positive self image in tournaments

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INTERNATIONAL ECONOMICREVIEW

Vol. 51, No. 2, May 2010

POSITIVE SELF-IMAGE IN TOURNAMENTS

BY LUIS SANTOS-PINTO1

University of Lausanne, Switzerland

This article analyzes the implications of worker overestimation of productivity for firms in which incentives take

the form of tournaments. Each worker overestimates his productivity but is aware of the bias in his opponents self-

assessment. The manager of the firm, on the other hand, correctly assesses workers productivities and self-beliefs

when setting tournament prizes. The article shows that, under a variety of circumstances, firms can benefit from worker

positive self-image. The article also shows that worker positive self-image can improve welfare in tournaments. In

contrast, workers utility declines due to their own misguided choices.

1. INTRODUCTIONThis article is related to a recent strand of literature in economics that studies the welfare

consequences of behavioral biases. The article focuses on the welfare implications of workeroverestimation of skill when firms use tournaments to provide incentives.2

The article derives two main results. First, it shows that, under a variety of circumstances, firmscan benefit from worker positive self-image if they wisely structure prizes in tournaments. Thearticle argues that in order to do that, firms should take into account how self-image changesworkers incentives to exert effort. This finding is consistent with the idea that some partiesinvolved in a contract might gain when other parties are not fully rational.

Second, the article finds that moderate levels of worker positive self-image can improve

welfare in tournaments. This happens when (i) workers have increasing absolute risk aversionand (ii) positive self-image lowers the prizes that the firm needs to pay for workers effort. Thisresult is consistent with the theory of the second best.

In this article, a worker with a positive self-image overestimates his productivity of effort buthas an accurate assessment of his cost of effort and his outside option. The firm correctly assessesworkers productivities and self-beliefs. Each worker is aware that his opponents perception ofability is mistaken but thinks that his own perception is correct. Thus, the firm and each workerhold divergent beliefs about the workers productivity.3

Worker positive self-image has two effects in tournaments for fixed prizes. First, it makesparticipation in tournaments more attractive to workers than it should actually be. Since in

a tournament higher prizes are paid to workers who produce higher output, a worker with a Manuscript received April 2006; revised August 2008.1 This article was started during my Ph.D. dissertation in University of California, San Diego. I am grateful to my

advisor, Joel Sobel, the editor, and two anonymous IER referees. I also thank conference and seminar participants at

University of Montreal, Universidade Nova de Lisboa, University of California, San Diego, University of East Anglia,

University of Innsbruck, and the 22nd Annual Congress of the European Economic Association in Budapest. An

important part of this work was undertaken when I was at Universidade Nova de Lisboa. I gratefully acknowledge

financial support from Praxis XXI, Fundac ao Calouste Gulbenkian, and INOVA. Please address correspondence to:

Lus Santos-Pinto, Faculty of Business and Economics, University of Lausanne, Quartier UNIL-Dorigny, Internef 535,

1015 Lausanne, Switzerland. Phone: +41 021 692 3658. E-mail: [email protected] .2 Tournaments are one of many forms of providing incentives in firms. Managers are involved in promotion tourna-

ments: vice presidents compete to be promoted to president and senior executives competeto become CEO. Salespeople

are often paid bonuses that depend on their sales relative to those of the other salespeople in the firm.3 In the standard tournament literature, all parties are assumed to hold identical and accurate beliefs regarding the

distribution of output induced by workers effort choices


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positive self-image will overestimate the probability that he will attain a high prize. This effectof worker positive self-image is favorable to the firm.4

Second, worker positive self-image can change workers incentives to exert effort. A workerwhothinksthatheismoreablethanothersmaythinkthatifheworkshardertheincreaseinutilityassociated with a higher probability of success more than compensates the increase in disutility

associated with higher effort. It could also be argued that positive self-image reduces effortprovision by workers. A worker who overestimates his probability of winning the tournamentmay think that by reducing effort the decrease in disutility of effort more than compensates thedecrease in utility steaming from a lower probability of success.5

The first main finding of the article is that firms can be better off with a positive self-imageworkforce if they wisely structure prizes in tournaments. This result holds under the followingcircumstances: (1) worker risk neutrality, (2) worker risk aversion and complementarity betweenself-image and effort, and (3) worker risk aversion, substitutability between self-image andeffort, and moderate impact of self-image on effort.

