HETEROGENEITY IN CONTESTS
SÉRGIO O. PARREIRAS♯ AND ANNA RUBINCHIK♮
Abstract. We analyze a contest (all-pay auction) in which all n
contestants have independent uniformly distributed valuations for
the prize with the support between zero and a contestant-specific
upper bound (1/ci ). Such a contest has a unique Bayesian-Nash
equilibrium, which is derived in terms of the primitives, the profile
of lower bounds for the marginal cost of effort, or cost parameters,
(ci )ni=1 . Heterogeneity of contestants is measured by the spread
of cost parameters of the active participants (i.e., those who place
positive bids with positive probability). In the equilibrium (1) the
heterogeneity has no effect on the highest effort, the distribution
of which is fully determined by the average of the cost parameters,
(2) the more heterogeneous contestants are, the lower is the total
expected effort, and the lower (in the first-order stochastic dominance order) is the distribution of the minimal effort.
J.E.L. Classification numbers. D44.
Keywords: All-pay auctions, homogeneity, group composition.
1. Introduction
A setting with several agents competing for a prize by making irreversible investments captures the essential features of numerous “competitive environments” around us. For example, the agents could be
employees from the same department fighting for the only spot available to be promoted, runners in a city race striving to be the first to
Date: September 9, 2016.
We would like to thank Dan Kovenock for the detailed comments and for their
suggestions the anonymous referees as well as Ella Segev and participants of the
Game Theory seminar at the Technion, Michigan State University, University of
Queensland, the conference participants of the Operations Research Society of Israel, and of the Israeli Economic Association. The first author wishes to thank
CEMS-Kellogg-NWU for its hospitality. The scientific responsibility is assumed by
the authors.
♯
Dept. of Economics, UNC at Chapel Hill. [email protected].
♮
Dept. of Economics; University of Haifa. [email protected].
1
2
S. O. PARREIRAS AND A. RUBINCHIK
reach the finishing line, firms investing in R&D in order to win an exclusive right to sell a new product. The objective of the competition
organizer is not always the same. It can be to maximize the total effort
invested by all agents as in the first example, the minimal effort as in
the second one, and the top investment as in the last example. The
setting is known in the literature as an all-pay auction or contest and
its variations have been studied extensively.1
In this paper we fix the prize and vary the pool of contestants.2 Our
objective is to study its effects, in particular, the impact of heterogeneity of the contestants’ pool on the maximal and the minimal efforts as
well as on the total expected effort, or simply, the expected revenue.
To do so we make several simplifying assumptions: a contestant’s effort
fully reflects her performance, the utility is linear both in the value of
the prize and in the cost of the exerted effort.3 However, given our
objective, the competitors can not be identical, thus the traditionallystudied symmetric auction framework is not suitable for our purposes.
We work with the asymmetric all-pay auction: not only are our competitors distinct, these differences also can be commonly observed. In
addition, we believe that it is important to let contestants remain uncertain about the qualities of the rivals, as is often the case in reality.
Another reason to choose auctions with private information is our focus on heterogeneity: in a complete information all-pay auction only
the two strongest contestants participate actively, while the rest stay
out, unless several strongest bidders are identical. Thus, to detect the
effects of heterogeneity on behavior of active participants (and to have
more than two of those), one has to allow for some private information
in an all-pay auction.
In this model, conditionally on all the commonly observable information the value that contestant i attaches to a prize is an independent
random variable in the eyes of his rivals. Furthermore, to derive a
closed-form solution,4 we assume that each of these random variables,
indexed by i, has a uniform distribution with support [0, 1/ci ] for some
1
Cf. Konrad (2011) for the overview.
There is some connection between our results and the literature studying the design
of contests, and, in particular the prize structure see ft. 13.
3
The linearity in costs assumption can be relaxed though, see page 27 for details.
4The advantage of getting a closed-form solution for equilibrium is evident: we can
study the relevant properties of the equilibria directly, without relying on a local
2
HETEROGENEITY IN CONTESTS
3
parameter ci > 0, indicating the observed differences between the contestants, i ∈ {1, . . . , n}.
Although one might view as restrictive the assumptions that the
lowest point in the support is zero, and that the distributions are uniform, both can be viewed as appealing. Indeed, due to the linearity
of the utility of each agent in value, it can be divided by the agent’s
value from receiving the prize and still represent the same preferences
over the outcomes for each given type of contestant, thus yielding a
standard re-interpretation of the model.5 After the normalization, an
agent’s private information is about his marginal cost of effort, which
equals the inverse of the value. The marginal cost of effort then has
a Pareto (cumulative) distribution Pi : x 7→ 1 − cxi on [ci , ∞]. So, the
common lowest value of zero implies that there is no a-priori upper
bound on the cost, which, arguably, is not a bad assumption: one can
never underestimate the laziness of a rival.6 The second assumption
implies that the marginal distribution peaks at the lower bound, ci :
rivals believe it rather likely that the realized cost is not too far from
ci , which makes it a more meaningful ‘description’ of an agent in this
model.
These lower bounds of the cost distributions will be referred to as cost
parameters. Observe that the cost parameters represent the first-order
stochastic dominance (FOSD) order of the corresponding cumulative
distributions, meaning ci > cj implies that Pi ≻FOSD Pj , in which case
we refer to agent i as an ex-ante weaker agent.
In this setting we fully characterize equilibrium of the all-pay auction
game and proceed with the comparative statics. There are two basic
findings. First, as one would expect, an improvement in costs, i.e., a
decrease in all cost parameters, improves an outcome of the contest
(measured as the top, the bottom or the expected average effort of the
active participants). Second, none of the outcomes is positively related
to heterogeneity of the pool of contestants: a spread in cost parameters,
approximation as in, e.g., Ryvkin (2009) who examines binary elimination contests
with noisy outcomes.
5See Konrad (2011).
6Whenever in the original model the realized value for the prize is zero, the corresponding effort cost is +∞.
4
S. O. PARREIRAS AND A. RUBINCHIK
keeping the average of the parameters constant, lowers the average expected effort, decreases (in terms of stochastic dominance) the minimal
effort, and has no effect on the distribution of the top effort.
1.1. Related literature. For a detailed overview of the literature on
contests and all-pay auctions, we refer the reader to the recent book,
Konrad (2011). There are several closely related strands of this literature.
First one relates to the equilibrium characterization of the underlying games. A solution to an asymmetric information all-pay auction game with two contestants is provided in Amann and Leininger
(1996). Our proof of equilibrium uniqueness rests on the results from
Li and Parreiras (2015) who borrow some ideas from Lebrun (1999).
Identifying the set of active contestants (who exert strictly positive
effort with strictly positive probability in equilibrium) is based on
Parreiras and Rubinchik (2010).
Second, our comparative statics relates to the literature on excluding
(or handicapping) better contestants and giving a head start to weaker
ones. Indeed, the notion of heterogeneity we use here is based on Lorenz
order, which, roughly, ranks higher a wider-spread (“unequal”) profile
of values while keeping their average the same. We pick this definition
to formally distinguish between the effect of such a spread from the
“level” effect: increasing the average valuation of the object in an allpay auction or the quality (effectiveness) of competitors in a contest
should increase the revenues, and in most classical settings does. Note
that eliminating extreme contestants creates a more homogenous (less
spread) pool of potential competitors, however it might also change
their average quality. It has been noted in the literature that such
measures can be revenue enhancing, which might sound surprising,
provided it decreases the average quality of active participants.
The revenue-maximizing auction in Myerson (1981) handicaps stronger bidders. Baye, Kovenock, and de Vries (1993), introduce the “Exclusion Principle” suggesting that excluding the strongest bidder from
participating in a complete information all-pay auction increases revenue when the remaining bidders are sufficiently homogeneous and the
strongest bidder is an outlier. In addition, the authors show that
HETEROGENEITY IN CONTESTS
5
shrinking the gap between the top two valuations increases the revenue – this too can make valuations “more equal”, while changing the
average valuation of the prize. Moreover, in that setting, the characteristics of the other players do not matter, only two will typically
participate actively. There is also a series of papers on caps in political
lobbying cf. e.g., Che and Gale (1998), presented as an all-pay auction,
showing that the caps might have the undesired effect of increased
overall spending by the competing lobbyists. However, the result does
not extend onto the Tullock contests, as follows from Fang (2002).
Nevertheless, the direct effect of changes in heterogeneity might still
be present even for that format: Ostreiher, Pruett-Jones, and Heifetz
(2012) analyze “a surprising negative correlation between the number
of feedings that a nest received and the overall weight gain in the nest”
of the Arabian babblers. It is explained, both theoretically — using
evolutionary stable equilibrium in a Tullock’s contest, and empirically
— by greater intensity of fighting between siblings in those nests where
hatching times were closer, hence where the young birds are more alike
(closer in weight). Thus, even in the animal kingdom homogeneity can
intensify competition.
