ANAND J. SHUKLA - UCSB's Department of...

TOURNAMENTS: THEORY AND EVIDENCE

ANAND J. SHUKLA

Introduction

In this paper, I synthesize the literature in tournaments and present essential

theory and evidence. A tournament can be described as a game where there are

prizes and the participants are measured according to their relative rank among

other participants and the prizes are awarded accordingly. A simple example from

a personal experience is the case of a clothing store having a contest among its

sales representatives to sell pants and the one who sells the highest number is

awarded a prize. Of course, the applications of tournaments spans far beyond

just selling pants and has some very important bene�ts.

In this paper, I study the e�ciency of tournaments as pay schemes compared

to piece rates (under di�erent settings) and also as mechanisms to increase e�ort

among agents. Both of these pay schemes come from a root problem and that

is costly monitoring. The problem is that workers may shirk and if monitoring

is relatively inexpensive then the best pay scheme is an input-based pay such as

periodic wage. In many settings this may not be viable and so we must consider

piece rates or tournaments, which are output-based pay schemes.

The principal question: Are tournament based pay schemes e�cient and under

what conditions? This question was �rst studied by Lazear and Rosen (1981),

which has provided the theoretical framework for most of the work that has been1

TOURNAMENTS: THEORY AND EVIDENCE 2

done in this subject. Over the years, there has been much progress on the empir-

ical side of these questions and the results have yielded some interesting policy

implications. I will cover extensively evidence on the basic theory and uneven

tournaments. I will also brie�y cover self-selection, information (or feedback),

risk-taking, and moral hazard, and how these environments a�ect the e�ciency

and performance of tournaments.

Theoretical Framework

The seminal paper by Lazear and Rosen (1981) provides the general framework

for the tournament theory that has been studied in its original settings or vari-

ations of it over the past few decades. They study tournaments and piece rates

(pay based on per unit of output/action) as pay schemes and determine their

e�ciency under di�erent settings (risks and shocks).

Risk-neutral agents

First, consider the case where there are two risk-neutral agents and no common

shocks. There are two cases studied here, one where the workers are paid on a

piece rate, and the other where workers are paid on their relative rank, in prizes.

On the �rm's side, the production is additively separable in workers' outputs and

the manager is risk-neutral. Also, we will assume that the markets are competitive

so we have a �xed price V per unit. The model looks like the following:

The output agent j produces

qj = µj + εj

where µj can be thought of as the individual e�ort, or investment, or a measure

of skill, and εj is the random shock the agent j gets, which is independently


distributed from the shocks that other agents get, as mentioned before. The prize

for highest output is W1, while the second place prize is W2.

Proposition 1: Under risk-neutral agents and no common shocks, piece rates

and tournaments both yield pareto e�cient outcomes.

The proof is given in the Appendix section, but it is rather simple to understand.

The agent and the �rm will maximze their ex-ante utility function and pro�t

function, respectively. It is easy to see that under competitive markets, the price of

output equals the marginal product of labor (this is the piece rate), which is equal

to the marginal cost. Under the tournament case, since the prizes (W1 > W2) are

predetermined, the investment by each agent is also predetermined. The �rm must

set W1−W2 such that the agents play a Nash-Cournot game. The agents simply

maximize their expected payo�s using P as their probability of winning. Also,

since the shocks are identically distributed, the players play a Nash-Cournot game

where each of their investment is maximized at the intersection of their reaction

functions. This intersection corresponds to both of their investment being the

same and therefore the probability of winning for each player to be the same as

well. As mentioned before, the �rm must set the prizes optimally so that such

investment occurs (C ′(µi) = (W1 −W2)g(0) ). Again we have price equal to the

marginal cost.

Risk-averse agents

The above conclusion doesn't hold true under the case of risk-averse agents.

I will not provide an extensive proof. The piece rates do provide a slight im-

provement over the tournaments when there are no common shocks. Intuitively,


risk-averse people generally don't like the bimodal nature of tournaments, espe-

cially when the spread of the prize is large. However, the previous statement does

not apply under certain conditions such as when initial wealth is high and/or

there are common shocks, where relative pay may in fact yield greater e�ort than

piece rates. I will not go into great detail as these results are not fully proved.