When workers are risk neutralthe firm can counter any impact of positive self-image on effortby changing the prize spread (the difference between the winners prize and the losers prize)while keeping total prizes fixed. Since a positive self-image worker overestimates the probabilityof winning the tournament, the firm can change the prize spread and reduce total prizes.

When workers are risk averse and self-image and effort provision are complements (higherself-image increases workers perceived marginal probability of winning the tournament for alleffort levels) the firm can get more effort for a fixed prize structure or the same amount of effortwith lower prizes. This result is valid under very general conditions.

Matters are not so straightforward when workers are risk averse and self-image and effortprovision aresubstitutes. In this case, the firm may not be able to counter the unfavorable impactof positive self-image on effort by raising the prize spread while simultaneously reducing totalprizes. The reason is worker risk aversion implies that workers must be compensated for an

increase in the prize spread. However, the article shows that the firm is better off with a positiveself-image workforce if workers are risk averse and higher self-image only leads to a moderatereduction in the perceived marginal probability of winning the tournament for all effort levels.

The article also shows that firms can be better off with a positive self-image workforce whenworkers are risk averse and the relation between self-image and effort is nonmonotonic. Thishappens when output is either exponentially or normally distributed. These two examples alsoillustrate how the relation between effort and self-image depends on effort levels, perceptionsof skill, and technology.

The second main findingof thearticleis that moderatelevels of positiveself-image can improvewelfare in tournaments. This happens when workers have increasing absolute risk aversion and

positive self-image lowers the prizes that the firm needs to pay for workers effort.The intuition behind this result is as follows. If workers have increasing absolute risk aversion

and hold accurate perceptions of skill, there is undersupply of effort, that is, the effort levelchosen by the firm will be less than the effort level that maximizes welfare. If positive self-imagelowers the prizes that the firm needs to pay for workers effort, then the firm prefers to select ahigher effort level with a positive self-image workforce than with an accurate one. However, ifworker positive self-image is too high welfare might decrease since either workers might shirk orthe firm might decide to select an effort level that is greater than the one that maximizes welfare.Thus, moderate levels of worker positive self-image improve welfare because they reduce theundersupply of effort problem caused by increasing absolute risk aversion.

4 Santos-Pinto (2008) shows that positive self-image workers place more value in a contract with a wage-incentive

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POSITIVE SELF-IMAGE IN TOURNAMENTS 477

This result is consistent with the theory of the second best. According to this theory, introduc-ing a new distortionworker positive self-imagein an environment where another distortionis already presentworker increasing absolute risk aversioncan increase welfare. Of course,welfare does not always rise when workers have biased beliefs. It was clear from the previousparagraph that high levels of positive self-image might reduce welfare when workers have in-

creasing absolute risk aversion. The article also shows that positive self-image always reduceswelfare when workers are risk neutral.

Evidence from psychology and economics shows that most individuals hold overly favor-able views of their skills.6 This tendency is also present in workers self-assessments of perfor-mance in their jobs. Myers (1996) cites a study according to which: In Australia, 86 percent ofpeople rate their job performance as above average, 1 percent as below average. Baker et al.(1988) cite a survey of General Electric Company employees according to which: 58 percent ofa sample of white-collar clerical and technical workers rated their own performance as fallingwithin the top 10 percent of their peers in similar jobs, 81 percent rated themselves as falling inthe top 20 percent. Only about 1 percent rated themselves below the median.

Entrepreneurs, currency traders, and fund managers have also been shown to overestimatetheir skills. Oberlechner and Osler (2004) find that 75% of currency traders in foreign exchangemarkets think they are better than average. Similarly, Brozynski et al. (2006) find that fundmanagers hold overly positive views of their relative performance.7

Two experiments provide direct support for the notion that tournaments attract individualswho overestimate their skills. Camerer and Lovallo (1999) consider a market entry game wheresubjects payoffs are based on rank, which is determined either randomly or through a test of skill.They find that there is more entry when relative skill determines payoffs, which suggests thatindividuals overestimated their ability to do well on the test relative to others. Park and Santos-Pinto (forthcoming) show that players in real world poker and chess tournaments overestimatetheir performance and are willing to bet on their overly positive perceptions of skill.

