HIAS-E-34

The Advantage of Dual Discrimination in Lottery
Contest Games
Yosef Mealem
The School of Banking & Finance, Netanya Academic College, Netanya, Israel, 42365

Shmuel Nitzan
Department of Economics, Bar-Ilan University, Ramat Gat, Israel, 52900
Hitotsubashi Institute for Advanced Study, Hitotsubashi University

Takashi Ui
Department of Economics, Hitotsubashi University

September 2016

Hitotsubashi Institute for Advanced Study, Hitotsubashi University
2-1, Naka, Kunitachi, Tokyo 186-8601, Japan
tel:+81 42 580 8604
http://hias.ad.hit-u.ac.jp/
HIAS discussion papers can be downloaded without charge from:http://hdl.handle.net/10086/27202
https://ideas.repec.org/s/hit/hiasdp.html

All rights reserved.

The Advantage of Dual Discrimination in Lottery Contest Games1
byaYosef Mealem , Shmuel Nitzanb and Takashi Uic

Abstract
In a simple class of contest games, the designer can combine two types of
discrimination: a change of the contestants’ prize valuations subject to a balancedbudget constraint (direct discrimination), as well as a bias of the impact of their efforts(structural discrimination). Applying dual discrimination, the designer reduces(increases) the higher (lower) prize value up to a mimimal (maximal) level, butsuitably increases (reduces) the corresponding prize share. Our main result establishes
that in some cases this dual discrimination is advantageous and can yield almost the
maximal possible efforts - the highest valuation of the contested prize. The efforts inour setting can therefore be larger than those obtained under alternative contests with
optimal structural discrimination. This is true in particular with respect to the
optimally biased simple N-player lottery, Franke et al. (2013). In contrast to the main
findings in Franke et al. (2014a, 2014b), in our setting, efforts under the simple lottery
are not necessarily smaller than those under an optimally biased N-player all-pay
auction. Finally, the exclusion principle noted in Baye et al. (1999) – the eliminationof the strongest player - is not valid under dual discrimination.

JEL Classification: D70, D72, D74, D78
Keywords: contest design, dual discrimination, direct discrimination, balancedbudget-constraint, structural discrimination.
a

The School of Banking & Finance, Netanya Academic College, Netanya, Israel, 42365
yossime1@bezeqint.netb
Department of Economics, Bar-Ilan University, Ramat Gan, Israel, 52900 and Hitotsubashi Instituteof Advanced Study, Hitotsubashi University, nitzans@biu.ac.ilcDepartment of Economics, Hitotsubashi University, Tokyo, Japan, oui@econ.hit-u.ac.jp

1

The second author is grateful to HIAS for enabling the completion of this study.

1. Introduction
In the vast contest literature that has numerous applications (internal labor market
tournaments, promotional competitions, R&D races, rent-seeking, political and public
policy competitions, litigation and sports), the most commonly assumed contest
success function (CSF) is the simple lottery proposed by Tullock (1980), see Konrad(2009) and references therein. In two-player contests, for x1 ≥ 0 , x2 ≥ 0 , and δ > 0 ,this simple logit functions take the form:
(1)

⎧ x1
⎪
p1 ( x1 , x2 ) = ⎨ x1 + δx2⎪0.5⎩

if x1 + x2 > 0
if x1 = x2 = 0

Usually, x1 and x2 are interpreted as the contestants’ efforts. However, p1 has two
possible interpretations. It can be interpreted as contestant 1’s winning probability of
an indivisible prize or as his share in a divisible prize. In turn, the winning probabilityof contestant 2 or his share in the prize is equal to p2 = 1 − p1 . Henceforth, we use thesecond interpretation, as in Corchon and Dahm (2010), Franke et al. (2013), Lee andLee (2012), Warneryd (1998). Nevertheless, although under this interpretation there isno uncertainty in the model and the contestants compete on the certain shares of adivisible prize, we preserve the terms “contest” and “CSF”. The asymmetry between
the impact of the contestants’ efforts is captured by the parameter δ > 0 .
In our extended setting, we do enable the contest designer to control δ , as firstsuggested in Lien (1986, 1990) and later by Clark and Riis (2000). This means thatthe designer can apply structural discrimination that affects the contestants’ shares in
the contested prize (the same efforts may yield different shares, depending on thevalue of this parameter). By (1), a reduction in δ increases the bias in favor ofcontestant 1, who is assumed to be, with no loss of generality, the more motivatedcontestant (the one with the higher prize valuation). Furthermore, 0 < δ < 1 ( δ > 1)
implies a bias in favor of contestant 1 (contestant 2). When δ = 1 the contest is fair.
The empirical relevance of such discrimination in contests with a logit CSF is
thoroughly discussed in Epstein et al. (2011) and Franke (2012). Franke et al. (2013)
have justified structural discrimination on the grounds that it lends itself to a very
appealing competitive-market interpretation. For a recent survey of discrimination incontests, see Mealem and Nitzan (2016).
Fang (2002) considered the unbiased simple class of N-player lottery contests
1

