Principal competitors' tournament

Abstract

The static model presented in this paper formalizes accumulated knowledge and introduces new results from tournament theory. The model's methodology facilitates more accurate and in-depth analysis with regards to sorting competitors, imperfect information, high informativeness that corresponds to the all-pay auction, risk aversion, and cooperative effort. The model methodology also facilitates analyzing non-fixed prices tournament, and misestimation of competitors' abilities by the principal unenforceable contracts. The theory of moves provides an important additional insight.

Full text

1
Principal competitors’ tournament
Shai Danot
Last update 15 Jun 2021
Abstract
The static model presented in this paper, formalizes accumulated knowledge and introduces new
results from tournament theory. The model's methodology facilitates more accurate and in-depth
analysis with regards to sorting competitors, imperfect information, high informativeness that
corresponds to the all pay auction, risk aversion and cooperative effort. The model methodology
also facilitates analyzing non-fixed prices tournament, misestimation of competitors' abilities by
the principal unenforceable contracts. The theory of moves provides an important additional
insight.
Key words: tournament theory, income inequality, moral hazard, informativeness, winning
probability function.
JEL codes: C72, D33, J41, M55
1. Introduction
This article extends Lazear and Rosen (1981) (LR in the sequel) by assuming that the tournament
designer cannot identify with certainty the true winner of the tournament. In that line, better
information about the winner's identity is beneficial but costlier. It is also assumed, contrary to
classical tournament models, that the designer of the tournament and the competitors are rent
maximizers and so have conflicting goals.
The first paper that relaxed the main LR assumption was, O'Keeffe et al (1984), OVZ henceforth.
This meant that there was no need to add a source of noise to the competitors' effort or, output
could be a deterministic function of effort. The model presented in this paper relaxes the OVZ
and LR assumption that the tournament designer or the principal earns zero rents and unlike OVZ
has a well-defined winning probability function.

2
The model presented here also extends Hirshliefer's (1989) explicit analysis of the strategic
behavior of competitors in a contest with a difference winning probability function. The use of a
suitable winning probability function enriches the analysis of tournaments with the winning
probability function tool that is already used in contests.
1
For example, as the wining probability
function is well defined it is possible to reach accurate equilibrium properties.
2
The special properties of the wining probability function, which is stochastically derived in the
most natural way, facilitates the revelation of equilibrium properties without resorting to the usual
way (Clark and Riis 1998a; Clark and Riis 1998b) of maximizing a competitor's utility while
assuming all others are exerting equilibrium effort. This is important since essential equilibrium
properties cannot be revealed in the standard way (first best equilibrium effort, iterated elimination
of strictly dominated strategies, viability of the Nash equilibrium).
Demougin and Fluet (2003), using a variant of the LR setup, model limits in the principal's capacity
to extract rent through a minimum wage. In contrast, this paper assumes that the contracted
monitoring intensity between the principal and the competitors is not enforceable by an arbitrator
or court. Under this condition, the principal can extract rent even if he doesn't have any bargaining
power except his option to shirk, i.e. to reduce his monitoring efforts.
The distinction between identical and non-identical competitors generically replaces, the ex-ante
and ex-post concepts referring to competitors, who are not too different than what was perceived
ex-ante. Those concepts enable for example, to analyze equilibrium properties even if the principal
1
The difference between Hirshliefer's (1989) analysis, which is a classical analysis of a contest,
and the one presented here is that Hirshliefer analyzes a contest with no principal and two
competitors with linear cost functions. In a sense, his analysis resembles that of two animals
fighting for a given resource (the difference is that animals have convex cost functions). In a
tournament, the principal can design the wages for the winner and the losers, which is very
different than contests. Furthermore, the second prize is zero in a contest, even in the presence of
a principal who designs it, as in auctions.
2
Unlike the ratio winning probability function, the difference winning probability function used
in this paper is almost symmetric with two competitors, for all range of parameters, and
competitors may win after exerting minimal effort. Notwithstanding, the stochastic derivation of
the winning probability function guarantees that it is a difference function in efforts and a ratio
function in quantities.

3
ex-ante, is not very well informed about the competitors abilities.
The model main analysis is limited to two prizes. This is so as the winning probability function is
more complex with more prizes and becomes harder and more tedious. Even so, the freedom of
the principal to decide on the prize gap makes it more general than a mere contest or auction.
Furthermore, it facilitates expanded simple analysis of multiple prizes in a noisy environment.
The first proposition formalizes the needed properties for the existence of a unique IESDS
equilibrium effort, (IESDS stands for, iterated elimination of strictly dominated strategies) and
identifies wages for the winner and the loser according to agreed division of surplus between the
principal and the competitors under contracted fixed wages. A corollary follows that easily
identifies how wage difference change in order to maintain first best equilibrium effort. Next,
extensions to the model are analyzed. In particular, equilibrium properties under incomplete
information, high informativeness that corresponds to the all pay auction, risk aversion and
cooperative effort are analyzed.
Following, wages for the winner and losers that depend on ex-post total exerted effort are
identified. Furthermore, the consequences of under and over evaluation of the competitors' abilities
are evaluated. The third proposition show that first best tournament surplus may be unattainable
when the principal can shirk from his contract obligations as a monitor. dynamic non-standard
analysis of the game based on the theory of moves show that first best effort may not be exerted
even for identical competitors unless the principal cannot punish the competitors. Complementary
analysis follows each of the propositions' proofs.
2. The Model
At the initial stage of the game, the principal and the competitors bargain and sign a contract with
the following parameters
*12
( , , )k W W
. The monitoring intensity,
*
k
, a decision variable is known
to the competitors and enforceable by them at any level of supervision. This assumption is relaxed
later. The prizes' size are determined based on total anticipated effort by the principal and the

4
bargaining power of the two parties.
3
It is assumed for simplicity that monitoring intensity by the
principal equals his informativeness.
After signing the contract, two or more competitors compete for a monetary prize or wage
1
W
by
exerting vector of efforts
*

. The losers receive
21
WW
.
As in Rosen (1986), it is assumed that output is a deterministic function of effort.
4
The cost of producing effort

for competitor
i
is
( ) , 2
l
i
a
Cl
l



. Note that the cost
function is convex i.e.
( ) 0, ( ) 0CC

 

and that
(0) 0, (0) 0CC


.
Competitor 1 with cost function
1() l
a
Cl



is stronger than competitor 2 with cost function
2() l
b
Cl



if
ba
.
5
3
Effort is not specified in the contract. However, anticipated equilibrium effort

exists and
would have been part of the contract if the principal knew the specification of the competitors' cost
functions. This is so since the specification of the cost function affects equilibrium effort.
Proposition 1 and 2c demonstrate how anticipated effort is calculated.
4
Competitor's good luck in a single tournament could be seen in this model as performing like a
better competitor, see definition in the main text. Similarly, good (bad) conditions for both of them,
which is modelled explicitly in Nalebuff and Stiglitz (1983), could be viewed here as competitors
performing better (worse) than their perceived ability. Proposition 2c deals formally with this
issue.
5
The competitors' inventive compatible first order condition, characterized in (2), dictates that for
1l
increasing
W
doesn't increase effort. Secondly, Hirshleirfer (1989) showed that only mixed
Nash equilibrium exist in this case for a positive effort. Thirdly, the competitors' first best effort,
*argmax Va

  

, has no interior solution (this is untrue for a more general maximization
problem, see equation (3) in the main text) as it is the maximum effort for
Va
. If
2l
the
marginal cost function is concave, which means that increasing the wage difference increases
effort less and less. Proposition 1b endogenizes further a convex marginal cost function.

5
The winning probability function, or WPF henceforth, for competitor
i
is
( , )
ij
P

. It obviously
has to satisfy
0
i
P



and
0
ji
P




.
For given equilibrium effort, the informativeness of the WPF is denoted by
K
. This variable is
used in the following definitions.
: has the following properties
i
competitorof he winning probability functionIt is assumed that t
1)
( , / , ) ( / , )
i j l i j l
P K k P k
     
  
such that
,l i j
2)
12
( , / , , ) ( , / , , )
i j i j l i j i j l
P K k P K k
           
      
. Such that
12
,0kk


.
The first property states that the winning probability function is a difference probability function
in efforts i.e. for two competitors, the wining probability depends only on the competitors' output
difference. The second property states that for better informativeness the probability that the
outperforming competitor wins is larger.
The logistic distribution maintains those qualities. The WPF and the density of the WPF are:
1 1 2 1
12
1 2 2 1 1 1 2
exp{ } 1 exp{ ( )}
,
exp{ } exp{ } 1 exp{ ( )} (exp{ } exp{ })
k P k k
Pk k k k k
  
      

  
    
.
For three contestants or more,
3n
, the multi logistic WPF is
1
2
1
12
1
11
22
exp{ ( )}
1,
1 exp{ ( )} (1 exp{ ( )})
in
i
i
i n i n
ii
ii
kk
P
P
kk


   








   


.
Informativeness is defined as the probability of choosing the right winner or the accuracy of the
winning probability function,
AP
in short. For example, the winning probability of the better
competitor with
12
( , , ) (1.01,0.99,4)k


equals the winning probability of the better competitor
when
12
( , , ) (5.05,4.95,0.8)k


. More accurate winning probability is equivalent to higher chance
of pointing the right winner. Thus,
(1)
*
max , 1..
i
AP K i n

  
.
In this example, AP is 4 for ex-ante identical competitors. If we measure informativeness ex-post
AP is 4.04. Also, when the competitors are ex-ante non-identical AP is 4.04.