Adverse selection

Consider the simple case of two di�erent types of agents, who know their type,

but the principal doesn't:

There are two leagues, a, which yields the most e�cient outcome given the high

ability people, and league b, which yields the highest outcome given low ability

people. The problem is league a is strictly preferred since the revenue function

for league a yields a strictly higher payo�. This leads to the lower type b contam-

inating the league a, which results in a mixed bag of agents competing against

each other. To be precise, these agents don't sort themselves into their particular

leagues. It is intuitive to see why such mixing would lead to ine�ciencies if the

proportion of agents in each league is unequal. One type overinvests and the other

type underinvests.

Remarks and Extensions

In sum, since it may be less costly to measure the relative performance than

individual output, tournaments yield an e�cient solution, where wages are given

as prizes, and not necessarily in terms of the agents' marginal products. There

is one behavioral concern and that is, depending on the risk aversion, the agents

may well be satis�ed with the lower prize under common shocks, discouraged by

their relative chances of winning. The question here is how risk-averse are these


agents and how does it change their percieved costs. This would only occur if

agents didn't fully understand the fact that other agents are of the same type.

Well, it's a bad day outside, I'm not going to be able to sell much, I might as well

not work today and just get the lower prize. By symmetry of both players, we

may actually have a worse outcome than the piece rates.

Another crucial remark is that the tournament is a strategic game that involves

the agents maximizing expected values and �nding Nash solutions. Generally

speaking, this may not be robust in real settings as most people do not think as

strategically as conjectured.

Evidence

Testing the Theory

One of the �rst papers to test Lazear and Rosen's theory was by Bull, Schotter,

and Weigelt (1987). Since the theory required some restrictive assumptions (risk-

neutrality, independent random shocks, etc.), the test was done as a laboratory

experiment. The authors tested both the piece rate scheme and the tournament

scheme using the original model, where the cost function took this form:

C(e) = e2

cwhere c was some positive value that the authors decided upon (2000

for the piece rate, and 10,000 for the tournament for example).

So the parameters chosen by the authors are: c, M, m, and a, where M refers

to the large prize and m refers to the small prize; a is the bound on the random

shock, ε ∈ [−a, a]. Given these parameters, the authors calculate the predicted

e�ort levels (decision numbers) by the theoretical equilibriums. Table 1 in the

Appendix shows the parameters chosen.


The authors did ten experiments with one piece rate scheme and others being

variations of the tournament game. In the tournaments, participants were paired

with each other (but were not told whom they were paired up with), and were

asked to provide a decision number (which can be viewed as a e�ort level) between

0 and 100. After the decision number is chosen, the administrator goes around

with a basket of balls with numbers on them representing the random shock.

That is, the random ball picked by the participant has a number on it between

−a and a, inclusive. The participants then note this number down, put back the

ball in the basket and add the number to their decision number to get the total

output. In the case of the piece rate, the experimenter calculated the exact piece

rate needed to get an average decision number of 37. The experiments are divided

as follows: the �rst three experiments study the tournament setting with di�erent

cost and prize structure; Experiment 1 is the base case with 37 as the predicted

e�ort level. Experiment 2 and 3 study the variations in the costs and prizes to

check for the robustness of the tournament setting itself. Experiment 4 goes over

the asymmetric cost function, where one agent is given a higher cost than the

other Experiment 5 through 8 basically test how the the players use information

(the opponent's e�ort levels) to maximize their payo�s, instead of using strategy

to uncover the opponents e�ort levels (they are playing the �eld in a sense and

not the other players); experiment 10 is the piece rate case and experiment 9 is

simply the experiment 1 repeated 25 times instead of 12 rounds to see if it reduces

the variance.