The all-pay and Tullock’s contests, in which the designer can alter
the odds to win by favoring some contestants, i.e., weighting their bids
differently, are compared in Franke, Kanzow, Leininger, and Schwartz
(2014) (see also additional references to the literature on heterogeneity
in full information contests in the introduction of that paper). The
revenue-maximizing format there is the all-pay auction, which is biased
so that the two top players, the only participants, get the same expected
payoff of zero, cf. also Baye, Kovenock, and de Vries 1996. The result
holds for “all levels of heterogeneity” (not necessarily measured the
same way as here). However, when the full information assumption is
dropped and the designer can not favor competitors, the comparison is
not clear, as is shown in Einy, Haimanko, Moreno, Sela, and Shitovitz
(2013).
The optimal set of individual weights in the full information Tullock
contests derived in Franke, Kanzow, Leininger, and Schwartz (2013),
implies that the playing field should be somewhat “leveled” by the designer who, by changing the odds through those weights, encourages
6
S. O. PARREIRAS AND A. RUBINCHIK
otherwise-drop-outs to actively participate. Surprisingly, the set of active participants in the Tullock contest derived in Hillman and Riley
(1989, Proposition 5) bears close resemblance to our set of active contestants, see our Lemma 1 and Proposition 1 below. Possible connections between the two models are interesting to study but this is beyond
the scope of the current paper.
Coming back to the literature on the outliers, Kirkegaard (2013)
shows that combining handicaps for the strong and favoritism for the
weak can improve revenue in a two-bidder all-pay auction with private
values if, in particular, there is enough asymmetry ex-ante, i.e., if the
relative advantage of the strong bidder is pronounced enough at the
start. In sequential all-pay auctions studied in Segev and Sela (2011)
the effect of the head start on the revenue depends on whether the head
start is multiplicative or additive.
There is a large literature on contest design (Moldovanu and Sela,
2006; Moldovanu, Sela, and Shi, 2012; Xiao, 2013) which studies the
optimal prize structure. As we aim to disentangle the effects of the
composition of the pool of contestants from other factors, we consider
the prizes as given.
In a complete information contest Xiao (2013) shows that it always
pays to separate contestants with different costs of effort by allocating
them into separate contests for any objective that can be presented as
a weighted sum of expected effort levels exerted by the competitors.
This echoes our Proposition 4, which, demonstrates that in our setting, too, it is never optimal to mix weak (with high lower bound of
cost distributions, or cost parameter) with strong contestants (with
the low cost parameter). The formulation of the problem solved in our
Proposition 4 looks simpler: the number of weak and strong potential
contestants is equal, the prizes are fixed, each contestant eventually
has to be allocated to one of the two possible contests and the goal of
the designer is simply to get the highest total expected effort. Notice
however that although we require all the participants to be active, and
such a requirement is also imposed in Xiao (2013), it has very different
implications in the two models. In our case (with incomplete information), this simply requires the participants to be not too different
HETEROGENEITY IN CONTESTS
7
ex-ante (i.e., it imposes a restriction on the ratio of their cost parameters), while in Xiao (2013) it requires the prize structure to be tuned
just right to motivate all the players to exert effort. In the standard allpay auction with a single prize, the result in Xiao (2013) fails to hold.
More importantly, our proof clearly indicates the role of heterogeneity
(Lorenz ranking of cost parameters profiles) in generating the result
and we conjecture that it can be generalized to settings with more
than two tiers of potential competitors. Our approach, based on the
celebrated result of Hardy, Littlewood, and Pólya (1929), might also
help to identify effects of heterogeneity for very different environments
provided the objective can be written as a sum of convex functions of
individual characteristics.
2. The uniform model
Consider a first-price all-pay auction with n risk-neutral, privatelyinformed asymmetric bidders. Given her privately observed valuation
vi ∈ Ti = [0, 1/ci ] for a prize, each bidder i ∈ {1, . . . , n} chooses her
bid, bi ∈ S = R+ , and the highest bidder wins the object. Each bidder
is an expected utility maximizer and her payoff is vi − bi in case she
wins the object and −bi if she does not and hence is risk neutral. Any
two bidders k, l share a common belief that the valuation of their rival
i 6= k, l is a random variable Vi with uniform distribution on [0, 1/ci ] ⊂
R+ , a contestant-specific support, i.e., Pr(Vi ≤ x) = Fi (x) = ci x for
x ∈ [0, 1/ci ] with ci > 0.7 Valuations are independent and we denote
their joint distribution by Hc , where c = (ci )ni=1 is the parameter profile
of the game. For convenience and without loss of generality, we order
the bidders by their parameters: ci ≤ ci+1 , i = 1, . . . , n − 1, so bidder
1 is the strongest and the last one, n, is the weakest (in the first-order
stochastic dominance sense).
One can also view the all-pay auction as a contest, and we adopt this
view in what follows. Now bidders are contestants, bids are effort levels,
which fully determine the contestant’s output, and the owner of the
highest output wins. Moreover, contestants utility over outcomes will
be different, but by linearity (in value and bid), it still represents the
same preferences as the bidders in the auction had. Contestant i gets
7Of
course, Fi (x) = 0 for x < 0 and Fi (x) = 1 for x > 1/ci .
8
S. O. PARREIRAS AND A. RUBINCHIK
ui = 1 − vi−1 bi if she wins the prize and ui = −vi−1 bi if she does not.8
Here, vi−1 is the (realized) marginal cost of effort, whose lower bound is
the parameter ci . Note that each ci in the profile c identifies individual
i’s cost distribution (referred to in the introduction as Pi ) and thus a
profile c contains the ex-ante (commonly observable) characteristics of
all potential contestants. Several possible ways to generalize the model
are discussed in the conclusions.
Our first goal is to present an equilibrium of the Bayesian game Γc =
hS n , (Ti )ni=1 , Hc , (ui )ni=1 i indexed by the profile of the cost parameters,
c, the primitives of the game.
We start by introducing an additional notation. All of its elements
can be easily computed from the primitives and will be used in the
formal statements that follow as convenient shortcuts.
def
Notation 1. I = {i : (i − 1)ci <
i
P
def
def
cj }, I = #I, C =
j=1
def ci
.
C
κi =
Remark 1.
(i) I ≥ 2, as (i−1)ci <
i
P
I
P
ci
i=1
I−1
, and
cj for i = 1, 2 for any profile
j=1
of parameters 0 < c1 ≤ c2 ≤ . . . ≤ cn , so C is well-defined.
(ii) The order of agents by cost parameters ci is preserved by κi :
c1 ≤ c2 ≤ · · · ≤ cn ⇐⇒ κ1 ≤ κ2 ≤ . . . ≤ κn ,
so κi is a relative cost parameter, which measures how strong
(from an ex-ante point of view) a contestant is relative to the
others.
In Proposition 1 below we establish that I is the set of what we
will later call “active participants”, which is a subset of the full set
of potential contestants, {1, 2, . . . , n}. The following lemma provides
convenient ways of identifying an element of I.
Lemma 1. κi < 1 ⇐⇒ i ∈ I ⇐⇒ i ≤ I.
Proof. First, we establish the second equivalence, i ∈ I ⇐⇒ i ≤ I.
Let m = max{i : i ∈ I} be the index of the contestant with the highest
cost parameter among those contestants that are in the set I. By
8A
payoff in such a contest can be −∞ (if vi = 0 and bi 6= 0), and we adopt the
convention that the payoff is zero whenever bi = 0, so, in particular, ∞ × 0 = 0.
HETEROGENEITY IN CONTESTS
9
P
construction, (m − 1)cm < m
j=1 cj . The cost parameters are ordered,
P
so for any i < m, ci ≤ cm and hence (i − 1)cm + (m − i)cm < ij=1 cj +
Pm
Pi
Pm
=⇒
=⇒ (i − 1)ci + (m − i)cm <
j=i+1 cj
j=1 cj +
j=i+1 cj
Pi
(i − 1)ci <
j=1 cj which is equivalent to i ∈ I. The number of
elements in I is I, so m = I, and therefore the set I is the set containing
the first I indexes only, {1, . . . , I}.
Next, we show i ≤ I ⇐⇒ κi < 1. By the previous argument,
κI < 1. Then the statement follows by remark 1.(ii).
Proposition 1. The strategy profile b = (bi )ni=1 , bi : [0, c1i ] → R+ ,

1

 (ci v) 1−κi
if i ∈ I
bi (v) =
C

0
otherwise
is a Bayesian-Nash Equilibrium of the game Γc .
Proof. Assume contestants follow the strategy profile b. The cumulative distribution of effort of contestant i ∈ I, as function of b ∈ R+ , is

(Cb)1−κi if b ≤ C −1
Gi (b) = Fi (b−1
(b))
=
(2.1)
i
1
if b > C −1 .
P
P
By lemma 1 and notation 1,
i∈I κi =
i≤I ci /C = I − 1, hence
P
i∈I (1 − κi ) = 1. As a result, the winning probability of exerting
effort 0 ≤ b ≤ C −1 , is

(Cb)κi if i ∈ I
Y
def
Wi (b) =
Gj (b) =
(2.2)
Cb
if i ∈
/ I,
j∈I,j6=i
It follows that given the strategies of the rivals, the winning probability
for any contestant
with
index i > I is Cb, thus the payoff is linear in
1
b: Ui (b|vi ) = C − vi b. By Lemma 1, i > I implies κi > 1, which is
equivalent to C < ci , and so Ui (b|vi ) < 0 for all b > 0 and vi ≤ 1/ci .