6
This WPF (winning probability function) is stochastically derived according to a realization of the
signal
i i i
qq


that the principal observes,
exp( )
ii
qk


.
6
If
12
kk
,
1
()
i
qk
is more
informative than
2
()
i
qk
, or the probability of a principal's mistake in identifying the winner is
smaller (this also corresponds directly to the first order dominance concept).
7
8
6
Real output is
i
q
. The random variable
i

that distorts real output to signal is distributed
according to
..
~ ( )
i i d
iSIEXP

. This random variable has a density function
2
( ) exp( ) [ 0]
h z I z
zz



(
I
equals to 1 when
0z
and zero otherwise). Setting
1
70


calibrates
i

, such that
( ) 1
i
E


. The SIEXP density function remains nonetheless highly
asymmetrical with the mode to the left of the expectation.
7
The expression
exp( )
ii
qk


is intuitive, as for identical
i

the observed difference between two
close efforts is bigger when informativeness is bigger. As the realizations of
i

are unique for
each contestant, for a higher
k
, the true ordinal rank is noticed by the principal with higher
probability.
8
The proof that
i

is distributed as claimed is in Jia (2008) and was posed as an exercise in
Hirshleifer and Riley (1992), for the Tullock function. Modifying the proof for the logistic function
is almost immediate since the logistic function is a private form of
1
()
()
i
n
j
j
g
g




. To reach the result
one must define
exp( )
ii
qK


instead of
i
i
q


as in the Tullock function. The Jia proof is good
for non-identical contestants as well (Jia used identical contestants is because he proved the
theorem for efforts and not for quantities). Note that, for the winning probability function to be
logistic in efforts the difference in the competitors' abilities could be modelled only via different
effort cost functions. This is so because, when identical efforts exerted by two competitors produce
different individual outputs the winning probability function is Tullock (1980) in quantities but
not logistic in efforts.

7
Graph 1: Two densities of logistic winning probability of competitor 1 that depict principal's
different levels of informativeness when the competitors are identical
This graph provides visual aid for the following auxiliary propositions.
Auxiliary proposition: For a given equilibrium effort, the informativeness of
()PK
or AP,
increases monotonically with the increase of
K
. See footnote 8 for a very simple proof.
The competitors cost function (and the link between effort and productivity of all the competitors)
is common knowledge among themselves. This assumption is only made for simplicity is standard
in complete information static game. Incomplete information about the opponents cost function is
introduced later. When the principal decides who wins he knows less about total production than
the competitors, who, at this stage, know exactly how much they and their rivals produced. This
assumption
The size of the competitors' effort vector is calculated exactly as that of any vector. The relation
of the effort vector and its individual efforts components is stated the production function.
Each competitor solves
12
max (1 ) ( )PW P W C

  
to find an optimal effort
*

, which is the
principal's incentive compatible constraint. The winning probability function is
P
.
Each competitor participates if
12
(1 ) ( )PW P W C u

   
, the principal's incentive rational
constraint.
First order condition of this incentive problem shows that bigger wage difference

8
(2)
*
12 *
( ) ( )
P
W W C





  

increases effort when
()C

is convex, as in LR. Effort increases
also with informativeness as higher informativeness increases
*
P





. When
(2a)
*
( ( ) ( )) 0U C d

  




is maintained as well, the competitors will exert optimal rather
than minimal effort.
9
The expression of marginal utility is a bit confusing, as the competitors are assumed to be risk
neutral. They are risk neutral in money but not in effort and so
() P
UW


   
.
(2a) is a global incentive compatible competitor's condition or a no shirking condition, compared
with (2), the local incentive compatible optimum. This condition is equivalent to the no shirking
condition from OVZ,
*
12
2( ) ( )
2
WW
W C C


  
for two identical competitors and is more restrictive
than the participation problem if
2()W C u


( it is always beneficial to exert the minimal effort

because the logistic WPF density function is positive at
0


and so marginal cost is lower
than marginal utility up to this effort) .
10
Expressed as it is here it helps proving proposition 1b and
demonstrates graphically how tournament surplus maximizing equilibrium is reached.
9
Notice that the local second order conditions for 2 identical competitors is
22
12
23
1
()
exp{0} (1 exp{0}) 0
(1 exp{0})
PK


 


and therefore, second order conditions are always negative
for linear or convex marginal cost
10
It can also be expressed as
*
( ( ) ( )) ( ( ) ( )) ( ( ) ( ))U C d U C d C U d



        

     
    
  
,
which is descriptive of the WPF logistic function. Since effort is by definition non-negative, the
correct interpretation of
0()Ud



 

is that the utility of exerting zero effort is positive and
increases with the decrease in the principal's informativeness. The no shirking condition term is
taken from Shapiro and Stiglitz (1984), which is also an incentive compatible condition in a
different strand of literature.

9
The following two conditions characterize equilibrium. They are mathematically equivalent to
second order condition. Unlike soc, they can be visualized graphically.
i)
**
( ) ( ) , , , 0
i i i i i i i
UC
          


      
.
ii)
**
( ) ( ) ,
i i i i i i
UC
     


   
.
The two conditions, (i) and (ii) state that marginal utility is higher than marginal cost below and
lower above equilibrium effort. The limits posed in i), are for identical competitors as marginal
utility is always lower than marginal cost in the range
i
  

for them, see graph 4a and 4b
for visual aid.
The cost of information to the principal
()AP


.
11
Informativeness benefits society since better competitors are winning. The function

is convex
in the accuracy of
P
.
First best effort that maximizes total tournament surplus
()

is,
12
*, .. 1 2
argmax 1 ( ) ( ) .. ( ) ( )
nnn
Q C C C AP
  
    
      
.
Total output,
1
( .. , )
n
Q f K


,
0 1..
i
fin

  

and the price of
Q
is normalized to 1.
(3)
*12 *
( , .. )
( ) ( ) 1..
i
n
i
ii
ff AP
C i n
AP AP
   

  
    
   
.
12
11
Informativeness decreases with the number of competitors. The competitors' and principal's
characteristics, the competition environment and the monitoring technology affect the
specification of

.
12
If
f
AP AP




there is a difficulty in solving this optimization problem because
*
i
AP



is
dependent on the optimal effort. Solving this problem require guessing first the optimal effort
vector and engage in trial and error simulation until the searched effort solution solves (3).
Furthermore, total output
f
has to be symmetrical (but not necessarily additive) so that
competitors with identical cost functions are identical.

10
*
i
AP k



from (1). Proposition 2c show why the principal can calculate first best effort, when
Q
is additive, despite not knowing the exact specification of the cost function (The principal estimates
the specification of the production function).
It is assumed that the principal maximizes
()

if his rent is non-zero considering that maximizing
()

is mathematically identical to maximizing a fraction of
()

.
For competitors who are indifferent to the level of monitoring intensity,
(4)
*argmax ( ) ( )
K
k f AP AP


From first order condition we get
**
f
AP AP
k k k k





.
Also,
1
*.. 1 1
argmax ( .. , ) ( ) .. ( )
nnn
f K C C

    
  
,
*
( ) 1..
ii
i
fC i n



   

.
Surplus maximizing effort equals the competitors' maximizing effort as in LR.
Graph 2: Optimal monitoring intensity.
For example, when
1 2 1 2
11
( , ) ln lnf q q q q
KK

,
12
Q


as in LR and informativeness or
sorting has no value. For,
12
()QK


, sorting is important. Further analysis regarding
monitoring averse competitors and non optimal monitoring see appendix 1.

