Division of a Contest with Identical Prizes∗
Munetomo AndoCIRJE, Faculty of Economics, University of Tokyo
7-3-1, Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
E-mail: ando@e.u-tokyo.ac.jp

First version: August 20, 2001
This version: September 3, 2003

Abstract
This paper studies an economic contest with identical prizes. We
consider the eﬀects of division of the contest. When the contest designer divides the contest symmetrically, each participant competes ineach assigned division. The main result is that division is sometimes
proﬁtable for the contest designer, in the sense that division brings
about increases in lower ability agents’ (and occasionally higher ability agents’) eﬀorts in exchange for decreases in other agents’ eﬀorts.
We also study an application to educational attainment. The result
helps to explain why it is diﬃcult to ﬁnd class size eﬀects in empiricaldata.
JEL classification numbers: D44, D72, D82, I21, J31.
Keywords: Contest design; all-pay auction; ineﬃcient allocation ofprizes in contests; educational production; class size eﬀects.

∗

An earlier version of this paper was entitled “Dividing a Contest with Identical Prizes
into Plural Contests,” and was prepared for Summer in Tsukuba 2001 (August 22-24).
This paper was also presented in the 2002 Spring Meeting of the Japanese Economic
Association. The author gratefully acknowledges helpful comments and suggestions from
Noriyuki Yanagawa, Daisuke Oyama, two anonymous referees, and the editor, Hideshi
Itoh. The author acknowledges Grant-in-Aid for JSPS Fellows.

1

1

Introduction

In a large literature, economic contests have been studied under many situations. Well known applications are job promotion in internal labor markets,
rent seeking activities, and R&D rivalry. 1 Many studies assume the contest
structure to be given and study the equilibrium strategies of participants.
However, contest design problems from the viewpoint of the contest designer
have attracted much attention recently. Gradstein and Konrad (1999) compared multistage contests with simultaneous contests. Clark and Riis (1998)
studied complete information contests with identical prizes and compared
simultaneous distributions of prizes with sequential distributions of prizes.
Moldovanu and Sela (2001) studied the optimal allocation of prizes and provided the condition for a contest with multiple positive prizes to be optimalfor the contest designer.2
The aim of this paper is to study the eﬀects of division of a contest.3 An
example of division is the following. Consider a contest with four participants and two identical prizes. If the contest designer divides the contest,
each participant competes with one of his fellow participants for one prize.
If he does not divide the contest, each participant competes with his three
fellow participants for two prizes. Thus, the contest designer can determine
whether each participant competes in the whole contest or in each assigned
division.
In the real world, we can observe many phenomena that can be regarded
as division of a contest with identical prizes. Examples include the following:• A large company has several departments and sections. Peers compete
for promotion in each division.
• In the realm of politics, there exist various election systems, including the small electoral district system, the medium-sized system, andthe large-sized system. Here, we can regard such elections as divided
contests.
• In many schools, grading is conducted in each classroom with grading
on a curve.4
Now, our questions are “what does division bring about?” and “when andwhy does the designer of a contest divide the contest?”
1

An initial study on the economic analysis of contests has been conducted by Lazear
and Rosen (1981).
2
Further literature reviews are provided in Section 1.1.
3
An independent work by Moldovanu and Sela (2002) considered closely related topics
to ours. The contest analyzed in their paper is detailed in Section 1.1.
4
Classroom education with relative evaluations has contest aspects. Students study to
get the higher grade, say, “excellent.” In public elementary schools in Japan, grading had
been conducted in each classroom with grading on a curve until the school year 2001.

2

To answer these questions, we study the following contest game. The
principal, the contest designer, hires n risk neutral agents for one period andassigns a task for each of them. There exist k(< n) identical prizes. Eachagent outlays his eﬀort to get a prize. There are k winners, each of whom
obtains one prize. We assume that the numbers n and k are exogenously
given. Diﬀerent agents have diﬀerent abilities, which aﬀect the marginal
costs of their eﬀorts. Each agent’s ability is his private information. Abilities
are independently and identically distributed. The distribution of abilities iscommon knowledge for the principal and the agents. The eﬀort level of eachagent is observable by all players at the end when the agents have alreadychosen their eﬀort level.
The principal can divide this contest symmetrically into some contests.5
More precisely, the principal can divide agents into a(> 1) groups randomly,
and then each agent competes in each assigned division. Here a(∈ A) is acommon divisor of n and k, and A is the set of common divisors. In eachdivision, which consists of n/a participants and k/a prizes, each agent whose
eﬀort level is no lower than the k/a-th place gets a prize. The principal
chooses the optimal contest structure a∗ ∈ A according to his objective.
Our results are as follows: First, we identify the symmetric BayesianNash equilibrium strategy of the contest game. The equilibrium eﬀort levels
are strictly increasing with respect to abilities. Note that the contest gameconsidered here is isomorphic to an all-pay auction. Consequently, we canuse several existing results in the literature on auction theory to study ourcontest game.
Second, we demonstrate that division brings about increases in lower
ability agents’ (and occasionally higher ability agents’) eﬀorts in exchange
for decreases in other agents’ eﬀorts. The fundamental reason why division
makes diﬀerences in the equilibrium behavior of each type is that division
makes room for an ineﬃcient allocation of prizes. An ineﬃcient allocationmeans that the winners are not chosen in the order of the agents’ abilities. If
the principal divides a contest, it is possible that the eﬀort level of a winner
in one division is smaller than that of a loser in another division. Due to thefact that the agents use the same symmetric equilibrium strategy in every
division, the above phenomenon implies an ineﬃcient allocation of prizes.
Thus division changes each type’s probability of winning, while it does notchange the ratio of the number of participants to the number of prizes, andconsequently, the equilibrium behavior of each type is changed by division.
Third, we show that division decreases the expected eﬀort per agent
under a certain regularity condition on the distribution of abilities (the deﬁnition of regularity is described in Section 4.2). This is a direct consequenceof the result of the optimal multi-object auction studied by Maskin andRiley (1989).
5

We assume that n and k are not mutually prime.