The results are given in Table 2 in the Appendix. Looking at the piece rate

(experiment 10), one can see that the average is pretty close to the predicted value


of 37, which tells us that the subjects are maximizing their payo�s given the par-

ticular cost function. Under the even tournaments, where the cost functions are

the same for the paired agents, the results show the averages are on par with the

predicted theoretical values. The variance of the tournaments is quiet high, even

under full information (experiment 6,7, and 8), though still statistically insignif-

icant from the predicted value. The full information case gave the participants

full information about the other player's decision number to see if there was any

strategic di�culty in getting the predicted value, but the reduction in variance

was not much, which says this needs further explanation. In the case of the un-

even tournaments, where the cost of one agent is higher than the other's, we see

that the theory underestimates the e�ort level. Both the disadvantaged and the

advantaged exerted higher e�ort than what was predicted. This is another area

of further exploration.


The case of the high variance in this model is reduced dramatically when sorting

is allowed as we will see later. The high e�ort exerted by the disadvantaged group

in the uneven tournament is still under debate. A possible explanation is that

this type of setting (lab experiment given a utility function) may not be the best

at determining worker e�ort when there is cost asymmetry. There might be some

di�culty calculating the cost. A real-work experiment would perhaps yield better

results to this situation.

Uneven Tournaments

This section goes over the particular importance of uneven and unfair tour-

naments. Since most real-life cases are uneven, the results from Schotter and


Weigelt (1992) have much to o�er for policy-making. The authors try to answer

the question: Is there an equity-e�ciency trade o�? That is, by handicapping

the disadvantaged and equalizing the playing �eld, is there any loss to e�ciency?

They found that for the most part there is not a trade-o�.

The authors did lab experiments in the same sense as Bull, Schotter, and

Weigelt (1981). The participants had to choose a decision number between 0 and

100. The uneven tournaments are de�ned as follows:

Ui(p, e) = u(p)− c(e), where p is simply the payment

Uj(p, e) = u(p)− αc(e), where α >1

The payment to agent i is M , if yi > yj + k, and m < M o.w.

where k is a constant that indicates that j is favored if k > 0.

Three cases were studied in the experiments:

Unfair tournaments- α = 1 and k > 0; this can be thought of as prejudice in

the work place, where one person is favored over another.

Uneven tournaments- α > 1 and k = 0;

A�rmative action- α > 1 and k > 0; In this case, we are helping the disadvan-

taged (the high cost agent) by favoring him.

The authors �nd that under unfair tournaments, the mean of the e�ort is higher

than what is predicted for both agents. The pro�ts of the administrator are also

higher, but still less than the symmetric case, when there is no favoring. In the

case of uneven tournaments, where costs are asymmetric (α = 2 and α = 4),

the means are on par with the the theory, though there is a higher variation

among the disadvantaged in the case when α = 4. The later part is particularly

interesting because agents either gave too much e�ort or dropped out completely.


In real settings, the dropping out case is often noted as one of the selling points

for a�rmative action.

These results are then put into the format of a policy. Equal opportunity: In

the case where one agent is favored, this policy aims to reduce such discrimination.

In this case, as shown in Table 3 (in Appendix), tournament output is increased

as equal opportunity laws are imposed. Notice if the discrimination is very high,

then the favored worker exerts a lot less e�ort, which is the main reason for the

lack of e�ciency.

A�rmative action: The case where one agent has a higher cost, the policy

aims to level the competition by favoring the disadvantaged (the high cost). The

results, shown in Table 4 (in Appendix), indicate that the e�ort level actually

decreases for both agents if a�rmitive action is placed, when the cost di�erences

are small (α = 2). These results are statistically not di�erent from the predicted

results. However, in the case when there is a big di�erence among the two agents

(α = 4), the results indicate an increase in e�ort in both agents. In the earlier

analysis, we saw that when there is a huge cost di�erence, many agents tend to

drop out. These results indicate the a�rmative action policy will reduce or even

eliminate this behavior for the disadvantaged.


There is clear evidence that decreasing favoring in the tournament setting in-

creases overall e�ort and output. Similarly, for the case when there is a huge

di�erence among the agents, a�rmative action laws do provide a substantial gain

in output and e�ciency. The authors do not mention it, but it would be interest-

ing to see why the e�ort of the advantaged agents increased in the case of high


cost di�erences, but not in the case of low cost di�erence, when a�rmative action

was put in place.