assuming that the discretionary power of the contest designer is restricted to the
exclusion of specific players. He showed that the exclusion principle, established inBaye et al. (1993) for an all-pay auction framework with complete information, is not
valid in an unbiased lottery framework. Franke et al. (2013) extend this result to thebiased lottery contest showing that it is never optimal for the designer to discouragestrong contestants from participating in order to enhance competitive pressure among
the remaining weaker contestants. Moreover, they have pointed out the existence ofan additional inclusion principle: some weak contestants, who are inactive in theunbiased case of Fang (2002), are encouraged to become active. This implies that intheir setting of structural discrimination, the designer will endogenously induce amore leveled playing field in comparison to the unbiased contest setting. Thisenhances competition and, in turn, increases the contestants’ exerted efforts. However,as long as N > 2 , it is not optimal for the designer to completely level the playing
ground so some weak contestants may still remain inactive, although at least threewill always be active.
More recently, Franke et al. (2014a, 2014b) have shown that an optimallybiased all-pay auction contest with structural discrimination and an optimal Headstarts all-pay auction always dominate the optimally biased lottery contest; it yields
larger efforts. This is in contrast to the outcome of the comparison between theunbiased versions of these contest models where the (unbiased) all-pay auction canyield less efforts when the exclusion principle applies (it is effort-enhancing to
exclude the player with the highest prize valuation from participation, but the two
active weaker contestants may expend less efforts than all the active players in thelottery contest). Franke et al. (2014a) show that when the designer can apply
structural discrimination, the exclusion principle of the all-pay auction becomesobsolete. The designer will always bias the all-pay auction such that the two strongest
players are active and, moreover, compete on equal terms (the strongest player is not
excluded, but his effectiveness is sufficiently weakened). All other players choose to
be inactive. Applying a less extreme discrimination in the lottery contest induces more
contestants (at least three) to participate. But the effect of increased competitivenessdue to a higher number of active contestants cannot offset the effect of reduced
competitiveness due to a less extreme discrimination and, consequently, the optimallybiased all-pay auction yields larger efforts than the optimally biased simple lottery.
When the designer applies head starts-discrimination, Franke et al. (2014b) have2

shown that the total efforts of N contestants, N ≥ 2 , are larger in an all-pay auction
than in a simple lottery.
In our contest environment, the designer’s ability to discriminate is enriched incomparison to Franke et al. (2013, 2014a, 2014b). In addition to structural
discrimination, i.e., the control of δ , the contest designer can affect the contestants’
incentives by directly changing their rewards in case of winning the contest, thereby
increasing or decreasing the gap between their prize valuations. In other words, thedesigner can manipulate the size of the divisible prize. Such a policy is usually basedon a “give and take” mechanism in case of winning, which is henceforth referred to as
direct discrimination. This form of discrimination has been recently introduced andstudied in Mealem and Nitzan (2014, 2016), focusing on its comparative applicationin an all-pay-auction relative to a logit CSF, disregarding the possibility of structural
discrimination.
A crucial element in this second type of discrimination is the balanced-budget
constraint faced by the contest designer. This constraint, which limits the design of the
optimal tax schedule, implies that when one contestant's winning a share of the prize
is subjected to a positive tax, the share of the prize won by the other contestant mustbe subjected to a negative tax, viz., the granting of a subsidy. The tax scheme consists
then of two numbers (negative and positive) that are added to the contestants' initialvaluations of the divisible prize. These numbers need to satisfy the requirement thatthe designer’s net expenditures are equal to zero in equilibrium. Of course, whether
the constraint is satisfied or not depends both on the applied structural and directdiscrimination; the former determining the contestants' shares in the prize and thelatter the actual modified values of the prize.
The dual discrimination setting significantly changes the results obtained inFranke et al. (2013, 2014a, 2014b). Most importantly, in a simple lottery with dual
discrimination, the maximal efforts can be increased to almost the initially highest
prize valuation.2 These efforts are larger than those obtained in Fang (2002), wherediscrimination is not allowed, and larger than the efforts obtained in Franke et al.
(2013, 2014a) where only structural discrimination is allowed (by Theorem 3.4 inFranke et al. (2014a), the efforts in the optimally biased simple lottery are less than2This is implied by our main result, assuming that the upper bound of the net value of the prize
approaches infinity and the lower bound of the net value of the prize approaches zero.

3the average of the two highest prize valuations). In our simple lottery contest withdual discrimination, total efforts can be larger than those obtained in any optimallybiased contest game and, in particular, in an N-player all-pay auction. This is incontrast to the main results in Franke et al. (2014a, 2014b).
Finally note that in a simple lottery contest with dual discrimination, evenwhen N > 2 , the maximal efforts can be increased to almost the initially highest prize
valuation, where only two contestants are active in equilibrium; one of them must be
the contestant with the highest prize valuation, but the second active player can be anyof the other contestants. This result differs from the result (Theorem 4.6) obtained inFranke et al. (2013), where at least three contestants are active in the equilibrium of
the simple N-player contest with just structural discrimination. Furthermore, theexclusion principle of Baye et al. (1993) is not necessarily valid in our extended
contest because the strongest player can be induced to be always active.
Interestingly, in the extreme case of our setting (see footnote 1), the individualwith the lower prize valuation is offered the illusion of competing on a very large
prize, albeit only a very small share of it can be won. The expected value of his prize
is nevertheless positive and in fact, almost equal to the initial prize valuation of his
rival, the individual with the higher prize valuation. The existence of effective
incentives that induce participation in the contest together with the existence of an
extreme illusion that results in efforts incurred by the individual with the lower prize
valuation is a distinctive interesting feature of our contest. This feature is manifested
in the examples presented in the next section.
2. Illustration of direct discrimination
Our model of dual discrimination in a simple lottery contest is of particular relevancein certain applications. To illustrate the plausibility of direct discrimination with abalanced-budget constraint in contests, we present two applications. In the first one,the contest designer typically engages in a certain activity (some well- defined task or
project) restricted to a certain budget. Although the budget is earmarked only for thisactivity, it can be used to manipulate and affect the incentives of the contestants (thecontractors) who compete for the outsourced project. But a designer who engages insuch manipulations and in particular, in discrimination, must satisfy the contest
balanced-budget constraint that we assume in order to ensure the overall budget
constraint is satisfied:4

