Optimal Prize Allocations in Contests
Zongwei Lu∗
Job Market Paper
October 13, 2016
Abstract
Given a fixed prize budget for a contest, what is the optimal prize allocation among contestants?
The answer depends on the objective of the contest designer, which typically is either to maximize the
total performance of all contestants or simply the champion’s performance. We try to shed light on this
question for both objectives in a standard model in which contestants are heterogeneous in skill and
exert effort to win a prize. We show that weak concavity of the reduced-form cost function leads to
optimality of single prize for both objectives, which generalizes the previous results in the literature. We
find a dual relationship between the cost function and the principal’s utility function (in particular, risk
attitude), which not only helps to provide intuition for the optimality but also directly provides results for
a principal with a different risk attitude. Surprisingly, with the traditional Cobb-Douglas functional form,
optimality of single prize continues to hold for arbitrary degree of convexity under maximal performance
objective when the number of contestants is three. On the contrary, if the reduced-form cost function
is piecewise linear, then it may be optimal to reward the runners-up if the function is convex enough.
When the number of prizes under consideration is two, there is an interesting relationship between the
two objectives. In the derivation of the results, a series of simple facts about the winning probability
functions is presented, which may be useful for future works in contest theory and multi-object auction
theory.
Keywords: Contests; Optimal architecture; Prizes; Incentives
JEL codes: D44; C78; L86; J31
∗ Department
of Economics, BI Norwegian Business School, [email protected].
I thank my supervisor, Christian Riis, for his superior supervision. I also thank Espen R. Moen, Etienne Wasmer, Steffen
Grønneberg, Eivind Eriksen, Plamen Nenov and seminar participants at Norwegian Business School for their kind and useful
discussions and comments. The usual disclaimer applies.
1
1
Introduction
Instead of contracts, contests may be used for procurements (either private or public), or more generally
incentivizing agents to exert effort, with the advantage of mitigating moral hazard problem (but may induce collusion). The contest designer, or the principal, typically has the freedom of choosing an arbitrary
multi-prize scheme (non-increasing in the rank) or simply a single prize, which immediately leads to the
following question: Should there be multiple prizes or just one prize? Or when should the principal set a
single prize and when multiple prizes? It turns out that this kind of question can be traced back to 1902
when Francis Galton asked:
A certain sum, say £100, is available for two prizes to be awarded at a forthcoming competition; the larger one for the first of the competitors, the smaller one for the second. How
should the 100 be most suitably divided between the two? What ratio should a first prize
bear to that of a second one? Does it depend on the number of competitors, and if so, why?
(Biometrika, Vol. 1, 1902.)
However, Galton (1902) did not specify the goal of the principal, which is a starting point of design.
There are typically two kinds of objectives in contests: either to maximize the total performance of
all contestants or the maximal performance from the champion. The former objective is prevalent in
sport tournaments, worker promotion, student competition (e.g., Spelling Bee, Math Olympiad) etc. For
example, the FIFA World Cup is wonderful because the performance of every team is wonderful. There is
no lack of examples of contests with the latter objective either. Research tournaments, sponsored either
by governments or private companies, are typically meant to find the most efficient solutions for practical
problems, e.g., vaccines, engines, standard of technological products, and so on. As a concrete example,
in 2006, Netflix announced a million dollar challenge for a recommendation algorithm (to recognize
customers’ preferences and recommend contents accordingly) that can improve the current one by 10%.
The emergence of internet even makes it possible for individuals to post questions on Q&A web sites and
reward the best answer (with coveted virtual coins) among the ones created by large amounts of talented
internet users.
In addition to the two objectives mentioned above, there are also some other popular objectives.
For example, in a party election of nominee for the presidential election, the objective of the principal,
i.e., the party members, is to select the most talented candidate. Moreover, in rent-seeking games, the
principal may be interested in minimizing the total performance (i.e., wasteful effort or investments, thus
a social loss). On the other hand, the study of the two common objectives can also be useful for these
objectives. For example, the minimization problem is just dual to the maximization problem. And in our
model, there could be two kinds of monotone equilibrium (but exclusive to each other), one increasing
and one decreasing. Thus, based on the condition for the direction of the monotonicity, one can decide
which contestant is the most talented one.
2
An important feature of many contests is that contestants typically are heterogeneous in the proficiency of the skill relevant to the contest. To compensate the disadvantage from the endowed low
proficiency, a contestant can exert more effort or make more investments, which is always costly. To
shed light on the question of optimal allocation of prizes, we study a model for contests in which contestants are heterogeneous in skill, exert effort to win a prize, and the cost of effort is not recoverable.
As a standard model, contestants’ types (skill) are of private information and independently, identically
distributed. Also, both the principal and the contestants are risk-neutral. 1
The principal has a fixed prize budget and considers only a reward scheme with fixed prizes for each
ranked winner. We assume that the principal observes the ordinal information of the performance of
the contestants. If the cardinal information about the performance is also observed, then it may allow
the principal to use a performance-based prize scheme. However, the performance-based prize scheme
may require an agreement on assessment of cardinality of performance among the contestants and the
principal. On the other hand, the minimum requirement for a fixed-prize scheme is only the agreement
on the ordinal assessment.
The basic feature of our model is that both skill and effort are allowed to enter both the production
function and cost function. Furthermore, the previous works typically specializes in one of the two
objectives. For example, Moldovanu and Sela (2001) study total performance objective and Chawla et al.
(2015) study maximal performance objective. In contrast, we study the two objectives together.
We first characterize the symmetric increasing equilibrium for the contest and provide a sufficient condition for the existence. The sufficient condition is not only technical but also economical. Ultimately, the
monotonicity of the equilibrium requires monotonicity of total cost in skill. For example, in an increasing
equilibrium, if the total cost of the performance of the champion is higher than another contestant at that
performance level, then that contestant can take over the gold medal by a marginally higher performance
but a lower total cost and consequently a higher net profit. And similar arguments holds for other prizes.
The sufficient condition implies that for any given performance level the total cost is decreasing in skill
(the type) and rules out those possibilities. Hence, in the equilibrium there is no profitable deviation for
a contestant with a lower cost efficiency to win a higher prize and similarly for high type contestants.
Assuming a separable reduced-form cost function, a sufficient condition is then given for the optimality of single prize for both objectives, which generalizes the results of the previous works. In particular,
Moldovanu and Sela (2001) study a similar and simpler model in which the number of prizes is limited
to two although there are N contestants and the principal enjoys the total performance. 2 Our work
shows that, for arbitrarily large numbers of contestants and prizes, the weak concavity (in performance)
of the reduced-form cost function is also valid for optimality of single prize under the total performance
1 The
risk-neutrality of the principal is not essential, as we discuss below.
they mention that the proof of the optimality of single prize in N -prize case is analogous to two-prize case, it is not
2 Although
clear how a complete proof based on their derivation in the two-prize case can be done.
3
objective. Moreover, the optimality of single prize for the maximal-performance objective is identified in
Chawla et al. (2015) with a linear model. Our result generalizes their result to non-linear production and
cost functions and explains why they only get optimality of single prize.
As in Moldovanu and Sela (2001), optimality of multi-prize allocation for the objective of total performance can be rationalized by strong enough convexity. Intuitively, if the cost function is convex enough,
then the marginal cost of performance for contestants with highest proficiency, who is the champion in
the equilibrium, is too high at the highest performance level. This implies that it is too costly to incentivize the champion to produce performance at the level induced by a single prize and it is “cheaper to
buy” marginal units of performance from runners-up. Hence, the optimality of multiple prizes under
total performance objective follows naturally since the principal does not care from whom the marginal
performance comes.