11
Proposition 1a: for
2n
, a unique, IESDS (iterated elimination of strongly dominated strategies)
equilibrium
*

exists for fully informed competitors and for
2n
, first best equilibrium
*

exists for fully informed identical competitors under the condition specified in proposition 1b.
13
14
15
13
One of the sufficient conditions for the existence of Nash equilibrium is the existence of convex
set of strategies for each player, which is identical to having a quasi-concave utility function. This
condition is not met (by any WPF) when a convex combination of two strategies (efforts) each of
them preferred to a third strategy(effort) is not necessarily preferred to that third strategy (effort).
It is not met here when
P



increases rapidly only near equilibrium, which indicates very high
informativeness of the WPF, while marginal cost increases gradually. In this instance, the no
shirking condition doesn't hold. Notice that it is not met here even if the no shirking condition
holds.
14
With more than 2 non-identical contestants second order conditions doesn’t always hold. For
example, if there are 3 non identical contestants with cost functions
2 2 2
( ,1.5 ,2 )
  
and
33
22
( ), ( )
11
2 3 2 3
K
f d K m


  
     
,
d
and
m
are parameters, equilibrium exists
with
* * * *
1 2 3
( , , , , , , ) (1,1,0.25,3.94,0.105,0.073,0.054)d m k W


and
* * * *
1 2 3
( , , , , , , ) ( 8,1,2,4.21,0.99,0.5,0.259)d m k W


. The wage differences were calculated
according to assumed efforts (0.11,0.08,0.05) in the first case and (1,0.8,0.5) in the second case.
The participation of the weakest competitor is slightly negative in the second case and all the
competitors earn very small rent in the first case while the principal earns zero rent. Even though,
tournament surplus was not fully maximized, the heuristics of setting wage difference is close
enough to it and reasonable since optimizing the wage difference is difficult. One may be interested
in comparing this example with Szidarovsky and Yakowitz (1977), that prove the existence of
Nash-Cournot equilibrium with multiple non-identical firms when the profit function is concave.
15
If the principal's maximum informativeness resides only in the region of
*

, as is expected in
the triangular or beta distribution, the equilibrium for identical competitors is an IEDS equilibrium.
See appendix 2 for a proof. This is true even for multiple competitors according to the probability
definition presented in appendix 2 since marginal cost is convex and the density reaches its peak
at equilibrium (unlike in the multi-logistic function). The marginal cost function must be convex
for the no shirking condition to hold in the triangular distribution (when marginal cost is linear,
(2a) is zero and equilibrium effort is not unique).

12
Proposition 1b: For
*
1
1
1
( 1)
thr
l
f
nl a
kn











there is no Nash equilibrium with identical
competitors.
The proof of proposition 1b and a numerical example is relegated to appendix 1. The principal
knows
thr
k
if he knows
*
f



. Proposition 2c shows that this is true if
n
Q
is additive and linear.
Otherwise, the principal cannot know
thr
k
, which means that he has to be extra careful not to pass
thr
k
in those cases. One can conclude that in such cases, informativeness is sufficiently lower than
the maximum to ensure the existence of equilibrium.
Proof for 1a: firstly, a Nash equilibrium exists if competitor's
i
optimal strategy is to exert
*
ˆii


when the other competitor or competitors exert
*
ˆii



according to condition i) and ii) . This
applies also to multiple non-identical competitors.
Secondly, we need to prove that exerting
*

is an IEDS strategy for two non-identical competitors.
Assume that player 1 is stronger. This means that there exists
0


such that
* * * * * * *
1 2 1 2
( / ) ( , ) ( ) 1,2
ii
i
P
U W C i
      



      

with marginal utility at maximum for both
competitors when
**
12 0
  
  
(when the competitors are identical
0


).
Assume we start with strategies
1 2 1 2 ˆˆ
ˆ ˆ ˆ ˆ
( , ) ,
      
  
.
When
*
22
ˆ


, exerting effort
11
ˆ
  

infinitesimal higher is a better strategy, for competitor
1, than
1
ˆ

because by such a change competitor's 1 effort will not surpass his "temporary"

13
maximum as the logistic density function is continuous. Temporary maximum is defined such that
increasing
1

further is a worse strategy for competitor 1.
We continue in elevating the improved strategy until reaching a local maximum for player 1 and
then elevating and improving the second competitor strategy. We proceed in this manner and
converge to
**
12
( , )

. Similar reasoning applies when
2
*
2
ˆ


. This process takes place if Nash
equilibrium exist. Q.E.D
Non-identical competitors generically cannot exert first best equilibrium effort since an
infinitesimal reduction in the strength of a competitor doesn't not necessarily reduce his
equilibrium effort according to
*
*()
ii
fC





.
16
Identical competitors could behave as non-
identical if the principal is preferring one over the other(s) and this fact is common knowledge.
Thus, discrimination adversely affects production as well as sorting.
Note that IEDS equilibrium is not behaviorally identical to simple dominant strategy equilibrium
since common knowledge about the competitors rationality is necessary to reach equilibrium.
Furthermore, the competitors must be able to compute it. Thirdly, equilibrium is not viable in the
Kalai (2019) sense with more than 2 competitors, which means that if one competitor deviates
from it, the others must change their calculation as well.
The elimination of strategies for non-identical competitors is essentially similar to the
p
beauty
contest game, in which after eliminating iteratively weakly dominated strategies only a small
minority act in accordance with the Nash equilibrium that is zero.
17
Notwithstanding, unlike in
the guessing game the domination is strong and identical competitors can reach the conclusion of
exerting first best effort since their rational interest is to maximize total surplus. One may still ask,
16
Moreover, if one of the competitors has a higher chance of surviving longer as a winner, than
the wage that he perceives as a winner is higher than that of his competitor, and so, otherwise
identical competitors, will exert effort lower than first best.
17
One of the papers that analyze how players actually behave in this game, is Nagel (1995).
Footnote 5 in her article explains one of the possible processes of eliminating weakly dominated
strategies. Only a small minority state the Nash equilibrium of this game.

14
why non-identical competitors exert what seems to be an optimal effort even in many cases of
large tournaments despite the non-viability of the Nash equilibrium?
We can apply Aumann (2019) reasoning of rule rationality to a qualifying race with 8 runners. If
a very good runner drops, runners whose chances to win increase should increase their efforts. If,
however, there was a runner who competed with the runner for the winning prize he may face less
competition and thus decrease his effort. This is compatible with the new equilibrium. Note also,
that runners who are only slightly affected by the drop of this runner still exert their former effort,
which means that the equilibrium is almost viable for them with respect to the dropped runner.
Moreover, players learn if the game is played often enough (one of the first papers on learning is
Brown (1951) who suggested a simple non-Bayesian approach on how to play). Furthermore, each
competitor's decision on how much effort to exert may entail aspects beyond those of the outlined
game, the competitors may be susceptible to outside influence or guidance. Part of footnote 18
elaborates more on this. A multi prize static tournament could at least take into account the above
aspects.
The intuition for proposition 1b is simple. From (2) we get that as
*
P



increases
*
W
decreases. But when
*
W
is small enough the competitors prefer to exert zero effort.
For example, if
12
, ( )QC
   

  
,2n
,
4
thr
kk
the probability of competitor 1
winning when
12
1.01, 0.99


is 0.52. When
12
1.1, 0.9


, it is 0.69 and when
12
1.5, 0.5


, it is 0.982. In many settings, such as sports, most competitors don't shirk
despite much better informativeness. When
3
()C


,
8
thr
kk
, the winning probabilities
are more realistic (0.54,0.832,0.999). Here, the increased convexity enabled to increase
thr
k
,
because equilibrium effort is 1.
Generally, adding a kink just below equilibrium effort so that it is much steeper for higher efforts
increases
thr
k
. This is reasonable for repeated tournaments such as we see in competitive
swimming and running. For example, the cost for a world champion in 100 meters to run 10.3 is
very small. In tournaments that are less informative such as workers competing for promotion in

15
firms, the cost of producing not too far from equilibrium effort is much higher.
18
Increasing
thr
k
by adding a kink to the cost function near equilibrium enables also to reach first best
*
k
, which
increases tournament surplus.
Proposition 1c: for identical competitors the wages in equilibrium are
**
12
**
ˆˆ
1 1 ( ) 1 1 ( )
,
nn
n C C
W Q W Q
n n n n n n
PP
   
   




     


.
ˆ

is the expected profit or the
principal's rent and
*

is optimal equilibrium effort.
Proof: by solving the following equations we get the expression of
12
,WW
. The first describes the
competitors' incentive problem and the second describes the link between wages and total output.
*
12
*
12
( ) ( )
ˆ
( 1) n
PW W C
W n W Q






  


    

18
Since people physically exert efforts only in a range and as increasing effort beyond a certain
threshold is very costly a kink in the marginal cost function is reasonable. In that line, exerting
excess effort in a single competition may harm the competitor in the long run which is a reason for
the cost function to be steeper near equilibrium. Additionally, the principal may face more
difficulties in monitoring unfair competition such as damaging opponents near equilibrium, or he
may decide to enable it near equilibrium due to other reasons. In those cases, it is certainly a best
response for hawks to gain unfair advantage when the chances of getting caught is minimal..
Akerlof and Kranton (2000) model behavior conformity in lower class social groups through
sanctions. The punishment force of the opposing group makes the cost function steeper. In Kandel
and Lazear (1992) the group punishes shirkers as they model it in peer partnerships. The similarity
in both environments is that the group punishes the individual for deviating from group interests
in the pursuit of his self-interest. Acting rationally against private information is modelled in
Bikhchandani et al (1992) by learning from others past actions. Using the same logic here, the
competitors can believe that they have a steeper cost function than they actually have (beyond
some threshold) by observing the behavior of others. The latter mechanism doesn’t involve group
sanctions and is less efficient as the informational cascade is broken once a competitor acts
differently from its predecessors. In a tournament setup it may characterize competitors who are
not part of an inner group but watch those who belong to this group compete.

