ N . The anonymity axiom implies that the coefficients
α i > 0 from corollary 2 are identical for every contestant.
4. Conclusion
This paper presents an axiomatic characterization of the CSF while allowing for many
outcomes in the contest. Specifically, the possibility of a draw is allowed so that any group of the
contestants can end up in a “shared victory” of the contest. In a special case of a simple pairwise
contest it is possible to generalize the logit CSF axiomatized by Skaperdas (1996) and Clark and
Riis (1998) to allow for the possibility of a draw. The obtained logit CSF with the possibility of a
draw is characterized by the “independence of irrelevant alternatives” axiom and the “relative
homogeneity of winning probabilities” axiom. However, the property of homogeneity (either in
16
relative or absolute winning probabilities) cannot be sustained in a general case when there are
more than two contestants unless a restriction on the contest outcomes is imposed. An example of
such a restriction is when either only two or only three or only all contestants together can end up
in a draw.
The logit CSF with the possibility of a draw is shown to possess the following properties.
The probability that all contestants end up in a draw is the highest when all contestants contribute
zero effort and it strictly decreases in the effort of every contestant. The probability that any
group of two or more, but not all, contestants shares the prize increases in low effort of its
members and decreases in high effort of its members. The probability that any contestant wins
the contest unilaterally strictly increases in his or her effort.
The CSF with the possibility of a draw is a useful model in many economic situations
where the ending of competition without either side winning is a plausible outcome. The CSF
axiomatized in this paper can be employed instead the conventional logit CSF to analyze the
paradoxical results in the literature on the imperfectly discriminating contests such as Tullock
(1980) “overdissipation of rents” or the non-existence of pure strategy Nash equilibrium in highly
discriminating contests (e.g. Blavatskyy 2004). The CSF with the possibility of a draw is
continuous at zero and therefore it is more convenient as an analytical tool than the conventional
logit CSF, which is discontinuous at zero. However, for the contest with more than two
contestants the CSF with a possibility of a draw is characterized only in general monotone
functions (if the structure of contest is unrestricted). A useful extension of this work is to find an
intuitively appealing property that allows to specify these monotone functions while preserving
the rich structure of the contest outcome.
17
References
Baik, K. (2004) “Two-player asymmetric contests with ratio-form contest success functions”
Economic Inquiry, 42, 679-689
Baye, M. and Shin, O. (1999) “Strategic behaviour in contests: comment” American Economic
Review, 89, 691-694
Blavatskyy, P. (2004) “Why the Olympics have three prizes and not just one” IEW Working
Paper #200 http://www.iew.unizh.ch/wp/iewwp200.pdf
Clark, D. and Riss, C. (1996) “On the win probability in rent-seeking games” Discussion Paper in
Econoimcs E4/96, University of Tromso
Clark, D. and Riss, C. (1998) “Contest success functions: an extension” Economic Theory, 11,
201-204
Dixit, A. (1987) “Strategic behavior in contests” American Economic Review 77, 891-898
Gradstein, M. and Konrad, K. (1999) “Orchestrating rent seeking contests“ The Economic
Journal, 109, 536-545
Hillman, A. and Riley, J. (1989) “Politically contestable rents and transfers” Economics and
Politics, 1, 17-39
Hirshleifer, J. (1989) “Conflict and rent-seeking success functions: ratio vs. difference models of
relative success” Public Choice, 63, 101-112
Lazear, E. and Rosen, S. (1981) “Rank Order Tournaments as Optimal Labor Contracts” Journal