My article is an additional contribution to the growing literature on the impact of behavioralbiases on markets and organizations. The article is closely related to papers in that literaturethat study the impact of biased beliefs on the employment relationship.8

Hvide (2002) shows that a worker can gain from overestimating his skill if that improves hisbargaining position against the firm (the outside option). The firm is made worse off by theworkers positive self-image. Benabou and Tirole (2003) show that if a firm is better informedabout a workers skill than the worker, effort and self-image are complements, then the firmhas an incentive to boost the workers self-image by offering low-powered incentives that signaltrust to the worker and increase motivation. Gervais and Goldstein (2007) find that a firm isbetter off with a team of workers who overestimate their skill when there are complementarities

between workers efforts.My article shows how firms can design prize structures in tournaments to take advantage of

workers inflated self-perceptions of skill. This finding stands in contrast to those in Hvide (2002)and does not rely on the assumption of complementarity between workers efforts present inGervais and Goldstein (2007). My article also shows that moderate levels of worker positiveself-image can improve welfare in tournaments. This result is in line with findings in Waldman(1994) and Benabou and Tirole (2002).

6 Positive self-image is a staple finding in psychology. According to Myers (1996), a textbook in social psychology, on

nearly any dimension that is both subjective and socially desirable, most people see themselves as better than average.7 O fid d i i lf i i d i i h l i fi i l k l d


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2. SETUP

In this section, I incorporate worker positive self-image into a generalized version of Nalebuffand Stiglitzs (1983) rank-order tournament model. The timing of the model is as follows: (1)the firm chooses the optimal prizes; (2) workers observe the realization of a common environ-mental shock (this may be interpreted as an uncertain factor specific to one activity but thataffects all workers within that activity similarly); (3) workers choose simultaneously the optimallevel of effort after observing the realization of the common shock and the prizes chosen by thefirm; (4) the output of each worker is determined by the workers effort choice, the commonshock, and an idiosyncratic shock specific to each agent and distributed independently acrossagents; (5) the firm observes the workers ranking in terms of output; and (6) the firm awardsthe prizes to the workers according to their ranking.9

Throughout the article, attention is restricted to tournaments played between two workers.10

Let Ui (yi , ai ) denote worker is von NeumannMorgenstern utility function, which is assumedto be increasing in income, yi , and decreasing in effort, ai , with ai Ai = [0,), i = 1, 2. LetU represent the utility of an outside option. Worker is output, qi , is a stochastic function of his

effort, in the sense that each level of effort induces a distribution over output

Gi (qi | ei (ai , )),(1)

i = 1, 2, where ei (ai , ) is a measure of worker is productivity and is the common shock. Aworkers productivity is assumed to be strictly increasing in effort, but marginal productivityis subject to diminishing returns to effort. The cumulative distribution Gi (qi | ei ) is assumed tosatisfy the monotone likelihood ratio condition, that is, for qi2 > q

i1 and e

i2 > e

i1

gi qi2 ei2gi

qi2 ei1 >

gi qi1 ei2gi

qi1 ei1 ,(2)

i = 1, 2, where gi (qi | ei ) is the density function of Gi (qi | ei ).11 Worker is perception of hisproductivity is given by ei (ai , i , ), where i R+ parameterizes worker is degree of positiveself-image. So, from worker is perspective, each level of effort induces a distribution over output

Gi (qi | ei (ai , i , )),(3)

i = 1, 2. Worker i has a positive self-image if Gi (qi | ei (ai , i , )) first-order stochasticallydominates Gi (qi

|ei (ai , )) for all ai

Ai . That is, for any effort level of worker i, worker

i thinks that he is more likely to produce higher levels of output than he actually is. WhenGi (qi | ei (ai , i , )) Gi (qi | ei (ai , )) we say that worker i has an accurate self-image and leti = i .

Worker is mistaken beliefs of productivity influence behavior through worker is perceivedprobability of winning the tournament. For a given output level of worker j, say qj, worker iperceived probability of winning the tournament is given by

Pr(Qi qj) = 1 Pr(Qi qj)= 1 Gi (qj | ei ).

9 By assumption, in a tournament, the firm is not able to observe workers effort choices. This introduces the element

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Thus, worker is (unconditional) perceived probability of winning the tournament is given by

Pi (ai , aj, i ) =

[1 Gi (qj | ei (ai , i , ))]gj(qj | ej(aj, ))dqj,(4)

j = i, i = 1, 2. We see that (1), (2), (3), and (4) imply that worker is perceived probability ofwinning the tournament is increasing in worker is effort choice, decreasing in worker js effortchoice, and increasing in worker is self-image.