3Fourth, we show that division is sometimes proﬁtable for the contest designer. This is the main result of the present paper. We analyze the optimal
contest structure under the following objective of the principal. The principal maximizes his proﬁt and the proﬁt is the sum of each agent’s output.
Here each agent’s output may not be linear in his eﬀort. We show that,
under the regularity condition, division is never proﬁtable for the contest
designer if the function that transforms each agent’s eﬀort to output is linear. However, division is sometimes proﬁtable for the contest designer if thefunction is concave or convex, even if the regularity condition holds.
The result in the linear case is straightforward from the third result.
Since each agent’s output is linear in his eﬀort, the maximization of total
output is equivalent to the maximization of total eﬀort. Therefore, we canapply the results of optimal auction design to the study of optimal contest
design. Speciﬁcally, under the regularity condition, an ineﬃcient allocationof prizes (i.e., dividing) is not proﬁtable. However, in non-linear cases, themaximization of total output is no longer equivalent to the maximization of
total eﬀort. Therefore, we cannot apply the results of optimal auction tothe optimal contest design under a non-linear setting. Furthermore, if theobjective of the principal is not additively separable in each agent’s output,
the proﬁt maximization for the principal is not achieved by the maximizationof total eﬀort. Thus, generally, the optimal contest design is diﬀerent fromthe optimal auction design, so that the division of a contest can be proﬁtableeven if the regularity condition holds.
Our results have an implication to the study of educational attainment.
The relation between educational attainment and class size is studied in alarge literature. It has been expected that class size reductions yield an
increase in educational productivity. This positive eﬀect is called the classsize eﬀect. However, many empirical studies6 conclude that there exists no
signiﬁcant class size eﬀect.
Our discussion helps to explain why it is diﬃcult to ﬁnd class size eﬀect
in some countries. Suppose that the students are graded on a curve andgrading is conducted in each classroom as in Japanese public elementary
schools (until the school year 2001). Under such a grading method, weconclude that class size reductions yield a decrease in the expected eﬀort
per student. This topic is detailed in Section 6.
The rest of this paper is organized as follows: In the next subsection, we
describe related literature. In Section 2, we present the model. In Section
3, we identify the symmetric equilibrium strategy. In Section 4, we study
the eﬀects of division. In Section 5, the principal’s contest design problem isstudied. In Section 6, an application to educational attainment is described.
Section 7 concludes.

6

See the references in Hoxby (2000) and Lazear (2001).

4

1.1

Related Literature

The contest game considered in this paper can be regarded as a variationof the all-pay auction with incomplete information. Recently, Amann andLeininger (1996) and Krishna and Morgan (1997) studied the all-pay auction
and the war of attrition. The former analyzed an asymmetric all-pay auction
and the latter analyzed an all-pay auction with aﬃliated types.
However, the contest game considered in this paper and the all-pay auction have some diﬀerences. In the all-pay auction, each bidder’s bid is equivalent to his payment as well as the seller’s revenue. Therefore, the revenue
maximization for the seller is achieved by the maximization of the sum ofbidders’ bids. However, in situations of contests, the relation between eachagent’s eﬀort and the principal’s proﬁt may not be linear. As a consequence,the principal’s optimization problem is probably not the maximization of total eﬀort, and then it is more complicated than that in the all-pay auction.7
Singh and Wittman (2001) studied the contest in which each agent’s
output is non-increasing returns to his eﬀort. Additionally, it is assumed
that the participant with the highest output need not win. They described
the properties of optimal contests and show that, for an open interval oftypes, an optimal contest uses the rule that the agent with the highest
output wins.
The present paper and Moldovanu and Sela (2001, 2002) considered
not only non-increasing returns cases but also increasing returns cases.
Moldovanu and Sela (2002) considered closely related topics to ours. The
contest analyzed in their paper is the following. The contest designer has agiven amount of money for prize. He can split the money to several equal
prizes, as well as he can split the participants among several sub-contests.
The authors compared the performance of such split schemes to that of
grand winner-take-all fashion.
As in ours, the form of the function that transforms each agent’s eﬀort to
output plays the central role in their analysis. Notice that our terminology
is diﬀerent from Moldovanu and Sela (2001, 2002). Our concavity of output
functions corresponds to their convexity of cost functions.
Moldovanu and Sela (2002) concluded that, only if the cost function isconvex, split-the-prize contests or split-the-contestants contests can be more
proﬁtable than the grand winner-take-all contest.
Note that the present paper compares a contest with n participants andk prizes to divided contests. In our setting, under both concave and convex
output functions, division can be proﬁtable for the contest designer.

7

There also exists the following diﬀerence. In the all-pay auction, there is no social
cost since the total amount of money is unchanged. However, in contests, the total cost ofagents’ eﬀort and the principal’s proﬁt are not the same. Thus, economic contests involvea social cost, as in situations of the war of attrition.

5

2

The Model

In this section, we describe the ingredients of our model. Consider an economic contest with n risk neutral agents and k(< n) identical prizes. The
numbers n and k are exogenously ﬁxed. Each agent i decides his eﬀort ei
to get a prize. Eﬀorts are outlaid simultaneously and independently. All
agents’ eﬀort levels are observable by the principal and all agents at the end
when the agents have already chosen their eﬀort level. There are k winners,
each of whom obtains one prize. The common monetary value of each prize,v, is common knowledge.
Each agent has diﬀerent ability, which aﬀects his marginal cost of eﬀort.
The ability of agent i is denoted by θi and this is his private information.
Here, high θ means high ability and low marginal cost. Abilities are drawn
independently from an interval [0, 1] according to the distribution function
F that has a continuous and everywhere strictly positive density function
f .8 The distribution function is common knowledge for the principal and allagents. The agent i’s cost function is ei /θi .9 Notice that, in this contest, theonly uncertainty for each agent is his opponents’ ability levels. The principal
does not know any of his agents’ abilities ex ante.10
The payoﬀ of agent i is, v − ei /θi if he gets a prize, and −ei /θi if he doesnot. Each agent chooses his eﬀort in order to maximize his expected payoﬀ,
given his own ability, the shape of distribution function F , the number of
opponents, the number of prizes, and the value of the prize.
The principal can determine the organizational structure of the contest.
If the principal wants to, he can divide this “n participants and k prizes”
contest symmetrically into plural contests, when n and k are not mutuallyprime. Let A be the set of common divisors of n and k, and a representative
element of A be denoted by a. The control variable of the principal is thenumber of division a ∈ A only, where a = 1 implies not-dividing. Here,
since the principal does not know his agents’ abilities ex ante, he randomly
assigns his agents to each division. In each division, each agent whose eﬀort
level is no lower than the k/a-th place gets a prize. The eﬀect of division is
studied in Section 4 and the principal’s contest design problem is studied in
Section 5.