16
The solution to the equations is the result. When
ˆ0


wages are maximized and if the competitors
just meet their participation constraint the principal's profit is at maximum. As we saw earlier, the
rational principal prefers first best tournament surplus, as long as he doesn't get zero rent, and so
he chooses to decrease wages rather than increase effort beyond first best.
19
Example: with
2
2 1 2, ( ) 0.5QC
   
  
,
2k
,
( ) 0, ( ) 0f K K


, tournament surplus is
1


and
*42Wk
  
induce first best equilibrium effort. With equal division of the surplus

between the principal and the competitors
12
1.75,W 0.75W  
. When we increase the informativeness
parameter to
4k
,
12
1.25,W 0.25W
. See appendix 3 for more details. Note that negative wage
is relative to the alternative utility and not an absolute term. Only when the contract is enforceable
and the competitors' employment market is competitive as if the competitors offer the terms of the
contract to the principal, they gain all the tournament surplus as in LR. See additional analysis in
proposition 3.
Corollary 1a: for given
12
,WW
and identical competitors
**
13
NNN


  
and wage difference has
increase according to
2
*
*
( 1)
f
n
WnK




, in order to maintain first best equilibrium effort.
20
19
This statement is certainly not always true outside the model. Moldovanu and Sela (2001), for
example, analyze tournaments with a given purse. In this case, the tournament designer
maximization of rent means maximization of participants' efforts. Secondly, the behavioral
principal may not maximize tournament surplus if given, say, only 20 percent of the rent.
20
McLaughlin (1988) analysis of multiple contestants is cumbersome as he used the LR setup
which was the only one available at that time.

17
This is true for the multi logistic distribution since
2
*
*
( 1)
f
n
WnK




increases in
n
to maintain
first best equilibrium effort. The expression for
*
W
is derived in the proof of proposition 1b in
appendix 1. A proof for different types of distributions is in appendix 2.
3. Minor Extensions
3.1 Incomplete information
Corollary 1c: When uncertainty about the rival's productivity is introduced
*
W
increases to
maintain equilibrium effort. Specifically, if the two competitors are identical, there is a common
prior over the range
[ , ]
LH

according to distribution
F
about the rival's productivity, there is
common knowledge about this prior shared by the competitors and the principal is aware of the
competitors' beliefs.
21
Proof: we want to prove the best response for
*
2


is
*
1


.
22
If competitor 2 exerts
** 1
2*
f
C








, competitor 1 will exert
** 1
11 *
ˆˆ f
C
   








as
12
** ** ** **
1*
( / ) ( )
f
UC
   



  

by the definition of the
21
When there is a common prior belief on the competitor's productivity over the range
[ , ]
LH

according to a
F
distribution the no shirking condition is less restrictive since
2
W
is lower and
12
2
WW
remains the same.
22
The growth in
W
is the weighted average, according to
F
, of the needed increase of effort to
reach first best equilibrium. There is no simple expression for
W
since finding the reduced
equilibrium effort requires solving complex equations. However, equilibrium effort calculated
numerically is reduced only slightly in the logistic function when
[ , ]
LH

is limited to
10%
and is quite robust to the level of informativeness
K
. See appendix 3 for a visual demonstration.

18
difference function. The best response of competitor 1 is
*
1
ˆ


because
22
** * **
11
ˆˆ
( / ) ( )UC
    

  
and
*1 2 1
( ) ( / )CU
   


.
When
** *
2


, we know that
** ** ** *
1 1 2
( / ) ( )UC
   


. Since,
* ** * * **
1 2 1 2
( / ) ( )UC
    

  
and
()C


increases in

while
**
12
( / )U


maximum is
*
()C


means, that marginal utility will not cross marginal cost when
*


. Thus,
competitor's 1 best response for
2
** *


is smaller than
*

.
The bigger the spread
[ , ]
LH

the lower is equilibrium effort. From here we can conclude that,
W
increases to sustain first best effort as an equilibrium effort.
23
For visual aid see appendix 3.
Q.E.D
Decrease of
i
a
in the cost function of competitor
i
monotonically increases equilibrium effort
*
i

. Thus, the use of
[ , ]
LH

is equivalent to
[ , ]
HL
aa
. The corollary is not true for non-
identical competitors because effort can increase rather than decrease when uncertainty is added.
For example, when the weak competitor is much weaker, the stronger competitor exerts higher
effort. See graph 6 of appendix 3 for a visualization.
Assuming common knowledge, the corollary is also true for self-ignorance about exerted effort
because each competitor doesn't know how much his rival produces relative to his own output.
First best effort cannot be achieved even for identical competitors when there is a violation of the
23
From the analysis of the triangular density, presented in appendix 2 we know that contrary to
the logistic distribution if the rival exerts effort
*
ˆ


, then
*


is a best response.
However, since the marginal cost function must be convex to maintain unique equilibrium, the
decrease in best response effort when the rival exerts
*
ˆ


is bigger than the increase in effort
when the rival exerts
*
ˆ


and so equilibrium effort decreases assuming uniform distribution of
off uncertainty about the other competitor. This means that
*
W
increases. Notice though that
contrary to the corollary
*
W
doesn't increase for every distribution
F
of uncertainty about the
other competitor.

19
common knowledge assumption. For example, if the first competitor knows, or believes to know,
exactly how much his competitor produces. In this case, he exerts effort above first best and sorting
may be impaired. Such common knowledge violations coupled with imperfect information about
the nature of the WPF, are additional reasons for
*

or
*
q
not to be contracted.
24
3.2 Accommodating all pay auction
One may ask, why is there a connection between the accuracy of effort measurement and the cost
function of the competitors? The answer is that perfect measurement will make the competitors
want to exert a bit more effort than "planned" equilibrium, and then a bit more, and more. Only a
pathologic cost function that poses sufficient cost on exerting an additional infinitesimal effort is
an equilibrium in this case. The limit is identical to the LR limit of zero noise or its surrounding
neighborhood which, their framework cannot analyze properly.
But this limit case is exactly the all pay auction, in which with two players, there is only a mixed
strategy symmetric equilibrium, as for any pure strategy that is presumably winning the competitor
is better of exerting a higher effort.
With workers competing for promotion the basic interpretation of both games is that the players
know each other strength. However, in the all pay auction interpretation, a mixed strategy could
be interpreted as a player who knows his type, but doesn’t know who he competes against, except
knowing the population distribution which is similar to that of the mixed strategy equilibrium of
the opponent.
24
For a subtler example, assume that the principal believes that the competitors beliefs are common
knowledge as in the corollary. Each competitor believes that his rival produces according to
symmetrical
F
over the range
[ , ]
LH

. Furthermore, each competitor
i
believes that his rival
j
believes to know better than
F
how much
i
produces. Thus, each competitor
i
believes that
his opponent
j
produces according to a distribution more informative than
F
(characterizing
i
's), i.e. produce more than first best
*

. Thus, actual equilibrium effort will be lower than
*

even after correcting
*
W
according to the corollary.

20
Firstly, if
2
W
is regarded as an entrance fee where
2
W
is negative, or upfront payment to the
competitors where it is positive, it could be regarded as sunk cost and thus the tournament is just
like an all pay auction.
Calculating the mixed strategy equilibrium by adopting the calculation by Maynard-Smith (1974)
of a second price all pay auction to a first price all pay auction is easy:
12
0
12
0
( ) ( ( )) ( ) ( ) ( )
( ( ( )) ( ) ( ( ) ( )
E W W c p x dx c p x dx
W W c p x dx c p x dx


 
 
  
   



    
    


The expectation of the bid

consists of two terms in a symmetric game. The first term in the left
equation corresponds to the other player bidding below

according to his mixed strategy
x
and
above

in the second term. The payment for the bidder is
12
()W W c


in the first and
()c


in the second. Since payment in mixed equilibrium is identical for all values, we can add the second
equation just as Maynard-Smith did. From here we get,
00
12
( ) ( ) ( ) ( )
()
p x dx p x dx cc
WW
  
  
 


  

or
12
()
() ()
c
pWW




. If
()c


, the
competitors bid uniformly in the range
[0, ]V
and get zero surplus since
0
is in the support in the
regular all pay auction and linear cost function. If
( ) , 2
k
ck
k