In order to be able to compute equilibria when workers hold mistaken beliefs I follow Squin-tanis (2006) approach and assume that (1) the manager of the firm correctly assesses workersabilities and self-beliefs, (2) each worker is aware that his opponents perception of ability ismistaken, and (3) each worker thinks that his own perception of ability is correct.

Worker is ex post monetary income from taking part in the tournament is given by

yi

= yL if q

i < qj

yW otherwise

,

j = i, i = 1, 2, where yL is the losers prize and yW the winners prize, with yL < yW. Workeris interim perceived expected utility (the utility after having observed the realization of thecommon shock but before the realization of the idiosyncratic shocks) is given by

Vi (ai , aj, i , yL, yW) = Ui (yL, ai ) + Pi (ai , aj, i )[Ui (yW, ai ) Ui (yL, ai )]

j = i, i = 1, 2, and worker is ex ante perceived expected utility is given by

E[Vi

(ai

, aj

, i

, yL, yW)],

j = i, i = 1, 2, where the expectation is taken with respect to the common shock.The firm is assumed to be risk neutral and to be concerned exclusively with the maximization

of profits, that is, the difference between expected benefits and compensation costs:

(q, y) = E[Q1 + Q2] (yL + yW).

My analysis will focus on the monopsonistic firm, that is, a firm selling its product in a com-petitive output market but that has considerable influence in the input (labor) market.12 Thefirms problem is to find the optimal wages for the winning and the losing parties, (y

L, y

W) and

the optimal effort choice for the workers, (a1, a2), subject to the constraint that the latter beimplemented as a Nash equilibrium between the workers by the chosen incentive scheme andthe constraint that each worker receives an ex ante perceived expected utility that is at least hisreservation utility. Thus, the firm solves

maxyL,yW

E

i=1,2

qigi (qi | ei )dqi

(yL + yW)

s.t. ai arg maxaiAi

Vi (ai , aj, i , yL, yW), i = 1, 2,

E[Vi (ai , aj, i , yL, yW)] U, i = 1, 2.

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In the implementation problem the firm, for any arbitrary effort pair (a1, a2), chooses the pair(yL, yW) that minimizes the firms implementation cost, C(a

1, a2, 1, 2), subject to the incentivecompatibility and the participation constraints. In the effort selection problem the firm choosesthe effort pair (a1, a2) that maximizes the difference between expected benefits and implemen-tation cost. I will use this decomposition to characterize the impact of positive self-image on

tournament outcomes.

3. THE SPECIALIZED MODEL

This section specializes the model and shows that an equilibrium exists. I consider a specialcase of Nalebuff and Stiglitzs (1983) model.

First, I assume that workers are weakly risk averse and have identical von NeumannMorgenstern utility functions additively separable in income and effort, that is

Ui (yi , ai ) = u(yi ) c(ai ),

where u and c are twice differentiable with u > 0, u 0, c > 0, c > 0, and c(0) 0. Second,I assume that workers have the same degree of positive self-image, that is, 1 = 2 = . Third,I assume that there is no common shock to simplify the analysis. Fourth, workers perceivedstochastic production function is given by

Qi = ei (ai , ) + i , i = 1, 2.

where Gi is the distribution function of i , gi its density, with gi symmetric, E(i ) = 0, andE(i j)

=0 for i

=j. I also assume that workers perceived stochastic production functions are

identical. Worker is perceived probability of winning the tournament function P(ai , aj, ) isgiven by

P(ai , aj, ) =

[1 Gi (ej(aj) ei (ai , ) + j)]gj(j)dj.

Fifth, I assume that P(ai , aj, ) is twice differentiable in ai , and differentiable in aj and . Thus,worker is perceived expected utility is given by

Vi (ai , aj, i , yL, yW)

=u(yL)

+Pi (ai , aj, )

u

c(ai ),(5)

where u = u(yW) u(yL). Worker i maximizes his perceived expected utility by choosing anoptimal effort level, taking workerjs effort level, and prizes as given. Notice that, for each effortlevel selected by worker j, worker i may either choose a positive effort level or a zero effortlevel (shirk). Thus, worker i solves

max

maxai >0

[P(ai , aj, )u c(ai )], P(0, aj, )u c(0)

.

