Parham HolakoueePHDBA 279B

Spring 2012

Cognitive Biases and Contestant Over-Exertion

I. Introduction

Cognitive biases can induce deviations from rational strategic behavior in contests.

I am interested in exploring the implications of contest structures in which predictable

deviations from rationality are pervasive and systematic. Equilibrium outcomes in the

contest theoretical literature routinely assume players’ probabilistic expectations of all

relevant information equal the true probabilities. However, experimental research – and

some empirical observations -- demonstrates over-dissipation by contest participants with

effort levels exceeding the predicted equilibrium. Incorporating predictable deviations

from rationality into the theoretical models may provide some insight regarding a

possible source of the discrepancy between equilibrium predictions and experimental

observations.

II. Employment Tournament

In an employment tournament context, the employment contract can be analyzed as

a contest designed by the employer to maximize its expected payoff. The employer will

compensate its employees via a tournament, in which payoffs are contingent on relative

performance if arrangement maximizes expected employer rents.

In this tournament context, the employer (“Organizer”) will set the rules of the

tournament and its employees (“Contestants”) will choose effort levels in response to

these contest parameters. We will assume that the Contestants are vying for a single prize

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that is set equal to the net present value of earning partnership status within the firm.

I am interested in analyzing how the strategy of the Organizer and Contestants is

influenced by a systematic increase in each Contestant’s subjective probability of

winning the contest while the true probability of winning remains unchanged. While the

subjects carry a subjective, irrational belief regarding their likelihood of winning, the

Organizer is aware of both the true probability of winning and of the Contestants’ bias.

A. Excessive Effort

The experimental literature on contests reveals that individuals frequently exceed

the effort levels predicted by theoretical equilibrium analysis (Davis and Reilly, 1998).

These results are even more pronounced in contests with noise. The experimental

literature and empirical observations of effort levels exceeding equilibrium predictions

can possibly be reconciled with the theoretical literature if we take account of systematic

cognitive biases among Contestants. Specifically, overconfidence can be incorporated

into the models to account for excessive effort levels corresponding to subjective

perceptions of the probability of winning that exceed the true probability.

1. Overconfidence

There is substantial psychological research demonstrating the pervasive tendency to

overestimate the likelihood of success relative to one’s peers. Alpert and Raiffa (1982),

Buehler, Griffin, and Ross (1994), Weinstein (1980) and Kunda (1987) find that people

believe good things happen more often to them than to their peers. Langer and Roth

(1975), Weinstein (1980) and Taylor and Brown (1988) find that people are overly

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optimistic about their own ability as compared to others.

2. Overconfidence Among High Achievers

There is reason to believe that this over-optimism may be even more pronounced

among certain populations. It could be illuminating to focus our attention on tournaments

in which the pool of prospective Contestants and those among the pool selecting into the

tournament yield Contestants with overconfidence levels exceeding that in the general

population. Specifically, graduates of top ranked graduate schools earning positions in

the most selective investment banks, consulting firms, and law firms would be expected

to place at the high end of the overconfidence distribution.

The prospective candidates to these sought-after positions have been consistently

successful relative to their peers as indicated by their ability to earn a position at a top-

ranked graduate program. To be considered for a position in a prestigious firm, they have

likely excelled even when competing with high-caliber peers. Moreover, they have likely

self-selected into these more competitive endeavors and thus have a strong belief in their

ability to compete and win. In addition, by selecting into a competitive position in a firm

in which only the top performers are promoted to higher positions, they are once again

demonstrating their high level of competitiveness, and presumably, overconfidence.

Recent hires at a top-ranked financial, law, or consulting firm are likely to be

particularly susceptible to falling prey to the bias of overconfidence. Being accustomed to

consistently achieving all goals they have set their mind to, they are likely to maintain a

firm conviction that they will achieve their goal of being a star within this new

employment context as well.

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B. Firm Rent Extraction Due to Employee Overconfidence

If indeed this group is systematically overconfident, firms (contest Organizers) may

have an incentive to utilize that misperception to extract rents from these budding

superstars. The firm, as contest Organizer, can benefit from this overconfidence by

providing a salary largely contingent on relative achievement. If the Contestants’ average

subjective belief regarding the likelihood of winning the tournament exceeds the true

probability – and the firm is aware of both the true probability and of this systematic

deviation of the subjective probabilities – the Organizer can extract rents by providing

compensation contingent on winning the contest.