, then
1
12
() ()
k
pW W k




and the maximal effort is determined by that
max
0
( ) 1p




. Notice that
similar to the pure strategy equilibrium, increasing
W
increases expected effort and zero is not
in the support for a convex cost function.
Similarly, an approximation to this equilibrium persists for a symmetric imperfect principal's
informativeness regarding who bids most. For the sake of simplicity and ease of exposition,

21
assume that the for a bid difference less than

, the principal has a 10% probability of being wrong.
We can write the expected utility of each competitor as,
1 2 1 2
0
12
( ) (( ) ( )) ( ) 0.9 (( ) ( )) ( ) 0.1 ( ) ( )
0.1 (( ) ( )) ( ) 0.9 ( ) ( ) ( ) ( )
E W W c p x dx W W c p x dx c p x dx
W W c p x dx c p x dx c p x dx
   
   
   
   
    
  





      
    
  
  
Assuming that
()p

stays the same as in the complete informativeness case, if the principal's
informativeness is symmetric, as assumed in the equation, then
()E

is similar to the complete
informativeness case for small

. Since the logistic distribution is practically symmetric for high
informativeness,
()p

indeed stays almost the same.
We could compare this mixed strategy equilibrium of complete or incomplete informativeness to
that of corollary 1b where the competitors don’t know who they are competing against. The same
could be applied here. Each competitor is 'competing' against a distribution of the population which
is identical to the distribution of the mixed strategy equilibrium (since any distribution works for
corollary 1b we can 'borrow' the population distribution of the all pay auction to that of the pure
strategy equilibrium). This means that each distribution has what we might call a "population cost
function", since for different distributions the average and variance effort are different.
But each competitor has his own unique cost function. To find out how much effort to exert
optimally, each competitor maximizes his own surplus or profit, which is
12
0
max ( ) ( ) ( )
competitor
W W p x dx c


  

or
12
( ) ( ) ( )
competitor
c W W p

  
. Thus, we have an
interesting interpretation of this mixed strategy which is actually a first best pure strategy by each
competitor.
25
3.3 Risk aversion
25
This pure strategy interpretation of mixed strategy is somewhat different than those that appear
in Rubinstein (1991).

22
Corollary 1b: reaching near first best equilibrium requires higher wage gaps or higher
informativeness for risk averse competitors. Informativeness is more restrictive if
**
0 1 2 0 1 2 0 2 0 2
0.5 0.5 ( ) ( ( 0.5 0.5 ( ))) ( )w W W C U CE w W W C w W U w W

          
.
Summary of proof: first order condition for risk averse competitors indicates, that effort is smaller
for given wage difference compared with risk neutrals, see appendix 4 for details and an example.
Increasing informativeness increases effort according to the risk averse first order condition.
Participation constraint for multiple identical competitors is
**
0 1 0 2
11
( ( )) (1 ) ( ( ))U w W C U w W C u
nn

        
. That means that risk averters participate
only if wage difference is relatively small or if
u
is negative.
Clearly,
**
0 1 2 0 1 2
( 0.5 0.5 ( )) 0.5 0.5 ( )CE w W W C w W W C

      
, which is the right term of
the no shirking condition for risk averse and risk neutral respectively. If
0 2 0 2
()U w W w W  
,
which is the no shirking condition's left term, it is more restrictive for risk averters. If
0 2 0 2
()U w W w W  
, and the following holds
**
0 1 2 0 1 2 0 2 0 2
0.5 0.5 ( ) ( ( 0.5 0.5 ( ))) ( )w W W C U CE w W W C w W U w W

          
the no
shirking condition is still more restrictive. Q.E.D
If the spread is high enough, the competitors are less willing to participate than risk neutrals and
so must be given higher sum of wages. But to avoid a weak player from exerting minimal effort,
the principal cannot give too much rent to the competitors.
3.4 Cooperative effort
In the next corollary, we add a cooperative or damaging effort.
( , )
i i j
qf


,
0, 0
ij
ff




,
i

and
j

are individual and cooperative or damaging efforts. If the effort vector has a cooperative
dimension,
0
j
f



.

23
Corollary 1d: if both
W
and the chosen winner is determined by the size of the competitors'
effort vector,
*
j

is positive in equilibrium (the production function is borrowed from Stefanec
(2012)).
The principal has to be able to observe a signal of the size of the effort vector to determine the
winner's identity. Seeing a signal of output is not enough since part of the output is due to the effort
of the other competitor.
26
As noted by Dye (1984), the fact that the principal compares the size of the competitors' effort
vector means, that informativeness decreases with the increase in the effort vector dimension. That
also means, according to (2), that wage gap increases if the principal wants to sustain first best
equilibrium effort. All other things equal, non-individual effort increases rather than diminishes
the wage spread contrary to Lazear (1989), when the principal can take it into account.
27
It is noted in footnote 18 that the principal allowing damaging opponents near equilibrium may be
one of the explanations to the kink in the competitor's cost function. If differentiated punishment
of unfair competition is exercised, i.e. less near equilibrium and farther apart from equilibrium,
26
Stefanec (2012) analysis states that the first order conditions of the incentive problem are
,
P C P C
WW
   
   
   
   
. Differentiating both equations with respect to

and cooperative

we get
2 2 2 2 2 2
2 2 2 2 2 2
**
0, 0
C P C C P C
WW
WW
P P P P
   
     

   

     
 
 
     
     
   

   
for two
identical competitors. However, separating the incentive problem to two parts assumes that the
principal doesn’t take into account total effort size. Thus, if the spread is too low the competitors
will exert smaller than first best effort. For example, when
,
ij
f f C C
   
   

   
the optimal spread
is zero and the competitors will not exert any effort.
27
In Lazear (1989) the principal doesn't punish for damaging opponents. If he did, hawks would
not have won tournaments so easily in his model, damaging opponents unfairly would have been
used more scarcely and the optimal wage gap would have increased.

24
and if the competitors rationally utilize it only near equilibrium, the Lazear (1989) result of smaller
optimal wage gaps in the presence of adversarial misconduct could still hold, also because the
principal wants to control it by lowering the wage gap in order to lower it thus gaining maximum
efforts from the competitor's.
4 Extension
4.1 Prizes that depend on actual exerted effort
A mechanism of prizes that depend on total output could deter collusion to decrease output and
alleviate misevaluation of the competitors by the principal. The contract specifies
*
( , )Wk
, a
fixed rent that the principal gets and wages that depend on total exerted effort. This kind of
tournament should not be applied when sorting is more important than production and so first best
*
k
is likely to be small (the variance in the transformation of efforts to output is high) compared
with a regular tournament.
28
It is not preferred also when the competitors are risk averse. Risk
averters also prefer that the principal pay would depend on total effort exerted, which is not
analyzed here. The next proposition proves that
W
is smaller for this type of tournament with
identical competitors.
Proposition 2a: the wages for the winner and the losers are
2*
12
*
( 1) ( )
n
QnC
Wn n n
P




   


,
*
22
*
( 1) ( )
n
QnC
Wn n n
P




   


if
12
( 1) 1 n
W n W Q

     
.
29
30
Proof: each competitor solves
1 1 2 2 1 2
max * ( .. ) (1 )* ( .. ) ( )
in n i
p W p W C
      
        
.
28
If the principal is risk averse he bears the risk of
()AP


since production is at least
partially unknown in advance.
29
Gangadharan et al (2013) proved for two identical competitors in the basic LR setup that wages
are reduce by 50 percent, with a different kind of proof.
30
The tournament described here is somewhere between a linear piece rate and a regular
tournament.

25
Foc is
12
(1 ) ( )
i
i i i
W W P
P P W C

  
   
    
  
. The wining probability is
1
Pn

.
By solving the following two equations similar to that of fixed prizes one gets
12
,WW
.
12
12 12
*
( 1)
11
( ) ( ) ( )
n
i
i i i
W n W Q
W n W P W W C
nn



  

   

   

   
  

**
12
12
( 1)
1 ( ) 1
nii
WW
n
Q n C n
WPP
n n n n
   









   


and
**
12
22
( 1)
1 ( ) 1
24
nii
WW
n
QC
WPP
n n n n
   







   


.
The derivative of
12
( 1)
n
Q W n W

   
which is a constant is zero.
First order condition gives
12
( ( 1) )
n
ii
Q W n W

   


. Plugging
n
i
Q



in
12
,WW
gives the result.
Notice that second order condition is identical to (2) when
n
Q
is linear in efforts since the first
two SOC terms cancel each other. Q.E.D
Wage gaps are smaller by
1
100 n

percent than those of fixed prizes since increased individual
production increases prizes by
1
100 n

percent. Smaller wage gaps mean that the no shirking
condition is more restrictive.
31
31
With two identical competitors the no shirking condition,
*
*
( ) (0)
2
WCC


is equivalent
to
*
*
*
()
4( ( ) (0))
PC
CC






compared to
*
*
*
()
2( ( ) (0))
PC
CC






in the fixed prizes
tournament, and so it is more restrictive. Thus, maximum informativeness is reduced by half.