The global incentive compatibility condition is satisfied if both the level of positive self-image as

well as the variance of the idiosyncratic shocks are not excessively high.13

If the level of positiveself-image is very high, the worker may think that his probability of winning the tournament isso high that he is better off by shirking


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Assumingthat the global incentive compatibility condition is satisfied we can study the relaxedoptimization problem

maxai >0

P(ai , aj, )u c(ai ).

The first-order condition for this problem is given by

Pai (ai , aj, )u = c(ai ),

and the second-order condition by

Pai ai (ai , aj, )u c(ai ) < 0.

The second-order condition can be satisfied under a variety of conditions. For example, it issatisfied when the perceived probability of winning is increasing and concave in own effort, thatis, Pai ai < 0.14 The second-order condition is also satisfied if Pai ai (ai , aj, )u < minai c(ai ).15

Let e(, yL, yW) denote the workers simultaneous effort choice subgame.

PROPOSITION 1. If the symmetry and differentiability assumptions hold, the global incentivecompatibility condition is satisfied, and Vi is strictly concave in a i , then there exists a unique

pure-strategy symmetric Nash equilibrium of e(, yL, yW) with ai > 0, i = 1, 2.

The proof that a Nash equilibrium exists relies on the classical existence result due to Debreu(1952), Glicksberg (1952), and Fan (1952). The assumptions that the workers expected utility isdifferentiable and strictly concave in own effort guarantee that there exists a unique equilibriumin pure strategies. The assumption that the global incentive compatibility condition is satisfiedrules out a pure-strategy equilibrium where workersshirk. The symmetry assumptionsguaranteethat the equilibrium is symmetric.

In the unique symmetric pure-strategy Nash equilibrium the first-order condition of the rep-resentative workers optimization problem becomes

Pa (a, )u = c (a) .(6)

Equation (6) is the analogue of Nalebuff and Stiglitzs cornerstone equation of tournamentsbut now modified to take into account the presence of worker mistaken beliefs of ability. Ittells us that in equilibrium workers should increase their effort level up to the point where

the perceived marginal benefit of doing sothe perceived marginal probability of winning thetournament times the utility differential between winning and losingequals its incrementalcost to the marginal disutility of effort.

The relation between self-image and effort can be obtained from (6).16 Effort and self-imageare complements (substitutes) if, for a fixed prize structure, higher self-image increases (reduces)workers effort. Differentiation of (6) with respect to self-image gives us

a

=

2V/a

2V/a2= Pa(a, )u

Paa (a, )u c(a).(7)

14 This condition is satisfied ifGiiee(q|e) > 0, that is, if there are stochastically diminishing returns to effort. See Koh(1992).

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The denominator in (7) is the second-order condition and is negative. Since the utility prizespread is always positive the relation between effort and self-image only depends on the sign ofPa(a, ), that is, how self-image influences workers perceived marginal probability of winningthe tournament.

Thus, self-image and effort are complements when Pa(a, ) > 0forall a, that is, when a higher

self-image increases workers perceived marginal probability of winning the tournament for alleffort levels. In contrast, if a higher self-image decreases workers perceived marginal probabilityof winning the tournament for all effort levels, Pa(a, ) < 0 for all a, self-image and effort aresubstitutes. Since the sign of Pa(a, ) is jointly determined by effort levels, perceptions of skill,and technology (the distribution of output given effort), so is the relation between self-imageand effort.

4. RISK-NEUTRAL WORKERS

If workers are risk neutral, we have (up to an affine transformation) u(y) = y for all y. In thiscase, the firms per worker implementation problem is given by

minyL,yW

1

2(yL + yW)

s.t. Pa (a, )y = c (a)yL + P()y c (a) U,

where y = yW yL. In the solution to this problem the participation constraint is alwaysbinding; otherwise it would be possible to implement the same effort level at a lower cost(by reducing yL

+yW while leaving yW

yL unchanged). Solving the incentive compatibility

constraint and the participation constraint for the optimal losing and winning prizes, we obtain

yL = U+ c (a) P()

Pa (a, )c (a) ,(8)

yW = U+ c (a) +1 P()Pa (a, )

c (a) .(9)

Adding up (8) and (9) and diving by 1/2 we have that per worker implementation cost is given

by

C(a, ) = U+ c (a) P() 1

2Pa (a, )

c (a) .(10)

Let T() denote the tournament game with positive self-image workers and T() denote thetournament game with workers who have accurate perceptions of skill. Let denote the levelof positive self-image that makes the worker indifferent between shirking and exerting effort. Iuse (10) to prove my next result.