This is somewhat analogous to the lottery contest discussed in Section III.

However, the primary source of divergence between the subjective and true probability of

winning is not innumeracy but the overconfidence of the Contestants.1 In this context,

even if the Contestant has full and accurate information regarding the objective

probability of winning, their overconfidence leads them to believe they have a higher

probability of winning than the true, objective probability for a given Contestant.

C. Example: Law School Graduates

For a concrete example of this contest, we can analyze the behavior of recent law

school graduates working as 1st year junior associates at a large law firm (this is an

example I was (un)fortunate enough to experience first-hand and so have some

familiarity with). For purposes of this contest analysis, we can break down the total salary

1It is certainly possible that innumeracy can play some role in making a contestant carry an unrealistic probability of her winning among a large pool of contestants.

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to the associate as Current Salary and Tournament Salary. The Current Salary is simply

an amount the employer pays each associate for work in the current period. The

Tournament Salary is equal to the net current period value of the expected salary of

becoming a partner at the firm for the expected period a lawyer remains a partner minus

the anticipated outside option to the associate for this period (we assume the outside

option is always less than the partner salary and uniform across all Contestants). After

determining the total compensation, the firm and associate decide on the optimal

allocation between Current Salary and Tournament Salary. In this contest, the players are

competing only with respect to the tournament component of the salary.

The prize B can be assumed to incorporate all non-monetary rewards of being

selected as a partner at the law firm (non-pecuniary value of winning, prestige, etc.). B is

equal to the present value of all net expected returns from becoming a partner.

1. True Probability Versus Subjective Probability

A further assumption we will make with respect to the Tournament Salary is that

the firm has an accurate perception of each associate’s probability of earning a

partnership position: (T)Pr. This assumption is in accord with our setup since we presume

the firm has ex ante determined that exactly one associate out of the n associates will be

selected as a partner.2 Whether this probability for any individual Contestant i fluctuates

over time from the perspective of the firm is not critical to the analysis. The firm knows it

will award one prize to the n associates; its expected tournament payment, B, is fixed.

However, the associates have subjective beliefs regarding their expected probability

2 In fact, any objective observer without a subjective weight placed to any individual contestant will have a symmetric expectation regarding the probability of winning for each player i.

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of winning which deviate from the true probability; the Contestants’ belief diverges with

that of the firm regarding the monetary value of the tournament component of the salary.

We will assume that, due to the overconfidence of the associates, each associate i’s

subjective probability of winning exceeds the actual probability. This higher probability

is attributed to the Contestant i believing every unit of her effort is worth double the

effort of that of all other players j ≠ i. We will assume that each associate i is unaware of

the symmetric “bloated” expected tournament salary of her competitors j ≠ i. We will

also assume that each Contestant i believes all other players j ≠ i do not know i believes

her effort is worth more than that of the other players. We will assume further that this

higher IA(Pr) of the associates is symmetric across each player i.

Associates will prefer to receive a greater proportion of their salary as Tournament

Salary since the probability weight associate i places on her probability of winning

exceeds the firm’s expected probability that associate i will win the tournament: (IA)Pri >

(T)Pr for all i. Analogously, firms will prefer to compensate associates with Tournament

Salary since the expected payout from firm to associate is lower with Tournament Salary

relative to Current Salary when associates hold subjective probabilities of winning

exceeding the true probabilities.

D. Employment Tournament Model with Overconfidence

We can set the parameters of this employment tournament contest as follows:

• Contestants N = {1, 2, 3, 4}

• Cost of Effort: Ci (xi) = xi

• Each Contestant has the same cost function equal to the total units of

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effort expended.

• These effort units, xi, are equal to each dollar of Current Salary sacrificed

in exchange for Tournament Salary (described below).

• Prize = B

• One Contestant will receive the entire prize.

• Each Contestant i has the same belief regarding the value of the prize.

• The Organizer (firm) also has the same valuation of the prize.