1. Municipal projects. A municipal authority is conducting a tender for a divisible
project such as urban development including development of a sewage system,
roads, sidewalks and gardening. Two companies compete for a share in the project.
The municipal authority is restricted to a budget allocated, for example, by thefederal government. Although the budget is allocated only to urban development,
it can be used to influence the incentives of the competing contestants by applying
the two possible modes of discrimination. In order to satisfy the overall budget
constraint, a designer who resorts to structural and direct discrimination must alsosatisfy the assumed balanced budget constraint.
The next application describes a situation where the balanced-budget
constraint is due to a different reason. The constraint is no longer related to a fixedbudget which is at the disposal of the designer for the purpose of carrying out aparticular project. It is due to the fact that the two competing contestants are (at least
partly) controlled by a parent company. The parent company may not prevent
competition between its two subsidiaries by custom or by the law. However, despite
the existing competition, the parent company still has the ability to enforce someoverall financial discipline as well as the power to ensure that the designer's strength
in manipulating the companies is limited. The control of the parent company on its
two subsidiaries and its power in dealing with the designer, given the conflict ofinterests among them, explains its success in enforcing the balanced-budget
constraint:
2. Portfolio distribution between two investment houses. In the capital market, thecommission rate charged by an investment house is usually inversely related to thesize of the customer’s investment. Suppose that the average commission rate in
the industry is z and that a large client (e.g., a pension fund of some large
employer) is interested in distributing its portfolio between two investment houses
that are subsidiaries of some parent company. Despite the affinity among the two
investment houses, they compete in the market.3 The first investment house has anestablished reputation while the second smaller house is relatively less known.
Given the importance of the large customer, the preservation of the reputation of
the first investment house (contestant 1) implies that it assigns a higher value to2

Two such competing investment houses, Psagot and Ofek, were subsidiaries of the leading Bank in
Israel, Bank Leumi. Another example of two such companies is Gadish and Tagmulim, that were two
subsidiaries of another major Israeli bank, Bank Hapoalim.

5the investment of the large customer (the employer’s pension fund). Often, such apension fund prefers to invest a large share of its portfolio in the reputable andusually larger investment house and this enables obtaining a commission rate
lower than z . This implies that the pension fund actually “taxes” the larger andmore reputable investment house relative to z . On the other hand, the investment
house with the lower reputation usually receives the smaller share of the portfolio.
However, the commission rate they charge are higher than z . The balancedbudget constraint is satisfied because of the market forces; The pension fund isinterested in reducing the commission rate, and the parent company of the two
investment houses is interested in increasing the commission rate.
3. The setting
In our contest there are two risk-neutral contestants, 4 the high and low benefit
contestants, 1 and 2. With no loss of generality, the initial prize valuations of thecontestants, n1 and n2 , satisfy the inequality n1 ≥ n2 or k =n1≥ 1 and that the contest
n2

designer is assumed to have full knowledge of the contestants’ prize valuations. In
two-player contests, for x1 ≥ 0 , x2 ≥ 0 and δ > 0 , the simple logit functions take the
form:
(1)

⎧ x1
⎪
p1 ( x1 , x2 ) = ⎨ x1 + δx2⎪0.5⎩

if x1 + x2 > 0
if x1 = x2 = 0

and therefore p2 = 1 − p1 .
Direct discrimination via differential taxation of the contested prize that
affects the contestants’ actual prize valuations, n1 and n2 , is a pair of (positive or
negative) amounts, ε 1 and ε 2 that changes the prize valuations to (n1 + ε1 ) and(n2 + ε 2 ) where 0 < n ≤ ni + ε i ≤ n < ∞ (the lower and upper bounds of thecontestants' actual net prize are the parameters n and n ). We also assume that thecontest designer faces an ex ante expected balanced-budget constraint, that is, ε 1 andε 2 must also satisfy the requirement that:

3

The case of multiple contestants is dealt with later on.

6

(2)

p1ε 1 + p2ε 2 = 0 .
Given the contestants’ fixed prize valuations and the CSF, the function thatspecifies the contestants’ winning probability given their efforts, pi ( x1 , x2 ) , theexpected net payoff (surplus) of contestant i is:

E (ui ) = pi ( x1 , x2 )(ni + ε i ) − xi (i=1,2).

(3)

In the optimal contest design setting, the objective function of the contest designer is:(4)

G = x1 + x2 .