8

We assume that the ability space is [0, 1]. This restriction is only for the analyticalconvenience. We can preserve all our result in any non-negative abilities with bounded
support case.
9This is the key assumption that permits us to analyze the problem as a variation of
all-pay auction. We assume that e/0 = ∞ if e > 0. This means that the lowest ability
agent has an inﬁnite cost for eﬀort. Therefore, he never outlays eﬀort. This setting issimilar to Glazer and Hassin (1988)’s one.
10
Since we pay attention only to the symmetric equilibrium strategies that are strictlyincreasing with respect to abilities, the principal can ﬁnd his agents’ abilities by using theequilibrium strategies and agents’ eﬀort levels ex post.

6

3

Equilibrium Strategies of AgentsIn this section, we identify the symmetric Bayesian Nash equilibrium eﬀort
strategy in a contest with n participants and k prizes. First of all, we derive
the expected payoﬀ of an agent with ability θi under the situation where theopponents use the same eﬀort strategy β(θ). Here, we assume that β(θ) is
strictly increasing and diﬀerentiable.
Lemma 1. Suppose that all opponents use the same eﬀort strategy β(θ).
For the agent i with ability θi , his expected payoﬀ under the eﬀort level e isπ(e, β | θi ) = v


k 

n−1
t=1

t−1

(1 − F (g(e)))t−1 (F (g(e)))n−t −

e
,
θi

(1)

where g(·) is the inverse function of β(θ).
Proof. See Appendix A (all appendices are available upon request).
Given Lemma 1, we deﬁne the following.
Definition 1. A strategy β(θ) constitutes a symmetric equilibrium if∀θ, β(θ) = arg max π(e, β | θ).
e

In Proposition 1, we characterize an eﬀort strategy that constitutes asymmetric equilibrium.
Proposition 1. In a symmetric equilibrium, the eﬀort strategy is given by
n−1β(θ) = v(n − k)k−1
 θyf(y)(1 − F (y))k−1 (F (y))n−k−1 dy.
(2)×0

Proof. See Appendix A.
By using the proof of Proposition 1, we can easily conclude that the symmetric equilibrium is unique.
In the symmetric equilibrium, since β(θ) is strictly increasing, agent i’s
probability of winning (hereafter p(θi; n, k)) is equivalent to the probability
that his ability is no lower than the k-th place among n participants, thatis,p(θi ; n, k) =


k 

n−1
t=1

t−1

(1 − F (θi ))t−1 (F (θi ))n−t .

7

By using this notation, we can rewrite the eﬀort strategy that constitutes
the symmetric equilibrium (i.e., equation (2)) as follows: θyβ(θ) = v0∂p(y; n, k)dy.
∂yNote that, if v = 1, this equilibrium strategy is exactly the same as that of
the all-pay auction under the following situation: n potential buyers compete for k identical objects; each buyer demands at most one object; andvaluations are drawn independently from an interval [0, 1] according to the
distribution function F that has a continuous and everywhere strictly positive density function f .
In this section we identify the eﬀort strategy that constitutes the symmetric equilibrium. By using this strategy, in the next section, we study theeﬀects of division of a contest.

4

Division of a Contest

4.1

The Eﬀects on the Agents’ Equilibrium Behavior

The aim of this section is to study the eﬀects of division of a contest for
the agents’ equilibrium behavior. If the principal does not divide the contest with n participants and k prizes (hereafter (n, k)-contest), each agent
competes with his n − 1 fellow agents for k prizes. If the principal divides
the contest into a(= 1) contests, each agent competes with his n/a − 1 fellow agents for k/a prizes in each assigned (n/a, k/a)-contest. Here a is acommon divisor of n and k. In the rest of this paper, we use the notationβ(θ; n, k) as the eﬀort strategy that constitutes the symmetric equilibrium ina (n, k)-contest. To study the eﬀects of division, we will compare β(θ; n, k)
with β(θ; n/a, k/a).
In Proposition 2, the eﬀects of division of a (n, k)-contest is described.
Moreover, the comparison of β(θ; n/a , k/a ) with β(θ; n/a, k/a) is showed,
where a > a ≥ 1. This proposition tells us that there exist three andonly three patterns of change. In Case 1 and Case 2, division brings about
increases in lower ability agents’ eﬀorts in exchange for decreases in higher
ability agents’ eﬀorts. In Case 3, division brings about increases in lowerand higher ability agents’ eﬀorts in exchange for decreases in middle ability
agents’ eﬀorts.
Proposition 2. Division of a contest (or an increase in the number of divisions) changes agents’ equilibrium behavior. For a > a ≥ 1, there exist
three and only three patterns of change as follows:8Case 1: There exists a threshold θ̄ ∈ (0, 1),β(θ; n/a , k/a ) = β(θ; n/a, k/a) = 0,












β(θ; n/a , k/a ) > β(θ; n/a, k/a),β(θ; n/a , k/a ) = β(θ; n/a, k/a),β(θ; n/a , k/a ) < β(θ; n/a, k/a),

if θ = 0,

if θ ∈ (0, θ̄),if θ = θ̄,
if θ ∈ (θ̄, 1].

Case 2: There exists a threshold θ̄ ∈ (0, 1),β(θ; n/a , k/a ) = β(θ; n/a, k/a) = 0,
















β(θ; n/a , k/a ) > β(θ; n/a, k/a),β(θ; n/a , k/a ) = β(θ; n/a, k/a),β(θ; n/a , k/a ) < β(θ; n/a, k/a),β(θ; n/a , k/a ) = β(θ; n/a, k/a),

if θ = 0,

if θ ∈ (0, θ̄),if θ = θ̄,
if θ ∈ (θ̄, 1),
if θ = 1.

Case 3: There exist two thresholds that satisfy 0 < θ̄ < θ̂ < 1,β(θ; n/a , k/a ) = β(θ; n/a, k/a) = 0,
















β(θ; n/a , k/a ) > β(θ; n/a, k/a),β(θ; n/a , k/a ) = β(θ; n/a, k/a),β(θ; n/a , k/a ) < β(θ; n/a, k/a),β(θ; n/a , k/a ) = β(θ; n/a, k/a),β(θ; n/a , k/a ) > β(θ; n/a, k/a),

if θ = 0,

if θ ∈ (0, θ̄),if θ = θ̄,
if θ ∈ (θ̄, θ̂),if θ = θ̂,
if θ ∈ (θ̂, 1].

