Economics Letters 111 (2011) 185–187
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Economics Letters
j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / e c o l e t
The first-order approach in rank-order tournaments☆
Oliver Gürtler ⁎
Department of Economics, University of Cologne, Innere Kanalstraße 15 (3.25), D-50823 Cologne, Germany
a r t i c l e
i n f o
Article history:
Received 15 February 2010
Received in revised form 15 January 2011
Accepted 11 February 2011
Available online 18 February 2011
a b s t r a c t
The first-order approach in rank-order tournaments is addressed. It is demonstrated that the conditions given
in the literature are not sufficient to guarantee concavity of agents' objective functions. Additional conditions
are provided that ensure validity of the first-order approach.
© 2011 Elsevier B.V. All rights reserved.
JEL classification:
C72
D86
M52
Keywords:
Rank-order tournaments
First-order approach
Concave objective function
1. Introduction
In practice, many firms use incentive systems to reward relative
performance such as promotion tournaments and sales contests.
It is therefore not surprising that, following the seminal paper by Lazear
and Rosen (1981), a vast body of literature has emerged that analyzes
the properties and optimal design of so-called rank-order tournaments.1
Theoretical studies of rank-order tournaments typically use the firstorder approach. This means that the incentive constraints for contestants are replaced by first-order conditions for the contestants'
maximization problems. This procedure is justified by giving sufficient
conditions under which the first-order approach is said to be valid.
The aim of the current note is twofold. In a first step, it shows that
the conditions given in the literature are not sufficient to ensure
validity of the first-order approach. This is because each contestant's
objective function depends on prizes that are determined endogenously by the tournament designer in an earlier stage of the game.
Depending on how these prizes are chosen, it is simply not always
possible to ensure concavity of the contestants' objective functions.2
In a second step, the note provides alternative conditions under which
☆ I would like to thank Matthias Kräkel, Patrick Schmitz and Dirk Sliwka for their
helpful discussions.
⁎ Corresponding author. Tel.: + 49 221 4701450.
E-mail address: [email protected].
1
See Konrad (2009) for a survey of the literature on tournaments and contests.
2
It sometimes suffices that the objective functions are quasi-concave. With the
conditions imposed in the literature, however, quasi-concavity cannot be ensured
either. More generally, although the focus is on concavity in this paper, the qualitative
results can be extended if attention is restricted to the concept of quasi-concavity.
0165-1765/$ – see front matter © 2011 Elsevier B.V. All rights reserved.
doi:10.1016/j.econlet.2011.02.013
the first-order approach becomes valid. The conditions ensure that
the tournament organizer chooses prizes from a restricted set of
values. This helps to rule out the situations for which it is not possible
to guarantee that the objective functions are concave.
The note is related to papers by Rogerson (1985) and Jewitt
(1988). These papers consider the standard single-agent hiddenaction model and provide sufficient conditions under which the firstorder approach is valid. The current note adds to this literature by
performing a similar task in a setting in which attention is focused on
tournament contracts.
2. The problem
We consider a situation in which a female principal organizes a
tournament between two male agents, all of whom are risk-neutral
and have zero reservation value. In the first stage, the principal
chooses tournament prizes w1 and w2 (with w2 ≤ w1) for the winner
and the loser of the tournament, respectively. In the second stage,
agent i(i = 1, 2) observes these prizes and chooses his effort ei ∈Rþ .
By choosing effort, the agent produces output yi = ei + εi, which
accrues to the principal. εi is an idiosyncratic error term that is i.i.d.
according to the pdf f(⋅) and has a zero mean and variance σ2. Let G(⋅)
denote the cdf of the composite random variable ε2 − ε1 and let g(⋅)
be the corresponding pdf. Effort is costly for an agent and costs are given
by c(ei), where c(⋅) as an increasing and strictly convex function
satisfying the Inada conditions c′(0) = 0 and c′(ei) = ∞ for ei → ∞. The
agent with the higher output is declared the winner of the tournament.
When maximizing her profit, the principal has to regard the
incentive constraint for agent i. Since agent i wins the tournament
186
O. Gürtler / Economics Letters 111 (2011) 185–187
with probability G(ei − ej) (with j = 1, 2, i ≠ j), 3 this constraint is given
by
ei ∈ argmax w2 + ΔwG ê i −ej −cðê i Þ;∀ej ∈ Rþ ;
êi ∈Rþ
ðICÞ
where Δw : = w1 − w2. Following the work by Lazear and Rosen
(1981), this constraint is typically replaced by the corresponding
first-order condition. The main problem with this approach is that
the cdf G(⋅) may not be concave everywhere. In turn, it is not
guaranteed that the agents' objective functions are strictly concave.
The literature on rank-order tournaments tries to sidestep the
problem by assuming that the pdf g(⋅) is sufficiently flat and the cost
function c(⋅) is sufficiently convex. Lazear and Rosen (1981), for
example, state in footnote 2: “It is possible to show that a pure
strategy solution exists provided that σ2 is sufficiently large:
contests are feasible only when chance is a significant factor.”4
In this section, we show that the conditions given in the literature
are not sufficient to ensure validity of the first-order approach. The
second derivative of the objective function for agent i with respect to
his effort is given by
Δwg ′ ei −ej −c″ ðei Þ:
ð1Þ
If g(⋅) is rather flat and c(⋅) is very convex, g′(ei − ej) is rather
low, whereas c″(ei) is high. Therefore, it seems, as stated in the
literature, that the expression in Eq. (1) is always negative. Note,
however, that w1 and w2 are endogenous variables chosen by the
principal. In principle, she may choose these variables such that Δw
becomes very high and in the limit we may have Δw → ∞. If in this
case G(⋅) is not concave everywhere, i.e., if g′(ei − ej) is strictly positive
for some values of ei − ej, the objective function cannot be concave
everywhere. In other words, it is simply not possible that g(⋅) is
sufficiently flat or c(⋅) is sufficiently convex, because the (endogenously determined) prize spread could be so high that these two
conditions are overturned.