The designer selects δ , ε 1 and ε 2 . In this case, the two contestants maximize their
net payoffs:(5)

E (u1 ) =x1
(n1 + ε 1 ) − x1 and E (u2 ) = δx2 (n2 + ε 2 ) − x2 .
x1 + δx2
x1 + δx2

If the designer selects (ε 1 , ε 2 ) = (0,0) , then the optimal δ is δ =n1, and then2corresponding efforts are equal to G = 0.25(n1 + n2 ) , see Epstein et al. (2013).
Therefore, in the following, we assume (ε1 , ε 2 ) ≠ (0,0) and from (2) we get that

ε1ε 2 < 0 . Let⎛ n1 + ε 1 ⎞⎜⎜⎟n2 + ε 2 ⎟⎠
⎝
.
d=

(6)

δ

By the first order conditions (maximization of (5)),
(7)

x1* =

d (n1 + ε 1 )2(d + 1)and x2* =d (n2 + ε 2 )(d + 1)2

and, therefore,(8)

G = x1* + x2* =

(9)

p1 =

d (n1 + ε 1 + n2 + ε 2 )(d + 1)2d1
and p2 =d +1
d +1

and the balanced-budget constraint (2) takes the form
(10)

p1ε 1 + p2ε 2 =

d
1
ε1 +
ε2 = 0
d +1
d +1

or
(11)

dε1 + ε 2 = 0

7

By the balanced-budget constraint (11), d = −ε2ε. Substituting d = − 2 in (7) we getε1ε1εε ε (n1 + ε 1 + n2 + ε 2 ). By (6) and d = − 2 , we get that in2ε1(ε1 − ε 2 )that G = x1* + x2* = − 1 2equilibrium,⎛ n1 + ε 1 ⎞ ε 1⎟⎟⎝ n2 + ε 2 ⎠ ε 2(12)

δ = −⎜⎜and the positive expected net payoffs are:

u1* =

(13)

(n1 + ε1 )d 2 and u * = (n2 + ε 2 ) .
2(d + 1)2(d + 1)2

Therefore, the designer objective function is:ε ε (n1 + ε1 + n2 + ε 2 )(ε1 − ε 2 )2

G (ε1 , ε 2 ) = − 1 2

(14)

(

)

and his optimal strategy ε 1* , ε 2* solves:

Max G (ε1 , ε 2 )(ε1 ,ε 2 )∈C

(15)
where
(16)

{

}

C = (ε1 , ε 2 ) : n ≤ n1 + ε1 ≤ n, n ≤ n2 + ε 2 ≤ n, ε1ε 2 < 0

Proposition 1
Assume that n1 > n2 , n > 0 is sufficiently close to zero, and n is sufficiently large.
Then,) (n + n)(n − n)(n − n ) ,(n − n + n − n )n(n − n )(n − n )n(n − n )(n − n )
, x =
,
x =
(n − n + n − n )(n − n + n − n )n (n − n )n (n − n )
,u =
.
u =
(n − n + n − n )(n − n + n − n )(G (ε1* , ε 2* ) = G n − n1, n − n2 =

1

2

2

1

*
1

1

*
2

2

2

1

2

1

2

2

1

2

2

*
1

2

*
2

2

2

1

2

1

2

1

2

2

To prove the above, let
(17)

{

}

{

}

C1 = (ε1 , ε 2 ) : n ≤ n1 + ε1 ≤ n, n ≤ n2 + ε 2 ≤ n, ε1 < 0 < ε 2

and
(18)

C2 = (ε1 , ε 2 ) : n ≤ n1 + ε1 ≤ n, n ≤ n2 + ε 2 ≤ n, ε 2 < 0 < ε1 .
8Note that C = C1 ∪ C2 .
Lemma 1
Assume that n1 > n2 , n > 0 is sufficiently close to zero, and n is sufficiently large.
Then,

{(

)}

{(

)}
)

(a) arg Max G(ε1 , ε 2 ) = n − n1, n − n2(ε1,ε 2 )∈C1(b)⎧= n − n1 , n − n2
if n1 ≤ 2n2 − 3n ⎫⎪⎪arg Max G (ε 1 , ε 2 ) = ⎨ ⎧⎛ (n1 + n )(n2 − n )⎞⎫⎬(ε1 ,ε 2 )∈C2⎪⊆ ⎨ n − n1 , n − n2 , ⎜⎜ n − 2n + 3n , n − n2 ⎟⎟ ⎬ if n1 > 2n2 − 3n ⎪
2
⎝ 1
⎠⎭
⎩ ⎩
⎭

(

Proof of Lemma 1
We first show (a). Notice that∂G ε 2 {ε 1 (n + 3ε 2 ) + ε 2 (n + ε 2 )}=∂ε 1
(ε 1 − ε 2 )3(19)

∂G ε 1 {ε 2 (n + 3ε 1 ) + ε 1 (n + ε 1 )}=∂ε 2
(ε 2 − ε 1 )3

where n = n1 + n2 . Thus for ε1 < 0 < ε 2 ,∂G< 0 ⇔ ε 1 (n + 3ε 2 ) > −ε 2 (n + ε 2 )
∂ε 1

(20)