Clearly, then, contestant i > I is best-responding by choosing bi (vi ) = 0
in accordance with the suggested strategy profile.
Consider contestant i ≤ I with vi > 0. Her payoff, Wi (b) − vbi , is
strictly concave in b because Wi (b) is strictly concave, as 0 < κi < 1
(by Lemma 1). Moreover, the effort that maximizes the payoff is not
zero, as Wi′ (0) = +∞. Therefore the best reply has to be strictly
positive and has to satisfy the first-order condition, Wi′ (b) − v1 = 0.
10
S. O. PARREIRAS AND A. RUBINCHIK
The strategy profile b in the statement of the Proposition prescribes a
strictly positive effort for v > 0 and solves the first-order conditions,
as Wi′ (bi (v)) = v −1 for i ≤ I.
It remains to consider a contestant with zero valuation. According
to the suggested equilibrium profile, such a contestant exerts zero effort
and gets the highest attainable payoff, which is zero, cf. ft. 8, and hence
he is also best responding to the strategies of the rivals.
Remark 2 (Active Contestants). Proposition 1 establishes that I is the
set of (ex-ante) active contestants, i.e., those who, from ex-ante perspective, choose a strictly positive effort level, with a strictly positive
probability, {i : Pr[bi (Vi ) > 0] > 0}. As a result, by Lemma 1, the
inequality κi < 1 is a participation constraint for i: he is active if and
only if it is satisfied.
The parameter C then is an adjusted average of the cost parameters
of the active contestants.
Remark 3 (Weakly Active Contestants). Given any game Γc , if κi = 1
for some contestant i, then there exists another equilibrium, in which
bi (v) = 0 for v < 1/ci and bi (1/ci ) = 1/C, and the rest of the players
choose the same strategies as in Proposition 1. Thus, the equilibrium
is essentially the same as the one described in Proposition 1, i.e., both
equilibria induce identical distributions of effort levels. In this case, we
say that contestant i is weakly active and his participation constraint
binds.
Proposition 2. The strategy profile in Proposition 1 is the essentially
unique Bayesian-Nash equilibrium of the game Γc .9
Proof. The distribution of values Fi of every bidder i is uniform thus
satisfying the only assumption of Li and Parreiras (2015, Proposition
7), which requires Fiv(v) to be constant for any v from the support of
the distribution for any bidder in the all-pay auction with incomplete
information, thus the uniqueness by that Proposition.10
9The
adjective “essentially” used in Proposition 2 has exactly the same meaning as
in remark 3: any equilibrium of the game Γc generates the same bid distributions
as do the strategies described in Proposition 1.
10To find the equilibrium we had to solve the corresponding system of differential
equations (first order conditions), but this on its own does not constitute a proof
uniqueness, of course. The hard part of the proof is to assure the needed continuity
HETEROGENEITY IN CONTESTS
11
We now characterize the expected equilibrium revenue based on the
strategy profile in Proposition 1. Here, again, we use notation 1.
. Then, in equiCorollary 1. Denote by h the real function x 7→ 1−x
2−x
librium, the expected effort, Bi , of an active contestant i ∈ I equals
P
1
h(κi ), and so the expected revenue, Rc , equals C1 i≤I h(κi ).
C
Remark 4. As defined, the revenue Rc is a map from C × ( cCi )i∈I ,
and hence can be also viewed as a function of the primitives, the cost
parameters, c = (ci )ni=1 . Similarly, the expected effort Bi can be viewed
as a function of the primitives too.
R 1/c
Proof. By construction, Bi = 0 i ci bi (v)dv, where bi (v) is the equilibrium strategy of i. Using Proposition 1,
1/ci
2−κi
1
Z 1/ci
1−κi
1−κi
2−κ
1 1 − κi
ci
h(κi )
1 − κi 1−κi (ci v)
=
dv =
v i
=
.
Bi =
ci
C
C 2 − κi
C 2 − κi
C
0
0
As a result, the expected revenue is:
X
X 1 − κi
1 X
Rc =
Bi =
(2.3)
C −1 =
h(κi ).
2 − κi
C i≤I
i≤I
i≤I
Remark 5. Since h(1) = 0 and h is continuous on (0, 1], function Rc
can be alternatively written as
1 X
(2.4)
h(κi ).
Rc =
C i : κ ≤1
i
b def
Let I = max {i : κi ≤ 1}, then the average cost parameter can be expressed as
I
P
def i=1
C =
ci
(
I
P
i=1
ci ) + (Ib − I)C
Ib
P
ci
i=1
=
,
Ib − 1
Ib − 1
since κi = 1 if and only if ci = C. Thus, the two alternative ways to
write the expected revenue (expressions (2.4) and (2.3)) correspond to
I −1
=
of bidding strategies, verifying the Lipschitz conditions, and assuring the common
bidding range. The techniques developed for this purpose in Li and Parreiras (2015)
apply to a wide range of environments and so repeating the argument here would
have been redundant.
12
S. O. PARREIRAS AND A. RUBINCHIK
the two essentially identical equilibria:11 the one in remark 3 and the
one in Proposition 1 correspondingly. Clearly, both return the same
value for any profile of cost parameters.
Before continuing let us make a simple technical remark which will
be used multiple times in the following analysis.
Remark 6. The function h defined in Corollary 1 is concave whenever
the participation constraint is satisfied or binding. Indeed, its second
2
derivative, h′′ (x) = (x−2)
3 , is negative for any x < 2. Recall that the
argument of h is always below unity by the participation constraint
and is positive.
3. Revenue Maximization
Here we want to understand how ex-ante observable qualities, or
cost parameters, of contestants affect total expected effort, or expected
revenue. We start with the simple “level” effects and continue with the
effects of a spread in quality.
3.1. Decreasing costs boosts the expected revenue. As one would
expect, making contestants stronger has a positive effect on the revenue, cf. Baye et al. (1996) for a similar result for the full information
contests.
Corollary 2. The expected revenue, R(·) , is homogeneous of degree -1,
i.e., Rxc = x1 Rc for x > 0.
Proof. Multiplying every ci by x > 0 maps C to xC, hence κi are
mapped into themselves, and therefore the set of active participants I
is unchanged. The result then follows by corollary 1.
Second important factor is the “spread” of cost parameters.
3.2. Heterogeneity of contestants decreases the revenue. One
natural way to measure homogeneity of individual cost parameters,
(ci )i∈I , is to use the Lorenz order, introduced by Lorenz (1905), generalized (as majorization) by Hardy, Littlewood, and Pólya (1934) and
later used to assess risk (as second order stochastic dominance) by
Rothschild and Stiglitz (1970). More equal (homogeneous) vectors are
ranked lower by the Lorenz order.
11To
b
be precise, remark 3 and Prop. 1 describe 2I−I essentially identical equilibria.
HETEROGENEITY IN CONTESTS
13
Definition 1 (Lorenz order L on Rn ). Consider two vectors a, b ∈ Rn
such that their entries are ordered, i.e., a1 ≤ a2 ≤ · · · ≤ an and
P
P
b1 ≤ b2 ≤ · · · ≤ bn . Then a L b, if and only if, ki=1 ai ≤ ki=1 bi for
P
P
any k ∈ {1, . . . , n − 1} and ni=1 ai = ni=1 bi .
We will use the following (classical) result:
Theorem 1 (Hardy et al., 1929). Given any two vectors x, y ∈ Rn+ ,
P
P
inequality ni=1 f (xi ) ≥ ni=1 f (yi ) holds for any convex function f :
R → R, if and only if, the two vectors are Lorenz-ordered, x L y.
Definition 2. Contest A is more heterogeneous than contest B, if
and only if, cA L cB , where cA , cB ∈ Rn+ are their respective cost
parameters.
Our next Proposition shows that expected revenue, defined on a set
of feasible cost parameters, agrees with the Lorenz order defined over
this set: the more heterogeneous the contest, the lower the revenue.
Proposition 3. Pick any two contests A and B with cost parameter
profiles cA , cB ∈ Rn+ such that all contestants are active in both contests. Then, cA L cB implies that RcA ≤ RcB .
Pn k
1
Proof. Let C k = n−1
i=1 ci for k = A, B. The Lorenz order implies
A
B
the equality C = C , but then, by definition of κki = cki /C k , it also
yields κA L κB . By corollary 1, it is sufficient to show that the
P
Pn
B
latter ordering implies ni=1 h(κA
i ) ≤
i=1 h(κi ). By theorem 1 the
implication is true if h is concave, and it is by remark 6.