It is interesting to know whether optimality of multi-prize may ever hold for the objective of maximal
performance as well, either by the same convexity condition or else. The two objectives seem to be
very different. Specifically, as the discussion above suggests, cutting the first prize by a certain amount
and allocating it to runners-up may incentivize them to exert more effort. At the same time it seems
that the decrease in the first prize would typically reduce the performance of the champion. With total
performance objective, it may be beneficial for a principal to set multiple prizes, if the loss from the
champion (and possibly some other high ranked contestants) is over-compensated by the gain from the
other runners-up. This may also indicate that a multi-prize allocation is unlikely to be optimal for a
principal who enjoys only the performance of the champion. Such an intuition is confirmed by Chawla
et al. (2015) with a linear model.
To gain more knowledge, we consider the traditional Cobb-Douglas functional form for production
and costs. By limiting the number of contestants to three, we find that optimality of single prize still
stands under maximal performance objective, no matter how convex the cost function is, a result consistent with the rough intuition from above. This may sound surprising given the fact that the cost function
has been relaxed to be nonlinear, as compared to the linear model in Chawla et al. (2015), and it can
exhibit arbitrarily strong convexity. The result also shows that optimal prize allocation under maximal
performance objective is not obvious if one simply relies on a convexity argument as under total performance objective.
Interestingly, when we use a different functional form for the cost function, a piecewise linear function, optimality of multi-prize can then be established even under maximal performance objective. Hence
this result may suggest that the exclusion of multi-prize schemes in the Cobb-Douglas case may be due
to the fact that the reduced-form cost function, derived from the Cobb-Douglas production function and
cost function, has a constant elasticity in performance for all types of contestants.
A closer look at the expected revenue reveals an interesting fact that a change in the cost function
mirrors a change in the principal’s utility function (or risk attitude), which may provide more intuitive
4
explanation for the optimality of different prize allocations. For example, if the cost function is more
concave (convex), then it mirrors a more risk-taking (risk-averse) principal with the cost function unchanged. This explains the results intuitively if we take the linear cost function as a starting point. The
linear cost function mirrors a risk-neutral principal who, roughly speaking, puts the same weight on all
types of contestants and prefers single prize with both objectives. In the equilibrium, the high type contestants always produced more performance than low types. And the high type contestants also produces
more under single prize scheme than multi-prize schemes in expectation (by the optimality of single
prize under maximal performance objective). Hence, if the cost function becomes concave and it mirrors
a more risk-taking principal with the same linear cost function, then the principal still prefers single prize
since she cares more about the high output of the high type contestants. The case of more convex cost
function is just in the other direction where a more risk-averse principal cares more about the outputs
from the low-type contestants. If the principal is strongly risk-averse, then a multi-prize scheme may be
optimal for her.
Since the relationship is dual, a change in the principal’s utility function, in particular risk attitude,
also maps into a change in the cost function. Hence, the results we obtain can be directly applied by a
principal with a different risk attitude. Interestingly, a comparison between the constant elasticity case
and increasing elasticity case (piecewise linear case) shows that not only the risk attitude but also its
exact form is relevant for the optimality.
Recall from our first and third result that the optimality of single prize or multi-prize is valid for
both objectives under similar conditions, i.e., a weak concave or piecewise linear convex cost function.
Furthermore, we also find that in the Cobb-Douglas case if there are only three contestants, it is always
optimal to set a single prize under maximal performance objective and not necessarily optimal under total
performance objective. It seems that there is a connection between the two objectives in terms of optimal
prize allocations. By limiting the number of prizes to two, we find that there is an interesting relationship
between the objectives: If it is optimal to set a single prize under total-performance objective, then it is
also optimal to set a single prize for the maximal-performance objective; The contrapositive statement is
also true, i.e., if it is optimal to set multiple prizes for the maximal-performance objective, then it is also
optimal to set multiple prizes for the total-performance objective.
The optimal prize allocation problem has been deemed to be challenging and the previous works
typically concentrated on simple settings and single objective. For example, Moldovanu and Sela (2001)
restricts to two prizes and total performance objective. In particular, they conclude their work by the following quote also from Galton (1902): “I now commend the subject to mathematicians in the belief that
those who are capable, which I am not, of treating it more thoroughly, may find that further investigations
will repay trouble in unexpected directions”. For maximal performance objective, Chawla et al. (2015)
consider only a simple linear model. Moreover, the previous results are typically derived by advanced
but complex computation. We show that it is possible to have a unified approach to both objectives in
5
an even more general setting. As an additional contribution to the literature perhaps, we show that the
results in this paper can be smoothly derived by a series of simple facts about the winning probability
functions, which lead to a novel and useful finding of single-crossing property of the functions. 3 Together
with a simple single-crossing amplification lemma, the finding may be useful for future works in contest
theory and more generally multi-object (heterogeneous) auction theory.
The rest of the paper is organized as follows. A brief review of related literature is given for the rest
of this section. We describe the model in detail in section 2. In section 3 the equilibrium is characterized
and the sufficient condition for the existence is given. We then present the condition for the optimality of
single prize for both objectives in section 4. In section 5 We consider the Cobb-Douglas functional form for
both production and cost function and show that it is always optimal to set a single prize under maximalperformance objective when the number of contestants is limited to three. Thus, in this case there is no
role for convexity. In section 6 we show that it may be optimal to incentivize the champion by rewarding
the runners-up using a piecewise linear function. The relationship between the two objectives in terms
of optimal prize allocation is identified in section 7 for two-prize case. Finally, section 8 concludes.
Related literature
There has been a large literature in contest theory studying contests with homogeneous contestants but
noised performances following the pioneering work on rent-seeking by Tullock (1980). In that literature
the design problem is typically either to minimize or maximize the total performance.
The earlier works for contests with heterogeneous contestants typically take a certain prize scheme
as given and study the design problem from aspects such as contestant entry and the optimal number of
rounds.
An important insight from the literature on contestant entry, such as Nalebuff and Stiglitz (1983),
Taylor (1995), Fullerton and McAfee (1999) and Che and Gale (2003), is that allowing too many contestants to compete is typically not optimal. The reason is that if there are too many contestants then the
probability of winning any prize for each contestant is too low. Hence there will be a overall reduction in
efforts and the total performance (and maximal performance) is reduced.
Clark and Riis (1998) compare the total performance from simultaneous and sequential designs for
contests with multiple identical prizes and heterogeneous contestants. Moldovanu and Sela (2006) study
a two-stage contest which in the first stage contestants are allocated into multiple divisions and the
winners from sub-contests compete with each other in the second stage. There is a single prize in each
sub-contest and in the final (second stage). They compare the outcome from the two-round scheme to
the outcome from a single-round contest with a single prize for both total performance and maximal
3
Moldovanu and Sela (2006) also find a single-crossing property of bid functions when all prizes are equal, which serves for
their setting of a particular two-stage contest. Our finding complements theirs. Moreover, ours is more primitive and the usage may
be more flexible and general.
6
performance objectives.
Kaplan et al. (2002) and Kaplan et al. (2003) consider particular forms of performance-dependent
reward schemes and find qualitatively different behavior of contestants than fixed-prize schemes, under
incomplete and complete information. Cohen et al. (2008) study optimal performance-dependent reward schemes and they find that the optimal reward may be decreasing in performance and there is no
possibility for optimality of multiple rewards.
Considering an exogenous minimal performance threshold, Megidish and Sela (2013) compare the
outcome from a contest with a winner-take-all design to the outcome in a random contest in which
contestants with performances higher than the threshold share the same probability of winning the single
prize. They find if the threshold is too high then the random contest may induce more total performance
and maximal performance.
Given a single prize, Gavious and Minchuk (2014) study the ratio of the cumulative performance of
top k contestants to total performance of all contestants, which can be used a measure of welfare loss.
Chawla et al. (2015) obtain similar results for the maximal performance.
The most related to our work are Moldovanu and Sela (2001) and Chawla et al. (2015) which have
been introduced enough above and will be explained more below. In addition, Barut and Kovenock
(1998) and Glazer and Hassin (1988) study optimal prize allocation in complete information environment.
A more detailed and excellent survey for the literature in optimal prize allocation is provided by Sisak
(2009).