Self-selection

We learned in the paper by Lazear and Rosen that there is a problem of adverse

selection when agents allowed to choose between two leagues of tournaments.

There has been much research done in terms of entry decisions into tournaments:

Camerer and Lovallo (1989) found that there is excessive entry when they studied

MBA students and asked them to choose between relative pay and piece rate.

Given the sample of MBA students, it's no surprise that they are more competitve

and overcon�dent (and might I add, male dominated). Niederle and Vesterlund

(2007) study the gender di�erences in the entry decisions; they �nd that men

exhibit more con�dence given their abilities, and women tend to opt for piece

rates rather than tournaments.

Perhaps more important and de�nitely more relevant to our previous topics is

the paper by Eriksson, Teyssier and Villevel (2008). Bull et al. (1987) found that

there was large variance in tournament settings in the e�ort levels, and despite

introducing more information such as the opponent's e�ort levels, there was still

much variance (Table 2, experiment 5 and 6). Eriksson et al. has similar setup

to that of Bull et al. except that the participants are allowed to choose between

piece rates and relative-pay. They �nd that those who choose the tournament

provide, on average, higher e�ort. After running the experiments, they also did

a questionaire with a bunch of lottery decisions to elicit the risk aversion of each

participant. They found that risk averse agents tend to choose the piece rate


scheme. This sorting lead to two homogeneous groups, reducing the between-

subject variance dramatically (577 in Bull et al. (Table 2) to 101 in Erikkson et

al., Table 5) in the tournaments.

Remarks

One note on these results is that even though the piece rate and tournament

was designed to yield the same e�ort level and expected payo�, the tournament

yielded a higher e�ort level. It is understood that these subjects are less risk

averse than those who chose piece rate, but what is the explanation for choosing

higher e�ort levels?

Information (feedback)

Theory has shown us that the agents choose e�ort levels ex-ante by the amount

of payment, the randomness, and the cost function, but a revelation of the other

agent's e�ort should not have any e�ect on the decision of e�ort. In this section,

I consider the interim case, when information or feedback is given to one agent

about the other agent's performance. Eriksson, Poulsen, and Villeval (2008) do

an experiment in the lab with similar setup as Niederle and Vesterlund (2007),

where they give agents a simple task to perform, which actually requires some

e�ort. They study three levels of feedback: no feedback, halfway (discrete), and

continuous. They �nd that there is no e�ect on both piece rate and tournament,

when there is feedback (discrete) given, but they do �nd that when continuous

feedback is given, the underdog (the agent lagging behind) actually reduces e�ort.

Gill and Prowse (2009) �nd similar results. They also study in a lab using

a real-e�ort game (on the computer), but the di�erence here is that the two

agents did not go simultaneously, but one agent went �rst and then the e�ort


level given by this agent was shown to the second mover who then decided how

much e�ort to exert. They �nd agents who feel behind exert less e�ort, and call

it disappointment aversion. The opposite is found by Berger and Pope (2009).

They study basketball teams that are behind during half-time and �nd that they

actually exert more e�ort and increase their probability of winning. They also

use a lab experiment to back up this claim.


It is confusing to see how we can end up with both of these conclusions given

lab experiments, but there are crucial di�erences between the two studies and

their methods. As we learned in Eriksson et al. that the timing and the amount

of di�erence matters. Berger and Pope use the halfway time to give feedback (and

study only when teams lag behind a point or two), while suppose the feedback is

given at the last two minutes of the basketball game, and the di�erence between

winning team and losing team is a lot. Again from Eriksson et al. it is shown that

peer e�ects matter when there is continuous feedback, it is easy to see NBA games

have continuous feedback. Also, just the fact that one is in a team should have

some within team peer e�ect, this may be a key di�erence between the results

compared with Gill and Prowse (2009). Lastly, the nature of the tournament

matters: Gill and Prowse study a sequential game, while most studies are done in

a simultaneous format. From personal experience (selling clothes), I can say that

the Gill and Prowse result de�nitely holds, but then again timing, peer e�ects,

and the nature of the tournament matter, not to mention the sample itself (we

have seen risk-averse agents tend to drop out of tournaments).