Remark 7. Corollary 2 and Proposition 3 can be combined to compare
some cost parameter profiles based on the revenues they generate without the explicit calculation. Indeed, by the former, for any profile x,
the profile tx generates a lower revenue for any t > 1. The latter offers
a partial ordering of profiles that have a fixed sum, which, in case of
two-dimensional profiles, implies that the closer the profile is to the
diagonal, the higher the revenue. Figure 1 illustrates the ordering for
two-bidder contests.
While it is clear from Corollary 2 that the effects of improving quality
of contestants on expected revenue are unbounded (scaling down the
cost profile always generates a higher revenue), we want to illustrate
14
S. O. PARREIRAS AND A. RUBINCHIK
c2
c=t·x
c2 = c1
Rc < Rx
x2
Rc > Rx
x1
c1 + c2 = x1 + x2
c1
Fig. 1. A partial ordering of cost-parameter profiles.
that the effects of heterogeneity on revenues, on the other hand, is
bounded. For that we fix the quality of the contestants by requiring
the sum of costs parameters to be equal to some constant x > 0 and
find the highest and the lowest attainable revenue, Rc , as defined in
Corollary 1 .
Note that function Rc is continuous on a compact set
)
(
n
X
x
and
ci = x
Mx = c ∈ Rn+ : 0 ≤ c1 ≤ . . . ≤ cn ≤
n−1
i=1
for any x > 0, which makes it possible to find the extreme values of
Rc on this set. Moreover, Mx contains the set of feasible cost profiles
of active contestants (strictly positive cost parameters that satisfy the
participation constraint), more precisely, Mx is a closure of such a set.
Hence the extremum values of Rc on the set Mx might not always
correspond to an expected revenue from a particular contest, but they
will be the tightest bounds for a revenue from any contest, i.e., for any
possible combination of contestants whose cost parameters sum up to x.
We start by identifying the two solutions to the corresponding optimization problems.
n
P
x
n
ci = x .
Corollary 3. Let Mx = c ∈ R+ : 0 ≤ c1 ≤ . . . ≤ cn ≤ n−1 and
Then for any x > 0,
(i) ( nx , . . . , nx ) = arg maxc∈M Rc ;
i=1
HETEROGENEITY IN CONTESTS
15
x
x
, . . . , n−1
) = arg minc∈M Rc .
(ii) (0, n−1
Proof. By Proposition 3, the cost profile that maximizes the revenue is
the minimal one in the sense of the Lorenz order (L ). The minimal
profile, clearly, is the most equal one, ci = nx , and it also satisfies the
participation constraint, hence the first part of the claim is proved.
Similarly, by Proposition 3, the second part is equivalent to showing
x
x
that c0 = (0, n−1
, . . . , n−1
) is the L -maximal profile satisfying the
x
weak constraint cn ≤ n−1 . First, there is no L -higher profile, as
any such profile has to have its first component zero, and hence the
x
rest of the components must equal n−1
by the weak constraint, thus
0
such a profile coincides with c . Second, assume to the contrary there
is another profile c̃ 6= c0 that is L -maximal as well. The two are
unequal, therefore c̃1 > c01 = 0. Both are L -maximal, hence one of
the partial sums of this alternative profile should be smaller than that
of the initial one, i.e., there should be index 1 < k < n such that
Pk
Pk 0
k−1
i=1 c˜i <
i=1 ci = x n−1 , and because both sum up to unity there
x
should be j > k such that c̃j > n−1
thus violating the participation
constraint, a contradiction.
As a result, we can also provide tight (and easy to calculate) bounds
for the expected revenue from the primitives of the model, i.e., the
ordered profile of cost parameters, c.
i
P
Corollary 4. Take any profile c, let I = max{i : (i − 1)ci <
cj }
j=1
P
and C = ( j≤I cj )/(I − 1), as before. Then for any c that gives rise
to the same I and C, the expected revenue in contest Γc is bounded:
I 1
1
< Rc ≤
2C
I +1C
Proof. If all contestants have the same cost, ci = C(I−1)
, then κi =
I
I−1
1
I−1
and h( I ) = I+1 , with h as defined in corollary 1, which also
I
) and so the upper bound follows by
implies that the revenue is CI h( I−1
I
corollaries 2 and 3.
For the lower bound, we include the boundaries of the feasible set
as in corollary 3. Notice that c1 = 0 implies κ1 = 0 and, for 1 <
i ≤ I, ci = C implies κi = 1. Then, as h(0) = 12 and h(1) = 0, the
1
limiting revenue is limc→(0,C,...,C) Rc = 2C
, which is the infimum, again,
by corollaries 2 and 3.
16
S. O. PARREIRAS AND A. RUBINCHIK
These bounds indicate that the effect of heterogeneity is limited,
and, in particular, that any gain from homogeneity of contestants can
be offset by a big enough improvement in their quality.
Corollary 5. Consider two contests A and B with cost parameter profiles, cA and cB in Rn+ , such
that all contestants are active in both
P
def
n
ck
i=1 i
contests, i.e., ckn < C k = n−1
for k = A, B. A sufficient condition
for A to yield a higher revenue than B is 2C A < C B .
Proof. Recall, by corollary 1, total revenue in contest k = A, B equals
Pn
cki
1
k
k
i=1 h(κi ), where κi = C k and h defined as in corollary 1. Let
Ck
Pn
k
Hk =
⇐⇒ C B H A >
i=1 h(κi ), for k = A, B. So, RcA > RcB
C A H B ⇐⇒ (C B − C A )H A > C A (H B − H A ).
n
By Corollary 4, H B − H A is never higher than n+1
− 1/2 ≤ 12 and
H A > 21 . Hence for the required inequality between the revenues to be
satisfied, it suffices to assure that C B − C A > C A , which is equivalent
to C B > 2C A , as stated.
3.3. Sorting the contestants. Our next goal is to use previous results to consider the problem of allocating individuals to different contests. Again, since our focus is on composition of the group of competitors, we take the “contest architecture” as given. For the purpose
of this analysis we simplify the environment and restrict attention to
the case of “weak” and “strong” potential contestants only.
Definition 3. Contestants belong to the same tier if their cost parameters are the same. A strong contestant has cost parameter cL and a
weak contestant has cost parameter cH = tcL , where t > 1.
By corollary 2, the expected revenue is homogeneous in the profile
of cost parameters, so we can set cL = 1, as our main focus in the
next proposition are ratios of revenues. Thus, in case of two tiers, the
expected revenue becomes a function of a single parameter, t > 1, the
relative strength.
Remark 8. In a contest with n participants, of which k are strong
and n − k are weak, the adjusted cost parameter is C = k+(n−k)t
and
n−1
since after our normalization, the relative cost parameter of a strong
participant, C1 is always below that of the weak one, Ct , it is sufficient
HETEROGENEITY IN CONTESTS
17
to check the participation constraint of the latter, which then requires
either k = 1 or t < k/(k − 1).
Suppose that each tier has the same number of contestants and, consider the problem of assigning each contestant to one of two contests,
which must have identical prizes and an equal number of participants.
Should contestants from different tiers be assigned to the same contest?
The answer is negative, provided that the weak contestants are not ‘too
weak’ to actively participate, as the following proposition shows.
Proposition 4. Fix an even number n ≥ 2 of contestants. Let rk (t)
denote the revenue from the contest with n participants in which k are
strong and the rest (n − k) are weak and either k = 1 or t < k/(k − 1).
For any m ∈ {1, . . . , n − 1} and any t,
(i) the homogeneous contests generate a higher expected revenue
than mixed contests:
(3.1)
rn (t) + r0 (t) > rm (t) + rn−m (t)
(ii) the relative revenue gain of the homogeneous contests over fullymixed contests
rn (t) + r0 (t)
(3.2)
2rn/2 (t)
is increasing in the relative strength of contestants, t;
(iii) the gain from sorting is bounded from above:
(3.3)
r0 (t) + rn (t)
(n + 2)(n − 1)2
≤
.
rm (t) + rn−m (t)
n(n − 2)(n + 1)
Moreover, the bound is tight and is attained for fully-mixed
contests, m = n/2, in which the participation constraint binds,
t = n/(n − 2), that is:
n
n
r0 ( n−2
) + rn ( n−2
)
(n + 2)(n − 1)2
=
.
n
n(n − 2)(n + 1)
2rn/2 ( n−2 )
Proof. (i): When all contestants are of the same tier, their relative
does not depend on their tier and clearly, the
cost parameter κi = n−1
n
participation constraint, κi < 1, is always satisfied.
In mixed contests, however, κi can take two different values, κm
L
and κm
,
depending
on
contestant
i’s
tier
and
the
relative
strength
t.
H
18
S. O. PARREIRAS AND A. RUBINCHIK
Slightly abusing notation let
def
κm
L (t) =
n−1
m + (n − m)t
be the relative cost parameter κi (as defined in notation 1) of a strong
contestant when there are m strong and n − m weak contestants. Similarly let
def
κm
H (t) =
(n − 1)t
= tκm
L (t)
m + (n − m)t
be the κi for a weak contestant in a contest with the same configuration
of contestants.
