Sandbagging�

Matthias Kräkely

May 3, 2012

�I thank two anonymous referees and the editor, Leo Kahane, for helpful comments andsuggestions. Financial support by the Deutsche Forschungsgemeinschaft (DFG), grant SFB/TR15 (�Governance and the E¢ ciency of Economic Systems�), is gratefully acknowledged.

yUniversity of Bonn, Department of Economics, Adenauerallee 24-42, D-53113 Bonn, Ger-many, tel: +49 228 733914, fax: +49 228 739210, e-mail: m.kraekel@uni-bonn.de.

1

Abstract

Participants of dynamic contests sometimes play with the rules of the game by

withholding e¤ort in the beginning. Such behavior is referred to as sandbagging. I

consider a two-period contest between heterogeneous players and analyze potential

sandbagging of high-ability participants in period one. This sandbagging can be

bene�cial to avoid second-period matches against other high-ability opponents. I

characterize the conditions under which sandbagging leads to a coordination prob-

lem, similar to that of the battle-of-the-sexes. Moreover, if players�abilities have

a stronger impact on the outcome of the �rst-period contest than e¤ort choices,

mutual sandbagging by all high-ability players can arise.

Key Words: coordination problem; dynamic contest; heterogeneous contestants;

withholding e¤ort.

2

1 Introduction

In dynamic contests, sandbagging can take two di¤erent forms. First, a strong

player may withhold e¤ort1 in a current round in order to hide his strength. As a

result, the opponents�beliefs about the strong player�s ability are adjusted down-

wards, which leads to a reduction of the opponents�e¤ort choices in future rounds.

This form of sandbagging belongs to situations, where the same players meet in

repeated contests. Second, a strong player can withhold e¤ort to exploit the rules

of the dynamic contest. In particular, the player can voluntarily lose a current

contest in order to be matched with weaker opponents in the future. This kind of

sandbagging is aimed at the contest designer and not at the current opponents.

From the viewpoint of society, sandbagging is detrimental for two reasons � it

leads to an underprovision of e¤ort and to a possible misallocation of players to

higher-order contests.

There are several real-world examples for either kind of sandbagging. The

�rst class of sandbagging problems is based on signal jamming, where one player

chooses an unobservable action (e.g., e¤ort) to manipulate the beliefs of other

players regarding his true ability. In the case of sandbagging, a strong player tries

to appear as a weak contestant to make his opponents choose only low e¤orts

in the next round. Examples can be observed in sports when individual athletes

or teams try to win only with a small lead at the beginning of a tournament to

hide their true abilities until meeting stronger opponents in later rounds. At the

beginning of a military con�ict, one party can withhold military resources to make

the opposing party believe that one is easy prey in future battles. In poker, a

player that has a strong hand bets weakly to convince the other players to stay in

the game. In a litigation contest, one lawyer starts with rather weak arguments

to deter the lawyer of the other party from further collecting evidence.

The second class of sandbagging problems is not based on signal jamming.

Instead, the sandbagger withholds e¤ort to play with the rules of the game or

1E¤ort can take di¤erent forms. In sports, "e¤ort" consists of training intensity as well ase¤ort during the match. In military con�icts, "e¤ort" describes the military resources used bythe con�icting parties. In business and litigation contests, "e¤ort" is given by money and timeinvested in the competition.

3

even manipulate them. In a sales contest, individuals sometimes hold aside re-

alized sales when clearly having outperformed their opponents or when being a

clear loser in order to use these realized sales in the next-period contest. In many

(professional) sports, handicap systems are used for leveling out the playing �eld.

Here, sandbagging happens when individuals deliberately perform poorly in less

important contests to accumulate additional handicap for key contests in the fu-

ture. Such practice can be frequently observed in bowling and golf. In certain

racing series, drivers may prefer to underperform in qualifying rounds to obtain

a better starting position as handicap. Soccer tournaments like the FIFA soccer

world championship often consist of two parts � the group phase and the sub-

sequent knock-out phase. Typically, in the round of sixteen, group winners are

matched with second-best teams of other groups. A soccer team that wants to

be matched with the �rst instead of the second-best team of another group can

gain a strategic advantage by deliberately losing the �nal group match and be-

coming second instead of �rst in its respective group. In games like professional

chess, Go, billiards or BMX racing, players can voluntarily lose unimportant con-

tests to decrease their average rating and, thus, to start in the next contest in a

lower class. In this class, the sandbagger can more easily win against less stronger

opponents. However, such sandbagging has the drawback that winner prizes in

lower classes are typically smaller than prizes in higher classes. In dynamic career

contests, workers may prefer to underperform at some time to be assigned to a

less demanding career track in which they can easily outperform their opponents.

Similarly to the previous example, sandbagging leads to the disadvantage that the

player forgoes larger incomes at higher career tracks.

A well-known example for the second class of sandbagging problems, sometimes

referred to as "tanking", comes from professional team sports. A coach following

a tanking strategy makes his team underperform by resting star players, using

dubious tactics during a match, or even ask the players to deliberately lose a match.

Balsdon et al. (2007) show that teams that won the regular-season championship

in National Collegiate Athletic Association men�s basketball often signi�cantly

underperform in subsequent conference tournaments to increase their revenues in

4

an opportunistic way. Another kind of tanking can be observed in connection with

a draft, which is employed by many professional team sports leagues in order to

balance competition. A draft mechanism allows the least successful teams to be

the �rst who hire new talents so that they can catch up with the better teams in

the next season. A badly performing team can therefore pro�t in the long run to

perform even more badly for the rest of the season to manipulate the draft order in

a favorable way. Taylor and Trogdon (2002) use data from the draft lottery system

of the National Basketball Association�s and provide evidence for the existence of

such tanking behavior.

Several contest papers have discussed the signal-jamming argument of sand-

bagging.2 However, to the best of my knowledge there do not exist contest papers

on sandbagging as means of playing with the rules of the game. This paper aims

at narrowing this gap. The analysis will proceed in two steps. I start by consid-

ering a heterogeneous contest between a high-ability player and a low-ability one.

The contest winner is assigned to a high-prize contest in the second period (major

contest), whereas the contest loser is relegated to a low-prize contest (consolation

contest). In both second-period contests, there will be random matching with

an unknown opponent. The two �rst-period players only know a discrete type

distribution over the second-period opponents for each kind of match.

