DP 10-14
DISCUSSION PAPER
SERIES IN
ECONOMICS AND
MANAGEMENT
Competitive Careers as a Way to Mediocracy
Matthias Kräkel
Discussion Paper No. 10-14
GERMAN ECONOMIC ASSOCIATION OF BUSINESS
ADMINISTRATION – GEABA
Competitive Careers as a Way to Mediocracy∗
Matthias Kräkel†
Abstract
We show that in competitive careers based on individual performance
the least productive individuals may have the highest probabilities to be
promoted to top positions. These individuals have the lowest fall-back positions and, hence, the highest incentives to succeed in career contests. This
detrimental incentive effect exists irrespective of whether effort and talent
are substitutes or complements in the underlying contest-success function.
However, in case of complements the incentive effect may be outweighed by
a productivity effect that favors high effort choices by the more talented
individuals. Switching from wages-attached-to-jobs to pay-for-performance
will work against mediocracy if applied to top jobs, but may be detrimental
at lower career levels. The mediocracy problem will be aggravated if highability individuals decide to sandbag on lower career levels in order to avoid
strong opponents at higher levels.
Key Words: career competition; contest; mediocracy; sandbagging.
JEL Classification: D72; J44; J45; M51
∗
I would like to thank Oliver Gürtler, Daniel Müller, Petra Nieken, Anja Schöttner, the participants of the economics workshop of the Leibniz University of Hannover, in particular Hendrik
Hakenes, Vilen Lipatov and Georgios Katsenos, and the participants of the annual meeting of
the "Unternehmenstheoretischer und -politischer Ausschuss" of the German Economic Association (Verein für Socialpolitik), particularly Jürgen Eichberger, Dieter Pfaff, Andreas Pfingsten,
Kerstin Pull, Ulrike Stefani and Siegfried Trautmann, for helpful comments. Financial support
by the Deutsche Forschungsgemeinschaft (DFG), grant SFB/TR 15, is gratefully acknowledged.
†
University of Bonn, Adenauerallee 24-42, D-53113 Bonn, Germany, tel: +49 228 733914, fax:
+49 228 739210, e-mail: m.kraekel@uni-bonn.de.
1
"mediocracy = A society in which people with
little (if any) talent and skill are dominant
and highly influential."
1
(dictionary)
Introduction
Career systems that are not based on individual performance but on criteria like
seniority may end up in a situation where people with average or less than average
talent are assigned to key positions. At the level of society, the direct consequence
would be the emergence of a mediocracy. At first sight, one might suspect that
competitive career systems with performance-based job assignment should lead to
a significantly better outcome. However, in our paper we show that competitive
career systems have an inherent tendency to promote the least productive individuals, thus leading to mediocracy. The intuition for this result comes from the fact
that more productive people have better fall-back positions than less productive
ones when failing in the competition for top positions. Hence, highly productive people have only moderate incentives to win the competition for top jobs,
whereas individuals with low productivity have strong incentives to avoid their
rather unattractive fall-back positions.
We use a contest model to analyze competitive careers of heterogeneous individuals. At the beginning, all individuals have the chance to reach the top position
with the highest possible career income.1 However, during the career we have
losers and winners, where winners still compete for the top job but losers strive
1
Another motive for career competition is typically an increase in status. See Frank (1985)
on the role of status as motivation in individual careers. Moldovanu, Sela and Shi (2007) discuss
competition for status in a formal contest model.
2
towards less important positions in a so-called consolation match.2 Winning such
consolation match defines the fall-back position of an individual. The fact that
more productive individuals have better fall-back positions than less productive
individuals does not depend on the underlying contest-success technology: In any
kind of consolation contest, a more productive player can either achieve the same
winning probability at less effort costs or a higher winning probability at the same
effort costs compared to a less productive opponent. Hence, the expected utility of
an individual when participating in a consolation contest will always be positively
correlated with his productivity. As natural consequence, the individuals with the
lowest productivities have the highest incentives to climb the ladder in order to
avoid their respective fall-back positions.3
In the basic career model, we analyze a setting with effort and talent (or productivity) being substitutes in the contest-success function of an individual. Our
results show that near the top position — individuals only have to pass one hurdle for being assigned to the top job — the individuals with the worst fall-back
positions will be most likely to win the contest for the top position, due to the
incentive effect mentioned above. If individuals have to pass more than one hurdle
to reach the top, it will be crucial whether productivity has a higher impact on
the fall-back position or the future career and how dissipative the competition is
on different career levels. For example, if consolation contests are sufficiently dissipative (i.e., expected utility from participating in these contests is rather low),
2
See, for example, Uehara (2009) on consolation contests in a Japanese corporation. For a
theoretical analysis of consolation contests see Kiyotaki (2004).
3
Anecdotal evidence for such kind of incentive effect can be seen in the careers of Hollywood’s
most successful actresses and actors. Many of them either had very unattractive jobs before they
became superstars or orriginated from rather poor families (see, among many others, the CVs
of Sandra Bullock, Harrison Ford, Megan Fox and Hillary Swank). Starting from the hypothesis
that talent is uniformly distributed over all social classes of a society, the fact that so many
superstars in Hollywood have really low fall-back positions can most naturally be explained by
very strong incentives induced by the difference of superstar income and individual fall-back
position. Note that, of course, this possible anecdotal evidence is not based on low ability but
on low outside options or reservation utilities.
3
fall-back positions will be quite unattractive, but most unattractive for the least
productive individuals. In this situation, the less productive individuals are most
likely to reach the top.
In the discussion, we consider a career model where talent and effort are complements in the contest-success function. Again, there is a detrimental incentive
effect that makes winning of the top position by the least productive individuals
most likely. However, now an additional effect works into the opposite direction:
The larger an individual’s productivity the more effective will be the exertion of
effort to win the contest. For this reason, a highly talented individual may prefer to
spend high effort despite an attractive fall-back position. If this productivity effect
is dominated by the incentive effect, society will still tend to mediocracy. Otherwise, the most talented people will assert themselves in the career competition
and be assigned to the top position.
Besides the application to society in general, our model also offers insights for
competitive careers in a more concrete situation. For example, it can be applied to
career competition between politicians, who first struggle for being elected as local
leader of their political party. Thereafter, they may further compete for becoming
national party leader, governor or president of a certain state. As another example,
consider the case of internal careers within corporate hierarchies. These hierarchies
are often organized as internal labor markets with clearly structured career paths.4
Here, a manager first wants to be promoted to become division head. In further
career steps, the manager may be elected to the board of directors or even compete
for the position as chief executive officer.
Moreover, there are many associations that have local offices and a national or
even international headquarter (e.g., worker and employer associations). In these
organizations, individual careers typical proceed as follows: Individuals that suc4
See, for example, Baker, Gibbs and Holmström (1994).
4
cessfully managed their jobs in the local establishment get the chance of becoming
chief of the local office. Thereafter, they may be promoted to the national or international headquarter where they can succeed in further career steps up to the
top of the organization. There are also career ladders that include moves from
one organization to another. Consider, for example, the European Central Bank
(ECB). Often, individuals employed by the ECB come from national central banks
where they have climbed the ladder before. In all these examples, the individuals
that do not succeed to the top may win the consolation contest and become head
of the local establishment or climb an inferior ladder in the headquarter.