Proof. See Appendix A.
We provide examples. Each follows each case of Proposition 2, respectively.
Example 4.1. Suppose that the environment is n = 4, k = 2, v = 1.
• If F (θ) = θ2 , then β(θ; 4, 2) = 12(θ 5 /5 − θ7 /7), and β(θ; 2, 1) = 2θ3 /3.
In this case, the changes in eﬀort levels brought by division follow Case1.
• If F (θ) = θ, then β(θ; 4, 2) = 2θ3 − 3θ4 /2, and β(θ; 2, 1) = θ2 /2. In
this case, the changes in eﬀort levels brought by division follow Case2.
• If F (θ) = 2θ − θ2 , then β(θ; 4, 2) = 6(4θ 3 /3 − 7θ4 /2 + 18θ5 /5 − 5θ6 /3 +2θ7 /7), and β(θ; 2, 1) = θ2 − 2θ3 /3. In this case, the changes in eﬀort
levels brought by division follow Case 3.
These examples are depicted in Figures 1, 2, and 3. Figure 1 depicts
the case of F (θ) = θ2 , Figure 2 depicts the case of F (θ) = θ, and Figure 3
9

e
0.7
0.6

- H6;4,2L

0.5- H6;2,1L
0.4
0.3
0.2

0.1

0.2

0.4

0.6

0.8

0.6

0.8

q

1

Figure 1:

e
0.5

0.4

0.3

-+6;4,2/-+6;2,1/

0.2

0.1

0.2

0.4

Figure 2:

10

1

6

e
0.3

- +6;4,2/

0.25- +6;2,1/
0.2
0.15

0.1
0.05

0.2

0.4

0.6

0.8

1

6

Figure 3:
depicts the case of F (θ) = 2θ − θ2 . Horizontal axis is abilities and verticalaxis is eﬀort levels.
We brieﬂy describe the reason why division changes agents’ equilibriumeﬀort as described in Proposition 2. When the principal divides a contest,
winners are not always chosen in the order of the agents’ abilities, while winners are chosen in the order of the agents’ abilities if the principal does notdivide the contest. That is, division makes room for an ineﬃcient allocationof prizes.
An ineﬃcient allocation improves the winning prospect of agents withlower ability. 11 Because the probability of winning of agents at the left tail(note that the probability of winning of an agent with the lowest ability
is always zero) increases faster in this situation compared to the situation
with eﬃcient allocation, the possibility of ineﬃcient allocation induces moreaggressive eﬀort at the left tail.
For middle ability agents, division decreases the gradient of the probability of winning function p(θ), while division increases the gradient of p(θ)
for lower and higher ability agents. This reﬂects decreases in the gradient of
the equilibrium eﬀort function β(θ) for middle ability agents, and increases
in the gradient of β(θ) for higher ability agents.
11

Another explanation is as follows: Given the ratio of the number of participants tothe number of prizes, the winning prospect of lower ability agents is smaller in a larger
contest due to the law of large numbers eﬀect.

11

4.2

The Aggregate Eﬀect of Division

Now we turn our attention to the aggregate eﬀect of division. First of all,we deﬁne and consider a regularity condition on the distribution of abilities.
Definition 2. The distribution function F is regular if
J(θ) ≡ θ −1
ρ(θ)

is increasing, where ρ(θ) = f (θ)/(1 − F (θ)) is the hazard rate for F .
The regularity of distribution functions is satisﬁed if the hazard rateincreases or does not decrease too rapidly with θ. Many distributions ofabilities satisfy this condition, including the distribution functions displayedin Example 4.1.12
As mentioned before, the contest game considered in this paper is isomorphic to an all-pay auction. Moreover, this contest game satisﬁes theconditions of the revenue equivalence theorem: Bidders are risk neutral andhave unit-demand, and their types are i.i.d. Under the hypotheses of therevenue equivalence theorem and the regularity condition, Maskin and Riley(1989) showed that, in optimal auctions, the winners are chosen in the order of the bidders’ valuations (under an appropriate minimum allowable bidrule). As a direct consequence of the above result, we obtain the following
result.
Proposition 3. If the distribution function is regular, division of a contest(or an increase in the number of divisions) decreases the expected total eﬀortof the agents.
Proof. See the proof of Proposition 4 in Maskin and Riley (1989).
Proposition 3 implies that division decreases the expected eﬀort per agent.
This is the key ﬁnding that is used in Section 6.
To sum up, division changes agents’ behavior as described in Proposition
2, while it decreases the expected eﬀort per agent (under the regularity condition). Hence, in the case where the principal welcomes these consequences,
he divides the contest.

5

The Principal’s Contest Design

In this section, we demonstrate that division is sometimes proﬁtable for thecontest designer. We assume that the proﬁt function of the principal isγ(ei),π(e1 , . . . , en) =i
12

See Remark 8.1 in Maskin and Riley (1984) for a discussion on the regularity condition.

12

where γ(·) is the output function that transforms each agent’s eﬀort to his
output. We also assume that the function γ(·) is strictly increasing. We
consider the cases where γ(·) is a linear, concave, or convex function.13
The optimal contest structure is characterized by a∗ ∈ A, where 1γ(β(θ; n/a, k/a))f (θ) dθ.
a∗ ∈ arg max
a∈A

1

0

Here 0 γ(β(θ; n/a, k/a))f (θ) dθ is the expected output per agent in the(n/a, k/a)-contest.
Firstly we describe the case where γ(·) is linear. By using Proposition
3, we obtain the following.
Proposition 4. Under the regularity condition, if γ(·) is linear, division isnever proﬁtable for the contest designer.
Proof. When γ(·) is linear, the maximization of the sum of agents’ output
is equivalent to the maximization of the sum of agents’ eﬀort. From Proposition 3, division of a contest decreases the expected sum of agents’ eﬀort,
and therefore it is never proﬁtable for the contest designer.
In this linear case, the maximization of total output is equivalent to the maximization of total eﬀort, so that we can apply the results of optimal auction
design to the study of optimal contest design and an eﬃcient allocation ofprizes (i.e., not-dividing) is optimal.
Secondly we describe the case where γ(·) is concave.
Proposition 5. When γ(·) is concave, division of a contest can be proﬁtable
for the contest designer, even if the regularity condition holds.
Proof. See Appendix A.
An intuitive explanation of Proposition 5 is as follows: If the output function
γ(·) is a concave function of eﬀort, the principal prefers to have a low eﬀort
agent increase one unit of his eﬀort in exchange for a high eﬀort agent
reducing one unit of his eﬀort. Therefore contests that increase the eﬀort oflow eﬀort types and reduce the eﬀort of high eﬀort types are more likely to
be chosen under a larger degree of the concavity of the output function.
In this non-linear case, the maximization of total output is no longer
equivalent to the maximization of total eﬀort. Therefore, the results of
optimal auction design are not applicable to the study of optimal contest
design. That is to say, an ineﬃcient allocation of prizes can be proﬁtable,
even if the regularity condition holds.
The following example illustrates the situation with a concave output
function.
13

Other objectives of the principal are also convincing depending on the situation. We
will discuss the other objectives of the contest designer later.