26
For example, when
*12
()
2
Qk



, first order condition
*
**
_
1 1 1
()
4 2 2
fixed wages
P
k W C k




   

is identical to (2) or
*_1
2
fixed wages
PWk




.
That means that
*
P




is identical to that of the regular tournament.
But as (2a) is more restrictive in terms of maximum informativeness and as
K

is a priory small
compared to fixed wages, first best
*
k
and
*
P




is expected to be smaller. Thus, actual wage gaps
are higher than those implied by the proposition which means that the advantage of this tournament
is smaller than indicated with regards to decreased wage gaps.
32
Corollary 2a: if
12
0
()WW



, first order condition of the incentive problem is identical to the
case when
12
0
()WW



.
Proof: the competitors solve
12
( ) 1 2
max (1 ) ( )
WW PW P W C

  
. The FOC of the competitors'
incentive problem
*
*1 2 1 2
()
( ) ( )
PWC
W W W W





  

   
    
or
*
*()
PWC





 

.
If an increase in the sum of wages increases effort and the marginal cost curve is convex, increasing
wage gaps will have a smaller impact on effort (but not in the linear case) and since increased
effort is translated to increased output the difference should be noticeable. Still, past empirical tests
reveal that increasing the sum of wages hardly increases effort. See footnote 44 for more discussion
on this issue.
4.2 Misevaluation of abilities by the principal
32
Another potential disadvantage of this tournament is that the principal who sells the output,
may be able to cheat the competitors regarding the quantity sold.

27
Definition: the tournament designer misvalues identical competitors' abilities when estimating
their cost function as
()Cq
while
()Cq
is the actual cost function. If
( ) ( )C q C q q
they are
undervalued and vice versa.
Proposition 2b: the following result assumes ignoring surplus changes due to informativeness
adjustment: if the principal overvalues (undervalues) the competitors’ abilities he losses (gains)
and the competitors gain (lose) overall. Efficiency is reduced if
Q
is not additive and linear.
33
Proof: when the competitors are overvalued, actual effort exerted is lower than anticipated effort,
or
*


. As the principal is contractually obliged to
*
k
,
*4
P P K








and (2)
holds. The competitors exert effort
1f
C








, that is not first best as
*
( ) ( )CC



unless
*
ff






which happens when
Q
is additive and linear. Accordingly, ex-post
AP K


is lower than ex-ante AP for overvalued competitors. The principal loses,
1
1*
0
1
()
()
( ( ) ( )) ( ( )) _ __ _
()
f
fC
C
f
C C d C d initial surplus party share
f
C




    











  





   










The above is easily visualized in the following graph. The described size is the surplus that is lost
due to the principal's misevaluation.
34
Q.E.D
33
A decrease (increase) in equilibrium effort for given
*
k
decrease (increase) the accuracy of the
winning probability according to (1).
34
The competitors' initial surplus share is the part they get from the perceived tournament surplus
1()
*
0
( ( ))
f
CfCd













. Similarly, if the principal undervalues the competitors'

28
Graph 3: Gain and loss from overestimation for additive joint output
Taking into account also surplus decrease due to deviation of informativeness from its optimal
value, the principal prefers to estimate the competitors as less able if he still earns a positive sum.
Obviously, the principal gains less than what is depicted in the proposition and he may even lose.
The same applies to the competitors.
An unbiased regulator or an arbitrator may help reduce misevaluations. The game between the
principal and the competitors is not formalized here. Notice, that underestimation of the
competitors is somewhat reminiscent of a biased belief perception described in Heller and Winter
(2019).
35
Now we can formulate the following corollary.
Corollary 2b: for tournament with high informativeness relative value, misevaluation is
undesirable for both the principal and the competitors.
abilities
*


,
*
1
2
k
C





, for the additive output, portrayed in the graph. The competitors
lose what the principal gained in the reverse case.
35
The difference is that here the bias is in the characteristics of the players and not their strategies.
The reader may also be interested to know that the distorted perception of the competitors'
strategies described in this model cannot fulfill their theory's requirements of monotonicity and
continuity, as we are dealing with a stochastic distortion here, whereas their distortion is
deterministic in nature.

29
It would be interesting to analyze what happens when we relax the assumption that the contract is
fully enforceable. Assume
*12
()
2
n
Qk



and underestimation occurs, the principal lowers
k
, if the costs on monitoring were allocated in advance, since
**
( ) ( )kk
     
    
. Since
W
was planned for weaker competitors and
k
decreases and there is common knowledge that
this happens, the two factors that determine effort in (2) has opposite effects. Effort is not first best
but could be close enough. If overestimation occurs,
k
increases. Since
W
was planned for
stronger competitors, the two factors that determine effort in (2) has again opposite effects.
4.3 Unenforceable contract
In the following proposition the enforceable contract assumption is relaxed so that the principal
can shirk. In order to see what is the minimal rent that the principal gets from his option to shirk
we assume that the competitors set the terms of trade.
Proposition 3: Given competitors strengths,
1. First best
**
( , )k

is attainable if
*
kk
.
2. If
*
kk
, only second best
** **
( , )k

is attainable.
3. the principal may get a bigger part of the tournament surplus

compared to the enforceable
contract case.
Proof: Naturally,
* * * * *
ˆ ˆ ˆ
ˆˆ
( ( ), ) ( ( ), ) 0 ,k k k k k k
     
    
, i.e, tournament surplus decreases
when supervision deviates from its optimal value. From the competitors' first order condition (2),
**
0Kk
K

  

for given
W
.
If
*
kk
, we already know that the marginal cost function has a kink near equilibrium to maintain
equilibrium, and so it is steeper. Assume first that
()C


is linear with a kink. The function
()fK


is set according to (2)
2
( 1) ()
kn WC
n


 
, and so
*
K



equalls a smaller constant
near equilibrium. This means that the competitors' marginal cost of principal's deviation is

30
increasing with the principal's level of deviation. Note that first best effort doesn’t depend on
k
.
36
Ignoring loss from a decrease in effort, since
*
f AP AP Kk
AP K AP K

   
  
   
, if the principal gets full
rent, he doesn't deviate. Otherwise, lowering
*
k
decreases his cost by more than his loss of rent
because he gets only part of an incremental increase
*
*
k
f APdK
AP K
k





and bears all the cost of that
increase in term of informativeness cost (the benefit from deviation decreases with the principal's
rent share). Ignoring loss from a decrease in effort is possible if it is a sorting tournament where
the surplus from producing is small.
First best is attainable when
*
kk
because both parties lose by a principals' infinitesimal
deviation as a decrease in production is a major part of the surplus.
But even if sorting is the sole source of surplus, the principal deviation is partial. Thus, in any case,
the principal doesn't get all the surplus. In graph 2, for example, the horizontal curve
f
AP


, is
lowered, which lowers also the optimal
*
k
.
To prevent deviation by the principal from second best, the rent given to the principal guarantees
no deviation by equalizing his marginal benefit from further infinitesimal deviation with his
marginal cost from a downward deviation in monitoring.
37
The optimal deviation from first best
36
If
()C


is convex or,
1
( ) , 2
l
C a l



,
1
1
2
( 1) l
W K n
an


  



. For example, for
3l
,
2
1 ( 1)
2
Wn
K K an

  


. The marginal cost of deviation increases with the size of the deviation.
37
The competitors can reduce the principal's shirking in the following cases:
1. The principal knows that the probability that he will be caught cheating increases with the
magnitude of deviation, there is a punishment for being caught and cheating is costly.
2. The principal knows that the competitors can lower his utility by shaming him for deviating.

31
is such that infinitesimal deviation is neither beneficial nor harmful to the competitors. But finding
the optimal second best is not easy because high candidate for second best requires giving more
rent to the principal and a low candidate for second best generates less tournament surplus. Q.E.D
The lower
*
k
is the benefit to the principal from deviating is lower also because the cost to the
competitors is higher (there is no kink, or the kink is less steep). If the principal can convince the
competitors that he benefits from retaliation by lowering monitoring, in case of disagreement, he
may extract more rent.
38
Appendix 1 shows that first best monitoring for monitoring averse competitors is set according to
(5)
1( , )
( , ) ... n
C AP
C AP
f AP AP AP AP
AP K AP K AP K AP K





     
    
       
. Wage difference,
W
is set
according to (6)
**
*
*
( , ) ( , )P C AP C AP dAP
WAP d


  

  
  
  
to ensure first best equilibrium
effort. When the principal can shirk the optimal supervision is set by (5), or lower. It could be
lower when
*
kk
as discussed above. First best
**
( , )k

is attainable for a lower principal's
rent here since (5) guaranties that
**
( ) ( )
fkk
AP AP




.
39
3. The competitors incur a disutility from participating in a tournament that runs frequently if
past deviations occurred.
38
In, Bental and Demougin (2006) increased monitoring increases the principal's share in a non-
tournament setup; in, Bental et al (2012) increased monitoring reduces, rather than increases the
agent's effort.
39
The minimal rent that guaranties that the principal doesn't deviate downward ensures that he
doesn’t deviate upwards as the principal's optimal deviation is a monotone function of the
competitors' share of total rent.