In the applications mentioned before, those wage policies would work against
the problem of mediocracy that ensure highly talented individuals a significantly
higher career income at the top position than less talented candidates. In other
words, if wages are not attached to jobs but depend on the productivity of the
respective individual, the top position should be quite attractive for the most talented candidates. For example, top positions could be combined with performancebased incentive schemes so that highly productive individuals have sufficiently large
expected incomes when reaching the top. By this wage policy, the detrimental incentive effect stated above could be turned into a beneficial one. This possible
solution by using pay for performance at the top is addressed in the discussion
part of the paper. However, our results also point out that pay for performance at
lower career levels may aggravate the mediocracy problem.
Furthermore, there are situations where highly talented individuals can either
compete against other strong opponents for the top positions or play against a
significantly weaker field for less attractive jobs. If the pay (and status) gap
between the top positions and the less attractive ones is not too large and the
competition against other strong opponents really exhausting it may be optimal
for talented individuals not to strive towards the top positions. This effect in
5
competitive careers would further explain the deficit of top-class candidates for
top positions. Note that this sandbagging strategy may be optimal for either
high-potential individual so that they face a coordination problem: If there are
two highly talented players and both prefer to sandbag, then they will end up in
a consolation contest with strong competition. Of course, such failed coordination
would exacerbate the mediocracy problem. We discuss the sandbagging issue in
the final subsection of the discussion.
Our paper is related to the literature on contests and rank-order tournaments.
There are two widely used classes of contest-success functions — the ratio-form and
the difference-form contest-success function. The former one was introduced by
Tullock (1980) for the discussion of rent seeking, and generalized and axiomatized
by Skaperdas (1996). The latter one is used by rank-order tournament models in
labor economics that discuss job promotions and worker careers. Most of these
papers build on Lazear and Rosen (1981) and Nalebuff and Stiglitz (1983). A
dynamic tournament with subsequent career steps is analyzed by Rosen (1986).5
The basic model and most of the subsequent parts are based on Skaperdas’ (1996)
ratio-form contest-success function. The discussion of the sandbagging problem in
Subsection 4.3 uses a difference-form model to highlight the competitive effects in
a clearly laid out setting.
There are also parallels to the public economics literature on bad politicians.
This literature offers an explanation for why political leaders and presidents often have rather low qualifications or are of low quality. Contrary to our paper,
they do not consider a contest framework but follow either the political-agency
or the citizen-candidate approach (see, among many others, Caselli and Morelli
(2004), Messner and Polborn (2004), Mattozzi and Merlo (2007)). In the model
5
There are also contest papers that discuss the difference-form contest-success function in
a more general context; see Dixit (1987), Hirshleifer (1989), Baik (1998) and Che and Gale
(2000). Konrad (2009) offers a comprehensive overview of the most frequently used contestsuccess functions, applications of contest models and the most important results in contest theory.
6
by Caselli and Morelli (2004), bad and dishonest candidates have rather low opportunity costs and extract more money from their political positions. These two
effects make them more likely run for office. Messner and Polborn (2004) consider
a similar approach, but in their setting political candidates care for both their
salaries and the impact of the politician’s ability on the quality of the political
office. Mattozzi and Merlo (2007) show that, due to competition for highly talented individuals between the political sector and the lobbying sector, a political
party may prefer not to recruit the best candidates although they would accept a
contract offer. However, all these papers do not address the individuals’ incentives
and endogenous investments or activities during their careers, which take center
stage in our model.
Finally, the part on sandbagging is related to the ratchet effect in dynamic
principal-agent models (Baron and Besanko 1984; Freixas, Guesnerie and Tirole
1985; Laffont and Tirole 1988): Agents might prefer to withhold effort in the first
period in order to influence the incentive contract for the second period. High
effort and, hence, good performance in the first period may make the principal
adjust the performance standard upwards for the next period. In order to meet
the new high standard the agent has to exert significantly more effort. Since he
anticipates the adjustment of the standard, the agent optimally chooses low effort
in the beginning of the contractual relationship. The sandbagging problem differs
from the ratchet effect as the latter one does not lead to a coordination problem
between several agents.
The paper is organized as follows. We start with the analysis of talent and effort being substitutes in the contest-success function. Section 2 deals with the case
where individuals are near the top and only have to pass one hurdle to reach the
highest possible position. Section 3 considers a two-hurdle career model with productivity influencing both an individual’s fall-back position and his future career.
7
Section 4 discusses the main findings. Subsection 4.1 offers a robustness check.
Here, we analyze whether the mediocracy result also holds under the assumption
of talent and effort being complements in the contest-success function. In Subsection 4.2, pay for performance at top positions is analyzed as a possible solution
to the mediocracy problem. Subsection 4.3 addresses sandbagging of high-ability
players, which may exacerbate the mediocracy problem. Section 5 concludes.
2
One-Hurdle Career
2.1
Basic Model
We consider a career game between ≥ 3 risk neutral individuals (e.g., workers
of the same cohort entering a firm in a certain period). The structure of this game
is sketched in Figure 1.
[Figure 1]
The players start by simultaneously choosing productive activities (e.g., efforts)
≥ 0 ( = 1 ) to become the winner of the major career contest. This winner receives a high career income . The − 1 other players enter a consolation
match where they compete for less attractive positions. The winner of this consolation contest earns , whereas the − 2 losers get with 0 .6
We assume that the three career incomes are sufficiently large so that players
choose positive efforts in equilibrium.
In both, the major contest for and the consolation contest each player has
to bear the costs of his activity (e.g., disutility of effort). For simplicity, when
player exerts activity level let his costs be equal to . Moreover, in both
6
Note that in our model career incomes are exogenously given, but as we can see below the
general effects still exist in a situation where a career designer endogenously chooses incomes.
8
contests, players face the same contest-success function based on the one suggested
by Skaperdas (1996):7 If player chooses and the other players 6= choose
activity levels , then ’s probability of winning is given by
=
( − )
P
( − ) + 6= ( − )
(1)
with (·) as monotonically increasing impact function that is concave and strictly
positive.8 Hence, each player increases his winning probability by exerting more
effort, but the other contestants’ efforts as well as luck or measurement error also
influence the outcome of the contest. The parameter 0 indicates player ’s
productivity.9 The lower the value of the more productive will be the player.
If, for example, all players choose identical activity levels, the most productive
player will have the highest winning probability, whereas the player with the lowest
productivity is least likely to win the contest. The scaling parameter 0 is
identical for all contestants. As can be seen from the equilibrium outcomes below,
measures how dissipative the contest is (i.e., the larger the smaller will be an
individual player’s expected utility from participating).
We assume that productivity parameters 1 are common knowledge.
This assumption can be justified for at least two reasons. First, players typically
observe each others’ qualifications, which are positively correlated with productivity.10 Second, note that the common-knowledge assumption is only introduced
7
This assumption can be motivated by the fact that both contests take place between the
same individuals excepting the first-round winner, who is missing in the consolation round.
8
As the player’s activity is assumed to be productive, ( − ) may describe a worker’s or
a politician’s output.
9
Hence, effort and ability/productivity are substitutes in the players’ impact function. The
case of complements will be discussed in Subsection 4.1.
10
On the one hand, following human capital theory based on Becker (1962), investment in a
player’s skills rises his productivity, which is certified by a formal qualification. On the other
hand, according to Spence (1973), high qualifications can be a credible signal for corresponding
high ability.
9
for the contestants.11 These players often observe each other when working close
together at the same organization every day. For example, the two arguments
hold for a situation where workers compete for job promotion in internal labor
markets and for a situation where politicians compete for becoming the leader of
their political party.
Although we do not explicitly specify a welfare function, we assume that welfare
will be highest if the most productive player (i.e., the one with parameter value
min{ 1 }) passes the decisive hurdle by winning the major contest and is
assigned to the most important position, which is associated with career income
.