In the benchmark case of an isolated �rst-period contest without the exis-

tence of subsequent second-period contests, both the high-ability player and his

low-ability opponent would choose identical equilibrium e¤orts. I show that the

assignment to di¤erent second-period contests can make the high-ability player

sandbag by withholding e¤ort in the �rst round. Such sandbagging will be ra-

tional if saved e¤ort costs from avoiding another high-ability player in a possibly

homogeneous major contest exceed expected income losses from being assigned to

the consolation match. The corresponding sandbagging condition also shows how a

contest designer can choose the characteristics of the contest to avoid sandbagging.

A high winner prize for the major contest does not necessarily solve the sandbag-

ging problem since this winner prize also in�uences equilibrium e¤orts and, thus,

2See the related literature cited below.

5

e¤ort costs in equilibrium. However, optimally designing the distributions over

the second-period opponents�types is a very e¤ective device against sandbagging.

In a second step, I consider a situation with two heterogeneous contests that

simultaneously take place in period one. This setting is used to address the prob-

lem of endogenous matching in the second period. While �rst-period winners are

assigned to the major contest, the �rst-period losers are relegated to a consolation

contest. This more complicated setting renders a complete solution of the game

impossible but I can still characterize situations in which sandbagging happens.

In particular, there are cases in which the high-ability players face a coordination

problem similar to that of the battle of the sexes, since each high-ability player can

only pro�t from sandbagging if the other one prefers not to sandbag. Moreover,

if the impact of players�abilities dominate the impact of e¤ort on the outcome

of the �rst-period contests, even mutual sandbagging by both high-ability players

can be an equilibrium.

One immediate implication of my results is that the current rules for the soccer

world championship�s group phase should be redesigned in order to avoid sand-

bagging. According to these rules, the �nal matches within a certain group have

to take place simultaneously, which should restore suspense. However, in order to

prevent sandbagging, the �nal matches of each two adjacent groups whose best

and second best teams are seeded together within the round of sixteen should take

place at the same time. This redesign would come at a cost since customers would

like to visit several of these matches which is only possible if the matches proceed

sequentially.

My paper is related to the broad economic literature on contests and tourna-

ments, in particular to the theoretical work on sporting contests (e.g., Szymanski

2004, Késenne 2006, Dietl et al. 2008, Fort and Winfree 2009). Szymanski (2003)

and Konrad (2009) give comprehensive overviews of the theory of contests, with

Szymanski (2003) focussing on the economics of sporting contests. The cited work

does not address the sandbagging problem, but sandbagging has some parallels

to the discouragement e¤ect highlighted by Konrad (2009, pp. 189�191). My pa-

per is also related to the work on dynamic contests (e.g., Harris and Vickers 1987,

6

Konrad and Kovenock 2009, Gürtler and Münster 2010, Ederer 2010). I consider a

two-stage tournament model with a major contest and a consolation contest in the

second stage. Consolation contests have also been addressed by Kiyotaki (2004),

Uehara (2009) and Kräkel (2012). The signal-jamming argument for sandbagging

has been �rst suggested by Rosen (1986, p. 714) and, thereafter, theoretically

investigated by Amegashie (2006), Hörner and Sahuguet (2007), Zhang and Wang

(2009), Münster (2009). However, all these papers do not discuss sandbagging as

a kind playing with the rules of the contest.

There exists a rich literature on sandbagging in psychology (e.g., Gibson and

Sachau 2000, Gibson et al. 2002, as well as the work cited therein). Here, sand-

bagging is explained as a self-presentational strategy that is used by individuals

to reduce performance pressure. Low performance decreases future performance

expectations and, hence, "psychological costs" that are associated with high pres-

sure. Since the expectations of others are manipulated, there are parallels to the

above mentioned signal-jamming model in economics. However, the psychological

argument is also valid for situations in which an individual feels pressure to per-

form, but does not participate in a contest. Note that, in a competitive situation,

the sandbagging strategy especially applies to favorites, who are under the highest

pressure as a result of people�s expectations. Another argument for sandbagging

deals with the perception of performance: the creation of a low benchmark can

be bene�cial when future high performance is evaluated much more positive by

outsiders due to a comparison with the old benchmark. There are also paral-

lels between sandbagging and self-handicapping (e.g., bad preparation or drinking

before an exam). If an individual is uncertain about his or her own ability, self-

handicapping can be bene�cial by avoiding negative ability attributions in case

of a failure and generating even stronger positive ability attributions in case of

success.3

The paper is organized as follows. The next section introduces the basic model.

Section 3 characterizes the conditions for a sandbagging equilibrium in the basic

model. Section 4 considers the case of two simultaneous �rst-period contests,

3See Tirole (2002) for a respective behavioral economics model.

7

leading to endogenous matching of players in the second period and to a possible

coordination problem of the high-ability players. Section 5 concludes.

2 The Basic Model

I consider a contest game that lasts two periods.4 In the �rst period, there is a

heterogeneous match between a player with low ability aL and a player with high

ability aH (> aL). These abilities are common knowledge. Let �a := aH � aLdenote the ability di¤erence. Each player i (i = H;L) exerts non-negative e¤ort

ei to in�uence his score function

xi (ei) = h (ei) + ai + "i (1)

with h being monotonically increasing and concave and "i describing random

noise.5 Following the seminal paper by Lazear and Rosen (1981), I assume that

"H and "L are stochastically independent and identically distributed (i.i.d.). The

cumulative distribution function of "H�"L is denoted by G and the correspondingdensity by g. Let g be unimodal and symmetric about zero.6 Exerting e¤ort eileads to cost c (ei) for player i with c (0) = c0 (0) = 0 and c0 (ei) ; c00 (ei) > 0 for

ei > 0. Player H (L) is declared contest winner if xH (eH) > xL (eL) (xH (eH) <

xL (eL)). The contest winner receives income Y > 0, whereas the loser gets zero.

In the second period, the two players H and L are assigned to two di¤erent

contests �depending on the outcome of the �rst-period contest. The winner of

4The game is based on the di¤erence-form contest-success function used, e.g., by Lazear andRosen (1981), Dixit (1987), Baik (1998), Che and Gale (2000).

5In most of the contest literature, player heterogeneity is modelled in two di¤erent ways �either the players di¤er in the steepness of their cost functions, or di¤erent abilities enter asadditive parameters in the players�output or score function; see O�Kee¤e et al. (1984), Schotterand Weigelt (1992). The main e¤ect �equilibrium e¤orts decrease in the degree of heterogeneity�is qualitatively the same in both settings. However, only the second modelling allows for anexplicit equilibrium solution in a general framework. Therefore, I follow Meyer (1991), Fairburnand Malcomson (2001), and Hö­ er and Sliwka (2003) by using a setting with additive abilities.