32
Since equal division of rent between the principal and the competitors is a Nash bargaining solution
when both parties have similar risk characteristics and disagreements points, the above proposition
suggests a source of bargaining power in favor of the tournament designer.
4.4 A dynamic analysis based on the theory of moves
Under proposition 1a, identical competitors are unable to cooperate in exerting effort below first
best unless the OVZ condition doesn’t hold. This is not true according to analysis by the theory of
moves (see Brams and Ismail (2021) and the references therein for a detailed account of this
method, which is only briefly explained here). Assume that there are finite number of effort
strategies and as prescribed in the above model, the principal cannot refrain from payment if the
competitors exert lower than first best effort.
Starting with two effort strategies. Assume that the competitor exerting eventually high effort,
identical to first best, wins with 0.9 probability if the other is exerting a low effort and that exerting
high effort is beneficial as long as the other player exerts low effort. If both competitors start in
the low effort state, either by pre play communication or due to past experience, the players will
stay in this state. This is so because if a player is considering moving to the high effort state, he
knows that the other player will do it as well to avoid losing with high probability. In that case,
both will get the same reward and exert higher effort which is not profitable for either of them.
Thus, we have what Brams calls a non myopic equilibrium, which is not a Nash equilibrium either.
If both start in the high effort state, it wouldn't be beneficial for either of them to move to the low
effort state because the other competitor will not follow.
The other option involves the case in which only one start in a high effort state. If the high effort
competitor moves first, we end up in a low effort equilibrium. Whether this is the case depends on
nuance factors such as how long does it take for the principal to detect if one of the competitors
exerts high effort and the other doesn't. If both competitors have enough time on their hands
without worrying about the principal's monitoring at the, they may end up in the low effort
equilibrium. On the other hand, if the low effort competitor is afraid that the principal will notice
the difference soon enough and the other competitor is not in a hurry to decrease his effort,
equilibrium may result in high effort for both of them.

33
Extending the analysis to multiple efforts, where the principal can devoid payment bellow a certain
threshold equilibrium may result in that effort, which is naturally assumed to be below first best.
This is certainly true if both start at that threshold effort. The analysis is similar to the two efforts
case if both start at different levels of effort or if both start at first best effort.
5 Discussion
The principal's maximum informativeness may reside only in the region of anticipated planned
equilibrium as is elaborated in appendix 2. Distribution functions that match those criteria are, for
example, the triangular and the beta distribution. The propositions and corollaries with
modifications (except 2c) in this model are less valid as the maximum informativeness attribute is
not kept outside equilibrium, the difference attribute is not maintained, and they are unfit for non
identical competitors. Thus, they lose power if the principal has misvalued the competitors or if
the competitors don’t have complete information about their rivals, they are not stochastically
derived, proofs are less elegant and reaching further conclusions is more difficult or impossible.
This model applicable also for purchased output when money and effort can buy output in a
competitive market under certain conditions. If effort is used to buy output for a given budget and
people are subject to fatigue, convex marginal cost of effort applies also here.
Differences, in risk aversion as in abilities to damage opponents, if they indeed exist between men
and women may explain the lack of desire for women to compete and succeed in tournaments.
40
41
42
Rational doves who are not part of a group that can monitor and sanction competitors for
40
Dato and Nieken (2014) showed in a controlled experiment that women harm opponents less
than men even when they are equally able to do so. This may be due to greater harm exerted against
women if women utilize their damaging potential.
41
Lazear and Rosen (1990) claim that less presence of woman in management is due to their
comparative advantage in other professions compared to men. Only when women have superior
management abilities relative to male peers they prefer to engage in this trait. Niederle, and
Vesterlund (2007) research lab find that men are more prone to compete and that they are over
confident about their abilities compared to women.
42
When competitors' abilities are known to the competitors group and gaining unfair advantage,
by damaging opponents, is possible for “hawks”, all the hawkish contestants, possibly like past

34
damaging opponents are less inclined to participate. Repeated tournaments may yield better
informational results than a single tournament, but they are not always possible nor desirable, due
to time, cost constraints and negative side effects wielded by and on the participants, some more
than others.
6 Conclusions
The analysis in this paper may further explain the cost of using tournaments and partially explain
why they are prevalent.
43
Establishing the validity of tournament theoretic predictions in sports or
in other environments which are not hierarchical businesses is easier since the economic
environment is simpler.
44
Sunde (2003) do it with professional ATP tennis tournaments and
Ehrenberg and Bognanno (1990) with professional PGA golf tournaments. Accordingly, Main et
al (1993) and Erikson (1999) backing up predictions of tournament theory that higher wage gaps
lead to higher employees' productivity and business results ignore the fact that better business
results may reflect also an imperfect competitive market that is characterized by hierarchies with
events of the Tour the France, will do so to their rivals if they believe ex-ante that it is worthwhile
(it costs less for hawks to inflict harm on their rivals than to doves. see Lazear (1989) for a formal
definition).
43
Pendergast (1993) show in his article, that higher wage for a more demanding job, could be
explained as a mechanism for incentivizing employees to acquire skills that could not be
compensated directly. Tournaments may do the same. Pendergast claims, though, that tournaments
suffer from the severe competition embedded in them, whereas the mechanism that he describes
doesn't.
44
There is an identification problem in sports tournaments since bigger tournaments offer both
higher total prizes and increased differentials. As noted by Bognano (2012), the correlation
between the difference and the sum of pay is .97. Knoeber and Thurman (1994) tested to see if
competitors' effort in tournament are not affected by the sum of the prizes. They could not refute
this hypothesis (p-value of .22). Eriksson (1999) also found that there is only a weak connection
between the sum of prizes and effort. Thus, it is possible to conclude that wage gap incentive is
stronger than total higher wages incentives.

35
wage structure that is compatible with tournament models. This, considering other forces that
stress the efficiency of more equal pay within an organizational unit.
45
46
47
The seed of this paper is from part of my master thesis. Benjamin Bental's guidance and help is
appreciated. Remarks from Dan Peled, Anna Rubinchik and Todd Kaplan are also acknowledged.
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Appendix 1

38
If social imperative constrains the supervision level to be higher than the optimal, the tournament
attainable second-best effort,
*1
*
f
C








, is different than first best effort, according to
(4), since
f
AP AP




. Note that According to (2) optimal wage gaps increase in
*

and decrease
in
*
k
. These are two opposite effects on the wage gaps.
For competitors who are averse to the level of monitoring intensity first best informativeness is,
*1
argmax ( ) ( , ) .. ( , ) ( )
Kn
k f AP C AP C AP AP
  
    
.
(5)
1( , ) ( , )
... n
f AP C AP AP C AP AP AP
AP K AP K AP K AP K
  
       
    
       
. From here,
f
AP AP




and
so
**
monitoring_averse monitoring_neutral
kk
. According to (3) first best effort for monitoring averse
competitors,
*1
*
f
C









, is bigger than that of monitoring neutral competitors.
48
First order condition of the incentive problem for monitoring averse competitors is
(6)
**
*
*
( , ) ( , )P C AP C AP dAP
WAP d


  

  
  
  
. Since we have,
*
( , ) 0
C AP
AP



and
*
dAP k
d


the marginal cost function is steeper and so,
**
monitoring_averse monitoring_neutral


for given wage
difference.
48
When the competitors' cost function is
( , )CW


and
*
( , ) 0
CW
W




the competitors prefer
higher first best
k
compared with the normal case since increasing
k
lowers
W
according to
(2).

39
Proof of proposition 1b: From the multi logistic function
2
*
( 1)P K n
n




. From (2)
2
*
*
( 1)
f
n
WnK




and so,
*2*
**
() ( 1) ( 1)
ff
nn
P
U d dp
n K n K



 
 




  

. To calculate
*

we
compare marginal cost with marginal utility at first best,
*
1
() f
l
Ca
 




. Thus,
**
*
0
1
1111
1
( 1)
l
flfl
fa
a
na
l
ad
n K l








 




. From here we get that
*
1
11
( 1)
thr
fl
nl a
kn









. So,
1
*
f
C








is not an equilibrium for
*
1
11
( 1)
fl
nl a
kn









.
49
49
If we increase
k
by

and adjust
W
to be
2
*(1 )
( 1)( )
f
n
WnK





 
,
0


. We get
*
( ) (1 )
f
C





according to (2). If
*
1
11
( 1)
fl
nl a
kn









 
the increase in effort
produce equilibrium with higher
thr
k
. But,

40
With
2n
and identical competitors, first best equilibrium is unique since
* * *
1 1 2 2 2 12
** *
12 2
12
( ) ( ) ( )
... ...
nn
n
C C C
WPP P
     
 
 
  
       
 
 
is impossible. Q.E.D
The following explain the following graphs.
2
2 1 2, ( ) 0.5 , ( ) 0Q C K
    
   
, first best effort
is
*1


. From proposition 1b
4
thr
k
is the maximal informativeness to reach first best
equilibrium effort. In graph 4a since we cannot see the mass of the wining probability when
exerting zero effort, the area below marginal cost is bigger than the area above it. When
2
( ) 0.6C


,
4.8
thr
k
and
*0.83


. With
2
()C


,
6
thr
k
. One can see in graph 4b
that the area below marginal cost equals the area above it. This is so as the probability of winning
by exerting zero effort is zero.
**
_
11
11
11
(1 )
,,
( 1) ( 1)
new thr
ff
ll
n l n l
aa
k l n
nn




   


    
   
   

   
  

in the specified
range.