It is easy to verify that (m−1)t < m is, indeed, the participation constraint required for this case, in accord with remark 8 and the definition
of rm . Observe also, that with this composition of competitors in the
first contest, the other contest has m weak and n−m strong competitors
so, in addition (n − m − 1)t < n − m has to hold for full participation.
m
n−m
Both constraints impose upper bounds for t: m−1
, n−m−1
for any m 6= 1
in the first case and m 6= n − 1 in the second. The first upper bound
is smaller whenever m > n/2, i.e., it is sufficient to check participation
constraint for the group where weak contestants are a minority. We assume the constraints are satisfied for the rest of the proof of this claim.
By Corollary 1, the expected revenue from a contest with n ex-ante
h( n−1
), so
identical contestants with parameter x is n−1
x
n
1
1
h( n−1
(rn (t) + r0 (t)) = 1 +
(3.4)
).
n
n−1
t
Similarly, by Corollary 1,
(3.5)
(3.6)
m
mh(κm
rm (t)
L (t)) + (n − m)h(κH (t))
=
n−1
m + (n − m)t
m
m
h(κL (t)) + (1 − m
)h(κm
H (t))
n
= n
m
m
+ (1 − n )t
n
1−x
Further, since h : x 7→ 2−x
is concave on (0, 1) (cf., remark 6),
βh(x) + (1 − β)h(y) < h(βx + (1 − β)y) for any x, y, β ∈ (0, 1). In
n m
n
= m
κL (t) + (1 − m
)κm
addition, for any m ∈ {1, . . . , n − 1}, n−1
H (t).
n
Thus, for any m ∈ {1, . . . , n − 1},
(3.7)
h( n−1
)>
n
n
h(κm
L (t))
m
+ (1 −
n
)h(κm
H (t))
m
HETEROGENEITY IN CONTESTS
19
Combining identities (3.4)-(3.6) and inequality (3.7), one gets a sufficient condition for proving the claim: for any β ∈ (0, 1)
1+
(3.8)
1
1
1
>
+
.
t
β + (1 − β)t (1 − β) + βt
The latter holds, indeed, subtracting the right-hand side from both
sides of (3.8), yields an equivalent inequality
(1 − β) β (1 − t)2 (1 + t)
> 0,
t (β + (1 − β)t) ((1 − β) + βt)
which holds for any β ∈ (0, 1) as required.
n
. By
(ii): The participation constraint for m = n/2 implies t < n−2
Corollary 1, the expected gain from sorting (3.2) can be written as a
n
) to
map from t ∈ [1, n−2
h( n−1
)
(1 + t)2
n
n
n
t
h(κ 2 (t)) + h(tκ 2 (t))
L
L
(1+t)2
Clearly, t is increasing in t, and h( n−1
) is independent of t, so it is
n
def
n/2
left to show that the denominator of the last ratio, d(t) = h(κL (t)) +
n/2
h(tκL (t)), is decreasing in t.
To verify that d(t) is decreasing, consider the profile of relative cost
parameters in a contest where half of the participants are strong and
half are weak:
(3.9)
n/2
n/2
n/2
n/2
pt = (κL (t), . . . , κL (t), tκL (t) . . . , tκL (t))
|
{z
} |
{z
}
n/2 times
n/2 times
Recall that by construction, the sum of the entries of pt is n − 1. Also,
n/2
n/2
is decreasing in t and tκL (t) = 2t(n−1)
notice that κL (t) = 2(n−1)
n(t+1)
n(t+1)
n/2
n/2
is increasing in t. Fix t1 > t2 > 1, then κL (t1 ) < κL (t2 ) and
n/2
n/2
t1 κL (t1 ) > t2 κL (t2 ). Therefore, by definition of the Lorenz order
(cf. def. 1), pt1 L pt2 . Hence, theorem 1 and concavity of h (cf. remark
6) imply that d(t1 ) ≤ d(t2 ).
(iii): The proof proceeds in a series of steps. We first formulate auxiliary claims, next we construct the argument using those claims to support point (iii) and lastly, we provide the proof for the auxiliary claims.
As a first step we get an upper bound for the relative revenue gain
which stems from a lower bound for the revenue of mixed contests
rm (t) + rn−m (t). Here, as in claim (i), both participation constraints
20
S. O. PARREIRAS AND A. RUBINCHIK
have to hold, the one for k = m and the one for k = n − m number of
strong contestants. As before, for t to satisfy the pair of constraints, it
is sufficient to check the more stringent one, for the contest where the
weak participants are a minority.
Lemma 2. For any m ∈ {1, . . . , n − 1} and any t that satisfies t <
m
n−m
if m ∈ {1, . . . , n2 } and, in case n > 3, t < m−1
if m ∈
n−m−1
n
{ 2 + 1, . . . , n − 1}, the following inequality holds:
1 + 1t h( n−1
)
r0 (t) + rn (t)
n
(3.10)
.
≤
1
1
n
rm (t) + rn−m (t)
+ mt+n−m n+2
m+(n−m)t
The second step is to find the upper bound of the right hand side of
(3.10) in t and m. Note that the highest t can ever get while satisfying
n
the pair of participation constraints is n−2
.
n
Lemma 3. Define the function f on [1, n−2
] × {1, . . . , n − 1} by
(3.11)
f (t, m) =
1 + 1t
1
1
+ mt+n−m
m+(n−m)t
Then f is maximized at m = n/2 and t =
n
n−2
so that f (t, m) ≤
(n−1)2
.
n−2
Lemmata 2 3 imply that
(n − 1)2 n + 2 n−1
(n + 2)(n − 1)2
r0 (t) + rn (t)
≤
h( n ) =
.
rm (t) + rn−m (t)
n−2
n
n(n − 2)(n + 1)
Moreover, the bound is tight as it is attained at m = n/2 and t =
n/(n − 2), as claimed in point (iii).
Now we proceed by proving each of the two lemmata.
Proof of lemma 3. The function f is symmetric in m around m = n2
and it is maximized there for any given t. In addition, f is increasing in
(m−n)t2 +(m−n)
> 0. It follows
t, its derivative with respect to t being m
n
t2
(n−1)2
n
n
that f (t, m) ≤ f ( n−2 , 2 ) = n−2 .
Proof of lemma 2. Given equation (3.4), to prove the claim it is sufficient to establish the following inequality:
rm (t) + rn−m (t)
n
1
1
(3.12)
≥
+
n−1
m + (n − m)t mt + n − m n + 2
Here we use again the calculation of the revenue rm that appears in
the proof of claim (i), eq. (3.5). Using that equation to express the left
HETEROGENEITY IN CONTESTS
21
hand side of the inequality (3.12), the idea for the proof becomes clear:
we wish to find the same bound for both terms, mh(κm
L (t)) + (n −
n−m
n−m
m
m)h(κH (t)) and (n − m)h(κL (t)) + mh(κH (t)), or, simply a bound
m
for mh(κm
L (t)) + (n − m)h(κH (t)) for any m ∈ {1, . . . , n − 1} and any t
that satisfies the relevant participation constraint. Here is how we do it.
Take an arbitrary m 6= n/2, an arbitrary t > 1, and define the cor(t)) of relative
responding profile pt = (κm
(t), . . . , κm
(t), tκm (t) . . . , tκm
{z L } | L
{z L }
|L
m times
n−m times
cost parameters. Assume first that m < n/2.
We want to show that the profile pt is Lorenz dominated by the profile of relative cost parameters of n/2 weak contestants and n/2 strong
n/2
contestants with relative strength tm > 1 such that κL (tm ) = κm
L (t).
That is, we want to show that:
(3.13)
n/2
n/2
n/2
n/2
qtm = (κL (t), . . . , κL (t), tm κL (t) . . . , tm κL (t)) L pt .
{z
} |
{z
}
|
n/2 times
n/2 times
To explicitly obtain the value of tm , we solve:
n−1 2
n−1
=
⇔
n tm + 1
m + (n − m)t
(t − 1)(n − m)
.
tm =1 +
n
Notice that tm increases in t. Also, if t = 1 then tm = t = 1 but, as the
derivative of tm with respect t is (n − m)/n < 1, it follows that t > tm
for t > 1.
Before we establish the dominance relation, we must ensure that the
new profile can be generated by a contest with active participants, half
of which are weak and half of which are strong, i.e., that the participan/2
tion constraint is satisfied, κH (tm ) < 1. This is equivalent to imposing
n
the upper bound on tm : tm < n−2
. To verify that it holds, recall that
the initial profile is feasible and m < n2 , and so by assumption of the
n−m
. By the argument above, tm < t, thus it is sufficient
lemma, t < n−m−1
n
n−m
. This is indeed, true whenever m < n2 . So,
to show that n−m−1 < n−2
participation constraint is satisfied.
Now we can prove the Lorenz dominance (3.13) for any m < n/2
and any t > 1. Indeed, observe that the two profiles give rise to the
same partial sums up to element m and the partial sum generated by
n/2
κL (tm ) =κm
L (t) ⇔
22
S. O. PARREIRAS AND A. RUBINCHIK
the second profile falls short of the first one for any element above m
since 1 < tm < t and m < n/2.