∂G
> 0 ⇔ ε 2 (n + 3ε 1 ) < −ε 1 (n + ε 1 )∂ε 2Therefore, for ε 2 > 0 ,⎧⎛ε (n + ε 2 ) ⎞⎫⎟⎟ ⎬arg Max G (ε1, ε 2 ) = max⎨⎜⎜ n − n1,− 2ε1 ≤ 0
n
+
3
ε
2 ⎠⎭
⎩⎝

(21)
and for ε1 < 0,⎧⎫⎧⎛
ε1 (n + ε1 ) ⎞ ⎫⎟⎟ ⎬ if n + 3ε1 > 0⎪⎪min⎨⎜⎜ n − n2 ,−n + 3ε1 ⎠ ⎭arg Max G (ε1 , ε 2 ) = ⎨⎬⎩⎝ε 2 ≥0⎪⎪
if n + 3ε1 ≤ 0 ⎭⎩n − n2

(22)

(

)

Let ε 1 , ε 2 ∈ arg Max G (ε1 , ε 2 ). Then

(ε1 ,ε 2 )∈C1

9

⎧

(23)

ε 1 = max⎨n − n1,−⎩()

ε2 n +ε2 ⎫⎬ = n − n1
n + 3ε 2 ⎭because if ε 2 = n − n2 , then(n − n1 ) − ⎨− ε 2 (n + ε 2 )⎬ = (n − n2 )(n + n1 ) + n − n1 > 0

(24)

⎧

⎫

⎩

n + 3ε 2 ⎭3n − 2n2 + n1

when n is sufficiently large and if ε 2 = −(ε , ε ) = (0,0) or (− 0.5n,−0.5n) .
⎛ε (n + ε )⎞⎜ n − n ,−⎟ . However, because
1

1

1

)

(

)

(ε , ε ) = (n − n , n − n ) or

Therefore

2

⎜
⎝

(

ε1 n + ε1ε2 n +ε2and ε 1 = −, thenn + 3ε 1
n + 3ε 2
1

2

1

2

1

n + 3ε 1 ⎟⎠(25)

n + 3ε 1 = n2 − 2n1 + 3n < 0

(

) (

)

when n is sufficiently close to zero, we must have ε 1 , ε 2 = n − n1 , n − n2 .

()

We next show (b). Let ε 1 , ε 2 ∈ arg Max G(ε1, ε 2 ) . By a similar argument,(ε1 ,ε 2 )∈C2

(ε , ε ) = (n − n , n − n ) or ⎛⎜⎜ − ε n(n+ +3εε ), n − n ⎞⎟⎟ , but the latter solution is also
1

2

2

1

2

2

2

2

⎝

⎠

possible if n + 3ε 2 = n1 − 2n2 + 3n > 0 .

Q.E.D

Proof of Proposition 1
Since n1 > n2 , we have
(26)

(

)

2

2

(n − n2 ) n − n 2 (n + n − n1 − n2 ) > 0
G (n − n , n − n ) − G (n − n , n − n ) = 1
1

2

1

2

2

2

(n − n + n − n ) (n − n − n + n )
1

2

1

2

If n1 > 2n2 − 3n , then we can verify that

⎛ (n + n )(n2 − n )⎞lim G n − n1 , n − n2 − G⎜⎜ 1, n − n2 ⎟⎟n→∞⎝ n1 − 2n2 + 3n
⎠

(

(27)

)

2

=

2

n(2n1 + 4n2 ) − 9n + n1 (3n1 − 4n2 ) − n(7n1 − 4n2 ) − 9n + 2n1n2>>0
8n + 4n1 − 4n2
8n + 4n1 − 4n2

(

)

(

when n is sufficiently small. Therefore, we must have G ε1* , ε 2* = G n − n1 , n − n2
when n is sufficiently small and n is sufficiently large.

10

Q.E.D

)

In the next proposition, we obtain a sufficient condition for the values of n and n .

Proposition 2n2, and r > 4.5 . Then,r

Assume that n1 > n2 , n = n1r , n =2) n(1++ nr )r ,(G (ε 1* , ε 2* ) = G n − n1 , n − n2 =

2

1

2

n2
n1r 2
*
,
,
x =
x =
(1 + r )2 2 (1 + r )2
*
1

u1* =

n2 r
n1r
, u2* =.
2
(1 + r )(1 + r )2

Proof of Proposition 2
We show that r > 4.5 suffices the assumption that n > 0 is sufficiently close to zeroand n is sufficiently large.
We first consider the proof of Lemma 1, where (24) and (25) use thisassumption. The left-hand side of (24) is reduced to()⎧ ε 2 n + ε 2 ⎫ (n1r − n2 )(n1r 2 − 2n1r − n1 + 2n2 )(n − n1 ) − ⎨−⎬=r[(3r + 1)n1 − 2n2 ]⎩ n + 3ε 2 ⎭which is positive if r > 4.5 . The left-hand side of (25) is reduced tonn + 3ε 1 = (r + 3) 2 − 2n1rwhich is negative if r > 4.5 .
We next consider the proof of Proposition 1. It is enough to show that ifnn1 > 2n2 − 3n = (2r − 3) 2 and r > 4.5 thenr⎛ (n + n )(n2 − n )⎞G n − n1 , n − n2 − G⎜⎜ 1, n − n2 ⎟⎟ > 0 .
⎝ n1 − 2n2 + 3n⎠()