2.2
Consolation Contest
We start the analysis by solving the contest game of those − 1 players that have
failed in the major contest and, therefore, enter the consolation stage. Here, the
winner earns career income but the − 2 losers end up with the lower income
. Player chooses activity to maximize expected utility
g ( ) = + ( − )
( − )
P
−
( − ) + 6= ( − )
Since this objective function is strictly concave and, by assumption, career incomes
are sufficiently large to prevent a corner solution where neither player chooses
positive effort, the equilibrium is described by the − 1 first-order conditions
11
P
( − ) 0 ( − ) 6= ( − )
= 1
h
i2
P
( − ) + 6= ( − )
We do not have a contest designer, who shares the same information as the contestants. Note
that our results will not change when introducing an uninformed designer, who endogenously
chooses optimal tournament prizes but cannot observe individual productivities.
10
which can be rearranged to
−
1
h
i
i2 =
P
−1
0
( − ) ( − ) + 6= ( − )
=1 ( − )
hP
with player denoting an arbitrary opponent of player . In analogy, the first-order
condition of player 6= can be written as
−
1
h
i
hP
i2 =
P
−1
0 ( − ) ( − ) +
(
−
)
6=
=1 ( − )
Combining the right-hand sides of both equations yields
P
P
( − ) + 6= ( − )
( − ) + 6= ( − )
=
0 ( − )
0 ( − )
As both sides describe the same monotonically increasing function of − and
− , respectively, we have
− = − ∀ ( )
Hence, the higher the productivity of a player the lower will be his equilibrium
effort. Intuitively, a highly productive player prefers to save effort costs by choosing
a low activity level since activities and productivity parameters are substitutes in
the impact function (·). By inserting − = − ∀ ( ) in player ’s
first-order condition, the equilibrium activity level ∗ is described by
¶
µ
( − 2) ( − )
1
( − 2) ( − )
∗
+
⇔ =
= 0 ∗
( − )
( − 1)2 (∗ − )
( − 1)2
with (·) denoting the monotonically increasing inverse function of 0 . Equilibrium activity increases in the size of the career income difference − , because
11
any contestant earns at least in the consolation round. Furthermore, ∗ is nonincreasing and for 3 strictly decreasing in the number of contestants. This
effect can be labeled discouragement effect: Each player exerts less effort when
the number of opponents increases since his relative impact on the outcome of the
contest becomes smaller.
g ( ).
Finally, we insert equilibrium activities in the objective function
Thus, in equilibrium player ’s expected utility is given by
¶
µ
∗
( − 2) ( − )
−
∗
g
−
−
( ) = +
−1
( − 1)2
Now, we can see why measures the contest’s degree of dissipation. Note that, in
equilibrium, a player’s winning probability is always 1 ( − 1), irrespective of the
value of . However, a player’s equilibrium activity and, therefore, his effort costs
rise in . Consequently, the higher , the lower will be each player’s expected
utility from participating in the consolation match. Nevertheless, a player strictly
g ∗ (∗ ) decreases in ).
benefits from higher productivity (i.e.,
2.3
Major Contest
A player will earn the highest career income , if he passes the hurdle by winning
the major contest. In case of losing, he will be relegated to the consolation contest
g ∗ (∗ ). Hence, player ’s objective function in
associated with expected utility
the major contest can be written as
( − )
P
( − ) + 6= ( − )
Ã
!
∗
−
)
(
g (∗ ) 1 −
P
+
−
( − ) + 6= ( − )
d ( ) =
12
³
´
∗
∗
g
−
(
)
· ( − )
g ∗ (∗ ) +
P
−
=
( − ) + 6= ( − )
In analogy to the consolation contest, the first-order conditions of two arbitrary
players and are given by
and
g ∗ (∗ )
1
−
P
= 0
P
( − ) [ ( − ) + 6= ( − )]
[ =1 ( − )]2
(2)
∗
g (∗ )
−
1
P
= 0
P
( − ) [ ( − ) + 6= ( − )]
[ =1 ( − )]2
(3)
g ∗ (∗ )
g ∗ (∗ ) and
Hence, if , then we have
1
h
i
P
0 ( − ) ( − ) + 6= ( − )
1
h
i⇔
P
0 ( − ) ( − ) + 6= ( − )
P
P
( − ) + 6= ( − )
( − ) + 6= ( − )
0 ( − )
0 ( − )
P
which implies − − since [ () + 6= ( − )] 0 () is a
monotonically increasing function of . Let (1) (2) · · · () denote the
order of the players’ productivity parameters (i.e., player (1) is the most productive
one), ∗(1) ∗(2) ∗() the respective equilibrium efforts and ∗(1) ∗(2) ∗() the
winning probabilities in the major contest. Then we obtain the following result:
Proposition 1 The players’ winning probabilities in the major contest satisfy
∗(1) ∗(2) ∗() and the corresponding equilibrium efforts ∗(1) ∗(2)
∗() .
13
Proof. The first part immediately follows from the fact that implies
P
− − and that = ( − ) [ ( − ) + 6= ( − )] is
strictly increasing in − . The second part follows from − − ⇔
− − ( − ) 0.
Proposition 1 shows that the more productive a player, the less likely he will
win the major contest and the less effort he will choose. The intuition for the first
result comes from the players’ different fall-back positions in the major contest. If
a player has a large productivity (i.e., a small ), then he will also be a strong
player in the consolation match, which will guarantee him a large expected utility
g ∗ (∗ ) as a kind of fall-back position. This fact reduces his incentives in the
major contest so that we have a tendency to mediocracy where key positions
are filled by less productive individuals, leading to welfare losses. In other words,
participation in the consolation match is rather unattractive for the less productive
players so that they have very strong incentives to win the major contest.
Note that although we assume career incomes to be exogenously given, our
results of Proposition 1 as well as the following results will qualitatively hold under
endogenous incomes or prizes that are optimally chosen by a contest designer before
the competition starts. The career incomes or contest prizes of course influence
the levels of all players’ equilibrium efforts. However, they neither have an impact
on the players’ effort differences − in the consolation match nor an impact
on the ranking between − and − in the major contest. As the same
holds for the findings in the following sections our results are robust with respect
of exogeneity/endogeneity of the contest prizes.
The result on effort ranking ∗(1) ∗(2) ∗() stems from the intuition
before together with the fact that effort and productivity are substitutes in the
impact function. Hence, even if all players had identical fall-back positions, the
order ∗(1) ∗(2) ∗( ) would not change. Welfare was solely defined via
14
the career decision based on the outcome of the major contest. However, if the
activity levels ∗ ( = 1 ), which are productive by assumption, were also
important for welfare considerations, we might have a second source for welfare
losses since the most productive individuals choose the lowest activity levels.
3
Two-Hurdle Career
In this section, we consider the case where a player has to pass two career hurdles
to reach the top. This situation is described by Figure 2.
[Figure 2]
First, a player must assert himself in the organizational unit that he belongs to
(e.g., a division or affiliate of a corporation, a certain political party). Second,
when being successful in becoming head of the organizational unit (e.g., division
head or party leader), the player enters a higher-order contest to reach the top
position of his career path (e.g., CEO of the corporation or president of a state).
As crucial difference to the one-hurdle career, now a player has to succeed two
times before reaching the top and a player’s productivity has both an influence on
his fall-back position and an influence on his future career.