• There is a constraint on effort, m, and m < ¼(B). Therefore, every player will exert

effort equal to this maximum value m (Che and Gale 1998).

• Total effort in this contest is fixed at 4m. The “effort” component of the

contest is the decision by Contestants to opt for Tournament Salary in

place of Current Salary.

• All Contestants will choose to transfer the maximum amount m of

Tournament Salary in lieu of Current Salary as long as the expected payoff

exceeds that from Current Salary.

• Each Contestant i is overconfident about the relative effect of her total effort

relative to her competitors. Therefore, Contestant i believes xi = 2xj for all j ≠ i.

Each Contestant holds this belief regarding the relative impact of her effort

compared to her competitors.

• Every Contestant i believes all Contestants j ≠ i believe the effect of their effort xj is

equal to that of all other players.

• Every Contestant i believes all other Contestants j ≠ i do not know i believes her

effort is worth more than that of the other players.

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• Therefore, Contestants’ uniform Irrational Anticipated belief regarding the

probability of winning is: (IA)Pri = 2xi / Σj=1 to n xj, j = all players [1, …, n].

• (IA)Pri = 2m / 5m

• (IA)Pri = 2/5 (0.4)

• True probability of winning for each Contestant i: (T)Pr = xi / Σj=1 to n xj

• (T)Pri = m / 4m

• (T)Pr = 1/4 (0.25)

• Contestants are risk-neutral.

• Compensation from firm to employee can be paid as Current Salary or Tournament

Salary. For every $1 of Current Salary, employee can instead opt for an amount

$3.50 in Tournament Salary. The present value of this amount is $3.50 multiplied

by the probability of winning the tournament.

• The Tournament Salary yields a higher “irrational” expected payoff to a

risk-neutral Contestant than Current Salary as long as the Tournament

Exchange Rate is greater than $2.50.

• A Tournament Salary Exchange Rate lower than $4 yields a higher true

expected payoff to a risk-neutral Organizer.

• Within the Tournament Exchange Rate ranging [$2.50, $4], Contestant(s)

and Organizer are willing to exchange Current Salary for Tournament

Salary.

• While Contestants yield a higher irrational expected payoff, exchanging

Current Salary for Tournament Salary at any price lower than $4 yields a

negative true expected payoff to Contestants.

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• Organizer earns rents from Contestants at any Exchange Rate greater than

$2.50.

• With risk-neutral Contestants, Organizer should be able to implement a

Tournament Exchange Rate equal to $2.50 leaving Contestants indifferent

between Tournament Salary and Current Salary.

• If we: (1) relax the risk-neutrality assumption or (2) lower the degree of

overconfidence below the 2 to 1 ratio, Tournament Salary in lieu of

Current Salary can still yield rents to the Organizer at any Exchange Rate

below $4.

III. Lottery Contest

State lotteries provide an appropriate context to study more subtle sources of

deviations from rationality in contests: basic innumeracy and the tendency to overweight

very small probabilities.

The state lottery context illustrates how these predicable biases can yield rents to

the contest Organizer. In this contest, Contestants have an extremely small probability of

winning a very large prize. Since a defining feature of this form of lottery is that total

Contestant efforts outweigh total prizes, the lottery contest is always a negative

expectation game from the perspective of the Contestants (and positive expectation for

the Organizer). For purposes of clarity, a unit of "effort" here is set equal to the purchase

of a single ticket. The probability of winning with a single ticket multiplied by the

prize(s) is always less than the cost of a single ticket.

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A. Innumeracy

There is experimental and empirical evidence to indicate that individuals are

limited in their ability to understand the mathematical implications of extremely small

probabilities. Consequently, we may expect some level of pervasive irrationality when it

comes to implementing a strategy choice in the face of such extreme probabilities.

Distinguishing, for example, between a 1x10^(-5) probability of receiving a prize and the

identical prize but with a 1x10^(-10) probability of winning should induce a substantially

different optimal strategy. However, individuals are frequently incapable of appropriately

distinguishing between the expected value implications of these two probabilities – they

are often both placed under the broad category of “unlikely events” and there is no

meaningful change in strategy when facing one highly improbable event relative to

another.