Risk-taking and Moral Hazard


Bronars (1986) �nds that leading agents choose a low-risk strategies, while those

that are behind choose a high-risk strategy in a sequential. This is quite intuitive,

if one was to think about race car drivers who are in the lead, they tend not to

take any risky moves. This theory is supported by quite a few papers in the lab.

Nieken and Sliwka (2008), however, �nd the opposite, that leading agents do no

play it safe, instead they tend to imitate their opponents. There are explanations

I can think of that could determine most of the di�erence in these two results.

First, there is di�erence between a sequential game and a simultaneous game. The

leading agent again doesn't know the e�ort of the second mover in a sequential

game and therefore could play it safe. If one was to see that the there is a car

very close behind him, then he would drive a bit faster and more recklessly, and

so the distance between the two competitors also matters. Charness and Kuhn

(2009) suggest one application of the Nieken and Sliwka prediction would be on

mutual fund managers, who invest in the same risky assets.

Moral hazard in tournaments is studied mainly in the terms when reducing

the relative performance of the peers increases one's own probability of winning

(sabotage). Sabotaging is a big issue in tournaments, when monitoring is rela-

tively expensive. Due to the negative nature of this subject, it has mainly been

studied in the lab. One exception, however, is Carpenter, Matthews, and Schirm

(forthcoming); they do a real-work experiment where participants are asked to

type and print a paper, stu� it in an envelope, write the name and address, and

then put it in the output box. The sabotaging comes into play when everyone

is asked to do a peer review (how many envelopes? the quality?, etc.). They

�nd that participants do �miscount� the envelopes and do report negatively (all


evaluations were done by administrators also to cross reference). As a result,

there was actually a decrease in the e�ort and the envelopes produced among the

participants because heck, they were being miscounted anyways. In peer review

case, the piece rate performed much better than the tournaments, with far less

sabotage.

Remarks

In the beginning of the last paragraph, I mentioned sabotage as the main moral

hazard, but an additional case (and to the contrary) is collusion. If the workers

collude then again we may have an ine�cient outcome. The collusion can occur

as a way to equalize e�orts and reduce any large penalties to participants. One

example of this is by Bandiera, Barankay, and Rasul (2005); they study a �eld

experiment on a UK farm. The farm originally has the tournament-based pay

scheme and then switches to a piece rate. The fruit pickers who know each other

well tend to pick less fruit so as to not raise the average of the group. The result

was even more severe among the groups, where there was a strong friendship

among the fruit pickers. Collusion, however, is not a big issue as some other

papers point out, when participants don't know each other.

Conclusion

In sum, the theory and evidence are pretty close, however, there are a few

irregularities and concerns yet to be answered. One problem that we got from

our empirical evidence is that there is a huge variance among participants in a

tournament. Introducing information such as the opponent's e�ort level reduces

the strategic di�culty and makes the problem a maximization problem. This

reduces the variance a bit, but not much. Eriksson et al. show that when self


sorting is allowed there is a more homogeneous group and so variance decreases

immensely.

Uneven tournaments can be improved by a�rmative action, but only in the case

when there is a huge di�erence between the costs of the two agents. Equal oppor-

tunity laws are de�nitely welfare improving (favoring any agent is unproductive

and something that �rms should consider anyways).

Entering tournaments is relatively similar to what the theory predicts except

the case when we have overcon�dence (Camerer and Lovallo 1989), which is quite

common among men (Niederle and Vesturland 2007). Risk-averse people tend

to opt of tournaments (this is intuitive) and prefer piece rates. Thus, it is not

surprising that tournaments increase risk-taking (by nature of the game, it induces

agents to take high risk) even for those who are in the lead and have a high

probability of winning (Nieken and Sliwka 2008). The e�ects of information

(feedback) of other player's performance is contradictory. We have that agents

lagging behind lose motivation and reduce their e�orts and we also have the result

that agents increase their e�orts given this feedback. Many things are at play here

and we have identi�ed timing, peer e�ects, and the nature of the game as being

possible solutions to the di�erence in outcomes.