6Unimodality is often assumed in contest models; see, e.g., Dixit (1987), Drago et al. (1996),Hvide (2002), Chen (2003). For example, if "H and "L are normally (uniformly) distributed,the convolution g will be normal (triangular). If "H and "L are exponential, g follows a Laplacedistribution.

8

the �rst-period contest participates in the major contest M where he is randomly

matched with another player. This opponent has ability aH with probability pM 2(0; 1) and ability aL with probability 1� pM . After having entered contest M but

before choosing e¤orts, both players observe individual abilities. The two players

have the same score function (1) as in the �rst-period contest. The contest winner

(i.e., the player with the higher score) earns income YM = � � Y with � > 1, and

the loser gets zero.

The loser of the �rst-period contest is assigned to a consolation contest C.

Here, he meets an opponent of ability aH with probability pC 2 (0; 1) and an

opponent of ability aL with probability 1� pC . Analogously to the major contest,individual abilities become common knowledge before both players exert e¤orts,

each of the two players has the same score function (1), and the player with the

higher score is declared contest winner. This player gets income YC = � � Y with

� < �, whereas the loser of contest C earns zero.

Finally, let all players have zero reservation values in both periods, which guar-

antees participation in each contest. The game is solved by backward induction,

starting with the second-period contests M and C. The main concern is with the

players�equilibrium e¤orts (e�H ; e�L) in the �rst-period contest. Since without the

existence of second-period contests both players would exert identical e¤orts in pe-

riod one,7 the equilibrium (e�H ; e�L) will be called a sandbagging equilibrium, if the

high-ability player exerts less e¤ort than his low-ability opponent (i.e., e�H < e�L).

3 Solution to the Basic Model

In the second period, two players (the �rst-round winner and a randomly matched

opponent) meet in the major contest and two players (the �rst-round loser and a

randomly matched opponent) have to participate in the consolation match. Con-

sider either of these two contests � 2 fC;Mg with participants i and j havingtalents ai and aj (i; j 2 fH;Lg) competing for prize Y� 2 fYC ; YMg by choosing

7This claim can be immediately checked by the solution of the second-period contests, becausethe game ends after these contests.

9

e¤orts ei and ej, respectively. Let EU�ij (EU�ji) denote the second-round expected

utility of player i (j) from competing against opponent j (i) in contest � 2 fC;Mg:

EU�ij = Y� �G (h (ei) + ai � h (ej)� aj)� c (ei)

and EU�ji = Y� � [1�G (h (ei) + ai � h (ej)� aj)]� c (ej) :

I assume that an equilibrium in pure strategies exists and that it is characterized

by the �rst-order conditions8

Y�g (h (ei) + ai � h (ej)� aj) =c0 (ei)

h0 (ei)=c0 (ej)

h0 (ej):

Thus, if a pure-strategy equilibrium exists, it will be unique and symmetric with

both players choosing e¤ort level (Y�g (ai � aj)) where denotes the (monoton-ically increasing) inverse of c0=h0.9 Note that, in equilibrium, a player has e¤ort

costs c ( (Y�g (0))) when competing in a homogeneous match and c ( (Y�g (�a)))

when competing in a heterogeneous match. Let

�C� := c ( (Y�g (0)))� c ( (Y�g (�a))) > 0

denote a player�s additional equilibrium e¤ort costs in contest � from being matched

with an equally strong opponent instead of a weaker or stronger opponent, where

�C� > 0 follows from the unimodality of g.

In the �rst-period contest, both players H and L anticipate the possible out-

comes in the second period when choosing optimal e¤orts. Let EU��ij denote the

equilibrium expected utility of player i from competing against player j in second-

8The problem that the existence of pure-strategy equilibria cannot be guaranteed in generalis well-known in the contest literature; see, e.g., Lazear and Rosen (1981), p. 845, and Nalebu¤and Stiglitz (1983), p. 29. See Schöttner (2008) and Gürtler (2011) on an existence condition. Ifollow the usual procedure by assuming existence of pure-strategy equilibria and then describingtheir characteristics.

9Since c is strictly convex and h is concave, c0=h0 is a monotonically increasing function and,therefore, invertible.

10

period contest �. Then, player H maximizes

�Y + pM � EUM�

HH + (1� pM) � EUM�HL

��G (h (eH) + �a� h (eL))

+�pC � EUC�HH + (1� pC) � EUC�HL

�� [1�G (h (eH) + �a� h (eL))]� c (eH)

and player L

�Y + pM � EUM�

LH + (1� pM) � EUM�LL

�� [1�G (h (eH) + �a� h (eL))]

+�pC � EUC�LH + (1� pC) � EUC�LL

��G (h (eH) + �a� h (eL))� c (eL) ;

yielding the following result:

Proposition 1 In the �rst-period contest, a pure-strategy equilibrium will be a

sandbagging equilibrium (e�H ; e�L) with e

�H < e

�L, if and only if

(1� 2pC)�CC + (2pM � 1)�CM >

�G (�a)� 1

2

�(�� �)Y: (2)

Proof. See Appendix.

The right-hand side of condition (2) is strictly positive. It describes player H�s

worst-case second-period opportunity costs from sandbagging.10 If sandbagging is

successful and player H is assigned to contest C instead of contest M , the worst

thing that can happen is to meet another aH-player in contest C whereas player

L is matched with another aL-player in contest M . Now, player H would deeply

regret not to be assigned to contest M , because switching to contest M in this

situation would increase his winning probability by G (�a) � 12and the winner

prize by (�� �)Y .The left-hand side of condition (2) may be positive or negative. It is negative

if pC > 1=2 and pM < 1=2, which makes (2) impossible to be satis�ed. The

intuition for this �nding is clear, because the exclusive aim of sandbagging is to

avoid another aH-player. If, on the contrary, pC = 0 and pM = 1 then sandbagging

can be quite attractive for H, since he would avoid (meet) another aH-player for

10First-period opportunity costs from sandbagging amount to Y , but these costs cancel out inthe comparison of e�H and e�L as shown in the proof of Proposition 1.

11

sure by entering contest C (contest M). In that case, player H realizes gains

from sandbagging in form of saved e¤ort costs �he directly realizes a relative cost

advantage �CC from being matched with an aL-player in contest C and indirectly

gains cost savings �CM from not being matched with an aH-player in contest M .