Most of the Contestant’s attention is focused at the cost of a unit of xi (purchase

price of a single ticket) and the total prize B (jackpot prize). The Contestant is likely to

bucket together a wide range of very low probabilities of winning this jackpot in her

strategy decision. If, for example, the lottery were to increase the number of correct

numbers needed (out of say 50, without replacement) from 6 to 7, there is unlikely to be a

corresponding change in the Contestant’s strategy or willingness to participate.

Nevertheless, a rational Contestant should demand the jackpot be increased by a multiple

of 44 (e.g. $10,000,000 prize increased to $440,000,000) if the number of correct

numbers (out of 50) needed to win were increased from 6 to 7.

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B. Probability of winning independent of other Contestants’ effort

In the state lottery context, there is a slight deviation from the standard Tullock

model in that the probability of winning is not dependent on the total effort by all

Contestants. The probability of Contestant i winning is contingent solely on total tickets

purchased multiplied by the mathematical probability of a single ticket correctly choosing

the winning numbers. The expected number of winning tickets as a ratio of total tickets

will approximate the probability of winning and will approach this probability as the xi

approaches infinity. An added wrinkle in the lottery context – which we will ignore for

purposes of this analysis -- is that the prize is shared when multiple players have a

winning ticket; this would indirectly makes the prize dependent on the total tickets

purchased.

C. Prize Structure

The lottery contest structure is well-suited to yield high rents for the Organizer and

negative expected rents for all Contestants. From the perspective of an Organizer trying

to maximize rents, a very large prize and an extremely remote probability of winning is

precisely suited to take advantage of the cognitive limitations of prospective Contestants.

Breaking the single large payout into multiple, smaller prizes -- each with a higher

probability of winning -- would likely yield lower rents even when the expected payoff

per ticket remains unchanged.

It is worth acknowledging the possibility that Contestants obtain some utility

beyond the financial utility of the lottery prize. There could, for example, be

psychological utility from imagining the possibility of winning after purchasing a ticket

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even in the absence of ultimately carrying a winning ticket. Thus, the contest may be

utility enhancing for Contestants that obtain utility from imagining a win, making the

purchase utility-enhancing in spite of the negative expected monetary payoff.

Moreover, the payoffs may not be linear – as we assume in the model description

below – and thus the utility loss of the ticket price may be exceeded by the utility gained

from winning multiplied by the probability of winning. This would be the case, for

example, if there were almost no utility loss for negligible losses below a certain

threshold and dramatic utility gains when the payoff exceeds a certain, presumably life-

altering, amount. In either case, there may be some utility gain to the Contestant even

when there is a negative expected pecuniary payoff.

From the Organizer’s perspective, whether the participant’s decision to expend

effort is attributed to the innumeracy of the Contestant, some non-monetary source of

utility or a combination of these or other factors does not make a difference to the

Organizer’s strategy. The Organizer seeks to maximize expected rents by maximizing the

difference between total aggregate efforts and the total prize.

Lottery Organizers further utilize Contestant innumeracy by making the announced

prize equal to the total of payments to be paid by the lottery in aggregate over an

extended period. The actual net present value of the award is a fraction of the announced

prize. Nonetheless, many Contestants behave on the belief that the reward is the

announced amount. This detail further drives irrational behavior on the part of the

Contestant. One would expect that a prospective Contestant lacking a full understanding

of the true probability of winning is also likely not to have a full grasp of the time value

of money which makes the total amount paid over time quite distinct from the one-time

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payment of an equal aggregate amount.

D. Winning Ticket Probability

To illustrate a simple example, we can assume that choosing a winning lottery

ticket entails selecting four consecutive single digit numbers correctly, each from 0 to 9

inclusive – each draw is independent and the correct numbers must be selected in the

appropriate order. The true probability of a winning ticket is thus .0001. But we can

assume that the Contestants fail to accurately understand this probability of winning and

effectively interpret this as a probability of .001. This example of relatively larger

probabilities is used to simplify the problem. It is reasonable to assume that the

deviations from actual probabilities would be both greater in magnitude and more

pervasive with the significantly lower probabilities of winning in an actual state lottery.