2
The model
A risk-neutral principal initiates a contest in which there are N risk-neutral contestants. A total budget
PN −1
of $1 for the contest is divided into N − 1 prizes, denoted a1 , ∙ ∙ ∙ , aN −1 , i.e., n=1 an = 1.4 Without loss
of generality, we assume a1 ≥ a2 ≥ ∙ ∙ ∙ ≥ aN −1 ≥ 0.
The contestants differ in endowed proficiency of a certain skill relevant for the contest, which are
independent draws from a common atomless distribution F [s, sˉ] which has a density function f (s) > 0.
For convenience, we denote the inverse function of F (s) simply by s(F ) and similarly the inverse of F (x)
by x(F ).
By exerting effort e, a contestant with skill s incurs a cost c = c(s, e) and produces a performance
q = q(s, e). The q function is also called the production function or output function below. A contestant
can win at most one prize and the cost is not recoverable, i.e., the market value of performance and the
potential side-benefit of effort are normalized to zero. We assume qs0 > 0 and qe0 > 0. We also assume
4 It
is obvious that a positive amount of prize for the lowest rank is a waste of money since anyone ranked the lowest will earn
that prize and it does not provide any incentive. Mathematically, see footnote 5.
7
c0e > 0 but no assumption on the sign of c0s . The reason is that there can be two opposite effects when the
skill enters into the cost function. On one hand, a higher skill may imply a higher efficiency in producing
the performance. The efficiency improvement can be modeled in different ways, qs0 ≥ 0 or c0s ≤ 0 or both.
That is, qs0 ≥ 0 implies that for a given effort level a higher skill always leads to a better performance,
which seems natural, while c0s ≤ 0 implies that for a given effort level a more skilled contestant is more
efficient in cost-saving, either physically or mentally or both. On the other hand, a more skilled contestant
typically earns a higher hourly wage which implies the unit cost of effort is higher for him. Therefore,
we leave the sign of c0s undetermined. And it turns out that the crucial point is the sign of the partial
derivative of the reduced-form cost function derived from both the cost and production function, as we
will see below.
The principal observes and values only q (but not s and e separately) and ranks the performances
according to observed q. When the performance of each contestant is (equally) useful for the principal,
we say that the objective of the contest is to maximize the total performance; if the principal only enjoys
the performance from the champion and the performance of all other contestants are not useful, we say
that the objective of the contest is to maximize the maximal performance.
For a contestant with skill s, s is exogenous for him and a choice of effort e translates immediately
into a choice of q and vice versa. This implies that the contest can be viewed as an auction in which
contestants are bidders and their bids are their q which is received by the principal. Notice that here
bidders share known and common values for the prizes, i.e., the value of the nth prize is known and the
same across bidders. Because the cost is not recoverable, it can also be regarded as the payment from the
bidder. However, what a bidder pays is not necessarily the same as what the principal receives. Hence,
the contest may be thought as a generalized all-pay auction.
If a contestant is awarded $π, then the expected payoff for him is
u(s, e) = π − c(s, e)
Define v(s, e) ≡
q(s,e)
c(s,e) ,
and the equation above can be rewritten as
v(s, e)u(s, e) = πv(s, e) − q(s, e).
(1)
The inverse of v(s, e) can be interpreted as the average cost of the performance.
There are some works in the literature assuming that the average cost depends only on the skill level,
not the effort level, e.g., Chawla et al. (2015) and DiPalantino and Vojnović (2009), so that the contest
mirrors a standard all-pay auction. In that case, abuse the notation a little and write v(s, e) = v(s) and
(1) becomes
v(s)u(s, e) = πv(s) − q(s, e).
Maximizing u(s, e) w.r.t e is equivalent to maximizing v(s)u(s, e) w.r.t e because s is a constant for the
contestant. Also, a type s contestant determines his output level by choosing effort level e and vice versa.
8
Abusing notation a little again the equation above can be written as
v(s)u(s, e(q)) = πv(s) − q.
Hence, by such a way of modeling the contest is equivalent to an all-pay auction in which the bidders
have valuation v(s), their choice variable is q (the bid) and their payment is also q given the existence of
a symmetric increasing equilibrium.
If the average cost depends on the effort, maximizing u(s, e) w.r.t e is not equivalent to v(s, e)u(s, e)
and the contest does not mirror a standard all-pay auction and the results in all-pay auction theory do
not readily apply.
3
The increasing (in s) equilibrium
To start the analysis, we assume a symmetric increasing (in s) equilibrium exists by assuming that the
equilibrium performance function is increasing in s. And then we verify that indeed there is such a
equilibrium under a natural assumption on the reduced-form cost function introduced below.
In the equilibrium the probability of a contestant with skill s winning the nth prize, an , is
N −1
(1 − F (s))n−1 F (s)N −n ≡ Pn (F (s)).
πn (s) =
n−1
(2)
which implies
d
d
πn (s) =
Pn (F (s))f (s).
ds
dF (s)
It is clearest to derive the equilibrium if we do a transformation of the production function and cost
function. For a contestant with skill s, there is one to one mapping from e to q. Because by assumption q is
monotonically increasing in e, we can write e(s, q) as the effort required for a certain q. The corresponding
cost is then c(s, e(s, q)) ≡ C(s, q), which we call the reduced-form cost function . Because q = q(s, e(s, q))
implies that
∂e
∂q
= 1/ ∂q
∂e , we have
Cq0 (s, q) =
c0e (s, e(s, q))
.
qe0 (s, e(s, q))
In the equilibrium, effort is a function of skill, denoted by e(s). Then, abuse notation a little, the
equilibrium production function is given by q(s) ≡ q(s, e(s)). The incentive compatibility condition says
that in the equilibria,
s = arg max
ŝ
which implies
N
−1
X
N
−1
X
n=1
πn (ŝ)an − C(s, q(ŝ))
πn0 (s)an = Cq0 (s, q(s))q 0 (s).
n=1
9
(3)
Notice that the equilibrium production function is given in differential equation (3) implicitly, which in
turn gives the equilibrium effort function through the definition of the equilibrium production function
q(s, e(s)). Alternatively, we can also write
q(s; {an }) =
Z
s
0
N
−1
X
1
π 0 (x)an dx.
Cq0 (x, q(x; {an })) n=1 n
(4)
The following definition is repeatedly used in the proofs.
Definition 1 (Single-crossing). A function defined on a certain interval has single crossing property if it
crosses zero only once and from below. If function g1 crosses another function g2 only once and from
below, then we say g1 single-crosses g2 .
Lemma 1.
PN −1
n=1
πn0 (s)an ≥ 0 or
PN −1
n=1
Pn0 (F )an ≥ 0.
Proof. The proof uses the simple fact that
PN
Pn (F ) = 1 for any F , which simply says that any contesPN
tant can win one prize for sure if there are N prizes available. This implies n=1 Pn0 (F ) = 0. Combined
n=1
with the single-crossing property of Pn0 (F ), a discrete version of the amplification lemma in the next
section can be used to establish the statement because {an } is a non-increasing (in n) sequence. And the
statement simply says that in the increasing equilibrium, the expected gross pay-off for a contestant is
increasing in his type given any non-increasing sequence {an }. See Appendix A for details.
By assumption c0e > 0 and qe0 > 0, which implies Cq0 (s, q) > 0. Thus q 0 (s) > 0, which provides a
necessary condition for the existence of the equilibrium. For sufficiency, We resort to the single crossing
00
difference condition (from Athey (2001) or Milgrom (2004)) which is satisfied if −Cqs
(s, q(ŝ))q 0 (ŝ) > 0.
Because we are considering the increasing (in s) equilibrium, q 0 (∙) > 0. Hence, a sufficient condition for
the existence of the equilibrium is
00
Cqs
(s, q) < 0.
(5)
The condition says that the marginal cost of performance, hence the total cost and average cost, is lower
for a contestant with a higher proficiency of skill, for any given performance level.