The left-hand side is reduced to(r − 1)⋅ f (r)
4r (r + 1) [r (n1 − n2 ) + 2n2 ]
2

where

f ( r ) = r 3n1 (3n1 − 4n2 ) + r 2 n1 (n1 + 2n2 ) + rn2 (2n1 − 5n2 ) + n22
11nNote that n1 > (2r − 3) 2 and r > 4.5 implyrn3n1 − 4n2 > (2r − 9) 2 > 0rThus, f '' ( r ) > 0 and

f ' ( r ) > f ' (1) = 11n12 − 6n1n2 − 5n22 > 0
because n1 > n2 . Consequently,f ( r ) > f (1) = 4(n12 − n22 ) > 0 if r > 4.5 .

Q.E.D

It is straightforward to verify that comparative statics with respect to theparameter r yields the following results:

∂x1*
2n 2
∂x2*
2n1r
,
=−
<
0
=
> 0 and3∂r∂r (1 + r )3
(1 + r )∂G 2(n1r − n2 )=> 0 . The latter effect is due to the dominance of the positive effect∂r(1 + r )3of a change in r on the efforts of player 1 relative to its negative effect on the effortsof player 2. Also note that the equilibrium efforts are increasing in the prize valuationsof the two players, however these efforts are smaller than n1 as long as n1 > n2 and r
is finite.
3. The N – player contest
With dual discrimination total efforts cannot exceed the prize valuation of contestant
1. That is,
Proposition 3
Under dual discrimination, in equilibrium, n1 ≥ G .
Proof of Proposition 3
In equilibrium, the net payoff of every contestant under dual discrimination cannot be
negative, since otherwise he can improve his situation and secure a zero net payoff by
not taking part in the contest. In other words, for every contestant i,

ui = pi (ni + ε i ) − xi ≥ 0 . Summing over all the contestants, we get that
N

N

N

N

N

N

j =1

j =1

j =1

j =1

j =1

j =1

∑ u j = ∑ [p j (n j + ε j ) − x j ] ≥ 0 or ∑ (p j n j ) + ∑ (p jε j ) − ∑ x j ≥ 0 . Since G = ∑ x j
12

N

N

N

j =1

j =1

j =1

and ∑ (p jε j ) = 0 , ∑ (p j n j ) ≥ G . Since n1 ≥ n2 ≥,..., ≥ n N and ∑ p j = 1, it must beNtrue that n1 ≥ ∑ (p j n j ) ≥ G .5

Q.E.D.

j =1

Proposition 4
If N = 2 and n1 > n2 , then the contest designer can always set r such that theequilibrium efforts are sufficiently close to n1 . In this case x2* → n1 − n , u1* → n andu2* → 0 .
Proof of Proposition 4(

2) n + n r < n and lim G = n . Since G is

It can be verified that G ε 1* , ε 2* =2

1

(1 + r )2

1

1

r→∞

continuous and monotone in r, there exists r such that G is sufficiently close to n1 .
Q.E.D
By Proposition 3, we have shown that the contestants cannot be induced to exert
larger efforts than n1 . Therefore, Proposition 4 can be extended to the case of any
number of contestants N. In the more general multi-player contest, the designer has toreduce the stakes of N − 2 contestants to zero, making sure that contestant 1 with thehighest stake is not included among them. That is,5This result is also valid for unbiased contests or for contests with alternative modes of discrimination,
when discrimination is through modifying the efforts. The explanation is that in equilibrium, the net
payoff of every contestant cannot be negative, since otherwise he can improve his situation and securea zero net payoff by not taking part in the contest. In other words, for every contestant i,Nui = pi ni − xi ≥ 0 . Summing over all the contestants, we get that

j

j =1

N

N

j =1

j =1
#
"!

N

∑ u = ∑ (p n − x ) ≥ 0 or
j

j

j

j =1

N

N

j =1

j =1

∑ (p j n j ) ≥ ∑ x j . Since n1 ≥ n2 ≥,..., ≥ nN and ∑ p j = 1 , it must be true that n1 ≥ ∑(p j n j ) ≥ G .
G

13

Corollary 1
Given any number of contestants N, such that ∞ > n ≥ n1 ≥ n2 ,..., ≥ nN ≥ n > 0 , thecontest designer can always set r such that the equilibrium efforts are sufficiently
close to n1 .
Proof of Corollary 1
The proof is based on the following simple three-stage strategy that the designer
applies:
1. Stage 1: The designer selects a contestant j ∈{2,..., N }.
2. Stage 2: For any contestant i ≠ {1, j}, the designer chooses ε i = −ni . That is,
he reduces the initial prize valuations of N − 2 players to zero
3. Stage 3: Applying the dual discrimination strategy with respect to the twocontestants 1 and j, according to Proposition 4, the designer can induce efforts
that are almost equal to n1 .

Q.E.D.