PROPOSITION 2. If workers are risk neutral, then the firms profits are higher in T() than inT(), with < < .


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the prize spread allows the firm to neutralize any impact of positive self-image on incentives. Ifself-image and effort are complements (substitutes), then the firm should reduce (increase) theprize spread to reduce (increase) effort. The fact that positive self-image makes participationin the tournament seem more attractive to workers than it actually should be allows the firm toreduce prizes.17

5. RISK-AVERSE WORKERS

If workers are risk averse, the firms implementation problem is given by

minuL,uW

1

2[h (uL) + h (uW)]

s.t. Pa (a, )u = c (a)uL + P()u c (a) U,

where the firms control variables are utility payments (uL, uW), with uL = u(yL) and uW =u(yW), instead of monetary payments (yL, yW), and h = u1. I will use this problem to study theimpact of worker positive self-image on the firms profits when workers are risk averse for (i)complementarity, (ii) substitutability, and (iii) nonmonotonic relation between self-image andeffort.

5.1. Self-Image and Effort Complements. If self-image and effort are complements, then,for a fixed prize structure, positive self-image leads workers to exert more effort than theywould exert if they had accurate perceptions of skill. Furthermore, positive self-image relaxesthe workers participation constraint. This implies that the firm can implement the same ac-

tions with lower prizes or obtain more output for the same prizes. Thus, the firms profits arehigher in a tournament where workers overestimate their abilities and self-image and effort arecomplements than in a tournament with accurate workers.

PROPOSITION 3. If workers are risk averse and Pa 0, then the firms profits are higher in T()than in T(), with < < .

This result can be proved under very general conditions and does not depend on the particularassumptions of the specialized model. Appendix Section A.2 shows that one can use the theoryof supermodular games to show that if self-image and effort are weak complements and thereis a weak complementarity in workers effort choices, then the firms profits are higher with a

positive self-image workforce than with an accurate workforce.It is easy to find perceptions of skill and production functions that lead to a complementarity

between self-image and effort. For example, if output is uniformly distributed with support on[ai , ai + ], with > 0, each worker perceives his own output to be uniformly distributedwith support on [ai , ai + ], with < < , and the cost of effort function is sufficientlyconvex, then effort and self-image are complements.

5.2. Self-Image and Effort Substitutes. When self-image and effort are substitutes andworkers are risk averse, the firm may not be able to neutralize the unfavorable impact of posi-tive self-image on effort by raising the prize spread while simultaneously reducing prizes. This

happens because risk aversion implies that workers must be compensated for increases in the

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prize spread. However, we also know that, for fixed prizes, positive self-image workers find thetournament more attractive than accurate workers.

Now, consider the firms implementation problem. The firm selects prizes to induce a desiredlevel of effort subject to the individual rationality and incentive compatibility constraints. Thisopens the possibility that the firm, aware of workers positive self-image and of its unfavorable

effect in effort, may be able to choose a prize structure that implements the same effort levelthat the firm would like to implement if workers had accurate self-images and do it at a smallercost. My next result provides a condition under which the firm can do that.

PROPOSITION 4. If workers are risk averse and P()Pa (a,)1P() < Pa(a, ) < 0 for all a, then the

firms profits are higher in T() than in T(), with < < .

This result says that if workers are risk averse, effort and self-image are substitutes, and higherself-image leads to a moderate reduction in workers perceived marginal probability of winningthe tournament for all effort levels, then the firm is still better off with a positive self-imageworkforce than with an accurate workforce.

When workers are risk averse they dislike increases in the prize spread. This makes it costlyfor the firm to counter the unfavorable impact that positive self-image has on effort. However,if higher self-image leads to a moderate reduction in workers perceived marginal probability of

winning the tournament for all effort levels, P()Pa (a,)1P() < Pa(a, ), the firm can increase effort

by raising the utility prize spread while reducing prizes. The firm does this by reducing both thelosers and the winners prize in a way such that the reduction in the utility of the losers prizeis larger than the reduction in the utility of the winners prize. The reduction in prizes increasesfirms profits.