Theorem 1. Assume (5) holds. There exist a symmetric increasing equilibrium in the contest. The equilibrium performance function is implicitly given by (3).
00
If Cqs
(s, q) > 0, then a decreasing in s equilibrium exists. If we define t ≡ 1s , then it is an increasing
(in t) equilibrium. It is then intuitive that for any decreasing in s equilibrium there is a corresponding
increasing in s equilibrium with a different distribution. Since the analysis below does not rely on any
specific distribution functions, the results are also valid for the decreasing equilibrium.
10
4
Separable reduced-form cost function
The principal receives
R sˉ
0
q(s)f (s)ds from a contestant in expectation. Denote the skill of contestants by
S1 , ∙ ∙ ∙ , SN . Then, if the objective of the principal is total performance, the expected revenue is
Z sˉ
Z sˉ
E[q(S1 ; {an }) + ∙ ∙ ∙ + q(SN ; {an })] =
q(s1 ; {an })f (s1 )ds1 + ∙ ∙ ∙ +
q(sN ; {an })f (sN )dsN
s
=N
Z
(6)
s
sˉ
s
q(s; {an })f (s)ds.
On the other hand, the expected performance from the champion, i.e., the principal’s expected revenue
under maximal performance objective, is
E[q(F
1:N
; {an })] =
Z
sˉ
s
q(s; {an })N F (s)N −1 f (s)ds.
where F 1:N represents the first highest order statistics from N independent draws of distribution F . The
two revenue function can be represented together by 5
Z sˉ
Ri =N
q(s; {an })F (s)i f (s)ds
s
=N
Z
s
sˉ Z s
s
N
−1
X
1
0
π
(x)a
dx
F (s)i f (s)ds
n
Cq0 (x, q(x; {an })) n=1 n
where i = 0, N − 1. That is, R0 is the expected revenue under the total performance objective and RN −1
the maximal-performance objective.
The principal chooses {an } to maximize Ri . For tractability, we assume that the reduced-form cost
function C(s, q) is separable in s and q, either multiplicatively or additively. It will be clear that the
additive case is even simpler and our results in the multiplicative case are also valid in the additive case.
Hence, we assume that
C(s, q) =
1
B(q).
A(s)
By the assumption above, B 0 (q) > 0. The single crossing difference condition implies that A0 (s) > 0. The
differential equation for the equilibrium performance function becomes 6
A(s)
N
−1
X
πn0 (s)an = B 0 (q(s))q 0 (s),
n=1
5 If
there are N prizes, the revenue would be
Ri = N
0 (x) ≤ 0 and
Since πN
6 If
1
Cq0 (x,q(x;{an }))
Z
0
s
ˉZ s
0
N
X
1
0
π
(x)a
dx
F (s)i f (s)ds.
n
n
Cq0 (x, q(x; {an })) n=1
≥ 0 for any x and {an }, the optimal allocation for both objectives should always set aN = 0.
C(s, q) is additively separable, then the differential equation does not contain A(s). Since all our results do not depend on
the behavior of A(s), they are all valid as well for the additively separable C(s, q).
11
which can now be solved explicitly as
B(q(s)) =
Z
s
A(x)
s
N
−1
X
πn0 (x)an dx.
n=1
By the fact of monotonicity of B(∙), we have
q(s) =B −1
Z
s
A(x)
s
N
−1
X
n=1
πn0 (x)an dx .
Throughout the proofs of this paper, the following intuitive argument is repeatedly used, which we
state as a lemma without a proof.
Lemma 2 (Amplification lemma). On an interval (a, b), suppose g(F ) has the single-crossing property. Then
for a non-decreasing function h(F ) ≥ 0 , we have
Z
b
a
g(F )dF ≥ 0 =⇒
Z
b
a
g(F )h(F )dF ≥ 0.
If we define g(F ) = g1 (F ) − g2 (F ), then it is clear that the following is also true: if a function g1 (F )
single-crosses another function g2 (F ), then
Z
b
a
(g1 (F ) − g2 (F ))dF ≥ 0 =⇒
Z
b
a
(g1 (F ) − g2 (F ))h(F )dF ≥ 0.
The following lemma gives important insights for the optimality of single prize for both objectives.
Lemma 3. On the unit interval, except for the end points, for j ∈ {2, ∙ ∙ ∙ , N − 1}, as functions of F ,
• P1 (F ) − Pj (F ) has the single-crossing property, or P1 (F ) single-crosses Pj (F ).
• P10 (F ) − Pj0 (F ) has the single-crossing property, or P10 (F ) single-crosses Pj0 (F ).
• Let Fj∗ be the point at which P10 (F ) − Pj0 (F ) = 0 and F ∗∗ the point at which P1 (F ) − Pj (F ) = 0.
Then Fj∗∗ ≥ Fj∗ .
Proof. See Appendix B.
Figure 1 illustrates Lemma 3 nicely when N = 5.
In addition, there is a simple fact about the ex-ante probability of winning nth prize for a contestant:
for any n,
Z
0
1
Pn (F )dF =
1
,
N
(7)
which simply says that ex ante the probability of winning the nth prize for any contestant is N1 , equal
RF
for everyone. Moreover, N 0 Pn (F )dF is a valid distribution function. This fact, together with (7) and
RF
the single-crossing property of P1 (F ) − Pj (F ), implies that distribution N 0 P1 (F )dF stochastically
12
N =5
N =5
1
4
P1
P2
P3
P4
P1′
P5
P2′
P3′
P4′
P5′
3
2
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
F
-1
-2
-3
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
F
-4
Figure 1: Pn (F ) and Pn0 (F ) when N = 5, n = 1, ∙ ∙ ∙ , N.
dominates distribution N
F . Also, (7) implies
RF
0
Pj (F )dF on the first-order. That is,
Z
1
0
RF
0
P1 (F )dF ≤
RF
0
Pj (F ) dF for any
[P 1 (F ) − Pj (F )]dF = 0.
(8)
Then, combining the amplification lemma, Lemma 3 and (8), we have the following lemma.
Lemma 4. For a non-negative function V(F ) with V 0 (F ) > 0 on the unit interval, except for the end points,
RF
V(F ) P10 (F ) − Pj0 (F ) dF , as a function of F , has the single-crossing property. Moreover,
0
Z
1
0
Z
F
0
V(F ) P10 (F ) − Pj0 (F ) dF dF ≥ 0.
(9)
Proof. Let Fj∗ be the one defined in Lemma (3). The amplification lemma is used with the help of a new
function defined as follows,
Vˉj (F ) =


V(F )

V(F ∗ )
j
if F ≤ Fj∗
if F > Fj∗ ,
with which the reader may already be able to visualize the geometry of a complete proof. See Appendix
C for details.
By the amplification lemma again and Lemma 4, the following proposition is easy to prove.
Proposition 1. If B(q) is concave or linear, then it is optimal to set a single prize for both objectives of total
performance and maximal performance, i.e., for i = 1, N − 1,
Z sˉ
Z sˉ
q(s; {a1 = 1})F (s)i f (s)ds ≥
q(s; a1 6= 1)F (s)i f (s)ds.
0
0
Proof. The revenue of the principal is
Ri = N
Z
sˉ
s
B −1
Z
s
A(x)
s
N
−1
X
n=1
13
πn0 (x)an dx F (s)i f (s)ds.
(10)
Let
τj (ε) ≡
Z
sˉ
B −1
s
Z
s
s
N
−1
X
A(x) π10 (x)(a1 + ε) + πj0 (x)(aj − ε) +
πn0 (x)an dx F (s)i f (s)ds.
(11)
n6=1,j
Then because {a1 + ε, a2 , ∙ ∙ ∙ , aj−1 , aj + ε, aj+1 , ∙ ∙ ∙ , aN −1 } is just another admissible prize allocation
similar to {an }, without loss of generality, we can write
Z sˉ R s A(x) π 0 (x) − π 0 (x)dx
d
1
j
s
F (s)i f (s)ds.