3.1
Model Modifications
There are two organizational units, and , that consist of and members,
respectively, with ≥ 3 ( ∈ { }). In each unit , the members
compete for becoming unit head (level-I contests). These two contests are modeled
analogously to those in Subsection 2.1. Each member ∈ {1 } chooses
activity level ≥ 0 (at cost ), which influences his probability of winning the
15
level-I contest for unit head ,
=
( − )
P
( − ) + 6= ( − )
(4)
Again, indicates the productivity of the respective player, with lower values
corresponding to higher productivities. The parameter 0 measures the grade
of dissipation in the level-I contests.12 If player wins and becomes unit head he
will enter the higher-order level-II contest where he competes against the other
unit head to reach the top of his career. However, the − 1 losers are relegated
to a consolation contest in unit where the winner earns career income 0
and the − 2 losers get 0.
As outcome of the level-I contests, two players — the unit heads and —
enter the level-II contest. Here, they compete for the unique top position, which
is associated with career income . Whereas the winner of the level-II contest
is assigned to this top position, the loser gets a lower career income with
0 . Career incomes are assumed to be sufficiently large so
that the players exert strictly positive efforts in each contest. The contest-success
function for the level-II contest is again a Skaperdas-type function like (1) and
(4). The scaling parameter for measuring the degree of dissipation at level II is
denoted by 0.
As in Section 2, we assume that from a welfare perspective the player with
the highest productivity (i.e., the one with parameter min{1 | ∈
{ }}) should be assigned to the top position with career income .
12
Our results will not change qualitatively, if we define different parameters and for
the two units.
16
3.2
Level-II Contest
Let and denote the productivity parameters of the two unit heads, who
enter the level-II contest, and and the corresponding activity variables. In
the contest, the head of unit maximizes
( − )
−
Ω∈{} (Ω − Ω )
d ( ) = + ( − ) P
Proceeding in the same way as in Section 2, we find that, in equilibrium, activity
levels are described by
−
1
i2 =
( − ) 0 ( − )
Ω∈{} (Ω − Ω )
hP
=
1
( − ) 0 ( − )
yielding ( − ) 0 ( − ) = ( − ) 0 ( − ) and,
hence, − = − . Altogether, in the level-II contest the head of
organizational unit optimally chooses
∗
µ
−
=
4
¶
+
and gets expected utility
d ∗
µ
¶
−
−
−
−
( ) = +
2
4
We can see that in equilibrium the more productive unit head exerts less effort than
the other head, but has the same probability of being promoted to the top position
since the higher productivity completely outweighs the effort deficit. Thus, it is
pure luck whether the better head is assigned to the top job or not.
17
3.3
Level-I Contests
Consider the contest in organizational unit . The objective function of member
can be written as
´
³
∗
∗
d
g
− · ( − )
g ∗ +
P
( ) =
−
( − ) + 6= ( − )
If player wins, he will earn the expected utility from participating in the leveld ∗ . In case of losing, he will enter the consolation contest of his
II contest,
g ∗ . Hence, his fall-back position is
unit , where he receives expected utility
g ∗ and the extra utility from being successful at level I by
characterized by
g ∗ . From the first-order condition we obtain the following description
d ∗ −
of player ’s equilibrium activity:
g ∗
d ∗ −
1
hP
i
i2 =
0 (
−
)
(
−
)
6=
=1 ( − )
hP
(5)
To further characterize the equilibrium activities, we have to calculate player ’s
g ∗ . Applying the results for the consolation contest of Subfall-back position
section 2.2 yields
g ∗
(∗ )
¶
µ
( − 2)
−
−
=
− 1
( − 1)2
and, therefore,13
g ∗ = + − + ( − )
d ∗ −
2
− 1
¶
µ
¶
µ
−
( − 2)
−
+
4
( − 1)2
13
Recall that, by assumption, career incomes guarantee interior solutions for the equilibrium
∗
∗
g 0.
d −
activities. Hence, we must have
18
Hence, if then for two arbitrary members and of unit in the level-I
d ∗ −
g ∗
d ∗ −
g ∗ ⇔ . The first-order
contest we have
conditions (5) of players and can be written as
"
P
d ∗ −
g ∗
=1
#2 =
0
1
P
( − ) [ ( − ) +
( − )]
6=
( − )
and
"
g ∗
d ∗ −
P
=1
#2 =
0
( − )
1
P
( − ) [ ( − ) +
( − )]
6=
d ∗ −
g ∗
d ∗ −
g ∗ then
Thus, if
( − ) +
P
6=
( − )
0 ( − )
( − ) +
P
6=
( − )
0 ( − )
which implies − − , analogously to the findings in Subsection
2.3. Since this comparison holds for any two members of organizational unit ∈
{ } and since players have identical winning probabilities in the level-II contest,
we have proven the following result:
Proposition 2 If , then the most (least) productive player of each unit
∈ { }, i.e., the player with = min{ 1 } (with = max{ 1
}), has the lowest (highest) probability to reach the top position with career
income .
Proposition 2 shows that the mediocracy result of the one-hurdle career will
prevail under the assumptions of the two-hurdle career if the level-I contests are
more dissipative then the level-II contest. The intuition for this result is the
19
following. The higher relative to , the less attractive will be participation in
the consolation match for the players. Moreover, since expected utility increases
in the productivity of a player (i.e., decreases in ), the less productive a player
the stronger will be his incentives to avoid a consolation match and, hence, to pass
the hurdle that leads to participation in the level-II contest.
A situation with level-I contests being more dissipative than the level-II contest
seems quite realistic. For example, note that a single player is less visible in the
large contest against − 1 opponents on level I compared to the level-II contest
with two players. Therefore, it is rather difficult for an individual to stand up to
his opponents on level I compared to level II. In order to become winner on level
I, each player has to spend a lot of effort, which makes participation in the contest
costly and, hence, level-I contests quite dissipative.14
4
Discussion
The purpose of this section is twofold. First, we will check the robustness of the
main result on mediocracy. We will analyze whether the most able individuals
are still least likely to reach the top if activities and productivities are not substitutes any longer (Subsection 4.1) or if salary at the top job depends on the job
holder’s ability (Subsection 4.2). Second, we will address the sandbagging problem
mentioned in the introduction: High-ability individuals might prefer to sandbag
in order to avoid playing against a strong field (Subsection 4.3). This effect could
exacerbate the mediocracy problem.
14
Note that equilibrium efforts strictly increase in the dissipation parameters , and ,
respectively.
20
4.1
Activities and Productivities as Complements
So far we have assumed that activities ( ) and productivity parameters ( ) are
substitutes in the players’ impact function (·). This assumption drives part of
the previous results, in particular the findings that players with higher productivities (i.e., lower values of ) choose lower efforts in equilibrium. However, this
paper does not focus on effort choice but on the probability that the most productive player is not assigned to the top career position. In this subsection, we will
check the robustness of the finding that the correlation between productivity and
promotion probability of a player may be strictly negative.
For this purpose, we reconsider the one-hurdle career model and assume that
player ’s contest-success function is given by
ˇ =
³ ´
+
³ ´
P
6=
³ ´
(6)
with (·) denoting the same strictly positive, increasing and concave impact function as before. Again, the smaller the more productive will be the respective
player, and for the case of identical activity levels by all players the most productive one has the highest winning probability. However, comparison of (1) and
(6) shows that (besides skipping the dissipation parameter ) the contest-success
functions and ˇ differ significantly. Contrary to (1), now the activity variable
and the productivity parameter are complements in the sense that lower values of
make higher activity levels more effective.15
As in Section 2, we first solve the consolation contest game and then turn
g ( ) =
to the major contest. In the consolation contest, player maximizes
15
Activities
³ ´ and productivities are complements in the contest-success function in the sense of
0.