Interestingly, the winner prize of the �rst-period contest does not play a role

for the sandbagging condition (2). At �rst sight, this observation seems puzzling,

because not winning the �rst-period contest due to sandbagging is directly associ-

ated with the loss of this winner prize. However, the proof of Proposition 1 shows

that the winner prize of the �rst-period contest in�uences the e¤ort choices of H

and L in the same way so that this in�uence cancels out when comparing e�H and

e�L.

Condition (2) shows that a su¢ ciently high winner prize in the major contest

(i.e., a large value of �) does not necessarily work against sandbagging. On the one

hand, a high YM makes participation in the major contest quite attractive for player

H: the right-hand side of (2) increases in �, which works against sandbagging. On

the other hand, � also in�uences �CM on the left-hand side of (2) so that the

overall e¤ect of � is not clear. Let, for example, h (ei) = ei and c (ei) = 2e2i

( > 0). Then c0 (ei) =h0 (ei) = ei and, hence, (x) = x= so that �CM =�2Y 2

2

�g (0)2 � g (�a)2

�. A higher winner prize YM will intensify sandbagging if

@

@�((2pM � 1)�CM) >

@

@�

��G (�a)� 1

2

�(�� �)Y

�,

< (2pM � 1)�Y

�g (0)2 � g (�a)2

�G (�a)� 1

2

Thus, if the cost function is not too steep �a very steep cost function would erase

advantages from sandbagging since e¤ort costs in equilibrium would be negligible

anyway �a higher value of YM can aggravate the sandbagging problem.

The in�uence of the ability di¤erence �a is ambiguous, too. The higher �a the

larger are the expected opportunity costs of sandbagging (via G (�a)). However,

saved e¤ort costs from avoiding a match with another aH-player also increase in

�a (i.e., @�C�=@�a > 0, � = C;M , by the unimodality of g).

12

We obtain clear-cut results for the in�uence of the probabilities pC and pM , as

highlighted above. The impact of the shape of the cost function c on the left-hand

side of (2) is twofold. On the one hand, the steeper the cost function, the larger are

the cost savings from sandbagging, �C� (� = C;M), for given equilibrium e¤orts.

On the other hand, the steeper the cost function the �atter is , and hence the

smaller are the equilibrium e¤orts in each match. If equilibrium e¤ort levels are

very small anyway, the cost-saving advantage of sandbagging will be negligible.

For a wide range of convex cost functions, this second e¤ect dominates the �rst

e¤ect. Let, for simplicity, h (ei) = ei and c belong to the family of cost functions

c (ei) = c � emi =m with c > 0 and m � 2. Then, according to (2), there exists a

cut-o¤ value so that a sandbagging equilibrium exists if and only if c is smaller

than this cut-o¤:

c <

�g (0)

mm�1 � g (�a)

mm�1

�m�1 h(1� 2pC) �

mm�1 + (2pM � 1)�

mm�1

im�1Y�

m (�� �)�G (�a)� 1

2

��m�1 :

Intuitively, only if the cost function is not too steep, equilibrium e¤ort levels will

su¢ ciently di¤er in homogeneous and heterogeneous matches, and the cost-saving

advantage for player H from avoiding another high-ability player via sandbagging

can be su¢ ciently large.

4 Endogenous Matching

The comparative static results have shown that a contest designer has several

possibilities to in�uence sandbagging behavior. Following condition (2), the most

successful way to prevent sandbagging seems to be an appropriate choice of pCand pM . For example, the contest designer could create maximal uncertainty

about the opponents� types in contests C and M (i.e., condition (2) cannot be

satis�ed if pC = pM = 1=2). However, such policy is not possible in situations

with two simultaneous contests in period one combined with the natural rule that

the �rst-period winners are both assigned to the major contest M , whereas the

13

two �rst-period losers are matched together in the consolation contest C.11 In that

case, matching is generated endogenously, which can lead back to the problem of

sandbagging.

Consider such situation with two heterogeneous �rst-period contests A and B

between one aH-player and one aL-player, respectively. Let eij denote the e¤ort

choice of the ai-player in contest j (i = H;L; j = A;B) and �Gj := G(h (eHj) +

�a� h (eLj)) the winning probability of the aH-player in contest j. The objectivefunction of the aH-player in contest k (k = A;B; k 6= j) can be written as

�Y + �Gj � EUM�

HH +�1� �Gj

�� EUM�

HL

��G (h (eHk) + �a� h (eLk))

+��1� �Gj

�� EUC�HH + �Gj � EUC�HL

�� [1�G (h (eHk) + �a� h (eLk))]� c (eHk)

whereas the aL-player in contest k maximizes

�Y + �Gj � EUM�

LH +�1� �Gj

�� EUM�

LL

�� [1�G (h (eHk) + �a� h (eLk))]

+��1� �Gj

�� EUC�LH + �Gj � EUC�LL

��G (h (eHk) + �a� h (eLk))� c (eLk) :

The two �rst-order conditions read as follows:

�Y +

�EUM�

HH � EUC�HL��Gj +

�EUM�

HL � EUC�HH� �1� �Gj

��� �gk =

c0 (eHk)

h0 (eHk)�Y +

�EUM�

LH � EUC�LL��Gj +

�EUM�

LL � EUC�LH� �1� �Gj

��� �gk =

c0 (eLk)

h0 (eLk)

with �gk := g (h (eHk) + �a� h (eLk)). As c0=h0 is a monotonically increasing func-tion, in a pure-strategy equilibrium we will have eHk < eLk if and only if

Y +�EUM�

HH � EUC�HL��Gj +

�EUM�

HL � EUC�HH� �1� �Gj

�<

Y +�EUM�

LH � EUC�LL��Gj +

�EUM�

LL � EUC�LH� �1� �Gj

�;

which � by inserting for the second-period expected utilities from the proof of

11This scenario captures the career-contest example from the introduction as well as the sportscases where a sandbagger wants to decrease his average rating to start in the next contest in alower class.