13

Example 5.1. We consider a (8, 4)-contest with v = 1. Agents’ abilities
are uniformly distributed. The output function isγ(ei) = − exp{−αei } + 1,
where α > 0. The Arrow-Pratt measure (m(ei ) = −γ  (ei )/γ (ei )) of this
output function is equal to the constant α(> 0) at all ei . Here, a largerm(= α) implies that the degree of concavity is larger.14 In this circumstance,• If α = 1, a∗ = 1 (i.e., division is not proﬁtable).
• If α = 10, a∗ = 2.
• If α = 20, a∗ = 4.
Thus, if γ(·) is concave enough, division can be proﬁtable.
This example tells us that, given the value of each prize v, a larger α
is required for division to be proﬁtable. Note that, because there is linear
relation between v and ei (Proposition 1), we can also conclude that, for agiven α, a larger v is required for division to be proﬁtable. Figures 4 depicts
above example. Horizontal axis is α, and vertical axis is the expected output
per agent, which is denoted by x.
Lastly we describe the case where γ(·) is convex.
Proposition 6. Suppose that γ(·) is convex and the regularity condition
holds.
• If changes in agents’ equilibrium behavior by division follow Case 1 orCase 2 of Proposition 2, division is never proﬁtable.
• The fact that changes in agents’ equilibrium behavior by division follow
Case 3 of Proposition 2 is a necessary condition for division to be
proﬁtable.
Proof. We can prove this proposition with a similar method used in the
proof of Proposition 5, thus this proof is omitted.
An intuitive explanation of Proposition 6 is as follows: If the output function
γ(·) is a convex function of eﬀort, the principal prefers to have a high eﬀort
agent increase one unit of his eﬀort in exchange for a low eﬀort agent reducing one unit of his eﬀort. In Case 1 and Case 2 of Proposition 2, because
division decreases the eﬀort of high eﬀort types and increases the eﬀort oflow eﬀort types, it is never proﬁtable. Only in Case 3 of Proposition 2, division increases the eﬀort of high eﬀort types. Therefore the fact that changes
14

The Arrow-Pratt measure of curvature is usually used to compare risk attitude of
agents.

14

x
0.7
0.6
0.5

a 1

0.4
0.3

a 2

0.2

a 4

0.1

5

10

15

20

25

,

Figure 4:
in agents’ equilibrium behavior by division follow Case 3 of Proposition 2 isa necessary condition for division to be proﬁtable.
By using the fact that changes in agents’ eﬀort that are brought bydivision always obey Case 2 of Proposition 2 if abilities are uniformly distributed,15 we obtain the following as a corollary of Proposition 6.
Corollary 1. If abilities are uniformly distributed and γ(·) is convex, division is never proﬁtable for the contest designer.
The following example illustrates the situation where division is profitable for the contest designer under a convex output function.
Example 5.2. We consider a (4, 2)-contest with v = 100 under F (θ) =2θ − θ2 . The output function isγ(ei) = exp{ei } − 1.
In this circumstance, a∗ = 2. Thus, there exist circumstances that division
is proﬁtable for the contest designer under a convex output function.
In this section, we consider the optimal contest structure under a speciﬁc
objective of the contest designer and demonstrate that division is sometimes
proﬁtable for the contest designer.
15

The proof of this statement is provided in Appendix B.

15

As mentioned before, there exist many possible objectives of the principal
in contests. For example, the principal may desire to maximize the minimum
eﬀort among the participants in contests. This objective is suitable if the
production function of the principal is a Leontief production function, andmay be suitable if the contest designer attaches greater importance to thecloseness of contestants eﬀorts as well as a high average eﬀort (as in a sport
contest).16 Under this objective function, division is sometimes proﬁtable,
because it increases the eﬀort of low eﬀort types. For example, consider a(4,2)-contest with a Leontief production function. If F (θ) = θ2 , division is
proﬁtable for the principal, while division is not proﬁtable if F (θ) = θ or
F (θ) = 2θ − θ2 .
We can describe another situation where the contest designer is willingto divide a contest. Suppose that, for each agent, if his eﬀort exceeds a given
minimum requirement level e(> 0), a positive constant output is realized.
In this environment, the principal designs the contest structure to maximize
the expected number of participants whose eﬀorts exceed the minimum requirement level. Since division of a contest increases eﬀorts of lower eﬀort
types, we can conclude that division is proﬁtable for the contest designer if
the minimum requirement level is suﬃciently small.
In the next section, we apply the result to the class size puzzle.

6

An Application to Educational Attainment

The relation of educational attainment to class size is studied in a large
literature. It has been expected that class size reductions yield an increasein educational productivity. For example, since class size reductions yieldrelaxation of the congestion eﬀects in classrooms, education may become
eﬃcient. This positive eﬀect is called the class size eﬀect. However, many
empirical studies17 conclude that no signiﬁcant class size eﬀect exists. This
inconsistency is called the class size puzzle.
Lazear (2001) presented one of the reasons why it is diﬃcult to ﬁnd classsize eﬀect in empirical data. He presented a disruption model of educational
production and paid attention to the case where the class size is a choicevariable of each school. He concluded that there exists a positive correlation
between students’ abilities and the optimal class size and therefore it isdiﬃcult to ﬁnd class size eﬀect in empirical data.
Our discussion exhibits another explanation of the class size puzzle. Consider the following situation. There exists a school with n students in each
grade. Each student has diﬀerent ability, which aﬀects his additional costof study. We assume that students study to obtain the higher grade. Here
16

Singh and Wittman (1998) studied two player contests and compared the contest
designer’s optimal reward for winning under diﬀerent objectives.
17
One of the recent studies is Hoxby (2000).