2
1
21
+ ( − ) · ˇ − . The first-order conditions of two arbitrary players and
can be combined to
( − )
³ ´h ³ ´ P
³ ´i
³ ´i2 =
hP
−1
0
+
6=
=1
=
(7)
³ ´h ³ ´ P
³ ´i
0
+ 6=
which yields the following equilibrium outcome:
Lemma 1 In the consolation contest, if then (i) ˇ∗ ˇ∗ and (ii)
g ∗ (∗ ).
g ∗ (∗ )
Proof. Part (i) can be shown by contradiction. Suppose that
ˇ∗ ≤ ˇ∗ ⇔
∗
∗
(8)
From the first-order conditions (7) we obtain
h ³ ∗´ P
h ³ ∗´ P
³ ∗ ´i
³ ∗ ´i
+ 6=
+ 6=
³ ∗´
³ ∗´
=
0
0
Since [ () +
P
6=
(9)
³ ∗ ´
] 0 () is a monotonically increasing function of , (8)
and (9) can only be satisfied at the same time if , a contradiction. (ii)
Since player can always choose the same effort level as so that he has the same
g ∗ (∗ )
g ∗ (∗ )
effort costs but a higher winning probability we must have
in equilibrium.
Lemma 1 points out that, in the consolation match, more productive players
have higher winning probabilities and larger expected utilities than less productive
players. Result (ii) is also important for the major contest, where players compete
22
for the top position with income . Due to the positive correlation between
productivity and expected utility, more productive players have better fall-back
g ∗ (∗ ) in the major contest, leading to less incentives. This effect also
positions
works in the models with substitutes (Sections 2 and 3) and will be called incentive
effect.
d ( ) =
g ∗ (∗ )+( −
g ∗ (∗ ))·
In the major contest, player maximizes
ˇ − . The first-order conditions of two players and ,
g ∗ (∗ )
−
³ ´i
hP ³ ´i2 = ³ ´ h ³ ´ P
0
+
6=
g ∗ (∗ )
−
³ ´i
and hP ³ ´i2 = ³ ´ h ³ ´ P
0
+
6=
(10)
g ∗ (∗ ) imply
g ∗ (∗ )
together with
h ³ ´ P
h ³ ´ P
³ ´i
³ ´i
+ 6=
+ 6=
³ ´
³ ´
0
0
The inequality shows that solutions of type
(11)
are always possible. Such
outcomes coincide with the findings above where more productive players are less
likely to obtain the top career position. However, the parameters and in
the numerators of the two sides in (11) indicate that we cannot rule out solutions
with
if is sufficiently small and sufficiently large. This effect can
be labeled productivity effect. Hence, if activities and productivity parameters
are complements in the impact function, we will have two effects that work into
opposite directions. Coming back to our question regarding efficient assignment
at the top we obtain the following result:
Proposition 3 Consider the major contest for the top position with income
23
and let . If
g ∗ (∗ )
−
∗
g (∗ )
−
then ˇ∗ ˇ∗ .
(12)
Proof. Combining the first-order conditions (10) yields
³ ´
+
P
0
6=
³ ´
³ ´
i ³ ´ P
³ ´
h
∗
∗
g
− ( ) + 6=
³ ´
i
= h
0
g ∗ (∗ )
−
If (12) is satisfied, we will have
and, thus,
³ ´
,
+
P
0
6=
³ ´
³ ´
³ ´
+
P
0
6=
³ ´
³ ´
which implies ˇ∗ ˇ∗ for the winning probabilities in equilib-
rium according to (6). From Lemma 1(ii) we know that both sides of inequality
(12) are larger than one so that there exist parameter constellations for which (12)
holds and others for which (12) does not hold.
Condition (12) points out the two opposing effects. While the right-hand side
describes the incentive effect, the left-hand side characterizes the productivity effect. If the incentive effect dominates the productivity effect, more productive
players will have lower winning probabilities in the major contest than less productive ones. In this case, the mediocracy result under substitutes qualitatively
still holds for activities and productivities being complements.
4.2
Individual Salaries
So far, we have assumed that all players have the same career incomes as contest
prizes. This assumption is realistic for those cases where wages are attached to
24
jobs. Such wage policy can be often observed in politics and (public) bureaucracies.
Here, we have a clear bundle of tasks that is assigned to a certain job. These tasks
determine the job holder’s qualification as well as his salary. Moreover, wages that
are attached to jobs are one of the key assumptions within the concept of internal
labor markets.16 Finally, if workers’ performance signals are unverifiable, tying
wages to jobs is necessary for a firm to use job-promotion tournaments as credible
incentive schemes.17
However, there are also jobs with verifiable performance signals, allowing pay
for performance. In these cases, a more able player has a higher expected career
income at a certain position than a less able one. In this subsection, we will discuss
whether such individual salaries for the same job will change the main findings on
competitive careers and mediocracy. As in the previous subsection, we reconsider
the one-hurdle career model of Section 2. The only modification of this model is the
introduction of individualized career incomes or contest prizes in this subsection.
Hence, player has career incomes , and with ( = )
if player is more able than player (i.e., if ).
First, we can analyze the effect of individual career incomes on players’ expected
g ( ) ( = 1 ).
utilities from participating in the consolation contest,
Since, by assumption, individual career incomes increase in the players’ abilities,
participation in the consolation contest will become more attractive for high-ability
players than for low-ability ones. This effect would strengthen the mediocracy result of Section 2. However, since players’ equilibrium efforts react to individualized
career incomes, which influences both effort costs and winning probabilities, we
16
See Doeringer and Piore (1971), Williamson, Wachter and Harris (1975).
See Malcomson (1984, 1986). Without the self-commitment property of wages-attached to
jobs, the employer would always promote the worker with the lowest promised salary for the
vacant job in order to save labor costs. Since such opportunistic behavior is anticipated by
the workers, job-promotion tournaments can only create incentives if salaries are linked with
positions.
17
25
have to do a comparative-static analysis of the equilibrium efforts with respect
to career incomes. Let, for simplicity, = 3 so that two players remain in the
consolation contest, say players and . Moreover, let player be more able than
player . Player maximizes
g ( ) = + ∆
( − )
−
( − ) + ( − )
g ( ) = +∆
( − )
−
( − ) + ( − )
with ∆ := ( − )
Analogously, the objective function of player reads as
with ∆ := ( − )
Obviously, a player’s expected utility increases in the respective base income
or which would strengthen the mediocracy result because of . Effort
costs as well as winning probabilities depend on the players’ income spreads ∆
and ∆ . From the first-order conditions for the equilibrium efforts ∗ and ∗ we
obtain the following comparative static results (see Appendix A):
∗
0
∆
∗
T 0
∆
∗
T 0
∆
∗
0
∆
if
∗ − T ∗ −
if
∗ − T ∗ −
The results show that a player’s equilibrium effort will increase if his income spread
becomes larger. The reaction to the opponent’s income spread depends on the
initial situation. If, for example, the more able player is leading in the initial
situation in the sense of ∗ − ∗ − , then a larger income spread of his
opponent, ∆ , implies a further increase of ∗ . Intuitively, higher incentives for
player (via an increase of ∆ ) brings him back into the race so that the contest
26
becomes more intense. Technically, we obtain ∗ ∆ 0. As a consequence,
competition becomes more balanced which implies a higher effort by player as
well.