14

Proposition 1 �can be rewritten so that we obtain the following result:

Proposition 2 A pure-strategy equilibrium in contest k (k = A;B) will exhibit

sandbagging by the aH-player (i.e., e�Hk < e�Lk) if and only if

�Gj >1

2+

�G (�a)� 1

2

�(�� �)Y

2 (�CM +�CC)(j = A;B; j 6= k): (3)

Proposition 2 shows that an extreme sandbagging result h (e�LA) > h (e�HA)+�a

in combination with h (e�LB) > h (e�HB) + �a is impossible. Suppose that �Gj <

12,

i.e., h (eLj) > h (eHj) + �a. In this situation, the al-player is more likely to win

than the ah-player in contest j, but the same cannot be true in contest k, because

here the ah-player already leads by �a and, in addition, chooses more e¤ort than

his opponent according to (3). Thus, it is impossible that in both �rst-period

contests the two al-players have higher winning probabilities than their respective

opponents. Intuitively, if an ah-player anticipates that in the other contest the

less able player has a higher winning probability than the more able one, for three

reasons he prefers to enter the major contest and, thus, to win the �rst round.

First, by winning the �rst-period contest the ah-player would earn Y , whereas

losing is associated with zero �rst-period income. Second, the ah-player wants to

avoid being matched with the other ah-player in the next round, which would imply

high e¤ort costs, due to the homogeneous competition, and a rather low winning

probability. Third, in the major contest he can win �Y but in the consolation

contest only �Y < �Y .

The proposition also points to a non-trivial coordination problem of the players

in period one. Condition (3) shows that if in contest j (j = A;B) the high-ability

player has a high winning probability, it will be quite attractive for the high-

ability player in the other contest k 6= j to sandbag. Inequality (3) is more easilysatis�ed if, for given �Gj, the expression

�G (�a)� 1

2

�(�� �)Y takes small values

and �CM+�CC large values.12 If, for example, Y > 0 but �! � both aH-players

will face a serious coordination problem since winning either second-period contest

is equally attractive. Now a major aim of the aH-players is to avoid meeting one

12The intuition for this �nding is analogous to the intuition for the �nding of Proposition 1.

15

another in period two. The coordination problem is comparable to that of the

battle-of-the-sexes game since achieving a heterogeneous match in contest M and

directly earning �rst-period income Y is more preferable for an aH-player than

being matched with an aL-player in contest C. Note that Proposition 2 does not

rule out the possibility that even both high-ability players sandbag in period one.

In particular, if the right-hand side of condition (3) is close to 1=2, the condition

can be satis�ed for e�Hj < e�Lj.

However, even if parameterized functions are used to specify h, c, and G,

the complicated structure of the two simultaneous �rst-period contests does not

allow for an exact computation of the equilibria. Note also that, aside from the

aH-players, an aL-player may be interested as well to be matched with a weak

aL-opponent instead of a strong aH-player in period two. For these reasons, I

refer to the tractable case of binary e¤orts to check the robustness of the previous

conjectures. I do not consider all 24 = 16 e¤ort combinations but focus on the two

possibilities of sandbagging equilibria sketched in the paragraph before:13

Proposition 3 Suppose that, in the �rst-period contests, players can only decide

between e1 and e0 with h (e1) := h1 > 0 =: h (e0) and c (e1) := c1 > 0 =: c (e0).

(a) If

G (�h1 +�a) �1

2+

�G (�a)� 1

2

�(�� �)Y

2 (�CM +�CC)� G (h1 +�a) ;

there exist feasible parameter constellations leading to multiple sandbagging equilib-

ria (e�HA; e�LA; e

�HB; e

�LB) = (e0; e1; e1; e0) and (e

�HA; e

�LA; e

�HB; e

�LB) = (e1; e0; e0; e1).

(b) If

G (�h1 +�a) �1

2+

�G (�a)� 1

2

�(�� �)Y

2 (�CC +�CM);

there exist feasible values of c1 so that both aH-players sandbag in equilibrium (i.e.,

(e�HA; e�LA; e

�HB; e

�LB) = (e0; e1; e0; e1)).

Proof. See Appendix.13A detailed proof can be requested from the author. [See the additional material for the

referees in the appendix.]

16

The proposition con�rms that indeed there exist feasible parameter values lead-

ing to a coordination problem of the aH-players in period one. In these cases, the

impact of higher e¤ort (i.e., h1) must be su¢ ciently large so that the players can ef-

fectively in�uence the outcome of the �rst-period contests. If, however, the impact

of higher e¤ort is dominated by player heterogeneity (i.e., �a > h1), even both

aH-players may prefer to sandbag and rest on their lead �a, while both aL-players

choose higher e¤ort to assure themselves a su¢ ciently high winning probability.

5 Conclusion

The contribution of the paper to the economic literature is twofold. First, the

paper theoretically investigates a new kind of counterproductive behavior in con-

test games. The previous literature has analyzed the possibilities of sabotage (e.g.

Konrad 2000, Chen 2003, Münster 2007), cheating and doping (e.g., Kräkel 2007,

Gilpatric 2011, Ryvkin 2012), and sandbagging as a signal jamming device (e.g.,

Hörner and Sahuguet 2007, Münster 2009) in contests. The current paper shows

under which conditions sandbagging is used to in�uence the matching of players

in future contests within a dynamic contest game.

Second, the well-known ratchet e¤ect (e.g., Baron and Besanko 1984, Freixas

et al. 1985, La¤ont and Tirole 1988) has highlighted how withholding of e¤ort can

arise in a dynamic context when a single agent faces an incentive scheme that is

adjusted according to observed performance. In the current paper, I have shown

that withholding of e¤ort can also be rational if a certain agent competes against

other agents in a dynamic contest. Here, manipulating the composition of future

contests and not the manipulation of a principal�s wage policy is the major motive

to exert low e¤ort levels.

17

Appendix

Proof of Proposition 1

Computing the players��rst-order conditions gives

c0 (eH) =��Y + pMEU

M�HH + (1� pM)EUM�

HL

���pCEU

C�HH + (1� pC)EUC�HL

��h0 (eH) g (h (eH) + �a� h (eL))

and

c0 (eL) =��Y + pMEU

M�LH + (1� pM)EUM�

LL

���pCEU

C�LH + (1� pC)EUC�LL

��h0 (eL) g (h (eH) + �a� h (eL)) :

Thus, both players have the same marginal costs of exerting e¤ort, but potentially

di¤erent marginal returns, as indicated by the expressions in square brackets.