The explanation of our results is based on the following idea. First, assume without
loss of generality that j = 2 . On the one hand, the designer applies directdiscrimination in favor of contestant 2 by reducing the stake of contestant 1 (thecontestant with the initially higher prize value) to n by choosing ε 1 and increases thestake of contestant 2 (the contestant with the initially lower prize value) to n bychoosing ε 2 . On the other hand, in order to satisfy the balanced-budget constraint, thedesigner must create an appropriate bias in favor of contestant 1 by selecting δ , suchthat the balanced-budget constraint (10) satisfies equation (12):(12)⎛ n1 + ε 1 ⎞ ε 1⎟⎟⎝ n2 + ε 2 ⎠ ε 2

δ = −⎜⎜

Propositions 1 and 2 imply that dual discrimination, that is, ε 1 = n − n1 andε 2 = n − n2 , is an effective strategy for increasing efforts. These efforts can beincreased almost up to n1 , the initial higher prize valuation of contestant 1, provided
that r → ∞ .
Propositions 1 and 2 imply that when the designer applies the two modes of
discrimination, each type has a positive “added value” that enhances the exertion of14

efforts relative to the situation where the designer resorts to just structural
discrimination (Franke et al. (2013, 2014a, 2014b)). That is, the two modes of
discrimination are supportive or “complementing” - their combination can yield larger
efforts than those obtained by just structural discrimination. The advantage ofcombining these two types of discrimination relative to the use of just structural
discrimination is due to the distinctive features of the contribution of each of thesemodes of discrimination to the exerted efforts as described below.
(i) Direct discrimination sufficiently increases the initially lower prize valuation
while sufficiently reducing the initially higher prize valuation. This increases thesum of the contestants’ prize valuations to infinity when r → ∞ and makes the‘income effect’ (associated with a scheme that increases the sum of the final
stakes from ( n1 + n2 ) to ( n1 + ε 1 + n2 + ε 2 ) = n + n ) of this mode of discrimination
the dominant effect.6(ii) The maximal possible increase in the sum of the contestants’ prize valuations is
not the result of direct discrimination alone. It is rendered possible by structural
discrimination that makes sure that the balanced-budget constraint is satisfied.
Specifically, structural discrimination counterbalances the above ‘income effect’
by almost completely favorably discriminating contestant 1, ensuring that his prize
share converges to 1. The moderating effect described in (ii) is necessary to attainthe maximal efforts. While structural discrimination has a ‘second order’ effect onefforts that moderates the income effect of direct discrimination, it also enables
the dominance of this ‘first order’ income effect on efforts described in (i), namely,
the increase in efforts due to the increase in the sum of the contestants’ prize
valuations. The dominance of the effect of direct discrimination means that themore extreme this mode of discrimination, the higher the total efforts and thisrequires the extremity of structural discrimination.
Corollary 1 implies, in particular, that dual discrimination can be advantageous
relative to just structural discrimination in the 2-player all-pay auction or the simple
lottery contest.

6

For a clarification of the meaning of the income effect associated with direct discrimination, see thediscussion following Proposition 2 in Mealem and Nitzan (2014).

15

Corollary 2
If N = 2 and n1 > n2 , then there exists r such that dual discrimination yields efforts
that are larger than the efforts obtained in the optimally biased 2-player all-pay
auction(simplelottery)

contestwith

structural

discrimination.

That

is,

k + 1 + 2(k 2 + 1)
n2 + n1r 2
or
.
(
)
(
)
r
>
>
0
.
5
n
+
n
>
0
.
25
n
+
n
1
2
1
2
k −1
(1 + r )2
Note that our results imply that under the assumption that n is sufficiently
large and n is sufficiently close to zero ( r → ∞ ), the combination of the two modesof discrimination results in an outcome which is practically reasonable; the amounttransferred between the contestants is finite. When the actual payment to contestant 2(the tax taken from contestant 1) is equal to p2ε 2 → n1 , since δ converges to zero,that is, p2 converges to zero, it seems that contestant 2 has no incentive to compete.
But this is not the case because he plays a crucial role in yielding the almost maximal
possible efforts n1 . The designer sufficiently increases his prize valuation and by thatinduces him to exert efforts that are almost equal to n1 .
In the first example of a municipal project (see the examples discussed in
section 2), despite the fact that both modes of discrimination can be extreme, their
combined use still results in a balanced effect. The designer promises the small
company a very large value in case it receives the entire project. The designer, bystructural discrimination, ensures that the small company's share in the project issufficiently small, such that the large reputable company wins almost the entire
project. In this extreme and most effective case from the designer's point of view, thelarge company transfers the small company a reasonable finite amount which isalmost equal to its initial valuation of the entire project.
In the second example of portfolio distribution between two investment housesin the extreme case the tax-subsidy transfer between the two investment houses under
the optimal dual discrimination strategy is finite. In particular, by Proposition 1, it isequal almost to n1 , the value for investment house 1 of getting the entire portfolio of
the large customer. Again, this result is plausible not because of the good will of thecustomer, but because it ensures him that the two investment houses exert almost the
maximal efforts.
16