In general, it is not clear whether higher absolute career incomes for more able
players are accompanied by higher income spreads. However, the comparative
static results show what happens in this case. Suppose we switch from a situation
with wages-attached-to-jobs and ∆ = ∆ = ∆ to individual career incomes,
leading to ∆ ∆ = ∆ . This switch additionally motivates the more able
player and discourages the less able one; thus, ∗ goes up and ∗ goes down so
that both player ’s winning probability and his effort costs increase. The total
effect unambiguously leads to a higher expected utility of player :18 Starting from
∗ − = ∗ − and ∆ = ∆ in the wages-attached-to-jobs situation and
marginally increasing ∆ only leads to a reaction by player — he will increase his
effort whereas player will not change his effort since initially ∗ − = ∗ − .
Therefore, winning probability and effort costs of increase. Note that this reaction
g ∗ (∗ ), because player optimally reacts to an increase
yields a higher value of
in and and not changing ∗ at all would already lead to a higher expected
utility due to higher career incomes. The next marginal increase of ∆ results
into reactions by both players and since now ∗ − ∗ − . Player is
discouraged and reduces ∗ whereas increases ∗ . Player ’s reaction benefits
g ∗ (∗ )
since ’s winning probability increases. Player ’s reaction must increase
as well since it would not be rational otherwise. A third marginal increase of ∆
has the same implications as the marginal increase before and so on. Altogether,
the introduction of individualized career incomes that lead to and ∆
18
As an example, consider the case of a linear impact function ( − ) = − . For
this case, we can compute explicit solutions for the players’ equilibrium efforts. We obtain
∗ − = ∆ ∆2 [∆ + ∆ ]2 and ∗ − = ∆2 ∆ [∆ + ∆ ]2 . Inserting into ’s
∗
∆3
g (∗ ) = +
objective function gives
2 − , which is strictly increasing in
[∆ +∆ ]
and ∆ .
27
g ∗ (∗ ) and
g ∗ (∗ ), thus strengthening
∆ further increase the gap between
the mediocracy result.
In a second step, we can analyze the impact of individualized career incomes on
players’ behavior in the major contest. Here, individual incomes and with
have straightforward consequences. We obtain first-order conditions
similar to (2) and (3). As only difference now the left-hand sides are given by
g ∗ (∗ )][P ( − )]2 and [ −
g ∗ (∗ )][P ( − )]2 ,
[ −
=1
=1
respectively. Career incomes additionally motivate the more able player,
thus working against the mediocracy result. If individual career incomes at the
∗
∗
g (∗ ) −
g (∗ ), the mediocracy problem
top are so large that −
will be even reversed.
To sum up, whereas individual career incomes along the players’ career paths
may aggravate the mediocracy problem, individual career incomes at the top position dependent on the job holder’s ability or performance unambiguously work
against mediocracy. This finding clearly speaks against wages being attached to
jobs at the top of bureaucracies and governments. For example, it may be worthwhile thinking about pay for performance at top positions in politics that depends
on voter satisfaction being evaluated in regular time intervals.
4.3
Sandbagging
In this subsection, we consider another effect which may further exacerbate the
mediocracy problem: If a high-ability player expects to compete against other highability individuals when entering the major contest for the top job, he may prefer
not to participate in that contest. In particular, he could sandbag at the preceding
career stage to avoid promotion to the major contest. Such sandbagging has three
advantages. First, the high-ability player saves effort costs at the preceding career
stage. Second, he avoids strong opponents and, hence, a relatively low winning
28
probability in the major contest. Third, he avoids a rather homogeneous field in
the major contest, which would lead to strong competition and high effort costs.
If the income at the top job does not differ too much from the income at less
important jobs, sandbagging can be a rational strategy. However, since each of
the high-ability players may prefer to sandbag, these players face a non-trivial
coordination problem when they choose efforts at the preceding career stage.
To analyze the sandbagging problem, we consider a stylized career model with
two rounds and two contests in each round. The two contests in the first round
have heterogeneous players: In each contest, a more able player with talent is
matched with a less able one of talent ∈ [0 ). In the second round, the two
winners of the first-period contests enter the major contest. The winner of this
contest gets the top job with income , whereas the loser gets nothing. The
two losers of the first-period contests are assigned to a consolation contest for
less important positions. The winner of the consolation contest receives income
∈ (0 ), whereas the loser gets nothing.
In order to highlight the different strategic effects that may lead to sandbagging, in this subsection we switch from the ratio-form contest-success function to
the difference-form one.19 We assume that player ’s winning probability when
competing against player is now given by
ˆ = ( ( ) + − ( ) − )
with and denoting the players’ effort choices, whereas and describe
the players’ exogenous talents. (·) denotes the same positive, monotonically
increasing and concave impact function as in the basic model and (·) a cumulative
19
For the difference-form contest-success function see Dixit (1987), Hirshleifer (1989), Baik
(1998), Che and Gale (2000), and the literature on rank-order tournaments in labor economics,
based on Lazear and Rosen (1981) and Nalebuff and Stiglitz (1983).
29
distribution function with density (·) := 0 (·) that is symmetric around its
mean at zero (e.g., a normal distribution with mean zero).20 Player ’s winning
probability can be written as
ˆ = 1 − ( ( ) + − ( ) − ) = ( ( ) + − ( ) − )
We assume that the functions (·) and (·) are well-behaved so that equilibria in
pure strategies exist.21
We can solve the two-stage game by backward induction, beginning with the
major contest and the consolation contest in round two. The equilibrium outcomes
of these two contests are anticipated by the players in the two heterogeneous
contests in round one when deciding on efforts and, hence, on whether to sandbag
or not. Let ∆ := − , and let (·) denote the inverse function of 1 0 (·), which
is monotonically increasing since (·) is concave. Furthermore, call, for example,
¯ denote the -player’s winning
the two round-one contests and ; then let
probability in contest from the perspective of the two players in contest . In
other words, when the two heterogeneous players choose their efforts in one of the
round-one contests they know that the high-ability player of the other contest will
¯ The solution of the game has the following properties:
win with probability .
Proposition 4 In the round-one contests, either both -players are more likely
to win or in one contest the -player has a higher winning probability and in the
20
For example, in the rank-order tournament models considered in labor economics, contestants
and have performance signals = ( )+ + ( = ) with and as idiosyncratic noise
terms (e.g., denoting measurement errors) that are identically and independently distributed (see,
e.g., Lazear and Rosen 1981). The winning probability of player is given by prob{ ( )+ +
( )+ + } = ( ( ) + − ( ) − ) with (·) as cumulative distribution function of the
composed random variable − . If, for example, and are normally (uniformly) distributed,
then the symmetric convolution (·) := 0 (·) is normal (triangular) with zero mean.
21
As is well-known from the literature on difference-form contests, equilibria in pure strategies
will only exist if uncertainty about the outcome of the contest is sufficiently large (i.e., the density
(·) is sufficiently flat and | 0 (·)| sufficiently small); see, for example, Lazear and Rosen (1981),
p. 845, fn. 2.
30
other contest the -player. In the latter case, the condition
¶
¶
µ
µ
1
1
¯−
( − )
2 {[ ( (0)) − ( (∆))]
(∆) −
2
2
(13)
+ [ ( (0)) − ( (∆))]}
must be satisfied.
Proof. See Appendix B.
The proposition shows that it is impossible that both -players are more likely
to win in round one. Intuitively, if a -player anticipates that in the other contest
the less able player has a higher winning probability than the more able one, for
two reasons he prefers to enter the major contest and, thus, to win the first round:
first, he wants to avoid being matched with the other -player in the next round;
second, in the major contest he can win and in the consolation contest only
.