Comparing these expressions shows that playerH will choose less e¤ort than player

L in the �rst-period contest, if

�Y + pMEU

M�HH + (1� pM)EUM�

HL

���pCEU

C�HH + (1� pC)EUC�HL

�<�

Y + pMEUM�LH + (1� pM)EUM�

LL

���pCEU

C�LH + (1� pC)EUC�LL

�:

Inserting for

EUM�HH = EU

M�LL =

�Y

2� c ( (�Y g (0)))

EUM�HL = �Y �G (�a)� c ( (�Y g (�a)))

EUC�HH = EUC�LL =

�Y

2� c ( (�Y g (0)))

EUC�HL = �Y �G (�a)� c ( (�Y g (�a)))

EUM�LH = �Y � [1�G (�a)]� c ( (�Y g (�a)))

EUC�LH = �Y � [1�G (�a)]� c ( (�Y g (�a)))

and rewriting yields condition (2).

18

Proof of Proposition 3

(a) Multiple sandbagging equilibria:

Note that the same conditions for (eHA; eLA; eHB; eLB) = (e0; e1; e1; e0) to be an

equilibrium must also hold for an equilibrium (eHA; eLA; eHB; eLB) = (e1; e0; e0; e1).

Therefore, we have to consider only one of them. The expected utility of the aH-

player in contest A under (eHA; eLA; eHB; eLB) = (e0; e1; e1; e0) is given by

EUHA (e0) =�Y + EUM�

HHG (h1 +�a) + EUM�HL [1�G (h1 +�a)]

�G (�h1 +�a)

+�EUC�HH [1�G (h1 +�a)] + EUC�HLG (h1 +�a)

�[1�G (�h1 +�a)]

=�Y + EUM�

HL ��EUM�

HL � EUM�HH

�G (h1 +�a)

�G (�h1 +�a)

+�EUC�HH +

�EUC�HL � EUC�HH

�G (h1 +�a)

�[1�G (�h1 +�a)]

= [Y + EUM�HL � EUC�HH �

X�2fM;Cg

(EU��HL � EU��HH)G (h1 +�a)]G (�h1 +�a)

+EUC�HH +�EUC�HL � EUC�HH

�G (h1 +�a) :

If this player deviates to eHA = e1 his payo¤ switches to

EUHA (e1) = [Y + EUM�HL � EUC�HH �

X�2fM;Cg

(EU��HL � EU��HH)G (h1 +�a)]G (�a)

+EUC�HH +�EUC�HL � EUC�HH

�G (h1 +�a)� c1:

Hence, he will not deviate if

c1 � HA � [G (�a)�G (�h1 +�a)] (4)

with HA := [Y + EUM�HL � EUC�HH �

X�2fM;Cg

(EU��HL � EU��HH)G (h1 +�a)].

19

Under (eHA; eLA; eHB; eLB) = (e0; e1; e1; e0) the aL-player in contest A obtains

EULA (e1) = [Y + EUM�LHG (h1 +�a)

+ EUM�LL (1�G (h1 +�a))] (1�G (�h1 +�a))

+�EUC�LH (1�G (h1 +�a)) + EUC�LLG (h1 +�a)

�G (�h1 +�a)� c1

= Y + EUM�LL �

�EUM�

LL � EUM�LH

�G (h1 +�a)� c1

+ [EUC�LH � EUM�LL � Y +

X�2fM;Cg

(EU��LL � EU��LH)G (h1 +�a)]G (�h1 +�a) :

Deviation to eLA = e0 leads to

EULA (e0) = Y + EUM�LL �

�EUM�

LL � EUM�LH

�G (h1 +�a)

+ [EUC�LH � EUM�LL � Y +

X�2fM;Cg

(EU��LL � EU��LH)G (h1 +�a)]G (�a) :

The aL-player will not deviate if

c1 � LA � [G (�a)�G (�h1 +�a)] (5)

with LA := [Y + EUM�LL � EUC�LH �

X�2fM;Cg

(EU��LL � EU��LH)G (h1 +�a)].

Conditions (4) and (5) show that for an equilibrium (e0; e1; e1; e0) we must have

HA � [G (�a)�G (�h1 +�a)] � c1 � LA � [G (�a)�G (�h1 +�a)] : (6)

A non-empty interval for feasible values of c1 is guaranteed, if

HA < LA ,

EUM�HL � EUC�HH �

X�2fM;Cg

(EU��HL � EU��HH)G (h1 +�a) �

EUM�LL � EUC�LH �

X�2fM;Cg

(EU��LL � EU��LH)G (h1 +�a) : (7)

20

Inserting for the second-period expected utilities from the proof of Proposition 1

yields

G (h1 +�a) �1

2+

�G (�a)� 1

2

�(�� �)Y

2 (�CC +�CM): (8)

Next, we have to ensure that the players in contest B do not want to deviate

from (eHA; eLA; eHB; eLB) = (e0; e1; e1; e0) either. The payo¤ of the aH-player in

contest B under (e0; e1; e1; e0) is given by

EUHB (e1) =�Y + EUM�

HHG (�h1 +�a) + EUM�HL (1�G (�h1 +�a))

�G (h1 +�a)

+�EUC�HH (1�G (�h1 +�a)) + EUC�HLG (�h1 +�a)

�(1�G (h1 +�a))� c1

= [Y + EUM�HL � EUC�HH �

X�2fM;Cg

(EU��HL � EU��HH)G (�h1 +�a)]G (h1 +�a)

+�EUC�HH (1�G (�h1 +�a)) + EUC�HLG (�h1 +�a)

�� c1:

If the player deviates he will realize

EUHB (e0) = [Y + EUM�HL � EUC�HH �

X�2fM;Cg

(EU��HL � EU��HH)G (�h1 +�a)]G (�a)

+�EUC�HH (1�G (�h1 +�a)) + EUC�HLG (�h1 +�a)

�:

He will not deviate if

c1 � HB � [G (h1 +�a)�G (�a)] (9)

with HB := [Y + EUM�HL � EUC�HH �

X�2fM;Cg

(EU��HL � EU��HH)G (�h1 +�a)].

The expected utility of the aL-player in contest B under (eHA; eLA; eHB; eLB) =

21

(e0; e1; e1; e0) is described by

EULB (e0) = Y + EUM�LHG (�h1 +�a) + EUM�

LL (1�G (�h1 +�a))

+ [EUC�LH � EUM�LL � Y +

X�2fM;Cg

(EU��LL � EU��LH)G (�h1 +�a)]G (h1 +�a) :

If the player deviates to eLB = e1 he obtains

EULB (e1) = Y + EUM�LHG (�h1 +�a) + EUM�

LL (1�G (�h1 +�a))� c1+ [EUC�LH � EUM�

LL � Y +X

�2fM;Cg

(EU��LL � EU��LH)G (�h1 +�a)]G (�a) :

Deviation is not pro�table if

c1 � LB � [G (h1 +�a)�G (�a)] (10)

with LB := [Y + EUM�LL � EUC�LH �

X�2fM;Cg

(EU��LL � EU��LH)G (�h1 +�a)].