5. Conclusion
The enrichment of the “box of tools” of the contest designer, that is, allowing him toexercise dual discrimination instead of just structural discrimination clearly matters.
We have shown that both modes of discrimination are effective and therefore will be
used by the designer possibly yielding almost the largest possible efforts that are equal
to the initially highest prize valuation, when r → ∞ . When the extent of
discrimination is sufficiently high, the efforts in our setting are larger than thoseobtained in a biased lottery contest allowing just structural discrimination, Franke etal. (2013, 2014a, 2014b), While in the latter context, at least three players are active,
under dual discrimination, there are just two active contestants; the contestant with thehighest prize valuation and another contestant who can be any of the other contestants.
The exclusion principle noted in Baye et al. (1999) the elimination of the strongest
player - is thus not valid, not only in the unbiased lottery contest without
discrimination, as shown by Fang (2002), but also in our biased lottery contest withdual discrimination. The inclusion principle is, however, valid, as in Franke et al.
(2013); a weak contestant, who is inactive in the unbiased case of Fang (2002), can be
encouraged to become active. Finally, under our biased lottery contest with dual
discrimination, efforts can be larger than those under any alternative contest game and,in particular, an all-pay auction. This is in contrast to the main results in Franke et al.
(2014a, 2014b), where efforts in the optimally biased N-player all-pay auction contest
with structural discrimination and an optimal Head-starts all-pay auction are strictly
larger than those obtained in the optimally biased simple lottery contest.
References
Baye, M.R., Kovenock, D. and de Vries, C., (1993), "Rigging the Lobbying Process:An Application of the All-Pay Auction", American Economic Review, 83, 289294.
Baye, M.R., Kovenock, D. and de Vries, C., (1994), "The Solution to the Tullock
Rent-Seeking Game When R > 2: Mixed-Strategy Equilibria and Mean
Dissipation Rates", Public Choice, 81, 363-380.
Baye, M.R. and Hoppe, H., (2003), "The Strategic Equivalence of Rent-Seeking,Innovation, and Patent-Race Games", Games and Economic Behavior, 44(2),217-226.
17

Clark, D.J. and Riis, C., (1998), "Contest Success Functions: An Extension",
Economic Theory, 11, 201-204.
Clark, D.J. and Riis, C., (2000), “Allocation Efficiency in a Competitive Bribery
Game”, Journal of Economic Behavior and Organization, 42, 109-124.
Corchón, L. and Dahm, M., (2010), "Foundations for Contest Success Functions",
Economic Theory, 43, 81–98.
Epstein G.S., Mealem, Y. and Nitzan, S., (2011), "Political Culture andDiscrimination in Contests", Journal of Public Economics, 95(1-2), 88-93.
Fang, H., 2002, Lottery versus All-Pay Auction Models of Lobbying", Public Choice,
112, 351-371.
Franke, J., (2012), "Affirmative Action in Contest Games", European Journal of
Political Economy, 28(1), 115-118.
Franke, J. , Kanzow, C., Leininger, W. and Schwartz, A. , (2013), “Effort
Maximization in Asymmetric Contest Games with Heterogeneous Contestants”,
Economic Theory, 2, 589-630.
Franke, J., Kanzow, C., Leininger, W. and Schwartz, A., (2014a), "Lottery versus AllPay Auction Contests - A Revenue Dominance Theorem", Games and Economic
Behavior. Volume 83, 116–126.
Franke, J., Leininger, W. and Wasser, C., (2014b), "Revenue Maximization Head
Starts in Contests", Ruhr Economic Papers No. 524. TU Dortmund.
Fullerton, R.L. and McAfee, P.R., (1999), "Auctioning Entry into Tournaments",
Journal of Political Economy, 107(3), 573-605.
Hirshleifer, J. and Riley, J.G., (1992), "The Analytics of Uncertainty and Information",
Cambridge University Press, Cambridge UK.
Jia, H., (2008), “A Stochastic Derivation of the Ratio Form of Contest Success
Functions”, Public Choice, 135, 125–130.
Jia, H., (2010), "On a Class of Contest Success Functions", The B.E. Journal of
Theoretical Economics, 10(1).
Konrad, K., (2009), Strategy and Dynamics in Contests (London School of Economic
Perspectives in Economic Analysis), Oxford University Press, USA.
Lee S. and Lee, S.Y., (2012) "Prize Allocation in Contests with Size Effect throughPrizes", Theoretical Economics Letters, 2, 212-215.
Lien, D., (1986), “A Note on Competitive Bribery Games”, Economics Letters, 22,337-341.
18

Lien, D., (1990), “Corruption and Allocation Efficiency”, Journal of Development
Economics, 33, 153-164.
Mealem, Y. and Nitzan, S., (2014), “Equity and Effectiveness of Optimal Taxation inContests under an All-Pay Auction", Social Choice and Welfare, 42, 437–464.
Mealem, Y. and Nitzan, S., (2016), "Discrimination in Contests - A Survey", Review
of Economic Design, 20(2), 145-172 .
Munster, J., (2009), "Group Contest Success Functions", Economic Theory, 41(2),345-357.
Skaperdas, S., (1996), "Contest success functions", Economic Theory, 7, 283-290
Tullock, G., (1980), Efficient Rent-Seeking. In: Buchanan J.M, Tollison R.D, TullockG., (ed) Toward a Theory of the Rent-Seeking Society, College Station, TX,
Texas A. and M., University Press, 97-112.
Warneryd, K., (1998), “Ditributional Conflict and Jurisdictional Organization”. Journalof Public Economics, 63(3), 435-450.

19