of Political Economy, 89, 841-864
Moldovanu B. and Sela A. (2001) "The optimal Allocation of Prizes in Contests" American
Economic Review, 91, 3, 542-558
Morgan, J. (2003) “Sequential contests” Public Choice, 116, 1-18
Nalebuff, B. and Stiglitz, J. (1983) “Prizes and incentives: towards a general theory of
compensation and competition” The Bell Journal of Economics, 14, 21-43
Nitzan, Sh. (1991) “Collective rent dissipation” The Economic Journal, 101, 1522-1534
Nitzan, Sh. (1994) "Modelling Rent-Seeking Contests," European Journal of Political Economy,
10, 41-60
Nti, K. (1997) “Comparative statics of contests and rent-seeking games” International Economic
Review, 38, 43-59
Rosen, Sh. (1986) “Prizes and incentives in the elimination tournaments” American Economic
Review, 76, 701-715
Skaperdas S. (1996) “Contest Success Functions”, Economic Theory, 7, 283-90
Szymanski S. (2003) "The Economic Design of Sporting Contests" Journal of Economic
Literature, 41, 1137-1187
18
5. Appendix
Counterexample to Skaperdas (1996) and Clark and Riis (1998) when n = 2
Consider a contest with two contestants contributing efforts y1 ≥ 0 and y 2 ≥ 0 . There is
no possibility of a draw. The first contestant wins with probability p1 ( y1 , y 2 ) =
e
e
the second contestant wins with probability p 2 ( y1 , y 2 ) =
e
e
y1
y 2 +1
y1
y2 +1
y1
y2 +1
+e
y2
y1 +1
and
y2
y1 +1
+e
y2
y1 +1
. This CSF satisfies axioms
(A1)-(A5) in Skaperdas (1996):
•
Axiom (A1): p1 ( y1 , y 2 ) + p 2 ( y1 , y 2 ) = 1
•
Axiom (A2): p1 ( y1 , y2 ) is strictly increasing in y1 and strictly decreasing in y 2 ,
p 2 ( y1 , y 2 ) is strictly increasing in y 2 and strictly decreasing in y1
•
Axiom (A3): p 1 ( y1 , y 2 ) = p 2 ( y 2 , y1 )
•
Axiom (A4) holds trivially for any CSF satisfying axiom (A1) when there are only
two contestants
•
Axiom (A5): p i ( y1 , y 2 ) is only a function of y i , i ∈ {1,2}.
However, the above CSF is not of the form p i ( y1 , y 2 ) =
f ( yi )
, i ∈ {1,2}, given
f ( y1 ) + f ( y 2 )
by Theorem 1 in Skaperdas (1996). Therefore, this CSF is a counterexample that demonstrates
that axioms (A1)-(A5) are not sufficient for Theorem 1 in Skaperdas (1996) when there are only
two contestants.
Similar counterexamples can be constructed for the axiomatization of Clark and Riis
(1998). However, axiom (A4’) in Clark and Riis (1998) actually is not compatible with their
19
axiom (A2) (see footnote 3). For contests with two contestants, axiom (A4’) in Clark and Riis
(1998) implies that p 1 ( y1 ,0 ) = 1 , y1 ≠ 0 , and p 2 (0, y 2 ) = 1 , y 2 ≠ 0 . However, axiom (A2) in
Clark and Riis (1998) requires that p i ( y1 , y 2 ) is strictly increasing in y i , i ∈ {1,2}. If we relax
axiom (A2) so that p i ( y1 , y 2 ) is nondecreasing in y i , i ∈ {1,2}, the following CSF satisfies all
axioms in Clark and Riis (1998):
⎧ y1
⎪e y2
⎪
⎪
⎪
1
p ( y1 , y 2 ) = ⎨
⎪
⎪
⎪
⎪
⎩
y2
⎛ yy1
⎜ e 2 + e y1
⎜
⎝
⎞
⎟,
⎟
⎠
⎧ y2
⎪e y1
⎪
⎪
⎪
2
p ( y1 , y 2 ) = ⎨
⎪
⎪
⎪
⎪
⎩
y1 ≠ 0, y 2 ≠ 0
1,
y1 ≠ 0, y 2 = 0
0,
y1 = 0, y 2 ≠ 0
y2
⎛ yy1
⎜ e 2 + e y1
⎜
⎝
⎞
⎟,
⎟
⎠
y1 ≠ 0, y 2 ≠ 0
1,
y1 = 0, y 2 ≠ 0
0,
y1 ≠ 0, y 2 = 0
(Clark and Riis (1998) rule out the possibility y1 = 0, y 2 = 0 )
•
Axiom (A1): p1 ( y1 , y 2 ) + p 2 ( y1 , y 2 ) = 1, if y i > 0 then p i ( y1 , y 2 ) > 0 , i ∈ {1,2}
•
Axiom (A2): p1 ( y1 , y2 ) is nondecreasing in y1 and nonincreasing in y 2 ,
p 2 ( y1 , y 2 ) is nondecreasing in y 2 and nonincreasing in y1
•
Axiom (A4’): p 1 ( y1 ,0 ) = 1 , y1 ≠ 0 , and p 2 (0, y 2 ) = 1 , y 2 ≠ 0
•
Axiom (A6): p i ( y1 , y 2 ) = p i (λ ⋅ y1 , λ ⋅ y 2 ) , ∀λ > 0 , ∀y i ≥ 0 , i ∈ {1,2}
However, the above CSF is not of the form p i ( y1 , y 2 ) =
α i y ir
, i ∈ {1,2}, given by
α 1 y1r + α 2 y 2r
Theorem in Clark and Riis (1998). Therefore, this CSF is a counterexample demonstrating that
axioms (A1)-(A6) are not sufficient for Theorem in Clark and Riis (1998) when there are only
two contestants.