If we assume a $1 ticket and a .0001 probability of winning (0.01%) the appropriate

prize would need to be greater than or equal to $10,000 to induce a risk-neutral

Contestant to participate. For simplicity, we will assume all players are risk-neutral and

that the only sources of utility to all players are financial gains and losses.

With a risk-neutral Contestant that believes her probability of winning for each unit

of effort is .001, the Organizer can yield a profit by offering a prize in the range of

[$1,000, $10,000]. Alternatively, the Organizer could potentially yield even higher rents

by increasing the number of correct numbers needed for a winning ticket from four to

five. This decreases the probability of winning to .00001. With this contest, a $1 ticket

must yield a prize that is worth at least $100,000 to induce a rational, risk-neutral

Contestant to participate. However, if the Contestant perceives the probability of winning

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as any number greater than .00001, the Organizer can obtain rents from this Contestant.

E. Large Pool of Prospective Contestants

Another relevant factor in the lottery contest setup, is that the Organizer can obtain

rents as long as there are some potential Contestants with a perceived probability of

winning exceeding the true probability. Thus, even if most prospective Contestants can

determine the negative expected value of participating and opt out, the Organizer can

earn rents even if only a small minority of prospective Contestants participates.

Therefore, those individuals who fail to understand that their effort yields a negative

expected rent will select into the contest.3

In a state lottery contest, the entry fee is small and there are few restrictions on

entry. Consequently, the contest is available to a wide number of prospective Contestants.

By casting such a wide net, and because individuals can participate with a negligible

expenditure of effort ($1), the lottery can appeal to those most susceptible to falling prey

to innumeracy. If the lottery required a $100 investment but yielded a chance of winning

that were 100 times more likely, the total rents would likely decline dramatically –

although the expected payoff per dollar expended would remain unchanged.

F. Lottery Model with Contestant Irrationality

The proper way to model the lottery contest as a modified version of the classic

Tullock contest with irrational Contestant beliefs is by assessing a probability weight to

winning the contest that diverges from the true probability. We will make the following

3 We will set aside the participation of contestants who participate because they receive some non-pecuniary rent from the contest -- though the organizer is indifferent as to the motivation of the contestant.

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assumptions:

• Contestants N = {1, …, n}.

• Each Contestant, i, purchases a number of tickets = xi.

• Cost of Effort: Ci (xi) = xi (xi = number of tickets purchased).

• $1 per ticket (eg. i purchases 5 tickets: xi = 5).

• Prize = B

• Vi (B) = B for all i and also equal to B for the Organizer.

• Prize given to any player(s) who earn a winning ticket.

• Expected number of winning tickets will be contingent on the total tickets

purchased multiplied by the probability of winning.

• Contestants may win at most one prize (this assumption can be relaxed –

Organizer will yield higher rents if permitting multiple awards to a single

Contestant leads to a marginal increase in the number of total tickets

purchased which exceeds the amounts awarded as subsequent prizes to

winning Contestants).

• All Contestants without a winning ticket receive zero.

• Contestants and Organizer are risk-neutral.

• Probability of any single ticket being a winner = (T)Pr = 0.0001

• Contestants’ uniform Irrational Anticipated belief regarding the probability of a

ticket being a winner: (IA)Pri = 0.001, with (IA)Pri > (T)Pr for all i.

• For simplicity, each Contestant has the same (IA)Pr.

• Knowledge of other Contestant’s (IA)Pr is not relevant to player i’s

strategy since her payoff is not a function of the effort of any j ≠ i.

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• True Probability of i winning is dependent on the number of tickets purchased xi

multiplied by the probability of each ticket being a winner: pi = pi(xi,(T)Pr)

• pi’s are between 0 to 1 (cannot exceed 1 since each Contestant can win, at

most, one prize).

• Each Contestant has linear utility with respect to the cost of the ticket (xi)

and the value of the prize (B).