Assume now m > n2 . Consider the relative cost parameters profile
(t))
κm (t), . . . , t−1
κm (t), κm (t) . . . , κm
gtm = (t−1
{z m H } | H
{z H }
|m H
n/2 times
n/2 times
n/2
where tm solves κH (tm ) = κm
H (t). Next we show that gtm Lorenzdominates the original profile pt .
Indeed, partial sums from the top (n-th element) are never lower in
gtm than the sums in the original one, pt . Note also that the participation constraint is satisfied here by construction, since κm
H (t) < 1.
Now by theorem 1, we can claim that for any m < n/2 and any t > 1
there is tm that satisfies the participation constraint such that12
n
m
m
mh(κm
(3.14)
(h(κm
L ) + (n − m)h(tκL ) ≥
L ) + h(tm κL ))
2
and for any m > n/2 and any t > 1 there is tm that satisfies the
participation constraint such that
n
m
m
m
(3.15)
mh(t−1 κm
(h(t−1
H ) + (n − m)h(κH ) ≥
m κH ) + h(κH )).
2
n/2
n/2
Furthermore, when m = n/2 the sum h(κL )+h(tκL ) falls with t (the
profile with higher t Lorenz dominates the one with a lower t, cf. the
last paragraph of the proof of point (ii)), thus, setting tm at its upper
n/2
n−2
n
and calculating the corresponding κL = t−1
boundary, n−2
m = n , we
get the lower bound for the left hand side of both inequalities (3.14)(3.15), this bound being
n
n−2
n
n
n−2
h
+ h(1) = h
=
,
2
n
2
n
n+2
1−x
, as in Corollary 1. With this bound we obtain,
in which h(x) = 2−x
using equation (3.5), the lower bound for the sum of revenues in (3.12),
which completes the proof.
In section A.1 of the Online Appendix, we consider several extensions
of the two-tier environment analyzed here and illustrate that segregation by the cost of effort might be optimal for revenue maximization in
12In
what follows we omit the dependence of κ on t.
HETEROGENEITY IN CONTESTS
23
Total revenue
10
7
1
Mix 1 (4, 2) (2, 4)
Mix 2 (5, 1) (1, 5)
Fully mixed (3, 3) (3, 3)
Sorted (6, 0) (0, 6)
5
7
1
2
0
1
1/t
Fig. 2. Total revenues for Example 1.
slightly different environments as well, thus suggesting that the main
message of Proposition 4 may be robust.
We proceed with a simple example demonstrating the importance of
the participation constraint for Proposition 4.
Example 1. Here n = 6, and the overall expected revenue from the
two contests is plotted in Figure 2 as a function of relative strength of
the two tiers, t. Different lines correspond to different combinations of
weak and strong contestants in the two contests. A kink in each line
occurs at the boundary of the range where participation constraint is
satisfied, i.e., the area where t is sufficiently low, so that all participate.
For higher values of t two purely homogeneous contests (black line)
does not necessarily generate a higher revenue than other allocations of
competitors in the two contests. Notice also that the gain from sorting
is maximized at the point where the blue line kinks, thus demonstrating
Proposition 4(iii).
24
S. O. PARREIRAS AND A. RUBINCHIK
4. Alternative goals
In this section we illustrate how other criteria, that differ from the
revenue (total-effort) maximization, are affected by the composition of
the pool of contestants. Here we are back to the initial model where the
cost parameters of the contestants are arbitrary (positive numbers).
4.1. Maximizing the highest effort. The highest effort, i.e., the
winner’s effort level, maxj bj (Vj ), is a random variable and given our
assumptions its distribution is easy to calculate based on the primitives
of the model. We start with a corollary to Proposition 1.
Corollary 6. The distribution of the maximal effort in an equilibrium
of the game Γc is uniform on [0, C −1 ].
Proof. The cumulative distribution of the highest effort is
Y
Pr[max bj (Vj ) ≤ b] = Pr[bj (Vj ) ≤ b, ∀j ∈ I] =
Gj (b).
j
j≤I
As follows from the characterization of the essentially unique equilibQ
rium in Proposition 1 (cf. eq. (2.2) in the proof), j≤I Gj (b) = Cb for
0 ≤ b ≤ 1/C, hence the conclusion.
Corollary 6 implies that for contests A and B with adjusted average
cost parameters, C A < C B , the highest effort of contest A first-order
stochastically dominates the highest effort of B. Moreover, by remark
1.(i), a contest designer who wants to maximize the expectation of
any increasing function of the highest effort and is free to select the
participants as well, should choose the two strongest contestants (i.e.,
contestants 1 and 2 who have the lowest cost parameters) to compete
for the entire prize and exclude all the rest.
Moldovanu and Sela (2006) obtain a similar result for a symmetric
contest.13
4.2. Maximizing the lowest effort. In contrast to the highest effort, for the lowest effort (the weakest link) heterogeneity does play an
13If
the competitors are ex-ante identical, of course, there is no way to identify
the two best ones a-priori. The result is similar “in spirit”: in Moldovanu and Sela
(2006), in order to maximize the expected highest effort the designer is to award the
whole prize to the winner of the second stage tournament, where the two winners
of the first stage are competing against each other.
HETEROGENEITY IN CONTESTS
25
important role. Here again we derive the distribution of the lowest effort and compare the resulting distributions for different combinations
of contestants.
Let us start with a simple corollary to Proposition 1.
Corollary 7. An equilibrium distribution of minimal effort is G(1) (b) =
I
I
Q
Q
1−
(1 − Gi (b)) = 1 −
(1 − (Cb)1−κi ) .
i=1
i=1
Then we have a simple implication.
Proposition 5. Pick a profile of cost parameters c ∈ Rn such that all
participants are active in the contest Γc , so that cn < C. Let Gx(1) be
the distribution of minimal effort in contest with cost parameters xc.
Then for any x > 1, G1(1) first-order-stochastically dominates Gx(1) .
Proof. It follows then that scaling up the cost parameters profile c →
xc, with x > 1, as we argued in the proof of corollary 2, scales up the
adjusted cost parameter, C, and leaves the relative cost parameters
κi of every active participant i unchanged, thus it yields a first order
stochastically dominated minimal effort by corollary 7.
Next, as before, we explore the implications of the spread of cost
parameters.
Proposition 6. Consider two contests ΓcA and ΓcB in which the cost
parameters cA , cB ∈ Rn+ are such that in both contests all contestants
are active and ΓcA is more heterogeneous than ΓcB , i.e., cA L cB .
Then the distribution of the lowest effort in the equilibrium of ΓcB firstorder stochastically dominates the distribution of the lowest effort in the
equilibrium of ΓcA .
Proof. First remember, cA L cB is equivalent to κA L κB and
equality of the adjusted average (cost) parameters, C A = C B (as in the
proof of Proposition 3). Consider the auxiliary function, f : [0, 1) → R,
defined by f (t) = log(1 − x1−t ), with fixed parameter, x ∈ [0, 1). Set
x = Cb ∈ [0, 1) and because function f is concave and increasing, by
theorem 1,
X
X
f (κA
f (κB
i ) ≤
i )
26
S. O. PARREIRAS AND A. RUBINCHIK
Finally, by corollary7, the distribution of minimal effort is G(1) (b) =
PI
A
B
A
B
1 − exp
i=1 f (κi ) , and so G(1) (b) ≥ G(1) (b), where G and G are
distributions of minimal effort in contests A, B correspondingly.
5. Conclusions
We have analyzed a contest with risk neutral participants who have
only partial information about costs of efforts of their rivals and who
are different at the outset of the game. We further assume that the
valuations for the prize (winning) are distributed uniformly on [0, 1/ci ],
where ci is the contestant-specific characteristic observed by all the
players. We provide a full characterization of the unique equilibrium
of the game. The equilibrium is easy to calculate and has several
interesting properties.
First, we find that a contestant either participates actively, i.e.,
chooses an effort in an open interval or stays out of competing altogether, independently of the effort cost that she has. The participation threshold is fully determined by the relative ex-ante strength of
the competitors, based on the profile of cost parameters, c = (ci )ni=1 .
Thus, depending on the parameters, any subset of them can become
active in equilibrium, in contrast to the full information case where the
number of active participants is higher than two only in the knife-edge
case of identical competitors.
Second, we find that the expected revenue or total expected effort
in equilibrium agrees with the classical Lorenz order over the cost profiles of the active participants. This immediately implies that holding
the sum of cost parameters constant, their spread yields a decrease in
expected revenues, provided all the contestants are active. In other
words, heterogeneity of the competitors’ pool is detrimental to the total effort. However, the effect of heterogeneity is bounded: we show
that a sufficient increase in competitors’ quality (scaling down the cost
parameters) can neutralize it.