τj (ε) =
0 (q(s))
dε
B
s
The inverse function of distribution function F (s) is denoted by s(F ) and F (x) by x(F ). By change of
variable method,
Z
d
τj (ε) =
dε
1
0
1
B 0 (q(s(F )))
Z
F
0
A(x(F )) P10 (F ) − Pj0 (F ) dF F i dF.
Because A0 (x) > 0, A(x(F )) is increasing in F . By Lemma 4,
RF
0
(12)
A(x(F )) P10 (F ) − Pj0 (F ) dF F i also
crosses zero only once and from below. Moreover, since F i is non-decreasing, by the amplification lemma,
Z
1
0
Z
F
0
A(x(F )) P10 (F ) − Pj0 (F ) dF F i dF ≥ 0.
If B(q) is concave or linear, then it is obvious that
d
dF
B 0 (q(s(F ))) =
d
d
d
0
dq B (q(s(F ))) ds q(s) dF s(F )
≤ 0.
That is, the inverse of B (q(s(F ))) is non-decreasing in F and non-negative. By the same amplification
0
argument as above, we have
d
dε τj (ε)
≥ 0 and this completes the proof.
The proposition above generalizes the results in previous works. In Chawla et al. (2015) it is assumed
q = se, c = e. Hence B(q) = q, which is linear, in their model with maximal performance objective.
Proposition 1 shows that optimality of single prize under maximal performance objective is valid for any
separable reduced-form cost function with weak concavity in performance and consequently explains why
they only get single prize optimality. On the other hand, Moldovanu and Sela (2001) limit the number of
prizes to two for the total performance objective while our result generalizes to arbitrary large number
of prizes.
More interestingly, equation (10) reveals the dual relationship between the structure of the reducedform cost function and the principal’s utility function. In particular, when B(q) = q, the revenue is
simpliy
Ri = N
Z
s
sˉ Z s
s
A(x)
N
−1
X
πn0 (x)an dxF (s)i f (s)ds.
n=1
Comparing the equation above to (10) we can see that a change in the performance part of reduced form
cost function, say another B(q) 6= q, mirrors a change of the principal’s utility function into B −1 (q) while
leaving the cost function unchanged and hence the contest unchanged. For example, a more convex B(∙)
means a more concave B −1 (∙) which mirrors a more risk averse principal. With this observation, the
14
results in the proposition above can be intuitively explained, as in the introduction. On the other hand,
a more risk averse principal can also be mapped into a more convex B(∙). Hence, this observation also
provides results directly for the principal with different risk attitude.
5
Incentivizing the champion by rewarding the champion only
We already know from Moldovanu and Sela (2001) that it may be optimal to set multiple prizes under
total performance objective if the reduced-form cost function is convex enough in effort. It is interesting
to know wether it may also be optimal to use a multi-prize scheme to maximize the maximal performance,
i.e., to incentivize the champion by rewarding the runners-up.
In this direction, the role of convexity is more subtle, as we show below that if the reduced-form cost
function has a constant elasticity in performance then there is no role for convexity when the number of
the contestants is three.
First, rewrite
Cq0 (s, q(s))
c0 (s, e(s))
= e0
=
qe (s, e(s))
e(s)
0
c(s,e(s)) ce (s, e(s))
e(s)
0
q(s,e(s)) qe (s, e(s))
c(s, e(s))
ξc,e (s, e(s)) c(s, e(s))
≡
,
q(s, e(s))
ξq,e (s, e(s)) q(s, e(s))
where ξq,e represents the elasticity of output q w.r.t effort e and likewise for ξq,e . Thus if the primitive production function and cost function have constant elasticity in effort, then the marginal cost of
performance depends only on the average cost, up to a multiplicative constant.
Consider the traditional Cobb-Douglas form for the production function and cost function. Assume
c(s, e) = sα eβ ,
q(s, e) = sγ eδ ,
where β, δ > 0.
It is straightforward to verify that
1
C(s, q) = s−ω q σ ,
where ω ≡ γ( βδ −
α
γ)
and σ ≡
δ
β.
And we have Cq0 (s, q) =
(13)
1
1 −ω σ
00
q −1 , Cqs
(s, q)
σs
1
= − ωσ s−ω−1 q σ −1 . Thus
the reduced-form cost function has a constant elasticity in performance. By assumption Cq0 > 0. The
single crossing difference condition implies that ω > 0 and we assume it holds.7
Differential equation (3) becomes
sω
N
−1
X
πn0 (s)an =
n=1
which implies
q(s) =
7 There
Z
1
1
1
d
q(s) σ −1 q 0 (s) =
q(s) σ ,
σ
ds
s
x
0
ω
N
−1
X
n=1
πn0 (x)an dx
σ
.
(14)
is a danger that one can easily obtain a wrong conclusion of multi-prize optimality for some ω < 0 and N = 3 if one
mistakenly applies the increasing equilibrium (in s) performance function while the equilibrium is in fact decreasing.
15
The revenue under maximal performance objective is then
E[q(F
1:N
)] = N
Z
0
sˉ Z s
x
ω
0
N
−1
X
πn0 (x)an dx
n=1
σ
F (s)N −1 f (s)ds.
By Proposition 1, we know that if σ ≥ 1, then it is optimal to set a single prize. For 0 < σ < 1, we
show that the optimality of single prize still stands in the following proposition, assuming there are only
3 contestants.
Lemma 5.
PN −1 0
Pn (F )an
Pn=1
N −1
n=1 Pn (F )an
Proof. Because
P10 (F )
P1 (F )
=
<
P10 (F )
P1 (F )
N −1
F ,
and
d
dF
P (F )
PN −11
n=1 Pn (F )an
> 0.
the first inequality is equivalent to
F
N
−1
X
n=1
Pn0 (F )an <(N − 1)
N
−1
X
n=1
Pn (F )an .
For each n ≥ 2,
F Pn0 (F )an − (N − 1)Pn (F )an
=1 − n < 0.
N −1
n−2 F N −n a
n
n−1 (1 − F )
Therefore, the first inequality holds and the second one follows immediately since both P1 (F ) and
PN −1
n=1 Pn (F )an are positive.
Proposition 2. Consider the case with the reduced-form cost function given by (13). If there are only 3
contestants, it is always optimal to set a single prize for a principal with maximal performance objective, i.e.,
for any σ > 0,
Z
0
sˉ Z s
0
xω π10 (x)dx
σ
F (s)
N −1
f (s)ds ≥
Z
0
sˉ Z s
x
ω
0
2
X
πn0 (x)an dx
n=1
σ
F (s)N −1 f (s)ds.
Proof. See Appendix D.
The result above may sound surprising, given the fact that the cost function can exhibit arbitrary
degree of convexity in performance. In particular, Moldovanu and Sela (2001) have given an example of
optimality of multi-prize allocation under total performance with σ =
1
2,
our result shows the fact that
the qualitative result of optimal prize allocation problem under maximal performance objective can be
very different from the one under total performance objective.
16
6
Incentivizing the champion by rewarding the runners-up
We now show that if the reduced-form cost function is piecewise linear in performance, then it can be
optimal to have multiple prizes for the principal with either objective. Specifically, assume 8


ηq
0 ≤ q ≤ qj∗
B(q) =

θq − λ q > q ∗
j
(15)
where ηqj∗ = θqj∗ − λ such that λ = (θ − η)qj∗ . By convexity assumption we must have that θ > η > 0 and
λ > 0. Then
B (q) =
0


η
0 ≤ q ≤ qj∗

θ
q > qj∗
The expected revenue takes the form as in (10) and similarly we define a τj (ε) function as in (11). Then,
similar to (12), we have
Z F
1
0
0
A(x(F )) P1 (F ) − Pj (F ) dF F N −1 dF.