16other motivations to study are ignored. The monetary value of obtaining
each grade is the same among all students. For the present, we assume thatthe values are exogenously given. If the same grade is obtained, less eﬀort
is proﬁtable for students．
Students are graded on a curve and grading is conducted in each classroom.1819 To simplify, we assume that there exist two distinct grades, Pass
and Fail. Only k students will pass each year. The model of our contest
game is suitable for the above situation. 20 Here, we can regard a reductionof class size as an increase in the number of divisions.
Now, we apply our results to the above situation. Proposition 3 implies
that a reduction of class size decreases the expected eﬀort per student (under
the regularity condition). Note that, if a ﬁxed proportion of students passin each class, an increase in the number of classes decreases the degree of
selection eﬃciency. Clearly, if all students belonged to one big class, thek most able students pass. If the students are divided into several classes,
and if the proportion of passes to fails in each class is ﬁxed, an ineﬃcient
selection occurs. The reason is obvious, as too many students pass frompoor classes and too few from good classes.
Additionally, a reduction of class size may decrease the monetary value ofobtaining the higher grade. Grades are presumably valuable to the student
in so far as they are eﬀective signal of ability or eﬀort. The possibility ofineﬃcient selection weakens the value of the higher grade as a good signal,and then it may decrease the monetary value of obtaining the higher grade.
Obviously, a reduction of the value of prize involves overall reduction ofeﬀorts.
As discussed above, class size reductions may decrease educational productivity if grading is conducted in this manner.21 Hence, we suggest that,to solve the class size puzzle, we have to consider not just the relation between educational attainment and class size but the relation among educational attainment, class size, and the method of grading. Lazear (2001) said
“Japan has high test score and large class size.”22 This fact is consistent to18

Until the school year 2001, grading in Japanese public elementary schools had been
conducted in this manner. From the school year 2002, grading is conducted with absolute
scales.
19
In Japanese public elementary schools, each student belongs to assigned classroom
and most subjects are taught in the classroom as a unit. Moreover, in each classroom,these subjects are taught by one teacher in charge. Each class teacher grades his students
with a given method.
20
Virtually, there exists no literature that studies educational productivity by using
principal-agent models. Exceptions include Becker and Rosen (1992) and Betts (1998).
Becker and Rosen (1992) compared grading on a curve with grading on absolute scales.
Betts (1998) studied the relation of educational standards to inequality.
21
Obviously, these eﬀects do not appear if grading is conducted with absolute scales.
22
See footnote 15 of Lazear (2001). See also Wößmann and West (2002). They concluded
as follows: “While we ﬁnd sizable beneﬁcial eﬀects of smaller classes in Greece and Iceland,
the possibility of even small eﬀects is rejected in Japan and Singapore.”17

our ﬁndings.

7

Conclusion

In this paper, we described the eﬀects of division of a contest with identical prizes. The fundamental reason why division changes the equilibrium
behavior of agents is changes in each type’s probability of winning due tothe possibility of ineﬃcient allocation of prizes. Additionally, we showedthat division is sometimes proﬁtable for the contest designer, in the sensethat division brings about increases in lower ability agents’ (and occasionally higher ability agents’) eﬀorts in exchange for decreases in other agents’eﬀorts.
This paper considers speciﬁc objectives of the principal. The purpose of
the principal is assumed to be providing appropriate incentives to agents.
Naturally, we do not exclude other reasons why the contest designer divides
a contest. Let us consider a competition for promotion in a company. If
whether or not the promoted employee accumulated professional knowledge
is important, the competition for promotion may be conducted in each department. Another reason is that, if it is diﬃcult to compare employees
who were assigned to diﬀerent jobs, the competition for promotion may be
conducted among employees who took the same job. However, irrespectiveof the principal’s intention of dividing, division of a contest changes theparticipants’ behavior as described in this paper.

18

References[1] Amann, E., and Leininger, W. (1996). Asymmetric All-Pay Auctions
with Incomplete Information: The Two-Player Case, Games Econ. Behav. 14, 1-18.
[2] Becker, W. E., and Rosen, S. (1992). The Learning Eﬀect of Assessment
and Evaluation in High School, Econ. Educ. Rev. 11, 107-118.
[3] Betts, J. R. (1998). The Impact of Educational Standards on the Level
and Distribution of Earnings, Amer. Econ. Rev. 88, 266-275.
[4] Clark, D. J., and Riis, C. (1998). Competition over More Than One
Prize, Amer. Econ. Rev. 88, 276-289.
[5] Glazer, A. and Hassin, R. (1988). Optimal Contests, Econ. Inquiry 26,
133-143.
[6] Gradstein, M., and Konrad, K. A. (1999). Orchestrating Rent Seeking
Contests, Econ. J. 109, 536-545.
[7] Hoxby, C. M. (2000). The Eﬀects of Class Size on Student Achievement:
New Evidence from Population Variation, Quart. J. Econ. 115, 12391285.
[8] Krishna, V., and Morgan, J. (1997). An Analysis of the War of Attrition
and the All-Pay Auction, J. Econ. Theory 72, 343-362.
[9] Lazear, E. P. (2001). Educational Production, Quart. J. Econ. 116,
777-803.
[10] Lazear, E. P., and Rosen, S. (1981). Rank Order Tournaments as Optimal Labor Contracts, J. Polit. Economy 89, 841-864.
[11] Maskin, E., and Riley, J. (1984). Optimal Auctions with Risk Averse
Buyers, Econometrica 52, 1473-1518.
[12] Maskin, E., and Riley, J. (1989). Optimal Multi-Unit Auctions, in “TheEconomics of Missing Markets, Information, and Games” (F. Hahn,Ed.), pp. 312-335, Oxford Univ. Press, Oxford.
[13] Moldovanu, B., and Sela, A. (2001). The Optimal Allocation of Prizes
in Contests, Amer. Econ. Rev. 91, 542-558.
[14] Moldovanu, B., and Sela, A. (2002). Contest Architecture, mimeograph,
University of Bonn.

19[15] Singh, N. and Wittman, D. (1998). Contest Design and the Objectiveof the Contest Designer: Sales Promotion, Sporting Events, and Patent
Races, in “Advances in Applied Microeconomics,” Volume 7, pp. 139167, JAI Press.
[16] Singh, N. and Wittman, D. (2001). Contests Where There is Variationin the Marginal Productivity of Eﬀort, Econ. Theory 18, 711-744.
[17] Wößmann, L., and West, M. R. (2002). Class-Size Eﬀects in School
Systems Around the World: Evidence from Between-Grade Variation
in TIMSS, PEPG Research Paper 02-02, Harvard University.

20

8

Appendix A (Not for Publication)

Proof of Lemma 1. Given that the agent i’s ability is θ, his eﬀort levelis e, and all other agents j = i outlay eﬀort levels β(θj ), agent i’s expected
payoﬀ isπ(e, β | θ) = v × ( Prob(e > β(θj ), ∀j = i)
+ Prob(largest eﬀort among all j = i > e≥ second largest eﬀort among all j = i)..
.
+ Prob( k − 1th largest eﬀort among all j = i > e≥ kth largest eﬀort among all j = i) )e
− .
θ
Since we assume that β(·) is strictly increasing and diﬀerentiable, we candeﬁne g(·) as the inverse function of β(·). Then we can rewrite the above
equation as follows:
π(e, β | θ) = v × ( Prob(g(e) > θj , ∀j = i)
+ Prob(largest type among all j = i > g(e)≥ second largest type among all j = i)..
.
+ Prob( k − 1th largest type among all j = i > g(e)e− .
θ≥ kth largest type among all j = i) )Since the distribution function F is i.i.d., we obtain
π(e, β | θ) = v × ( (F (g(e)))n−1+ (n − 1)(1 − F (g(e)))(F (g(e)))n−2
..