• True Expected Payoff to i: πi(x1…,xn) = pi(xi, (T)Pr)vi (B) – Ci(xi)

• Which can be simplified to: pi(xi, (T)Pr)(B) – xi

• Irrational Anticipated Payoff to i: πi(x1…,xn) = pi(xi, (IA)Pr)vi(B) – Ci(xi)

• Simplified to: pi(xi, (IA)Pr(B) – xi

• Expected Payoff to Organizer: [Σi=1 to n xi] – [Σi=1 to n xi (T)Pr(B)(B)]

• As long as the prize = B is less than $10,000, the Organizer can extract

rents from the Contestants.

• With the subjective irrational belief, (IA)Pr, risk-neutral Contestants will

be willing to purchase tickets at any prize greater than $1,000.

• Within the range [$1,000, $10,000] Organizer and Contestants will

anticipate a positive payoff from the exchange of tickets for the purchase

price of $1 per ticket.

• Contestants will earn a negative true expected payoff from any prize less

than $10,000.

IV. Experimental Tests of Irrationality in Contests

A. Testing for Overconfidence

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I would be interested in testing and analyzing the hypotheses set forth above in an

experimental context. To investigate how overconfidence can yield contest over-exertion,

I would be interesting in conducting an experiment with MBA students at a top ranked

business school. It will be most telling to test these students in situations in which they

are given the option of a sure reward versus the opportunity to receive a higher reward

contingent on them outperforming their peers. If the experiment is thoughtfully carried

out, we may obtain some insight regarding the level of overconfidence among this

segment and how potential employers of such individuals can extract rents at the

overconfident employees’ expense.

1. Experimental Setting

A simple experiment that could measure the level of overconfidence among these

accomplished individuals would involve the following: the subjects would be students

from a top ranked business or law school. These are the candidates most likely to take on

high paying jobs in firms that typically implement a tournament-like structure. It is

widely believed that at least some portion of the compensation of such firms comes from

the possibility of earning the high financial rewards of being promoted to partnership or

an equity level position. Ideally, the subjects of the study would be those individuals who

are high achievers even within the graduate school class as these are likely to be the

candidates most likely to earn positions in the most prestigious firms.

The subjects will be given a mental task to complete. In an optimal experiment, this

task will correspond to the types of skills generally required to be successful in an

employment setting. A test measuring intelligence, cognitive ability and perhaps

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requiring a high level of effort and persistence will approximate these skills. Critically,

the subjects must be informed at the start of the experiment that the test will measure

these qualities; subjects will make the decision on whether to compete for a higher prize

on the basis of this information.

2. Prize Selection

The subject will then be offered the option of either taking a fixed prize plus a

bonus amount for each question answered correctly or a larger “competitive prize”

dependent on achieving the highest score on the exam out of a cohort of nine other peer

participants who have each also opted for the competitive prize. The subjects will be

informed that their competitors will be members from their class and that these will be

individuals who themselves also chose to take a shot at the competitive prize. The reward

for the competitive will be less than 10 times the value of the fixed prize plus the

maximum bonus. Therefore, in the aggregate, the subjects will earn a lower payoff from

the competitive prize relative to the fixed prize.

If subjects are overconfident about their relative ability and take a shot at the

competitive prize, the experimenter will be yielding a positive expected rent at the

expense of the subjects. The subjects would be better off (and the experimenter worse

off) if each candidate takes the fixed prize. We would be likely to observe the most

confident (and presumably overconfident) individuals opting for the competitive prize.

3. Risk Aversion

It would be critical to account for risk-aversion as this mitigates the impact of

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overconfidence on Contestant strategies -- although we may expect that overconfidence is

likely to be correlated with lower risk-aversion. Moreover, risk aversion itself may be a

salient factor in this analysis. If we were experimentally assessing whether individuals

are willing to take on a probabilistic salary contingent on relative performance, the fact

that risk aversion mitigates this preference would be expected to reduce any rents to the

experimenter. Therefore, it would be meaningful to assess how risk aversion influences

the preferences of these high-performing, ostensibly overconfident subjects.

Nevertheless, we are likely to get some insight regarding the level of overconfidence in

this group and how this overconfidence can lead to over-exertion.