0
0 B (q(s(F ))) 0
RF
We know from Lemma 4 that 0 A(x(F )) P10 (F ) − Pj0 (F ) dF has single crossing property on the unit
RF
interval (except the two end-points). Let Fj∗ be the point at which 0 A(x(F )) P10 (F ) − Pj0 (F ) dF = 0
d
τj (ε) =
dε
Z
1
such that for F ≷ Fj∗ we have
Z
Then clearly
and
Z
Fj∗
0
Z
1
Fj∗
Z
Z
F
F
0
F
0
A(x(F ))
0
A(x(F ))
P10 (F )
P10 (F )
−
−
Pj0 (F )
Pj0 (F )
dF ≷ 0.
dF F N −1 dF < 0,
A(x(F )) P10 (F ) − Pj0 (F ) dF F N −1 dF > 0.
Furthermore, let qj∗ = q(s(Fj∗ ))9 . Then we can write
Z Fj∗ Z F
d
1
0
0
A(x(F )) P1 (F ) − Pj (F ) dF F N −1 dF
τj (ε) =
dε
η 0
0
Z 1 Z F
1
0
0
+
A(x(F )) P1 (F ) − Pj (F ) dF F N −1 dF.
Fj∗ θ 0
8 There
qj∗ .
is a small technical issue here. The assumed piecewise linear function B(∙) has a kink at qj∗ , i.e., not differentiable at
Thus it may seem the differential equation (3) is not appropriate to describe the equilibrium q(s) function. However, one point
on a atomless support has zero measure. Hence, the issue is trivial. Moreover, one can always smooth the kink away by defining
a smooth function on a small segment containing qj∗ such that B(∙) is differentiable everywhere and the derivative is continuous.
And the segment can be arbitrarily small such that it does not affect the qualitative conclusion below. For the purpose of clear
exposition, we take the limiting case as legal.
9 Since we have an degree of freedom from λ, for any q ∗ , there is always a corresponding λ such that B(∙) is continuous at q ∗ .
j
j
17
Since both integrals are finite, for any given j, there exists a φ ≡
θ
η
large enough such that
d
dε τj (ε)
< 0.
We then have the following result.
Proposition 3. If the B(∙) function in the reduced-form cost function takes the form as in (15), then a
multi-prize allocation is optimal for both objectives when
θ
η
is large enough.
In particular, if there are only 3 contestants, then it is optimal to set a1 = a2 =
1
2
when
θ
η
is large
enough. Using the intuition based on a mapping into the principal’s risk attitude, the result above is
clearest. That is, with the piecewise B(∙), it is the same to have a principal with utility function B −1 (∙)
but linear cost function. When
θ
η
is large enough, the principal is so risk averse such that marginal utility
at q > qj∗ is constant and too small while marginal utility at q ≤ qj∗ is also constant but large enough.
In the equilibrium q ∗ is supplied by lower type contestants. Hence, it is most important to incentivize
those low type contestants to compete at a performance level small but toward qj∗ and consequently the
champion’s performance will be close to qj∗ . That can be simply implemented by having multiple prizes.
The results from the Cobb-Douglas case in the previous section and this piecewise linear case reveal
the fact that not only the convexity but also the elasticity of the reduced-form cost function is important
for the optimality prize allocations with maximal performance objective. In particular, the piecewise B(q)
has an increasing elasticity for q > qj∗ , always higher than the constant elasticity for q ≤ qj∗ .
7
The relationship between the two objectives when the number of
prizes is 2
An interesting observation from Proposition 1 and Proposition 3 is that the optimality condition of single
prize or multi-prize is valid for both objectives under the similar qualitative condition. In Proposition 2
we find that in the Cobb-Douglas case convexity is irrelevant for the optimal prize allocation for maximal performance objective when there are only 3 contestants, i.e., it is always optimal to set a single
prize. On the other hand, enough convexity may lead to optimality of multi-prize for total performance
objective even in that case. We may wonder if there is any relationship between the two objectives. For
example, does the optimality of single prize under total performance objective imply optimality of single
prize under maximal performance, or the opposite? In general, it is difficult to give an analytic conclusion. However, if we limit the number of prizes to two, then we can show that the following interesting
relationship between the optimal prize allocations for the two objectives holds.
Proposition 4. Suppose only two prizes are considered. If it is optimal to set a single prize when the objective
is total performance, then it is also optimal to set a single prize for maximal performance objective, i.e., for a
18
certain B(∙) function, if
Z
then
Z
sˉ
B −1
0
sˉ
B −1
0
Z
s
0
Z
s
0
Z
A(x)π10 (x)dx f (s)ds ≥
Z
A(x)π10 (x)dx F (s)N −1 f (s)ds ≥
sˉ
B −1
0
sˉ
B −1
0
Z
Z
s
A(x)
0
2
X
n=1
s
A(x)
0
2
X
n=1
πn0 (x)an dx f (s)ds,
πn0 (x)an dx F (s)N −1 f (s)ds.
On the other hand, the statement above implies that If it is not optimal to set a single prize when the objective
is maximal performance, then it is not optimal to set a single prize for total performance objective either.
Equivalently, if it is optimal to set two positive prizes when the objective is maximal performance objective,
then it is also optimal to set two prizes for total performance objective, i.e., if for some a1 ≥ a2 > 0,
Z
sˉ
B
−1
0
Z
s
0
A(x)π10 (x)dx
F (s)
N −1
f (s)ds <
Z
then there is some admissible ã1 ≥ ã2 > 0 such that
Z s
Z
Z sˉ
B −1
A(x)π10 (x)dx f (s)ds <
0
0
sˉ
B
−1
0
sˉ
B −1
0
Z
Z
s
A(x)
0
=⇒
1
B
−1
0
Z
0
1
B
−1
Z
Z
F
A(x(F ))
0
F
A(x(F ))
0
P10 (F )
P10 (F )
dF dF ≥
dF F
N −1
Z
1
B
−1
0
dF ≥
Z
Z
1
B
0
F
s
A(x)
0
2
X
n=1
A(x(F ))
0
−1
Z
n=1
F
Z
F
0
=a2
Z
A(x(F ))P10 (F )dF −
F
0
Z
F
A(x(F ))
0
X
2
n=1
2
X
n=1
Pn0 (F )an
A(x(F ))
0
F (s)N −1 f (s)ds,
πn0 (x)ãn dx f (s)ds.
X
2
By assumption a1 + a2 = 1. Define
χ(F ) ≡
πn0 (x)an dx
n=1
Proof. The first statement in the proposition is equivalent to
Z
2
X
(16)
dF dF
Pn0 (F )an
dF F N −1 dF.
Pn0 (F )an dF
A(x(F )) P10 (F ) − P20 (F ) dF
RF
Obviously, from Lemma 4 we know that χ(F ) has the single-crossing property. Because 0 A(x(F ))P10 (F )dF
RF
RF
P2
and 0 A(x(F )) n=1 Pn0 (F )an dF are both non-negative and increasing, this implies that 0 A(x(F ))P10 (F )dF
RF
P2
single-crosses 0 A(x(F )) n=1 Pn0 (F )an dF , which, by the monotonicity of B(∙) function, in turn implies
RF
RF
that B −1 ( 0 A(x(F ))[P10 (F )]dF ) single-crosses B −1 ( 0 A(x(F ))[P10 (F )]dF ). Hence by the amplification
lemma, the statement in (16) is true.
8
Conclusion
By studying a standard model with heterogeneous contestants in a private information environment, we
generalize the previous results for optimality of different prize allocations for two common objectives. In
19
particular, we find that the previous result on the optimality of single prize under maximal performance
objective is due to linearity of the reduced-form cost function. We also find that optimality of multi-prize
is also possible even under maximal performance objective. We highlight the dual relationship between
the cost function and the principal’s utility function, which provides more straightforward intuition for
optimality conditions and direct answers to the principal’s risk attitude. A comparison of different forms
of cost function illustrates the importance of the elasticity of costs for the optimal prize allocation. New
findings on the properties of the winning probability function may be useful for future works.
A future direction of extension may be to study the scenarios where the contestants are risk averse.