41
Graph 4a: maximal supervision for first best equilibrium effort in the logistic function for differ
rent pairs of linear marginal costs and critical thresholds
Graph 4b: maximal supervision for first best equilibrium effort in the logistic function for quadratic
marginal cost
Appendix 2
A different definition of the WPF for multiple identical contestants.

42
1) The density function of the WPF,
 
*
/
i j i
P





, presented to the competitors is conditional
on the rivals exerting equilibrium effort.
2) The density function reaches its peak at planned equilibrium effort.
A triangular density function that raises up to equilibrium and diminish thereafter satisfy those two
requirements.
Equilibrium for identical contestants.
The equilibrium is IEDS because if the rival exerts effort
*
ˆ


, exerting
* * * ˆ
     
   
is a best response, see graph 3 for visual aid with identical competitors.
When the rival exerts effort
*
ˆ


, exerting
* * *
ˆ
     
   
is best response.
50
A
similar process to that presented in proof of proposition 1a leads the competitors to exert
*

.
Graph 3 also shows that for
*thr
kk
the no shirking condition is not held with the triangular density
and so a Nash equilibrium doesn't exist.
50
**
12
( ) 0.5P
  

, and is calculated based on
*
1 1 1
0
P P P
d


  
  

  

. The expression for
1
P



is for
*
2


. Notice that the distribution is no longer symmetric in this case.

43
Graph 5: Proposition 1 with a triangular density
Graph note:
W
in this graph represents the first order condition ratio
C
WP



.
Calculating equilibrium and wage difference with a triangular WPF.
There are no monitoring cost,
22 12
12
, ( ) , ( ) (1 ) , 0
2
a
Q C C a


      


     
.
When
0


marginal cost is linear,
0


creates convexity in the marginal cost function. Surplus maximizing effort is
1
*12
( (1 ))a




. Triangle area formula leads to
**
11
1 2 1 2
1
11
( (1 )) ( (1 ))
22
PP
aa

   





      

. The slope of
1
P



is
12
12 12
1
12
( (1 )) ( (1 ))
( (1 ))
aa
a








.
Thus,
2*
12 1
1
( (1 ))
Pa

   


    

when
P



starts from the origin.

44
*
1
12
1
12
1( (1 ))
( (1 ))
C
Wa
Pa











    


. From here we check that
2 1 1 1 2 *
1 2 1 2 1 2
( (1 )) ( (1 )) ( (1 )) (1 ) ( )
P
U W a a a a C

  
         



  


            

, if
1a
using the fact that
1
*12
( (1 )) 1 1aa



    
. Thus, all the conditions in
proposition 1a hold.
When
0


and both sides don’t earn rent
12
( ) 1
2
WW
C


  
. Solving,
2*2
2
a
a


  
.
*
μ=μ
2
Pa



by triangle formula. The slope is
2
0.5 0.25
2/
aa
a
and
24
2
a
Ca
WPaa



   


. In this
case the wage difference is 4 times that of first best effort.
Proof for corollary 1c (for given
12
,WW
and identical competitors
**
13
NNN


  
.
12
,WW
is the
winner and loser's prizes) based on the triangular distribution for multiple contestants.
Proof:
12
,tt
is the area beneath the WPF density to the left and to the right of equilibrium. For
identical competitors
11
tN

and
*
1
0
P
td






. Even assuming
**
1NN



, it must be that
*
1
( / ) ( / )
i i i i
NN
PP
     




  

since
1
( / ) ( / )
i i N i i N
PP
    
  

. As
W
is a constant
11
**
NN
NN
PP
U W U W
   






    

and so equilibrium effort must decrease because
**
( ) ( )UC



in equilibrium.
Appendix 3
Exemplifying how dominant strategy Nash equilibrium is reached in the logistic WPF.

45
The competitor incentive problem is
()
PWC





. If
22
12
( ) 0.5 , ( ) 0.5C a C b b a
   
  
the
incentive problem for competitor 1 is
21 1
2
21
exp{ ( )}
(1 exp{ ( )})
W k k a
k
 

    

. After algebraic
manipulation, similar to that of footnote 10 in Hirshleifer (1989), one gets the following equations
1 2 1
2 1 2
2cosh{0.5 ( )} 2
2cosh{0.5 ( )} 2
k a W k V
k b W k V
  
  
     

     

This system of equations doesn't reveal the fact that no equilibrium exists above
thr
k
because it is
based only on the incentive problem of the competitors.
Graph 6: reaction curves of identical and nonidentical competitors
The horizonal curve is the best response of competitor 2 to competitor's 1 effort. For the parameters
1, 2bk
,
*42Wk
  
and
0.67,0.909,1,1.1,1.5a
the vertical reaction curves of
competitor's 1 are drawn.
51
If competitor 1 is weaker (stronger) equilibrium effort is decreased by
8% (16% ) when the ratio of the opponents strength parameter is
weaker
stronger
1 0.67 2
1.5 1 3
a
a  
.
51
A possible MATLAB command for the identical competitors' reaction curves is >>
ezplot('2*cosh(x-y)*x^0.5-2',[0 2 0 2] ); hold on >> ezplot('2*cosh(y-x)*y^0.5-2',[ 0 2 0 2]); X-
axis stands for
1

and Y-axis for
2

.

46
To see convergence to iterated elimination of dominated strategy equilibrium start with competitor
1 exerting
10


. Competitor two reacts optimally with
20.63


. When
20.63


competitor
1 reacts optimally with
10.9


. When
10.9


competitor 2 reacts optimally with
21


. If
competitor 1 starts with
12


the optimal response is
20.1


and convergence is similar.
Appendix 4
The risk averter competitor incentive problem is
(1) (1 ) (2)P U P U   
.
The abbreviations are,
12
(1) ( ( )), (2) ( ( ))U U W C U U W C

   
.
First order condition is,
{ (1) (2)} { (1) (2)} (2)
PU U P C U U C U

    
       

.
Optimal effort for non-identical competitors is
*
()
PU
CU








, Which extends the
identical competitors condition in LR. The symbols,
U

stands for
(1) (1 ) (2)P U P U

   
, and
(1) (2)U U U  
.
Thus, optimal effort for identical competitors is
0.5
() KU
CU




.
This means that when
U
is high
(2)U
is also high and effort is low. Thus, increasing wage gap
doesn't increase effort as in the risk neutral case.
For example, with a logarithmic utility function and,
12
, ( )QC
   

  
, we get the winner and
the loser utilities from proposition 2(a),

47
*
01 *
*02
**
0 1 0 2
*
12
()
ln ()
( ) 0.5 11
( ) ( )
ˆ2
w W C
w W C
Ck
w W C w W C
WW


















   

  


Since first best equilibrium is not likely to be feasible, it is possible to start with a lower effort and
check weather SOC hold. With
0 1 2
( , , , ) (3,1,1.55,0.15)k w W W 
*(0.88,0.972)


for the
logarithmic and square root utility respectively. Note that in order to reach a near first best
equilibrium effort of 1, the wage difference is 1.4 compared with 1.33 in the risk neutral case.
Decreasing
(2)U
could be achieved by increasing initial wealth, the losers wage or increasing
the sum of wages. It would be interesting to check the change in effort with respect to initial wealth.
22
2
0
(( (1)) ( (2)) )
()
U U U U U
w U U
  
     




  

.
The denominator is positive. The left and the right terms in the numerator term are negative so the
effect could be both negative and positive. But if
(2)U
is sufficiently small the expression is
positive. Increasing the sums of wages for the purpose of increasing effort poses a difficulty since
it increases the incentive of the competitors not to exert effort at all.
Second order condition for non-identical competitors is,
22
22 ( )
PP
U C U C U C U


     
      

, and for identical competitors,
2
( ) 0C U K C U C U
     
    

.
0U

and so
2
()C U C U C U C U
KC U U C U
       

  
    
  
   
.
So, there are two limits on maximum informativeness (this limit was much less restrictive in the
examples than the no shirking condition).