According to conditions (9) and (10), an equilibrium (e�HA; e�LA; e

�HB; e

�LB) = (e0; e1; e1; e0)

requires

LB � [G (h1 +�a)�G (�a)] � c1 � HB � [G (h1 +�a)�G (�a)] : (11)

Thus, we must have that

LB < HB ,

EUM�LL � EUC�LH �

X�2fM;Cg

(EU��LL � EU��LH)G (�h1 +�a) �

EUM�HL � EUC�HH �

X�2fM;Cg

(EU��HL � EU��HH)G (�h1 +�a) : (12)

Comparison with (7) immediately shows that (12) is equivalent to

G (�h1 +�a) �1

2+

�G (�a)� 1

2

�(�� �)Y

2 (�CC +�CM): (13)

22

Combining (8) with (13) gives

G (�h1 +�a) �1

2+

�G (�a)� 1

2

�(�� �)Y

2 (�CM +�CC)� G (h1 +�a) ; (14)

which is satis�ed as long as�G (�a)� 1

2

�(�� �)Y < �CM + �CC and h1 is

su¢ ciently large.

Recall that (14) only guarantees that HA < LA and LB < HB. However,

we must still ensure that conditions (6) and (11) hold for the same values of c1.

Thus, the intervals (HA[G (�a)�G (�h1 +�a)];LA[G (�a)�G (�h1 +�a)])and (LB[G (h1 +�a)�G (�a)];HB[G (h1 +�a)�G (�a)]) must overlap. SinceHA < HB and LA < LB �implying HA < LA < LB < HB �the proba-

bility mass G (�a) � G (�h1 +�a) must be su¢ ciently large and G (h1 +�a) �G (�a) su¢ ciently small to make the intervals overlap, i.e.,

LA[G (�a)�G (�h1 +�a)] > LB[G (h1 +�a)�G (�a)],�Y + EUM�

LL � EUC�LH�([G (�a)�G (�h1 +�a)]� [G (h1 +�a)�G (�a)])

>X

�2fM;Cg

(EU��LL � EU��LH)� (15)

fG (h1 +�a) [G (�a)�G (�h1 +�a)]�G (�h1 +�a) [G (h1 +�a)�G (�a)]g :

There are several ways to construct parameter constellations so that this condition

holds without violating (14). Suppose, for example, that G has a bounded support

and �a is near the upper bound so that G (�a) � 1 and g (�a) � 0 (recall thatg has a unique mode at zero).14 Then, condition (15) boils down to

�Y + EUM�

LL � EUC�LH�([1�G (�h1 +�a)])

>X

�2fM;Cg

(EU��LL � EU��LH) f[1�G (�h1 +�a)]g :

14Note also that condition (14) becomes G (�h1 +�a) � 12 +

(���)Y4(�CM+�CC)

� 1, which canstill be satis�ed.

23

Inserting for

�Y + EUM�

LL � EUC�LH�=�1 +

�

2� � [1�G (�a)]

�Y

� [c ( (�Y g (0)))� c ( (�Y g (�a)))]

��1 +

�

2

�Y � c ( (�Y g (0)))

and

X�2fM;Cg

(EU��LL � EU��LH) =�G (�a)� 1

2

�(�+ �)Y � (�CM +�CC)

� (�+ �)Y

2� [c ( (�Y g (0))) + c ( (�Y g (0)))]

yields(2� �)Y

2> �c ( (�Y g (0))) ;

which is clearly satis�ed for � < 2. To sum up, there exist feasible parameter

constellations for which multiple equilibria (e�HA; e�LA; e

�HB; e

�LB) = (e0; e1; e1; e0)

and (e�HA; e�LA; e

�HB; e

�LB) = (e1; e0; e0; e1) exist.

(b) Both high-ability players sandbag:

The expected utility of either aH-player in the two �rst-period contests under

(eHA; eLA; eHB; eLB) = (e0; e1; e0; e1) is given by

EUH (e0) = [Y + EUM�HL � EUC�HH �

X�2fM;Cg

(EU��HL � EU��HH)G (�h1 +�a)]G (�h1 +�a)

+EUC�HH +�EUC�HL � EUC�HH

�G (�h1 +�a) :

If an aH-player deviates to e1, he will obtain

EUH (e1) = [Y + EUM�HL � EUC�HH �

X�2fM;Cg

(EU��HL � EU��HH)G (�h1 +�a)]G (�a)

+EUC�HH +�EUC�HL � EUC�HH

�G (�h1 +�a)� c1:

24

Thus, deviation does not pay if

c1 � [Y + EUM�HL � EUC�HH �

X�2fM;Cg

(EU��HL � EU��HH)G (�h1 +�a)]

� [G (�a)�G (�h1 +�a)] : (16)

The expected utility of either aL-player in the two �rst-period contests under

(eHA; eLA; eHB; eLB) = (e0; e1; e0; e1) is described by

EUL (e1) = Y + EUM�LL �

�EUM�

LL � EUM�LH

�G (�h1 +�a)� c1

+ [EUC�LH � EUM�LL � Y +

X�2fM;Cg

(EU��LL � EU��LH)G (�h1 +�a)]G (�h1 +�a) :

Deviation to e0, yielding

EUL (e0) = Y + EUM�LL �

�EUM�

LL � EUM�LH

�G (�h1 +�a)

+ [EUC�LH � EUM�LL � Y +

X�2fM;Cg

(EU��LL � EU��LH)G (�h1 +�a)]G (�a)

is not pro�table if

c1 � [Y + EUM�LL � EUC�LH �

X�2fM;Cg

(EU��LL � EU��LH)G (�h1 +�a)]

� [G (�a)�G (�h1 +�a)] : (17)

Both conditions (16) and (17) together require that15

EUM�HL � EUC�HH �

X�2fM;Cg

(EU��HL � EU��HH)G (�h1 +�a) �

EUM�LL � EUC�LH �

X�2fM;Cg

(EU��LL � EU��LH)G (�h1 +�a),

G (�h1 +�a) �1

2+

�G (�a)� 1

2

�(�� �)Y

2 (�CC +�CM):

15Note that the �nal inequality analogously follows from (7) and (8).

25

References

Amegashie, J.A., 2006. Information transmission in elimination contests. Work-

ing Paper, Department of Economics, University of Guelph, Ontario.