To illustrate these insights we consider a problem of allocating an
equal number of weak and strong competitors (whose cost parameter
is either high or low) to two different contests with equal number of
competitors in each. The worst allocation is the full mix, where both
contests have an equal number of weak and strong competitors and
HETEROGENEITY IN CONTESTS
27
where they are as different from each other in strength as the participation constraint allows. The best allocation, of course, is the one
where weak are competing only against weak and all the strong are
competing in a separate contest. This result yields then the upper
bound on gains from sorting competitors by strength.
Further, we fully characterize the distribution of the minimal and the
maximal effort in such contests. The Lorenz order over the cost profiles,
surprisingly, agrees also with the first order stochastic dominance of the
minimal effort: more spread yields lower minimal effort. In contrast,
the distribution of the maximal effort depends only on the sum of the
cost parameters and on the number of active participants.
The model can be extended while preserving the results. First, one
could generalize the contest model by introducing the size of the prize
(π) as an additional parameter, so that the winning payoff would be
π − vi−1 bi : this parameter just changes the “units” of measurement,
while leaving all the comparisons intact. Also, it is possible to generalize the model by allowing for non-linear costs, so that winning and
losing payoffs would be 1 − vi−1 f (bi ) and −vi−1 f (bi ), where f ′ > 0 and
f (0) = 0. In this case, the characterization of equilibrium can be easily adapted by “changing the variable” of choice of the bidders to the
value returned by f (b), hence the equilibrium bidding strategy of an
1
1−κi
active contestant becomes bi (v) = f −1 (ci v)C
, while leaving the set
of active participants the same as in the linear costs case. The results
concerning minimal and maximal effort remain the same, while to preserve the results involving the total expected revenue, potentially, more
assumptions on f are needed. An easy example of the cost function for
which all the results are unchanged is f (x) = xθ with θ > 0.
Observe that although reading the previous literature mentioned in
the introduction one might agree that homogeneity increases competition,14 the focus there is on particular cases of decreasing “dispersion”
in abilities (such as head starts for the weak and handicaps for the
14Cf.
also Stracke (2013), who illustrates that the choice of the best multi-stage
contest format depends on heterogeneity (one type or two types of players) in a
four-player Tullock contest and Groh, Moldovanu, Sela, and Sunde (2012) for the
derivation of the optimal pairings of four players in binary elimination tournaments,
i.e., full information all-pay auctions with two players.
28
S. O. PARREIRAS AND A. RUBINCHIK
strong), and on different competition formats (Tullock, full information, etc.) besides, the effect of head starts, for example, differs (in
terms of equilibrium behavior) from the effect of the change in valuations, at least for the full-information case, cf. Siegel (2014). This
paper offers a way to think more systematically about heterogeneity
in contests and potentially can give rise to new experiments exploring
this issue.15 In addition, the connection between the essentially unique
equilibrium characterization in our set up and the solution of the full
information Tullock contest mentioned in the introduction might be
interesting to explore.
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Appendix A. Online Appendix
A.1. Complementarities and segregation. Here we reconsider the
problem of allocating 2n inputs (contestants), into two equal groups
so as to maximize the sum of the values returned by a differentiable
objective function f : Rn → R defined for all combinations of inputs
from a group (i.e., the group’s profile of cost parameters). We show
that sufficient conditions for segregation to be optimal require f to be
symmetric and to have positive cross derivatives (expressing complementarities between inputs). We then apply this result to our contest
environment where expected revenue function happens to possess these
properties at the diagonal, where all contestants are ex-ante identical.
HETEROGENEITY IN CONTESTS
31
Definition 4.
(i) The join of two vectors c, ĉ in Rn is their component-wise maximum, c ∨ ĉ = (max{ci , ĉi })ni=1 . Their meet is
c ∧ ĉ = (min{ci , ĉi })ni=1 .
(ii) A function f : Rn → R is supermodular if f (c ∨ ĉ) + f (c ∧ ĉ) ≥
f (c) + f (ĉ) for any c, ĉ ∈ Rn .
That segregation is optimal follows from the symmetry and supermodularity of f .
Lemma 4. Consider 2n contestants with given cost profile,
(c1 , .., c2n ) such that 0 < c1 ≤ c2 ≤ · · · ≤ c2n . Among all permutations
of contestants, ψ, the identity, ψ(i) = i, maximizes f (cψ(i) )ni=1 +
f (cψ(i) )2n
i=n+1 given f is a symmetric, super-modular function.
Proof. Take any permutation of inputs, and name its first n elements
a ∈ Rn and its last n elements, b ∈ Rn . Let a′ be a permutation of
a where the elements are ordered in an ascending order: a′1 ≤ a′2 ≤
· · · ≤ a′n and let b′ be a permutation of b with elements ordered in a
descending order: b′1 ≥ b′2 ≥ · · · ≥ b′n .
Note that every element of the meet, a′ ∧ b′ , is (weakly) below every
element of the join, a′ ∨b′ . Indeed, let j be the highest index i (between
1 and n) such that b′i ≥ a′i . (If there is no such j then a′i ≥ b′i for all i
and every element of the join, a′ = a′ ∨ b′ is above every element of the
meet, b′ = a′ ∧ b′ , so the claim is true.) As b′i is decreasing in i and a′i
is increasing in i, for all smaller-than-j indices, k ≤ j, the inequality
holds too: b′k ≥ a′k , whereas for all the bigger indices it is reversed by
construction. It follows that the join contains the first j elements of b′
and the last N − j of a′ , the meet contains the rest, and therefore all
the elements of the join are above b′j whereas all the elements of the
meet are below. This, in turn, implies that the join, a′ ∨ b′ contains the
n highest elements of the original ordered profile, c1 ≤ c2 ≤ · · · ≤ c2N ,
and the meet, a′ ∧ b′ , contains the lowest n.
By symmetry of f , f (b) = f (b′ ) and f (a) = f (a′ ), and, combined
with its super-modularity,
f (a) + f (b) = f (a′ ) + f (b′ ) ≤ f (a′ ∧ b′ ) + f (a′ ∨ b′ )
= f ((ci )ni=1 ) + f ((cj )2n
j=n+1
32
S. O. PARREIRAS AND A. RUBINCHIK
with c1 ≤ c2 ≤ · · · ≤ c2n . As the initial choice of a and b was arbitrary,
the conclusion follows.
Next step is to find a sufficient condition for super-modularity. Here
we use the formulation of the Topkis’s result by Milgrom and Roberts
(1990):
Theorem 2 (Topkis’s characterization theorem). Let [x, x] ⊂ Rn . Suppose that f : Rn → R is twice continuously differentiable on some open
subset of Rn containing [x, x]. Then f is supermodular on [x, x] if and
∂ 2 f (x)
≥ 0.
only if for all x ∈ [x, x] and all i 6= j, ∂x
i ∂xj
Combining the two statements, we get an easy sufficient condition
for the segregation to be optimal.
Corollary 8. Let [x, x] ⊂ Rn . Suppose that f : Rn → R is twice continuously differentiable on some open subset of Rn containing [x, x]. If
∂ 2 f (x)
≥ 0, then among all permufor all x ∈ [x, x] and all i 6= j, ∂x
i ∂xj
tations of 2n contestants (c1 , .., c2n ), ci ∈ R+ , the one that maximizes
f ((ci )ni=1 ) + f ((cj )2n
j=n+1 ) is the ordered one, c1 ≤ c2 ≤ · · · ≤ c2n .
A.2. The cross-derivatives of the expected revenue. Recall, by
construction, all ci are strictly positive (and finite).
def
Lemma 5. Assume I ≥ 2, let J = I − 1, and take κi as a map
i
(ci )ni=1 7→ J PJc+1
, then with c = (c, c, . . . , c),
c
j=1
j
κi J − κi
J2
> 0, and at c, it equals c(J+1)
2.
ci J
κ2i
−J
i
(ii) For i 6= j, ∂κ
< 0, and at c, it equals c(J+1)
=
−
2.
∂cj
Jci
2κ
κ2i
J(J−1)
i
=
−
(iii) For i 6= j, ∂c∂ i ∂c
2 c2 (J − 2κi ), and at c, it equals − c2 (J+1)3 .
J
j
(i) For all i ≤ I,
∂κi
∂ci
=
i
(iv) For i 6= j, k,
∂ 2 κi
∂cj ∂ck
=
2κ3i
J 2 c2i
> 0, and at c, it equals
Proof. Follows by a direct computation using κi (c) =
2J
.
c2 (J+1)3
J
.
J+1
Here, again we view the expected revenue as a function of costs.
Lemma 6. At any diagonal point, c = (c, c, . . . , c), the revenue cross2R
derivatives are positive: ∂c∂i ∂c
(c) > 0 for all i 6= j.
j
(2−κi )2 −2 ∂κi
∂Bi
1−κi κi
and
so
=
for all
2−κi ci
∂cj
(2−κi )2 ci ∂cj
(2−κi )2 −2 ∂ 2 κi
∂κi ∂κi
4
− (2−κi )3 ci ∂cj ∂ck for j, k 6= i.