4. Overachievers Overconfidence

It would also be meaningful to compare the proportion of subjects who choose the

competitive prize in the high-achieving group compared to the proportion choosing the

competitive prize among a broader group of subjects. Observing a higher relative

proportion of subjects choosing the competitive prize among the high-achieving group

would be a meaningful result. This is because both groups will be informed of the peers

they will be competing with for the competitive prize. The high achievers know that they

will be measured against other high achievers – classmates who have also opted for the

competitive prize. Therefore, a higher proportion of subjects opting for the competitive

prize in the high achieving group would be an indication not simply of a heightened

belief in one’s own competence but would be at least partly attributed to greater levels of

overconfidence among the high-achieving group.

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5. Survey Data Measuring Overconfidence

I would also be interested in accumulating survey data asking newly minted

employees at top firms with a tournament-like employment structure what they believe

their likelihood is of making it to partnership or an equity-level position. I would be

interested in comparing the average predicted probability of “making it” to the true

probability. We would expect that the subjective perceptions would exceed the true

probabilities. Further, we would expect the widest discrepancy between the true and

subjective probabilities in the most prestigious, competitive firms employing the highest

caliber individuals.

Consequently, we would expect that the share of compensation allocated to

tournament pay would be highest among the most selective and prestigious firms. We

would expect, for example, that Goldman Sachs or McKinsey Consulting would provide

the greatest proportion of compensation in the form of expected tournament salary. This

would be translated as a very high salary provided to those at the partnership ranks of the

firm and a relatively low probability of earning this position among the lower ranks,

along with a relatively lower current salary. While there are a host of other factors that

heavily influence the optimal employment contract from the perspective of the employer,

ceteris paribus, we would expect more tournament type salary in those firms employing

the most high-caliber (and thus most overconfident) employees.

Another factor that I believe would be relevant to the analysis here is measuring the

value placed by different individuals on “winning the tournament”. It is here that I

believe it would be worthwhile to test whether individuals differ not only with respect to

their perceived probability of winning the tournament but also with regard to the value

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they attribute to winning the prize: vi(B). Holding subjective probability of winning

constant, a Contestant will exert greater effort if she obtains greater relative utility from

earning the prize B. Therefore, firms implementing a tournament-oriented compensation

structure will also be attracting those individuals who place greater relative value on

winning the prize. Therefore, even if the monetary value of winning is the same to two

players, the utility gain can differ. This difference can yield a greater preference for the

tournament structure and greater relative effort expended in the tournament.

6. Trends in Compensation Structure of Selective Firms

A possible manifestation of a trend towards a larger proportion of tournament

salary is the changing dynamic in law firm partnership promotion over the past few

decades. There has been a persistent and accelerating trend in decreasing the proportion

associates who make it to the partnership ranks. We have also observed a dramatic

increase in the relative salaries of partners relative to associates. Therefore, the

probability of winning the tournament has declined while the reward for winning the

tournament has increased.

It is possible that this trend may partly be a response to overconfident associates

who continue to believe they have what it takes to make partner in spite of the declining

probability of winning the tournament. In my own personal experience, associates seem

more fixated on the lucrative rewards of making it to partnership and are less focused on

relative changes in the proportion of associates who make it to partnership.4

Consequently, firms can earn larger rents by providing a higher prize B while reducing

4 Partnerships have a strong disincentive to promote employees to partnership as this reduces rents to existing partners unless the value-added of the incoming partner exceeds the partnership salary.

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the probability of winning.

B. Capitalism

Taking the tournament context to a broader, societal level, we can make the

argument that a capitalist social hierarchy, as compared to a more socialist distribution of

wealth is more likely to persist in societies in which: (1) the individuals are relatively

overconfident in their expected probability of achieving high status, and (2) the value to

Contestants of “winning the tournament” is higher than is the case in more socialist

societies (i.e. these individuals receive higher relative utility from earning lucrative

tournament prizes). With these two ingredients, overconfidence and high relative

valuation of B, a capitalist social structure can be sustained. Thus, from a broader societal

level, a perception that anyone can make it to the top if they work sufficiently hard, can

help sustain a system in which individuals exert effort (presumably, over-exert effort) for

a scarce number of prizes.