However, it is possible that the players face a coordination problem with at
least two coexisting equilibria. In one equilibrium, the more able player has a
higher winning probability in the first round-one contest and the less able player is
more likely to win the second round-one contest. In the other equilibrium, we have
the opposite constellation with the less able player being more likely to win the first
round-one contest and the more able player the second round-one contest. For such
coordination problem to exist, it is necessary that condition (13) is satisfied. The
two sides of (13) highlight two effects that favor a coordination problem (given
¯ 1 ): According to the left-hand side of (13), − must be sufficiently
2
small.22 The smaller − the less attractive will be the major contest relative
to the consolation contest. For − → 0, every player is completely indifferent
between the two contests concerning the winner prizes. However, both types of
22
Note that for − → 0 the condition is clearly satisfied.
31
players have a major interest not to be matched with the same player type in the
second round, which is pointed out by the right-hand side of condition (13): From
the proof of the proposition we know that player chooses effort ( · ( − ))
( = ) when competing against player in the second round. Hence, if
two players of the same type are matched in one contest, their marginal winning
probability (0) ( (∆) = (−∆)) will be quite large since (·) is single-peaked
at zero, implying high efforts and, therefore, high effort costs ( · (0)). The
expression 2{[ ( (0)) − ( (∆))] + [ ( (0)) − ( (∆))]} describes
the players’ saved effort costs in case of successful coordination.
To sum up, if condition (13) is satisfied the players may primarily be interested
to coordinate their matches in the second round. In such situation, one -player
prefers sandbagging in round one to compete against a -player in the consolation
contest. Of course, if the coordination of the players fails they may end up in the
constellation where both -players sandbag and enter the consolation contest in
the second round — thus leading to mediocracy.
5
Conclusion
At first sight, one might expect that career contests perfectly correspond to the
well-known phrase "survival of the fittest". According to Darwin, there should be
a natural selection among heterogeneous individuals so that the best suited ones
will win the competition for reproduction. Relating to career competition, the
most talented or most productive players should win and be promoted to the top
positions within structured career paths. However, in our paper we show that,
contrary to the Darwinian view, the least productive players may have the highest
probability of winning career competition. The intuition comes from the fact that
the phrase "survival of the fittest" implicitly assumes that all individuals choose
32
the same activity level. Maybe, in biology this crucial assumption holds. In an
economic context, however, equilibrium behavior of heterogeneous players usually
differs. Since the least productive players have the lowest fall-back positions, these
individuals are strongly motivated to win career competition and, thereby, to avoid
their unattractive fall-back options. In this sense, we have a natural tendency that
the least productive players succeed. It is important to emphasize that under
identical activity levels our model would replicate the Darwinian outcome: The
individuals with the highest productivities would most likely win the career contest.
In our paper, we used a game-theoretic perspective to show how the detrimental effect of fall-back positions may lead to adverse career outcomes for society.
Switching to a contract-theoretic view would not qualitatively alter our results.
Allowing endogenously chosen, optimal career incomes would probably lead to a
change in the levels of equilibrium activities. However, as mentioned in Section
2 the ranking between the players’ activities and winning probabilities would not
change. Introducing reservation utilities for the players may even reinforce our
findings. If the most productive players have also the highest reservation utilities,
these players may prefer their outside options and decide not to participate in the
consolation contest. If the fall-back positions are now determined by the players’
different reservation utilities, we will have the same natural tendency that the most
productive players have the lowest incentives to succeed in a given career contest.
All these considerations are based on the assumption that the contest designer cannot use a mechanism to reveal the players’ types and then choose type-dependent
contest prizes to adjust individual incentives. However, as has been emphasized
by Malcomson (1984, 1986), identical prizes for all contestants are important if individuals only have unverifiable but observable performance signals. In that case,
under different prizes the principal would ex-post always claim that the player
with the lowest winner prize has performed best, thereby saving labor costs. Since
33
the players can anticipate such opportunistic behavior, contest incentives would
break down if prizes differ.
According to the Peter Principle, individuals are promoted as long as they
reach their level of incompetence. This observation was made by Peter and Hull
(1969). The outcome of the Peter Principle is based on two rules — currently good
performance is rewarded by job promotion and demotions are not possible. In the
subsequent economic work, the Peter Principle has been used as a synonym for
the misallocation of managers at (high) hierarchy levels. For example, Prendergast
(1992) explains such misallocation by the personnel policy of hiding good talents,
whereas the explanation of Lazear (2004) is based on temporary luck. In the
light of our model, misallocation at higher hierarchy levels can be explained by a
detrimental incentive effect that gives less talented individuals strong incentives
to assert themselves in competitive careers.
34
6
6.1
Appendix
Appendix A: Comparative Statics to Individual Salaries
The two first-order conditions lead to the following set of implicit functions:
∆ 0 (∗ − ) (∗ − )
−1 = 0
[ (∗ − ) + (∗ − )]2
∆ 0 (∗ − ) (∗ − )
− 1 = 0
2 :=
[ (∗ − ) + (∗ − )]2
1 :=
Let
¯
¯
¯
|| = ¯¯
¯
1
∗
1
∗
2
∗
2
∗
¯
¯
¯
¯
¯
¯
denote the Jacobian determinant, which is strictly positive since
1
∗
1
∗
2
∗
2
∗
00 (∗ − ) Ψ − 2 [ 0 (∗ − )]2
0
Ψ3
(∗ − ) − (∗ − )
= ∆ 0 (∗ − ) 0 (∗ − )
Ψ3
(∗ − ) − (∗ − )
= ∆ 0 (∗ − ) 0 (∗ − )
Ψ3
00 (∗ − ) Ψ − 2 [ 0 (∗ − )]2
= ∆ (∗ − )
0
Ψ3
= ∆ (∗ − )
with Ψ := [ (∗ − ) + (∗ − )]. Comparative statics yield
∗
∆
¯
¯
¯ 1 1 ¯
¯
−
1 ¯ ∆ ∗ ¯¯
=
|| ¯¯ − 2 ∗2 ¯¯
∆
¯ 0 ∗ − ∗ −
( ) ( )
¯
1 ¯¯ −
Ψ2
=
¯
|| ¯
0
∆ 0 (∗ − ) 0 (∗ − )[ (∗ − )− (∗ − )]
Ψ3
2
∆ (∗ − ) 00 (∗ − )Ψ−2[ 0 (∗ − )]
Ψ3
35
¯
¯
¯
¯0
¯
¯
∗
∆
¯
¯
1
1 ¯¯ − ∆
=
|| ¯¯ − 2
∆
1
∗
2
∗
¯
¯
¯
¯
¯
¯
0
¯= 1 ¯
¯ || ¯
2
¯
¯ − ∆
1
∗
2
∗
¯
¯
¯
¯
¯
¯
(∗ − ) − (∗ − ) ∆ 0 (∗ − ) [ 0 (∗ − )]2 (∗ − )
=
||
Ψ5
T 0
if
∗ − T ∗ −
Analogous results can be computed for ∗ ∆ and ∗ ∆ .
6.2
Appendix B: Proof of Proposition 4
In the second round, two players compete in the major contest for , whereas the
two other players compete in the consolation contest for . In the major contest,
player maximizes
· ( ( ) + − ( ) − ) −
and player
· [1 − ( ( ) + − ( ) − )] −
The two first-order conditions together lead to
· ( ( ) + − ( ) − ) =
0
1
1
= 0
( )
( )
¡
¢
Hence, if an equilibrium ∗ ∗ in pure strategies exists,23 it must be symmetric:
∗ = ∗ = ∗ with ∗ = ( · ( − )). Inserting in player ’s objective
function yields = ( − ) − ( ( − )) as ’s expected utility
from competing against in the major contest. In analogy, we obtain =
( − ) − ( ( − )) as player ’s expected utility from participating in
23
Recall that we assume well-behaved functions that guarantee existence.