(2−κi )2 ci ∂cj ∂ck
Proof. By corollary 1, Bi =
j 6= i. Then
∂ 2 Bi
∂cj ∂ck
=
HETEROGENEITY IN CONTESTS
33
By Lemma 5, evaluated at at the diagonal point, c = (c, c, . . . , c),
−2J(J 2 −2)
−2J(3J 3 +6J 2 −4)
4J 2
this derivative equals (J+2)
2 c3 (J+1)3 − c3 (J+1)(J+2)3 = c3 (J+1)3 (J+2)3 < 0,
where J = I − 1.
2B
(2−κi )2 −2 ∂ 2 κi
(2−κi )2 −2 ∂κi
i ∂κi
i
=
− (2−κ4i )3 ci ∂κ
for
Further, ∂c∂ i ∂c
2 c ∂c ∂c − (2−κ )2 c2 ∂c
(2−κ
)
∂cj ∂ci
j
i
i
i
j
j
i
i
j 6= i. By Lemma 5, evaluated at the diagonal point, c = (c, c, . . . , c),
4 +3J 3 +2J+4)
> 0.
this derivative equals 2J(2J
c3 (J+1)3 (J+2)3
So, for all i 6= j
∂ 2 Bi
∂ 2 Bi
2J 2
∂2R
(c) = (J − 2)
(c) + 2
(c) = 3
>0
∂ci ∂cj
∂cj ∂ck
∂ci ∂cj
c (J + 1)3
A.3. Sorting the contestants into groups.
Proposition 7. For any n there is ε > 0 such that for any combination of contestants (c1 , . . . , c2n ) with maxi,j∈{1,...,2n} |ci − cj | ≤ ε, sorting contestants by abilities into two groups of size n maximizes expected
revenue among all possible allocations of contestants in two equal-sized
groups.
Proof. By corollary 8 in appendix A.1 it is sufficient to show for that
for any point where all costs are equal, there is a neighborhood where
the cross-derivative of revenue (as a function of costs) remains positive. Indeed, first, by Lemma 6, the cross-derivative of the revenue is
strictly positive at any (diagonal) point where all the costs are equal,
∂ 2 R(c,c,...,c)
> 0, ∀i 6= j. Second, one can always choose a neighborhood
∂ci ∂cj
of a diagonal point to assure that the participation constraints hold.16
Finally, by corollary 1, the revenue is continuously differentiable on the
set of costs where the participation constraint holds, hence there should
be a neighborhood of any such diagonal point such that the sign of the
derivative is preserved and participation constraints are satisfied. Second, we consider the contests with only two agents.
Proposition 8. Consider four contestants with distinct abilities which
are divided in two groups of same size to compete (within the group).
The prize is the same for each contest. Assigning contestants with
16Recall
a contestant is active iff κi < 1, so with x the smallest cost (= mini {ci })
κi ≤
(x + ε0 )(n − 1)
x
< 1 ⇔ ε0 <
x + ε0 + (n − 1)x
n−2
34
S. O. PARREIRAS AND A. RUBINCHIK
highest abilities to one group and those with lowest abilities to the other,
maximizes expected effort.
Proof. The revenue is homogeneous (by corollary 2), so normalize the
cost of agent four to 1. Denote costs of other agents by x, y, z: 1 > x >
y, z > 0. We need to show that R(1, x) + R(y, z) ≥ R(1, y) + R(x, z).17
A real-valued function p can be chosen to have the same sign as the
difference R(1, x)+R(y, z)−(R(1, y)+R(x, z)) for any triple x, y, z, and
to be a polynomial with integer coefficients, p ∈ Z[x, y, z] by corollary
1. Hence it is sufficient to show the semi-algebraic set,
S = {(x, y, z) : 1 > x > y, z > 0 and p(x, y, z) < 0} ,
is empty, which is done using Basu, Pollack, and Roy’s cylindrical algebraic decomposition algorithm (2006). We provide the source code
of a Maple implementation online18 , and a visual depiction19 of S and
{(x, y, z) : 1 > x > y, z > 0} \S in Mathematica.
A.3.1. The 2 by 2 case. Here we again limit a contest to contain only
two participants and run two parallel contests with equal prize each.20
In addition, out of four potential contestants, two are ex-ante strong
and two are weak. The question is how to allocate the competitors to
the two contests in order to maximize the expected revenue. It simply
follows from Proposition 4.(i) in the paper that in this case the two homogeneous contests will yield a higher expected total effort than two
mixed contests, i.e., sorting by the cost parameter increases total effort.
One might argue that our ‘stronger’ competitor, the one with a lower
cost parameter, ci , is also more of an uncertain rival for other contestants, since the support of his distribution of effort costs includes the
supports of the weaker contestants (or the variance of valuations for the
17Note
that this requirement is weaker than increasing differences. Indeed, the
latter requires R(z, y) − R(z, x) > R(w, y) − R(w, x) for all x > y and for all w > z.
We need to show the inequality holds only for all x > y > w > z, which only
2
R(x,y)
implies ∂R(x,y)
≥ ∂R(w,y)
for x > y > w, but does not imply ∂ ∂x∂y
≥ 0, since it
∂y
∂y
is impossible to take the limits without disrupting the inequality x > y > w, unless
in the limit x = y = w, i.e., the derivative is evaluated at a diagonal, where all the
costs are equal. In fact, the revenue does not satisfy the increasing differences in
this case, as the cross derivative is negative, e.g., at (c1 , c2 ) = (0.8; 0.05), we thank
an anonymous referee for the latter example.
18http://www.unc.edu/ sergiop/seg.txt
~
19
http://www.unc.edu/~sergiop/seg.nb
20Recall that participation constraint never binds in this case, cf. remark 1.(i).
HETEROGENEITY IN CONTESTS
35
prize in the corresponding all-pay auction game is higher). Therefore,
it might not be obvious whether it is the uncertainty that drives the
sorting to be optimal, or the differences in the perceived strengths of
the opponents. Luckily, more is known about equilibria in contests with
two participants only, so we can illustrate that our result is immune to
the critique we just presented. Consider a slightly more general model
where individual valuations for the prize are distributed uniformly between ai and ai + t, for some ai , t > 0, i = 1, 2. So, now the contestants
vary by parameter ai , that shifts the support of the distributions of
prize valuations and keeps its variance constant in the all-pay auction
interpretation of the model. Note that for the contest model it implies
that the realized cost of effort of competitor i belongs to the interval
[ ai1+t , a1i ], the corresponding cumulative distribution of effort costs on
1
that interval then is x 7→ 1 − at − xt
and so the variance of the costs of
effort drops with ai . The proof below rests on the equilibrium characterization in Lu and Parreiras (2016), showing, in particular that the
Bayesian equilibrium is unique, so the expected revenue is well-defined.
The result here, again, is that two contests with ex-ante identical
participants (only weak in one and only strong in the other) generate a
higher expected revenue than two contests with different participants,
keeping the size of the prizes constant.
Proposition 9. Assume a1 > a2 ≥ 0, t > 0. Let r(a1,a2) be the (equilibrium) expected revenue when valuations for the prize of agent i = 1, 2
are distributed uniformly on [ai , ai +t]. Then r(a1 ,a1 ) +r(a2 ,a2 ) > 2r(a1 ,a2 ) .
Proof. According to Lu and Parreiras (2016, Prop 2.) (or Amann and Leininger
1996) the expected revenue in this case is
Z 1Z y
Z 1 Z y
λ2 (Q(x))dxdy +
(A.1) r(a1 ,a2 ) =
λ1 (Q−1 (x))dxdy
0
0
Q(0)
Q(0)
where
λi : z 7→ tz + ai ,
z ∈ [0, 1], i = 1, 2
and the function Q is implicitly given by
Z 1
Z 1
1
1
(A.2)
dx =
dx,
s λ1 (x)
Q(s) λ2 (x)
36
S. O. PARREIRAS AND A. RUBINCHIK
we for the time being we assume that Q(0) ≥ 0. So, the first step is to
R1
calculate the (tying function) Q. As s λi1(x) dx = 1t ln(t + ai ) + ln(st +
ai ) , equation A.2 implies
ln(t + a2 ) − log(tqs + a2 ) = ln(t + a1 ) − ln(ts + a1 ) =⇒
a2
t + a2 ts + a1
−
Q(s) =
t
t + a1
t
Notice that
Q(0) =
t + a2 a1 a2
−
> 0 ⇔ a1 > a2
t + a1 t
t
as we assumed. Second step is to calculate the expected revenue
using
ts+a2
t+a1
−1
−1
− at1 ,
equation (A.1) and the inverse Q of Q: Q (s) = t
t+a2
r(a1 ,a2 ) =
(t + 3a1 ) (t + a2 ) (2t + a2 + a1 )
t
=⇒ r(ai ,ai ) = + ai , i = 1, 2
2
3
6(t + a1 )
Then the difference in revenues between sorting the opponents and
mixing them in the two contests is
r(a1 ,a1 ) + r(a2 ,a2 ) − 2r(a1 ,a2 ) =
(a1 − a2 ) (a2 t + 5a1 t + 3a1 a2 + 3a1 2 )
3(t + a1 )2
and so is positive, as a1 > a2 > 0.