The implication of this foundation for a capitalist society is that: (1) if a sufficiently

large proportion of citizens (Contestants) believe their likelihood of winning the

tournament is low and their overconfidence dissipates, and (2) the subjective value placed

on “making it” and winning the tournament declines, the capitalist distribution may

encounter a challenge. Overconfidence and a belief among the masses that, with hard

work, they could achieve the ‘American Dream’ and earn a spot among the elite, along

with a high subjective value placed on how happy they believe they will be once they

have earned high status may be necessary ingredients for sustaining an inequitable

distribution of societal wealth.

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This would mean that preserving a top-heavy distribution of wealth requires

preserving the ‘illusion’ that anyone can make it to the top if they only work hard enough

along with a grandiose image of how wonderful life can be once you have achieved the

riches at the top.

C. Innumeracy Experiments

Lastly, I would be interested in laboratory experiments that demonstrate the level

of basic innumeracy within contest structures. For example, providing individuals with

the option of earning (a) $10 if they choose three numbers correctly with replacement

from 0 to 9 inclusive versus (b) earning $200 if they choose five numbers correctly with

replacement from 0 to 9 inclusive would clearly illustrate this pervasive innumeracy. If

people choose (b) over (a) they are clearly demonstrating that they are incapable of

assessing that the probability of winning is now 100 times less likely and so they should

only opt for this second option if the prize is at least 100 times larger (assuming risk-

aversion the prize would need to be even higher). I believe that when the probabilities

become increasingly small, people have a very difficult time distinguishing between

remote events, even when the discrepancy has very meaningful strategic consequences.

Contest Organizers in the real world are able to utilize this pervasive and predictable

innumeracy to extract rents from Contestants.

V. Conclusion

The goal of a contest Organizer is to maximize the aggregate efforts of

Contestants; the Organizer seeks to earn rents by implementing a contest in which total

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efforts exceed total prizes. We observe countless examples of contests in which the total

aggregate efforts (loosely defined) exceed the payouts and yield rents to the Organizer.

The labor and lottery contexts described above can be supplemented with a host of other

empirical and experimental observations of over-dissipation.

A poker tournament illustrates a simple example of a contest in which Contestants

willingly participate in a contest that predictably transfers rents from Contestant to

Organizer. Players pay buy-in amounts that exceed the expected payout per contestant

(total prize money divided by the total number of players). Players choose to participate

in the tournament in spite of the negative expected return because they believe their

probability of winning exceeds that of the average Contestant. Alternatively, innumeracy

leads Contestants to fail to understand the negative expected payoff of the contest.

Individuals focus their attention on the high value of the highest payouts and imprecisely

calculate how the low probability of winning reduces their expected payoff below the

entry fee.

It must be conceded that a player with superior skills relative to her competitors

may be obtaining a positive rent in this and other contests in which relative skill predicts

the likelihood of success. It is even conceivable that most Contestants are earning

positive rents from participating. Nevertheless, there must be at least one Contestant that

is earning negative utility. Irrespective of what motivates the Contestant’s decision to

participate, the Organizer earns positive rents as the aggregate expected utility to the

Contestants is negative.

If Organizers are aware of the factors that induce prospective Contestants to

participate in a contest in which they earn negative expected aggregate rents, they can

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utilize these factors to construct a contest that maximizes Organizer rents. The state

lottery provides a paradigmatic example of a contest which yields high Organizer rents

due to multiple predictable deviations from optimality: (1) systematic overweighting of

low probabilities, (2) salience of the high reward, (3) innumeracy with respect to the

mathematical likelihood of selecting all winning numbers, (4) appealing to a broad range

of potential Contestants, (5) low cost of entry, (6) innumeracy with respect to the prize

amount itself. In the tournament employment context, the firm is able to extract rents

from high achiever employees due to overconfidence that drives an irrationally high

subjective expected probability of winning the tournament.

Contest designers can benefit from understanding the contest features most likely

to induce irrationality and aggregate over-exertion among Contestants. Organizers can

utilize these insights to construct contests that maximize Organizer rents. Alternatively, a

deeper understanding of cognitive biases that cause deviations from optimality can help

potential Contestants overcome the inclination to participate in negative utility contests

and to engage in effort levels yielding negative expected rents.

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