36
the consolation contest.
In the first round, we have two heterogeneous contests, each one between a
-player and a -player. In each contest, the -player chooses to maximize
£
¡
¢
¤
¯ ·
¯ ·
+ 1−
( ( ) + ∆ − ( ))
£
¡
¢
¤
¯ ·
¯ ·
+ 1−
−
+ [1 − ( ( ) + ∆ − ( ))]
whereas the -player decides on to maximize
£
¡
¢
¤
¯ ·
¯ ·
+ 1−
[1 − ( ( ) + ∆ − ( ))]
¤
£
¡
¢
¯ · + 1 −
¯ ·
−
( ( ) + ∆ − ( ))
As first-order conditions we obtain
£ ¡
¢ ¡
¢ ¡
¢¤
¯ ·
¯ ·
−
−
+ 1−
=
( ( ) + ∆ − ( ))
1
0 ( )
and
£ ¡
¢ ¡
¢ ¡
¢¤
¯ ·
¯ · −
− + 1 −
=
( ( ) + ∆ − ( ))
0
1
( )
Using the symmetry of the density function (·) (implying (0) = 12 , (∆) =
(−∆) and (−∆) = 1 − (∆)) the expected utilities can be computed as
follows:
− ( (0))
2
= (∆) − ( (∆))
= =
= [1 − (∆)] − ( (∆))
37
with = . The winning probability of the high-ability player, ( ( )+∆−
( )), shows that for equal effort levels of both players the -player has always
a higher winning probability because of his "lead" ∆. Hence, if the -player is
more likely to win the contest, we must have that . From the two first-order
conditions we can see that this condition is equivalent to
¡
¢ ¡
¢ ¡
¢
¯ ·
¯ ·
−
−
+ 1−
¡
¢ ¡
¢ ¡
¢
¯ ·
¯ · −
− + 1 −
⇔
µ
¶
1
(∆) −
( − )
2
£
¤
¯ − 1 ([ ( (0)) − ( (∆))] + [ ( (0)) − ( (∆))])
+ 2
the inequality given in the condition. Note that the left-hand side is strictly
positive since (∆)
1
.
2
¯
If
1
,
2
the inequality cannot be satisfied since
the right-hand side is negative due to (0) (∆). In words, if in the other
contest the -player is more likely to win than the -player, the same cannot be
true in the given contest, because here the -player already leads by ∆ and, in
addition, chooses more effort than his opponent . Thus, it is impossible that in
both contests the two -players have higher winning probabilities. However, if
¯
1
2
and − is sufficiently large so that the inequality is violated then
¯
in both contests the -players are more likely to win. Finally, if
1
2
and
− → 0 then the inequality is clearly satisfied. Note that the inequality
can hold in either contest. Hence, if it is satisfied, we have at least two equilibria
where in one contest the more able player wins more likely and in the other contest
the less able player chooses more effort. If in the latter contest the lead ∆ of the
more able player is sufficiently small the less able player will have a higher winning
38
probability than the more able one.
39
References
Baik, K.H. (1998): Difference-Form Contest Success Functions and Effort Levels
in Contests, European Journal of Political Economy 14, 685—701.
Baker, G.P., Gibbs, M. and B. Holmström (1994): The Internal Economics of the
Firm: Evidence from Personnel Data, Quarterly Journal of Economics 109,
881—919.
Baron, D.P. and D. Besanko (1984): Regulation and Information in a Continuing
Relationship, Information Economics and Policy 1, 267—302.
Becker, G.S. (1962): Investment in Human Capital: A Theoretical Analysis,
Journal of Political Economy 70, Supplement, 9—49.
Caselli, F. and M. Morelli (2004): Bad Politicians, Journal of Public Economics
88, 759—782.
Che, Y.K. and I. Gale (2000): Difference-Form Contests and the Robustness of
All-Pay Auctions, Games and Economic Behavior 30, 22—43.
Dixit, A. (1987): Strategic Behavior in Contests, American Economic Review 77,
891—898.
Doeringer, P.B. and M.J. Piore (1971): Internal Labor Markets and Manpower
Analysis. Lexington.
Frank, R.H. (1985): Choosing the Right Pond. Oxford University Press: New
York, Oxford.
Freixas, X., Guesnerie, R. and J. Tirole (1985): Planning under Incomplete Information and the Ratchet Effect, Review of Economic Studies 52, 173—191.
40
Hirshleifer, J. (1989): Conflict and Rent-Seeking Success Functions: Ratio vs.
Difference Models of Relative Success, Public Choice 63, 101—112.
Kiyotaki, F. (2004): The Effects of a Consolation Match on the Promotion Tournament, Journal of the Japanese and International Economies 18, 264—281.
Konrad, K.A. (2009): Strategy and Dynamics in Contests, Oxford University
Press: Oxford, New York.
Laffont, J.-J. and J. Tirole (1988): The Dynamics of Incentive Contracts, Econometrica 56, 1153—1175.
Lazear, E.P. (2004): The Peter Principle: A Theory of Decline, Journal of Political Economy 112, S141—S163.
Lazear, E.P. and S. Rosen (1981): Rank-Order Tournaments as Optimum Labor
Contracts, Journal of Political Economy 89, 841—864.
Malcomson, J.M. (1984): Work Incentives, Hierarchy, and Internal Labor Markets, Journal of Political Economy 92, 486—507.
Malcomson, J.M. (1986): Rank-Order Contracts for a Principal with Many Agents,
Review of Economic Studies 53, 807—817.
Mattozzi, A. and A. Merlo (2007): Mediocracy, NBER Working Paper No.
W12920.
Messner, M. and M. Polborn (2004): Paying Politicians, Journal of Public Economics 88, 2423—2445.
Moldovanu, B., Sela, A. and X. Shi (2007): Contests for Status, Journal of Political Economy 115, 338—363.
41
Nalebuff, B.J. and J.E. Stiglitz (1983): Prizes and Incentives: Towards a General
Theory of Compensation and Competition, Bell Journal of Economics 14,
21—43.
Peter, L.J. and R. Hull (1969): The Peter Principle. New York.
Prendergast, C. (1992): The Insurance Effect of Groups, International Economic
Review 33, 567—581.
Rosen, S. (1986): Prizes and Incentives in Elimination Tournaments, American
Economic Review 76, 701—715.
Skaperdas, S. (1996): Contest Success Functions, Economic Theory 7, 283—290.
Spence, A.M. (1973): Job Market Signaling, Quarterly Journal of Economics 87,
355—374.
Tullock, G. (1980): Efficient Rent Seeking, in Buchanan, J.M., Tollison, R.D.
and G. Tullock (Eds.): Toward a Theory of the Rent-Seeking Society, A&M
University Press: College Station, 97—112.
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.
Williamson, O.E., Wachter, M.L. and J.E. Harris (1975): Understanding the
Employment Relation: The Analysis of Idiosyncratic Exchange, Bell Journal
of Economics 6, 250—278.
42
YH
winner
major contest
consolation contest: YM, YL
N−1
losers
Figure 1: One-Hurdle Career
YH
winner
loser: YM
winner A
organizational unit A
consolation contest A: YL
NA − 1 losers
Figure 2: Two-Hurdle Career
level-II contest
level-I contests
winner B
organizational unit B
consolation contest B: YL
NB − 1 losers
© Copyright 2024