20
Proof of theorem 1
Axiom 3 implies that the ratio
p 1 ( y1 , y 2 )
does not depend on y 2 and it is only a
1 − p 2 ( y1 , y 2 )
function of y1 , which for convenience can be written as
p 1 ( y1 , y 2 )
H 1 ( y1 )
, where
=
1 − p 2 ( y1 , y 2 ) 1 + H 1 ( y1 )
H1:R+→ R+ is an arbitrary function5. Writing a similar equation for the second contestant, we
obtain a system of equations (1).
⎧ p1 ( y1 , y2 )
H 1 ( y1 )
=
⎪
⎪1 − p 2 ( y1 , y2 ) 1 + H 1 ( y1 )
⎨ 2
2
⎪ p ( y1 , y2 ) = H ( y2 )
⎪⎩1 − p1 ( y1 , y2 ) 1 + H 2 ( y2 )
The solution of system (1) is p1 ( y1 , y 2 ) =
(1)
H 2 ( y2 )
H 1 ( y1 )
2
,
(
)
.
,
p
y
y
=
1
2
1 + H 1 ( y1 ) + H 2 ( y 2 )
1 + H 1 ( y1 ) + H 2 ( y2 )
From axiom 1 it follows that p 12 ( y1 , y 2 ) = 1 − p 1 ( y1 , y 2 ) − p 2 ( y1 , y 2 ) =
1
1 + H ( y1 ) + H 2 ( y 2 )
1
.
Axiom 2 implies that p1 ( y1 , y2 ) is strictly increasing in y1 , which can be true only if H 1 ( y1 ) is
strictly increasing in y1 . Similarly, axiom 2 also implies that p 2 ( y1 , y 2 ) is strictly increasing in
y 2 , which can be true only if H 2 ( y 2 ) is strictly increasing in y 2 . It is straightforward to verify
that functions p1 ( y1 , y 2 ) , p 2 ( y1 , y 2 ) and p12 ( y1 , y 2 ) specified above satisfy axioms 1-3. Thus,
axioms 1-3 are both necessary and sufficient. Q.E.D.
5
Function H (.) can take only nonnegative values because probability p ( y1 , y2 ) is nonnegative.
1
1
21
Proof of theorem 2
It follows from theorem 1 that
assumption that
p 1 ( y1 , y 2 ) H 1 ( y1 )
. Axiom 4 then is equivalent to the
=
p 2 ( y1 , y 2 ) H 2 ( y 2 )
H 1 ( y1 ) H 1 (λ ⋅ y1 )
, ∀λ > 0 , ∀y1 , y 2 > 0 . Let λ = 1 y 2 . We obtain then a
=
H 2 ( y 2 ) H 2 (λ ⋅ y 2 )
functional equation (2):
H 1 ( y1 y 2 ) = H 2 (1) H 1 ( y1 ) H 2 ( y 2 )
(2)
Since equation (2) holds for any positive y 2 by letting y 2 = y1 we establish that
H 2 ( y1 ) = H 2 (1)H 1 ( y1 ) / H 1 (1) , ∀y1 > 0 . Thus, we can rewrite equation (2) as (3):
H 1 ( y1 y 2 ) = H 1 (1) H 1 ( y1 ) H 1 ( y 2 )
(3)
( )
Let y1 = e x + z , y 2 = e x and f ( x ) = ln H 1 e x − ln H 1 (1) . Notice that function f (.) is
strictly increasing because function H 1 (.) is strictly increasing. Equation (3) then becomes
Cauchy’s functional equation f ( x + z ) = f ( x ) + f ( z ) , which has the unique solution f (x ) = rx ,
where r is constant. Using the definition of function f (.) we then obtain H 1 ( y1 ) = H 1 (1) y1r and
H 2 ( y1 ) = H 2 (1) y1r , ∀y1 > 0 . According to theorem 1 function H i ( y i ) is strictly increasing,
which implies that r and H i (1) are non-zero and of the same sign, i ∈ {1,2}. Additionally,
theorem 1 implies that function H i (.) is strictly positive on R + + . Therefore, r , H i (1) > 0 ,
i ∈ {1,2}. Finally, let α i = H i (1) , i ∈ {1,2}.