 .
n−1
+
(1 − F (g(e)))k−1 (F (g(e)))n−k )
k−1
e
−
θ

 k 
 n − 1et−1n−t
(1 − F (g(e))) (F (g(e)))
− .
= v×
t−1
θ
t=1

Thus expression (1) was derived.

21

Proof of Proposition 1. To simplify the notation, we deﬁnen−1(1 − F (x))t−1 (F (x))n−t .
Ψt (x) =t−1
Ψt (x) is the probability that x is the t-th largest value among n variables.
The ﬁrst order condition of the maximization problem of agent i (i.e.,
F.O.C. of the maximization of expression (1) with respect to e) is
v

k

t=1

Ψt (g(e))g (e) −

1
= 0.
θ

(3)

In the symmetric equilibrium, since g(e) = θ and g  (e) = (β  (θ))−1 , equation(3) is equivalent to
v

k

t=1

Ψt (θ)

k


1
1

−
=
0
⇔
β
(θ)
=
θ
v
Ψt (θ).
β (θ) θt=1

We can rewrite the above as follows:n−1
f (θ)(1 − F (θ)) k−1 (F (θ))n−k−1 .
β  (θ) = θ v(n − k)k−1Integration with the boundary condition (β(0) = 0) yields
n−1β(θ) = v(n − k)k−1
 θyf(y)(1 − F (y))k−1 (F (y))n−k−1 dy.
×

(4)

(5)

0

We can easily check the strict increase and diﬀerentiability of β(θ) by equation (4).
We will show that for any type of agents no one can gain by deviatingfrom the strategy β(θ). Since the ﬁrst order condition is satisﬁed in theequilibrium, for the agent whose ability is θ , ∂π(θ , e )/∂e = 0 is satisﬁed.
Here e = β(θ ). We have to check the following conditions.

∂π 
>0
if e < β(θ ) = e(θ , e)<0
if e > β(θ ) = e .
∂eSince the function π(θ, e) is continuous in e, the above expression implies
that π(θ, e) is maximized at e = β(θ). First we verify ∂π(θ , e)/∂e > 0 ife < β(θ ) = e . Take ê as an eﬀort smaller than the equilibrium eﬀort levele for the agent whose ability is θ . Here we have to check ∂π(θ , ê)/∂e > 0.
The ability type which takes ê in the equilibrium is denoted by θ̂. This22

means that β(θ̂) = ê. Since β(θ) is strictly increasing and ê < e , we obtainθ̂ < θ . Becausek1∂π(θ, e) = v
Ψt (g(e))g (e) − ,
∂e
θ
t=1

∂ 2 π(θ, e)/∂θ∂e = 1/θ2 > 0. Therefore, for any e, ∂π(θ , e)/∂e > ∂π(θ̂, e)/∂eis shown. Since the above is satisﬁed at ê, ∂π(θ , ê)/∂e > ∂π(θ̂, ê)/∂e. Since
the right hand side is zero by the ﬁrst order condition, ∂π(θ , ê)/∂e > 0 isshown. With a similar argument, the other side can be shown too. Therefore
expression (5) is the equilibrium strategy in a symmetric equilibrium.
Proof of Proposition 2. In this proof, for the expositional convenience,
we compare a (2n, 2k)-contest with a (n, k)-contest. The general case canbe proven in the same way.23
From Proposition 1,β(θ; 2n, 2k) − β(θ; n, k) θ
2n − 1yf(y)(1 − F (y))2k−1 (F (y))2n−2k−1 dy= v(2n − 2k)
2k − 10 θn−1
yf(y)(1 − F (y))k−1 (F (y))n−k−1 dy(6)−v(n − k)k−10 θyf(y)(1 − F (y))k−1 (F (y))n−k−1= v(n − k)
0
 



2n − 1
n−1
k
n−k
× 2
(1 − F (y)) (F (y))
−
dy.
(7)
2k − 1k−1
Since v(n−k) is a strictly positive constant, we have to pay attention only tothe sign of the rest of above expression. To see the shape of the integrand of
expression (7) in the interval [0, 1], we divide this function into three parts,y, f (y)(1 − F (y))k−1 (F (y))n−k−1 ,


 

n−1
2n − 1
k
n−k
−
,
and 2
(1 − F (y)) (F (y))
k−1
2k − 1and study their characteristics. First, y is a strictly increasing function andnon-negative in the interval [0, 1]. Second, since the density function f is continuous and everywhere strictly positive, f (y)(1 − F (y))k−1 (F (y))n−k−1 > 0
in the interval (0, 1). Third,




n−1
2n − 1kn−k−
2(1 − F (y)) (F (y))
k−1
2k − 123

The comparison of a (n/a , k/a )-contest with a (n/a, k/a)-contest is equivalent to thecomparison of a (n, k)-contest with a (tn, tk)-contest, where t = a /a > 1. The generalcase can be proven by replacing “2” of the present proof with “t”.

23

is strictly negative in the interval [0, y  ), strictly positive in the interval(y  , 1], where y  ∈ (0, 1) and(y  , y  ), and strictly negative in the interval1y  ∈ (y  , 1) are certain thresholds. Here, 0 Φ(y)dy = 0, which is statedlater, implies the existence of interval (y  , y  ).
From these characteristics, the shape of the integrand in the interval[0, 1] is (1) zero at y = 0, (2) strictly negative in the interval (0, y ), (3) zeroat y = y  , (4) strictly positive in the interval (y  , y  ), (5) zero at y = y  ,and strictly negative in the interval (y  , 1). Here, for the convenience ofnotation, we deﬁneΦ(y) = f (y)(1 − F (y))k−1 (F (y))n−k−1
 