Other extensions include studying risk-averse agents when the distribution of

the agent types is unknown. That is, we may have risk-averse agents being overly

satis�ed with the lower prize money and simply not even competing. The strategic

nature of tournaments in real life can lead to these types of results, where agents

don't necessarily know all the information or simply don't understand how to

use such information. We have seen the e�ciency gains from tournaments and


possible problems such as adverse selection or moral hazard, but it would be

interesting to see how we can having small teams could reduce these problems

while preserving the gains from a tournament system.


References

[1] Bandiera, Oriana, Iwan Barankay and Imran Rasul. �Social Connections and Incentives in

the Workplace: Evidence from Personnel Data� Econometrica, Vol. 77, No. 4 (July 2009a),

1047� 1094.

[2] Berger, Jonah, and Devin G. Pope. �Can Losing Lead to Winning?� unpublished paper,

The Wharton School, University of Pennsylvania, August 2009.

[3] Bronars, S. (1986): �Strategic Behavior in Tournaments� Texas A&M University.

[4] Bull, Clive, Andrew Schotter and Keith Weigelt (1987), �Tournaments and Piece Rates:

An Experimental Study�, Journal of Political Economy 95(1): 1-33. (February)

[5] Camerer, C. and D. Lovallo. �Overcon�dence and Excess Entry� American Economic Re-

view 89 (1999): 306-318.

[6] Carpenter, Je�rey, Peter Matthews and John Schirm. �Tournaments and O�ce Politics:

Evidence from a real e�ort experiment� American Economic Review forthcoming (available

as IZA DP No. 2972)

[7] Charness, G., and Peter Kuhn (2010), "Lab Labor: What Can Labor Economists Learn

from the Lab?" University of California Santa Barbara.

[8] Eriksson, Tor, Anders Poulsen and Marie-Claire Villeval (2008), �Feedback and Incentives:

Experimental Evidence� IZA Discussion Paper No. 3440.

[9] Eriksson, Tor, Sabrina Teyssier and Marie-Claire Villeval (2008), �Self-selection and the

E�ciency of Tournaments�, Economic Inquiry (47): 530-548.

[10] Gill, David and Victoria L. Prowse. �A Structural Analysis of Disappointment Aversion in

a Real E�ort Competition� IZA working paper no. 4356, October 2009.

[11] Lazear, E. and S. Rosen, "Rank-Order Tournaments as Optimum Labor Contracts", Jour-

nal of Political Economy, Oct. 1981.

[12] Niederle, Muriel and Lise Vesterlund (2007), �Do Women Shy Away from Competition? Do

Men Compete Too Much?�, Quarterly Journal of Economics 122(3): 1067- 1101. (August)

[13] Nieken, Petra and Dirk Sliwka (2010), �Risk-taking Tournaments: Theory and Experimen-

tal Evidence�, Journal of Economic Psychology.


[14] Schotter, Andrew and Keith Weigelt (1992), �Asymmetric Tournaments, Equal Oppor-

tunity Laws and A�rmative Action: Some Experimental Results�, Quarterly Journal of

Economics 107(2): 511- 539. (May)


Appendix

Proposition 1: Under risk-neutral agents and no common shocks, piece rates

and tournaments both yield Pareto e�cient outcomes.

Proof. qj = µj + εjwhere µj is the level of investment, a measure of skill and εj is mean zero i.i.d.Under Piece Rate:Worker maximizes:E[rq − C(µ)] = rµ− C(µ)r = C ′(µ), Similarly for �rm V = C ′(µ)Tournaments:Worker maximizes:(P )[W1 − C(µ)] + (1− P )[W2 − C(µ)] = PW1 + (1− P )W2 − C(µ)Assuming Nash-Cournot, symmetry: µj = µk

C ′(µi) = (W1 −W2)g(0),Firms (zero pro�t condition):V µ = (W1 +W2)/2P = 1

2implies the expected utility is V µ− C(µ)

Again marginal cost equals marginal social return V = C ′(µ)

�


TABLE 3

TABLE 4

TABLE 5

ANAND J. SHUKLA - UCSB's Department of...

Documents

Transcript of ANAND J. SHUKLA - UCSB's Department of...