As is well-known in auction theory that risk-aversion of bidders is challenging to deal with, studies for
risk-averse contestants will be confronted by the same degree of difficulty. We will be delighted to see
further progress in this direction.
20
Appendices
A
Proof of Lemma 1
First,
N
−1
X
πn0 (s)an = f (s)
n=1
N
−1
X
n=1
N
−1
X
dPn (F (s))
dPn (F (s))
≥ 0, if
≥ 0.
dF (s)
dF (s)
n=1
It is clear to see that
N
N
N
−1
X
X
X
N −1
Pn (F ) =
Pn (F ) = 1,
Pn0 (F ) = 0,
Pn0 (F ) ≥ 0
(1 − F )n−1 F N −n ,
n−1
n=1
n=1
n=1
0
since PN
(F ) ≤ 0 for any F . Also, for n ∈ {1, ∙ ∙ ∙ , N − 1},
N −1
0
(1 − F )n−2 F N −n−1 N − n − (N − 1)F .
Pn (F ) =
n−1
For any F , if and only if n > (N − 1)(1 − F ) + 1, Pn0 (F ) < 0. For a given F , let Pn0 ∗ (F ) be the first negative
PN −1 0
Pn∗ −1 0
element in the sequence {Pn0 (F )}N
n=1 . For a1 ≥ ∙ ∙ ∙ ≥ aN ≥ 0,
n=1 Pn (F )an =
n=1 Pn (F )an +
PN −1 0
Pn∗ −1 0
PN −1 0
PN −1 0
n=n∗ Pn (F )an ≥
n=1 Pn (F )an∗ +
n=n∗ Pn (F )an∗ = an∗
n=1 Pn (F ) ≥ 0. The argument above
is an application of the discrete version of the amplification lemma.
B
Proof of Lemma 3
First we show that P1 (F ) − Pj (F ) crosses zero only once and from below, i.e., P1 (F ) and Pj (F ) intercept
with each other only once within the unit interval.
where θ =
N −1 1/(j−1)
j−1
N −1
−
(1 − F )j−1 F N −j
P1 (F ) − Pj (F ) =F
j−1
N −1
N −j
j−1
j−1
=F
F
−
(1 − F )
j−1
≡F N −j F j−1 − [θ(1 − F )]j−1 .
N −1
(17)
≥ 1. Geometrically, F j−1 and [θ(1 − F )]j−1 both move along the same curve,
i.e., F j−1 . As F increases from 0 to 1, F j−1 increases from 0 to 1 and [θ(1 − F )]j−1 decreases from θj−1
to 0, both monotonically. Hence, they intercept each other only once and F j−1 is lower (higher) than
[θ(1 − F )]j−1 for low (high) values of F .
Second, it is straightforward to derive
N −1
0
(1 − F )j−2 F N −j−1 (N − j) − (N − 1)F ,
Pj (F ) =
j−1
N −1
0
0
N −j−1
j−1
j−2 N − j
P1 (F ) − Pj (F ) =(N − 1)F
F
−
(1 − F )
−F .
j−1
N −1
21
(18)
Define
G(F ) ≡
N −1
j−2 N − j
(1 − F )
−F
j−1
N −1
Then
N −j
N −1
N −j
j−3
1−
(1 − F )
+ (j − 1)
G (F ) = −
−F
j−1
N −1
N −1
0
Function G(F ) ≶ 0 for F ≷
on [0,
N −j
N −1 ]
and negative on
N −j
0
N −1 and G (F ) ≤ 0
N −j
j−1
[N
−1 , 1], while F
Thus, there is a unique point F at which
∗
0.
Finally, when
N −j
N −1 .
Hence, G(F ) is positive and decreasing
is always increasing and has a value zero at the origin.
d
P1 (F )−Pj (F ) = 0. And for F ≷ F ∗ , dF
P1 (F )−Pj (F ) ≷
P1 (Fj∗ ) − Pj (Fj∗ ) = 0, P1 (Fj∗ ) − Pj (Fj∗ ) ≤ 0. By the first result above, we have
d
dF
Fj∗∗ ≥ Fj∗ .
C
d
dF
for F ≤
Proof of Lemma 4
From (8),
Z
1
0
P1 (F ) − Pj (F ) dF = 0.
By assumption V 0 (F ) > 0. From Lemma 3 we know that both P1 (F ) − Pj (F ) and P10 (F ) − Pj0 (F ) have
the single-crossing property. Let Fj∗ be the point at which P10 (F ) − Pj0 (F ) = 0. Define a new function
Vˉj (F ) =


V(F )
if F ≤ Fj∗

V(F ∗ )
if F > Fj∗ .
j
Then by the amplification lemma,
Z
1
0
Vˉj (F ) P1 (F ) − Pj (F ) dF ≥ 0,
(19)
because Vˉj (s) is a non-decreasing function. Inequality (19) is equivalent to
Z
1
0
Z
F
0
0
0
ˉ
Vj (F ) P1 (F ) − Pj (F ) dF dF ≥ 0.
Recall the single-crossing property of P10 (F ) − Pj0 (F ) and the definition of Fj∗ .
When F ≤ F ≤ Fj∗ , P10 (F ) − Pj0 (F ) ≤ 0 and V(F ) ≤ V(F ), which implies V(F ) P10 (F ) − Pj0 (F ) ≥
Vˉj (F ) P10 (F ) − Pj0 (F ) . Hence, for any F ≤ Fj∗ ,
Z
F
0
Z
V(F ) P10 (F ) − Pj0 (F ) dF ≥
22
F
0
Vˉj (F ) P10 (F ) − Pj0 (F ) dF .
(20)
For any F > Fj∗ , we write
Z
Z
F
0
F
0
V(F )
P10 (F )
−
Pj0 (F )
dF =
Z
Z
0
0
ˉ
Vj (F ) P1 (F ) − Pj (F ) dF =
Fj∗
0
Fj∗
0
V(F )
V(Fj∗ )
P10 (F )
−
P10 (F )
Pj0 (F )
−
Pj0 (F )
dF +
Z
dF +
F
Fj∗
Z
V(F )
F
Fj∗
P10 (F )
V(Fj∗ )
−
P10 (F )
Pj0 (F )
−
dF ,
Pj0 (F )
dF .
When F > Fj∗ , P10 (F ) − Pj0 (F ) > 0 and V(F ) > V(F ), which implies V(F ) P10 (F ) − Pj0 (F ) >
Vˉj (Fj∗ ) P10 (F ) − Pj0 (F ) . Hence, for any F > Fj∗ and consequently for any F , inequality (20) holds.
Therefore, for any F and any j,
Z
F
0
V(F )
P10 (F )
−
Pj0 (F )
dF ≥
Z
F
0
0
0
ˉ
Vj (F ) P1 (F ) − Pj (F ) dF ,
which implies
Z
1
0
Z
F
0
Z
V(F ) P10 (F ) − Pj0 (F ) dF dF ≥
1
0
Z
F
0
Vˉj (F ) P10 (F ) − Pj0 (F ) dF dF
≥0.
That is, (9) is true.
Furthermore, it is obvious that
RF
0
V(F ) P10 (F ) − Pj0 (F ) dF first crosses zero at some point when F
increases from 0 to 1, say F̃j . We must have that for F < F̃j the integral is negative and P10 (F̃j )−Pj0 (F̃j ) >
0 by the single-crossing property of P10 (F ) − Pj0 (F ) and the non-negativity of V(F ).10 Thus for F > F̃j ,
RF
0
0
V(F
)
P
(F
)
−
P
(F
)
dF > 0 by the single-crossing property of P10 (F ) − Pj0 (F ), which completes
1
j
0
the proof.
D
proof of Proposition 2
The general statement (for arbitrary number of prizes) is true if for V(F ) ≡ (s(F ))ω , which is increasing
in F when ω ≥ 0,
Z
1
0
Z
F
0
σ
Z
V(F ) P10 (F ) dF F N −1 dF ≥
1
0
Z
F
0
V(F )
NX
−1
n=1
σ
Pn0 (F )an dF F N −1 dF.