2n − 1
n−1
k
n−k
× 2
(1 − F (y)) (F (y))−.
2k − 1
k−1
 y
 ỹ
1
Because 0 Φ(y)dy = 0,24 there exists ỹ ∈ (y  , y  ), 0 Φ(y)dy + y Φ(y)dy =
1
 ỹ
 y
0 and ỹ Φ(y)dy + y Φ(y)dy = 0. This is equivalent to 0 Φ(y)dy =1ỹ Φ(y)dy = 0.
The integrand of expression (7) can be written as yΦ(y). Since y isstrictly increasing and non-negative in the interval [0, 1], there exist threepossible cases as follows:
Case 1: ∃ȳ ∈ (y  , y  ),
 ȳ
 y
1
 y
0 yΦ(y)dy+ y yΦ(y)dy = 0 and ȳ yΦ(y)dy+ y yΦ(y)dy > 0,
Case 2: ∃ȳ ∈ (y  , y  ),
 ȳ
 y
1
 y
0 yΦ(y)dy+ y yΦ(y)dy = 0 and ȳ yΦ(y)dy+ y yΦ(y)dy = 0,Case 3: ∃ȳ ∈ (y  , y  ),
 ȳ
 y
1
 y
0 yΦ(y)dy+ y yΦ(y)dy = 0 and ȳ yΦ(y)dy+ y yΦ(y)dy < 0.
In Case 1, for all θ ∈ (0, ȳ), β(θ; 2n, 2k)−β(θ; n, k) < 0, and for all θ ∈ (ȳ, 1],β(θ; 2n, 2k) − β(θ; n, k) > 0. In Case 2, for all θ ∈ (0, ȳ), β(θ; 2n, 2k) −β(θ; n, k) < 0, and for all θ ∈ (ȳ, 1), β(θ; 2n, 2k) − β(θ; n, k) > 0. In Case 3,
 ŷ
1
 ȳ
there exists ŷ ∈ (ȳ, 1), 0 yΦ(y)dy = ȳ yΦ(y)dy = 0 and ŷ yΦ(y)dy < 0.
This is equivalent to, for all θ ∈ (0, ȳ), β(θ; 2n, 2k) − β(θ; n, k) < 0, for allθ ∈ (ȳ, ŷ), β(θ; 2n, 2k) − β(θ; n, k) > 0, and for all θ ∈ (ŷ, 1], β(θ; 2n, 2k) −β(θ; n, k) < 0. Thus we conclude that only these three cases are possible.
What is left to prove is to verify the existence of these three cases. The
existence can be proven by Example 4.1. Hence, we obtain Proposition 2.

24

R1
0

Since expression (6) is equivalent to v

Φ(y)dy =R 1∂p(y;2n,2k)0∂y

R 1 ∂p(y;n,k)dy − 0

∂y



Rθ
0

y ∂p(y;2n,2k)
dy − v∂yRθ0y ∂p(y;n,k)dy,∂ydy /(v(n − k)) = (1 − 1)/(v(n − k)) = 0.

24

Proof of Proposition 5. Let â ∈ A be the number of divisions whereâ > 1. From Proposition 2, there exists θ̂ ∈ (0, 1) such that β(θ; n/â, k/â) >β(θ; n, k) for all θ ∈ (0, θ̂). Moreover, there exists θ̄ ∈ (θ̂, 1] such thatβ(θ; n/â, k/â) < β(θ; n, k) for all θ ∈ (θ̂, θ̄).
Proposition 3 implies θ̂0(β(θ; n/â, k/â) − β(θ; n, k))f (θ)dθ
 θ̄

<
θ̂

(β(θ; n, k) − β(θ; n/â, k/â))f (θ)dθ.

(8)

Let ea and eb be distinct eﬀort levels that satisfy eb > ea > 0. If γ(·) is
strictly concave, γ(ea + x) − γ(ea) > γ(eb + x) − γ(eb), where x is a positive
constant. This implies that a lower ability agent’s additional eﬀort is more
eﬀective for the principal’s proﬁt than a higher agent’s, since the symmetricequilibrium strategy is strictly increasing. Hence, by division, the left handside of inequality (8) is relatively ampliﬁed. Therefore, we can conclude
that, if the degree of concavity of γ(·) is large enough, division becomes
proﬁtable.

9

Appendix B (Not for Publication)

In Proposition 7, we describe the eﬀort strategy that constitutes the symmetric equilibrium under the uniformly distributed abilities. This proposition
implies that, if abilities are uniformly distributed, the changes in equilibrium behavior that are brought by division follow Case 2 of Proposition 2,
because the equilibrium eﬀort level of an agent with the highest possible
ability is always v(1 − k/n).
Proposition 7. If abilities are uniformly distributed,kβ(θ; n, k) = v 1 −n
k−1  n
(1 − θ)t θn−t .
tt=0Proof. From Proposition 1 and F (θ) = θ,β(θ; n, k) = v(n − k)

n−1
k−1

 θ

25

0

y(1 − y)k−1 y n−k−1 dy.

(9)Using integration by parts, we obtainn−1β(θ; n, k) = v(n − k)
k−1


 θ
θ
1
1
k n−k
k
−
− (1 − y) (n − k)y n−k−1 dy× − (1 − y) y
k
k
0
0


v(n − k) n − 1(1 − θ)k θn−k
= −
k−1
k



n − 1 (n − k) θ+v(n − k)(1 − y)k y n−k−1 dy.
(10)
k−1
k
0

Since
n−k
(n − 1)!
(n − 1)!
×==(k − 1)!(n − k)!
kk!(n − k − 1)!



n−1
k


,

we obtainn − 1 (n − k) θ
(1 − y)k y n−k−1 dyv(n − k)
k−1
k
0

 θ
n−1
(1 − y)k y n−k−1 dy.
= v(n − k)k0
By using equation (11) andβ(θ; n, k + 1) = v(n − k − 1)

n−1
k

 θ
0

(1 − y)k y n−k−1 dy,

we can rewrite equation (10) as follows:
v(n − k) n − 1(1 − θ)k θn−kβ(θ; n, k) = −
k−1
k
n−k
β(θ; n, k + 1).
+
n−k−1
The above equation is equivalent tov
β(θ; n, k + 1) β(θ; n, k)−=n − (k + 1)
n−k
k



n−1
k−1

Since
1
k



n−1
k−1



1
=
n

26

 
n
,
k



(1 − θ)k θn−k .

(11)we obtaink−1 β(θ; n, t + 1)t=1β(θ; n, t)−n − (t + 1)n−t

β(θ; n, k) β(θ; n, 1)
−
n−k
n−1
 
k−1

v n
=
(1 − θ)t θn−t .
n t
=

(12)

t=1

Due to the fact that β(θ; n, 1) = v(n − 1)θn /n, equation (12) is equivalent
to
k−1

v
β(θ; n, k) vθn
−
=
n−k
n
n
t=1

 
n
(1 − θ)t θn−t .
tTherefore, we obtainβ(θ; n, k) = v

k−1

t=0

n−k
n

Thus expression (9) was derived.

27

 
n
(1 − θ)t θn−t .
t