Baik, K.H., 1998. Di¤erence-form contest success functions and e¤ort levels in

contests. European Journal of Political Economy 14, 685�701.

Balsdon, E., Fong, L., Thayer, M.A., 2007. Corruption in college basketball?

Evidence of tanking in postseason conference tournaments. Journal of Sports

Economics 2007 8, 19�38.

Baron, D.P., Besanko, D., 1984. Regulation and Information in a continuing

relationship. Information Economics and Policy 1, 267�302.

Che, Y.-K., Gale, I., 2000. Di¤erence-form contests and the robustness of all-pay

auctions. Games and Economic Behavior 30, 22�43.

Chen, K.-P., 2003. Sabotage in promotion tournaments. Journal of Law, Eco-

nomics and Organization 19, 119�140.

Dietl, H., Franck, E., Lang, M., 2008. Overinvestment in team sports leagues: A

contest theory model. Scottish Journal of Political Economy 55, 353�368.

Dixit, A., 1987. Strategic behavior in contests. American Economic Review 77,

891�898.

Drago, R., Garvey, G.T., Turnbull, G.K., 1996. A collective tournament. Eco-

nomics Letters 50, 223�227.

Ederer, F., 2010. Feedback and motivation in dynamic tournaments. Journal of

Economics and Management Strategy 19, 733�769.

Fairburn, J.A., Malcomson, J.M., 2001. Performance, promotion, and the Peter

Principle. Review of Economic Studies 68, 45�66.

26

Fort, R., Winfree, J., 2009. Sports really are di¤erent: The contest success

function and the supply of talent. Review of Industrial Organization 34,

69�80.

Freixas, X., Guesnerie, R., Tirole, J., 1985. Planning under incomplete informa-

tion and the ratchet e¤ect. Review of Economic Studies 52, 173�191.

Gibson, B., Sachau, D., 2000. Sandbagging as a self-presentational strategy:

Claiming to be less than you are. Personality and Social Psychology Bulletin

26, 56�70.

Gibson, B., Sachau, D., Doll, B., Shumate, R., 2002. Sandbagging in competition:

Responding to the pressure of being the favorite. Personality and Social

Psychology Bulletin 28, 1119�1130.

Gilpatric, S.M., 2011. Cheating in contests. Economic Inquiry 49, 1042�1053.

Gürtler, O., 2011. The �rst-order approach in rank-order tournaments. Eco-

nomics Letters 111, 185�187.

Gürtler, O., Münster, J., 2010. Sabotage in dynamic tournaments. Journal of

Mathematical Economics 46, 179�190.

Harris, C., Vickers, J., 1987. Racing with Uncertainty. Review of Economic

Studies 54, 1�21.

Hö­ er, F., Sliwka, D., 2003. Do new brooms sweep clean? �When and why

dismissing a manager increases the subordinates� performance. European

Economic Review 47, 877�890.

Hörner, J., Sahuguet, N., 2007. Costly signalling in auctions. Review of Economic

Studies 74, 173�206.

Hvide, H., 2002. Tournament rewards and risk taking. Journal of Labor Eco-

nomics 20, 877�898.

27

Késenne, S., 2006. The win maximization model reconsidered: Flexible talent

supply and e¢ ciency wages. Journal of Sports Economics 7, 416�427.

Kiyotaki, F., 2004. The e¤ects of a consolation match on the promotion tourna-

ment. Journal of the Japanese and International Economies 18, 264�281.

Konrad, K.A., 2000. Sabotage in rent-seeking contests. Journal of Law, Eco-

nomics, and Organization 16, 155�165.

Konrad, K.A., 2009. Strategy and dynamics in contests. Oxford University Press:

Oxford, New York.

Konrad, K.A., Kovenock, D., 2009. Multi-battle contests. Games and Economic

Behavior 66, 256�274.

Kräkel, M., 2007. Doping and cheating in contest-like situations. European

Journal of Political Economy 23, 988�1006.

Kräkel, M., 2012. Competitive careers as a way to mediocracy. European Journal

of Political Economy 28, 76�87.

La¤ont, J.-J., Tirole, J., 1988. The dynamics of incentive contracts. Economet-

rica 56, 1153�1175.

Lazear, E.P., Rosen, S., 1981. Rank-order tournaments as optimum labor con-

tracts. Journal of Political Economy 89, 841�864.

Meyer, M.A., 1991. Learning from coarse information: Biased contests and career

pro�les. Review of Economic Studies 58, 15�41.

Münster, J., 2007. Selection tournaments, sabotage, and participation. Journal

of Economics and Management Strategy 16, 943�970.

Münster, J., 2009. Repeated contests with asymmetric information. Journal of

Public Economic Theory 11, 2009, 89�118.

28

Nalebu¤, B.J., Stiglitz, J.E., 1983. Prizes and incentives: Towards a general

theory of compensation and competition. Bell Journal of Economics 14,

21�43.

O�Kee¤e, M., Viscusi, W.K., Zeckhauser, R.J., 1984. Economic contests: Com-

parative reward schemes. Journal of Labor Economics 2, 27�56.

Rosen, S., 1986. Prizes and incentives in elimination tournaments. American

Economic Review 76, 701�715.

Ryvkin, D., 2012. Contests with doping. Journal of Sports Economics. Online-

First: published on January 11, 2012 as doi:10.1177/1527002511427320

Schotter, A., Weigelt, K., 1992. Asymmetric tournaments, equal opportunity

laws, and a¢ rmative action: Some experimental results. Quarterly Journal

of Economics 107, 511�539.

Schöttner, A., 2008. Fixed-prize tournaments versus �rst-price auctions in inno-

vation contests. Economic Theory 35, 57�71.

Szymanski, S., 2003. The economics of sporting contests. Journal of Economic

Literature 41, 1137�1187.

Szymanski, S., 2004. Professional team sports are only a game. Journal of Sports

Economics 5, 111�126.

Taylor, B.A., Trogdon, J.G., 2002. Losing to win: Tournament incentives in the

National Basketball Association. Journal of Labor Economics 20, 23�41.

Tirole, J., 2002. Rational irrationality: Some economics of self-management.

European Economic Review 46, 633�655.

Uehara, K., 2009. Early or late promotion/screening? Empirical analysis of

career ladders for japanese white-collar workers using employees�list. Japan

Labor Review 6, 25�58.

29

Zhang, J., Wang, R., 2009. The role of information revelation in elimination

contests. Economic Journal 119, 613�641.

30