Substituting for the newly obtained function H i (.) in theorem 1 we obtain
p i ( y1 , y 2 ) =
α i y ir
1+ α y + α2 y
r
1 1
r
2
and p 12 ( y1 , y 2 ) =
1
1 + α y + α 2 y 2r
r
1 1
to verify that this CSF satisfies all axioms 1-4. Q.E.D.
22
, i ∈ {1,2}. It is straightforward
Proof of theorem 3
For convenient notation let us introduce the following family of functions. H N ( y N ) = 1 ,
H S (yS ) = f
S
( yS )
(1 − f ( y )) where
f S : R + → [0,1] is an arbitrary function used in axiom 3a.
S
S
S
It is possible then to rewrite axiom 3a as equation (4)
⎡
⎤
p S ( y N ) = H S ( y S ) ⋅ ⎢ ∑ p T ( y N )⎥ , ∀S ∈ Ω
⎢T ∈Ω
⎥
⎣⎢ S ⊆T
⎦⎥
because
∑ p (y ) = 1 − ∑ p (y )
T
T ∈Ω
S ⊆T
T
N
T ∈Ω
S ⊆/ T
N
(4)
due to axiom 1a. Equation (4) defines the probability
distribution p S ( y N ), S ∈ Ω recursively. Iterating every probability p T ( y N ) on the right hand side
of equation (4) we obtain equation (5).
⎡
⎤⎤
⎡
⎢
A
B
N
⎢
p ( y N ) = H ( y S ) ⋅ ∑ H ( y A ) ∑ H ( y B )...H ( y N )⎥⎥ ⋅ p N ( y N ) , ∀S ∈ Ω (5)
⎢ A∈Ω
⎥⎥
⎢B∈Ω
⎢⎣S ⊂ A
⎥⎦⎥⎦
⎢⎣ A⊂ B
S
S
Since equation (5) holds for any element of Ω it is possible to sum (5) over all elements
of Ω . On the left hand side of equation (5) we then obtain
∑ p ( y ) = 1 due to axiom 1a. It
S
S ∈Ω
follows then that p N ( y N ) =
N
1
⎡
⎡
⎤⎤
⎢
A
B
C
N
⎢
H ( y A )⎢ ∑ H ( y B ) ∑ H ( yC )...H ( y N )⎥⎥⎥
∑
⎢C∈Ω
⎥
A∈Ω
⎢⎣BA∈⊂ΩB
⎣⎢B⊂C
⎦⎥⎥⎦
. Plugging this result
⎡
⎡
⎤⎤
⎢
A
B
N
H ( y S ) ⋅ ∑ H ( y A )⎢ ∑ H ( y B )...H ( y N )⎥⎥
⎢ A∈Ω
⎢B∈Ω
⎥⎥
⎢
⎢
⎥⎦⎥⎦
⊂
⊂
S
A
A
B
⎣
⎣
in (5) we finally obtain p S ( y N ) =
, ∀S ∈ Ω . It is
⎡
⎡
⎤⎤
H A ( y A )⎢⎢ ∑ H B ( y B )⎢ ∑ H C ( yC )...H N ( y N )⎥⎥⎥
∑
⎢C∈Ω
⎥
A∈Ω
B∈Ω
⎢⎣ A⊂ B
⎢⎣B⊂C
⎥⎦⎥⎦
S
straightforward to verify that this CSF satisfies all axioms 1a-3a and therefore these axioms are
both necessary and sufficient. Q.E.D.
23