We conclude this section with a simple example illustrating the
k
main argument. Let εi ∼ U[− a, a] with a ≥ 0 and cðei Þ = ðei Þ2 with k N 0.
2
In this case, ε2 − ε1 follows a triangular distribution and we have
8
1
1 >
>
+
ei −ej ; for ei −ej ∈½−2a; 0
>
2
>
>
4a
< 2a
1
1 g ei −ej =
>
− 2 ei −ej ; for ei −ej ∈½0; 2a
>
>
2a
4a
>
>
:
0; otherwise:
ð2Þ
g′(ei − ej) is positive only if (ei − ej) ∈ [− 2a, 0]. Here, the expression in
Eq. (1) becomes
Δw
−k:
4a2
ð3Þ
As long as a and k are finite (which is very reasonable to assume in
practice), this expression becomes positive for large values of Δw. As
argued before, it is thus impossible to ensure that the agents' objective
functions are always concave.
3. The solution
As explained in the previous section, it is not feasible to ensure
concavity of each agent's objective function since this function
depends on variables determined endogenously by the principal. In
particular, the problem arises since we cannot exclude the possibility
3
4
Notice that g(⋅) is symmetric around zero and hence G(ei − ej) = 1 − G(ej − ei).
Note that a high σ2 corresponds to a flat pdf g(⋅).
that the principal chooses prizes such that the prize spread is very
large. In this section, we introduce two additional assumptions. These
assumptions allow us to place an upper bound on the size of the prize
spread. In turn, we can give a sufficient condition for the first-order
approach to be valid.
Assumption 1. The agents have finite wealth w. Thus, the loser prize
must satisfy w2 ≥−w.
Assumption 2. There exists a finite effort level ẽ such that sup ðê −cðê ÞÞb
ê∈Rþ
cðeÞ−e; ∀e N ẽ .
Assumptions 1 and 2 are not very strong and should be fulfilled in
many real-world settings. Assumption 1 states that agents do not have
infinite wealth so that the losing prize cannot become infinitely low.
This is obviously the case in practice. Assumption 2 states that for
excessive effort activity becomes very unproductive. Since agents
would become tired when exerting a very high degree of effort, this
again seems reasonable. Note that Assumption 2 holds for many
specifications of the effort cost function, for example, for the quadratic
k
function cðei Þ = ðei Þ2 used before.5
2
Lemma 1. The principal never implements ei such that ei N ẽ .
Proof. We denote the expected payoff for the principal by π and
for agent i by ui. To ensure participation of all players, the conditions
π ≥ 0 and ui ≥ 0 have to be met. Combining the conditions for all three
players yields
π + u1 + u2 ≥ 0:
Since the tournament prizes are pure transfers from the principal
to the agents, this condition can be rewritten as
e1 + e2 −cðe1 Þ−cðe2 Þ ≥ 0:
From Assumption 2, it then follows that e1 ;e2 ≤ ẽ .
□
Lemma 1 states that the implemented effort is never greater than
ẽ . This is intuitive. If an effort level greater than ẽ was implemented,
the principal would destroy too much value, which is not in her
interest.
Lemma 2. There exists a finite value Δw̃ = 2ðẽ + wÞ such that the
principal never chooses Δw N Δw̃ .
Proof. The principal's expected profit is given by
π = e1 + e2 −w1 −w2 = e1 + e2 −Δw−2w2 :
Using Assumption 1 and Lemma 1 we can calculate an upper
bound for this profit. This bound is
π̃ = 2ẽ −Δw + 2w:
π̃ is positive in the case
2ẽ −Δw + 2w ≥ 0⇔Δw ≤ 2ðẽ + wÞ:
Lemma 2 follows directly from this condition and the principal's
participation constraint.
□
To formulate our main result, let g̃ ′ N 0 denote the upper bound on
g′(ei − ej) and c̃ ″ N 0 the lower bound on c″(ei).
5
See the example at the end of this section.
O. Gürtler / Economics Letters 111 (2011) 185–187
Proposition 1. The first-order approach in the rank-order tournament model is valid if ẽ ;w and g̃ ′ are sufficiently low and c̃ ′′ is
sufficiently high.
Proof. The proof directly follows from Eq. (1) and Lemma 2.
□
Proposition 1 offers conditions under which the first-order
approach in rank-order tournaments is valid. Note that these
conditions are stricter than those referred to in the literature. This is
because restrictions are imposed not only on g̃ ′ and c̃ ″ , but also on ẽ
and w. The additional restrictions place an upper bound on the spread
of the prizes that the principal chooses. This helps us to eliminate the
problems identified in Section 2.
We conclude the section by revisiting the example
pffiffiffi described
1+ 2
and hence
above. It is easy to verify that in this example ẽ =
k
187
!
pffiffiffi
1+ 2
Δw̃ = 2
+ w . The agents' objective functions are strictly
k
!
pffiffiffi
1+ 2
+w
k
−k b 0. By assuming specific parameter
concave if
2a2
values, we can now ensure that the condition is always fulfilled (e.g.,
w = 0, k = 1 and a = 2).
References
Jewitt, I., 1988. Justifying the first-order approach to principal–agent problems.
Econometrica 56, 1177–1190.
Konrad, K.A., 2009. Strategy and Dynamics in Contests. Oxford University Press.
Lazear, E.P., Rosen, S., 1981. Rank-order tournaments as optimum labor contracts.
Journal of Political Economy 89, 841–864.
Rogerson, W.P., 1985. The first-order approach to principal–agent problems. Econometrica 53, 1357–1367.