Let
τj (ε) =
≡
10 If
Z
Z
1
0
1
Z
F
0
σ
N
X
V(F ) P10 (F )(a1 + ε) + Pj0 (F )(aj − ε) +
Pn0 (F )an dF F N −1 dF
n6=1,j
I(F ; ε)σ F N −1 dF
0
P10 (F̃ ) − Pj0 (F̃ ) ≤ 0, then
R F̃
0
V(F ) P10 (F ) − Pj0 (F ) dF < 0, which contradicts the definition of F̃ .
23
(21)
Then the inequality is true if for every j ∈ {2, ∙ ∙ ∙ , N },
dτj (ε)
=
dε
Z
1
σI(F ; ε)σ−1
0
From Lemma 4 we know that I(F ; ε)
Z
−1
F
0
RF
0
V(F ) P10 (F ) − Pj0 (F ) dF F N −1 dF ≥ 0.
V(F )
P10 (F )
−
Pj0 (F )
dF F N −1 , as a function of F , also
has the single-crossing property. Because I(F ; ε)σ is positive and nondecreasing in F , thus by the amplification lemma, the inequality is true in this case if
Z
1
I(F ; ε)−1
0
Z
F
0
V(F ) P10 (F ) − Pj0 (F ) dF F N −1 dF ≥ 0.
Or because {a1 + ε, a2 , ∙ ∙ ∙ , aj−1 , aj + ε, aj+1 , ∙ ∙ ∙ , aN −1 } is just another admissible prize allocation, abus-
ing notation a little, the inequality above is equivalent to
Z
1
0
From Lemma 5 we know
RF
0
RF
0
V(F ) P10 (F ) − Pj0 (F ) dF N −1
dF ≥ 0.
F
PN −1 0
V(F )
n=1 Pn (F )an dF
d
dF
P1 (F )
PN −1
n=1 Pn (F )an
(22)
> 0.
Recall that P1 (F ) = F N −1 . Therefore, by Lemma 3, equation (8) and the amplification lemma,
Z
1
0
RF
0
RF
0
Z 1
P10 (F ) − Pj0 (F ) dF N −1
P1 (F )
dF =
[P 1 (F ) − Pj (F )] PN −1
dF ≥ 0.
F
PN −1 0
0
n=1 Pn (F )an
n=1 Pn (F )an dF
(23)
Thus (22) is true if for every F , if
RF
0
RF
0
RF
V(F ) P10 (F ) − Pj0 (F ) dF
P10 (F ) − Pj0 (F ) dF
0
≥ R F PN −1
PN −1 0
0
V(F )
n=1 Pn (F )an dF
n=1 Pn (F )an dF
0
(24)
Consider the special case where there are only 3 contestants. Thus we only need to look at the prize
allocations with a1 +a2 = 1. In order to prove inequality (24) for this case, we introduce a useful theorem
from Wijsman (1985).11
Theorem 2 (Wijsman’s inequality). Let μ be a measure on the real line R and let fi , gi (i = 1, 2) be four
R
Borel-measurable functions: R → R such that f2 ≥ 0, g2 ≥ 0, and fi gj dμ < ∞ (i, j = 1, 2). If f1 /f2 and
g1 /g2 are monotonic in the same direction, then
Z
Z
Z
Z
f1 g1 dμ f2 g2 dμ ≥ f1 g2 dμ f2 g1 dμ.
If in addition,
11 The
R
f1 g2 dμ > 0 and
R
star is always shining in the sky!
f2 g2 dμ > 0, then
R
R
f2 g1 dμ
f1 g1 dμ
R
≥R
.
f1 g2 dμ
f2 g2 dμ
24
In our case, let
f1 = V(F ), g1 = P10 (F ) − P20 (F ), f2 = 1, g2 = P10 (F )a1 + P20 (F )(1 − a1 ).
Obviously,
R
f1 g2 dμ > 0 and
R
f2 g2 dμ > 0 sine we have already shown in Lemma 1 that
0 for any F . Then because f1 /f2 = V(F ) is non-decreasing, (24) is true if
d
P10 (F ) − P20 (F )
≥ 0.
dF P10 (F )a1 + P20 (F )(1 − a1 )
It can be verified that
d
dF
P10 (F ) − P20 (F )
0
P1 (F )a1 + P20 (F )(1 − a1 )
=
Therefore, when N = 3, inequality (21) holds if σ ≥ 0.
25
1
≥0
(a1 (3F − 1) − 2F + 1) 2
PN −1
n=1
Pn0 (F )an >
References
Athey, Susan (2001), “Single Crossing Properties and the Existence of Pure Strategy Equilibria in Games
of Incomplete Information.” Econometrica, 69, 861–889.
Barut, Yasar and Dan Kovenock (1998), “The symmetric multiple prize all-pay auction with complete
information.” European Journal of Political Economy, 14, 627–644.
Chawla, Shuchi, Jason D. Hartline, and Balasubramanian Sivan (2015), “Optimal crowdsourcing contests.” Games and Economic Behavior, –.
Che, Yeon-Koo and Ian Gale (2003), “Optimal design of research contests.” The American Economic Review, 93, 646–671.
Clark, Derek J and Christian Riis (1998), “Competition over more than one prize.” The American Economic
Review, 88, 276–289.
Cohen, Chen, Todd R Kaplan, and Aner Sela (2008), “Optimal rewards in contests.” The RAND Journal of
Economics, 39, 434–451.
DiPalantino, Dominic and Milan Vojnović (2009), “Crowdsourcing and all-pay auctions.” In Proceedings
of the 10th ACM conference on Electronic commerce, 119–128, ACM.
Fullerton, Richard L and R Preston McAfee (1999), “Auctioning entry into tournaments.” Journal of Political Economy, 107, 573–605.
Galton, Francis (1902), “The most suitable proportion between the value of first and second prizes.”
Biometrika, 385–399.
Gavious, Arieh and Yizhaq Minchuk (2014), “Revenue in contests with many participants.” Operations
Research Letters, 42, 119–122.
Glazer, Amihai and Refael Hassin (1988), “Optimal contests.” Economic Inquiry, 26, 133–143.
Kaplan, Todd, Israel Luski, Aner Sela, and David Wettstein (2002), “All–pay auctions with variable rewards.” The Journal of Industrial Economics, 50, 417–430.
Kaplan, Todd R, Israel Luski, and David Wettstein (2003), “Innovative activity and sunk cost.” International Journal of Industrial Organization, 21, 1111–1133.
Megidish, Reut and Aner Sela (2013), “Allocation of prizes in contests with participation constraints.”
Journal of Economics & Management Strategy, 22, 713–727.
Milgrom, P.R. (2004), Putting Auction Theory to Work. Churchill Lectures in Economics, Cambridge University Press.
26
Moldovanu, Benny and Aner Sela (2001), “The optimal allocation of prizes in contests.” American Economic Review, 542–558.
Moldovanu, Benny and Aner Sela (2006), “Contest architecture.” Journal of Economic Theory, 126, 70–
96.
Nalebuff, Barry J and Joseph E Stiglitz (1983), “Prizes and incentives: towards a general theory of
compensation and competition.” The Bell Journal of Economics, 21–43.
Sisak, Dana (2009), “Multiple-prize contests–the optimal allocation of prizes.” Journal of Economic Surveys, 23, 82–114.
Taylor, Curtis R (1995), “Digging for golden carrots: an analysis of research tournaments.” The American
Economic Review, 872–890.
Tullock, Gordon (1980), “Efficient rent seeking.” in:J.Buchanan et al.(Eds.), Toward a theory of the rentseeking society. Texas A & M Univ Pr, College Station, 1980.
Wijsman, Robert A (1985), “A useful inequality on ratios of integrals, with application to maximum
likelihood estimation.” Journal of the American Statistical Association, 80, 472–475.
27