If workers arerisk averseand higherself-image leadsto a large reduction in workers perceived

marginal probability of winning the tournament for all effort levels, Pa 0,for all a, then a SB() < a SB() a F B and welfare is higher in T() than in T(), with < 1/2, and his ex ante actual expected utility is smallerthan his reservation utility.

The example illustrates a general result that does not require a formal proof. From the per-spective of an outside observer, positive self-image workers are worse off by comparison with

accurate workers since the firm will pay them less than their reservation utility.24

8. DISCUSSION

The main implication of this article for hiring decision by firms is that, everything else equal,firms should have a preference for hiring workers who overestimate skill when they use tourna-ments to provide incentives. In other words, if two job applicants have the same productivities,preferences toward risk, cost of effort, and outside options, then the firm should hire the onewho holds the most positive view of his skill.

In settings where performancedepends on ability, positive self-image leads individuals to over-

estimate the probability of favorable outcomes. If this is the case, individuals should, on average,prefer incentive schemes featuring payments contingent on relative performance (e.g., rank-order tournaments or incentive schemes composed partly by fixed pay and partly by variablepay dependent on the magnitude of relative performance) to individualistic incentive schemes(e.g., fixed salary plans or piece rates).

In this article, the firm is a monopsonist in the market for workers services. This assumptionimplies that the firm can make a take-it-or-leave-it offer to the workers and get all the surplusfrom the employment relationship. This assumption is appropriate when there is a large pool ofworkersandasmallnumberoffirms.Ifthereisalargenumberoffirmscompetingfortheservicesof a few workers, then it would be better to assume that the firm chooses tournament prizes tomaximize workers expected utility given incentive compatibility and zero-profit constraints. In

this case, there is no impact of worker mistaken beliefs on the firms profits since the zero-profitconstraint implies that workers get all the surplus from the employment relationship.25

The article studies tournaments with two workers and two prizes to make the analysis simpler.However, the results obtained extend to tournaments with more than two workers and morethan two prizes. The model also assumes that the firm faces a homogeneous workforce in termsof productivity and self-image.26 These assumptions simplify the firms problem by looking onlyat a representative worker. If one of these assumptions is dropped the tournament becomesasymmetric but the main findings will hold.

24 This is consistent with field and experimental data that show that mistaken perceptions of risk lead to financial

losses. See Camerer (1987), Simon and Houghton (2003), and Malmendier and Tate (2008).25 The impact of worker overestimation of skill on worker utility when firms compete for workers services depends

on worker preferences toward risk. If workers are risk neutral and have accurate perceptions of productivity, the

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APPENDIX

A.1.PROOF OF PROPOSITION 1. In order to show that a pure-strategy equilibrium exists I need to

show that (a) worker is strategy set is nonempty, convex, and a compact subset ofR and (b)worker is expected utility is continuous in ai and aj and quasi-concave in ai . Let us start byverifying (a). Worker is effort belong to the set [0,), which is not compact. However, forai too large costs must dominate benefits, so these strategies are dominated. This follows fromthe assumption that costs are convex. So, in effect, worker is effort will belong to a set [0, ai ],with ai finite, which is a nonempty, convex, and a compact subset ofR. Let us now verify (b).The assumptions that u, c, and P are twice differentiable imply that worker i s perceived ex-pected utility function is continuous in ai and aj in the set [0, ai ]. The assumption of strictlyconcave of expected utility in ai for all ai [0, ai ] implies quasi-concave of expected utility inai [0, ai ]. Thus, since all the required conditions are satisfied there exists a pure-strategy Nashequilibrium.27 The strict concavity of the expected utility function implies that the pure-strategyequilibrium is unique. The assumption that the global incentive compatibility condition is satis-

fied rules out a pure-strategy equilibrium with zero effort. Finally, the equilibrium is symmetric.Suppose, by contradiction, there exists an asymmetric pure-strategy Nash equilibrium such thata1 > a2. Then, by the workers first-order conditions, we have

Pa1 (a1, a2, )[u(yW) u(yL)] = c(a1),(A.1)

and

Pa2 (a2, a1, )[u(yW) u(yL)] = c(a2),(A.2)

with

Pai (ai , aj, ) = eiai (ai , )

gi (ej(aj) ei (ai , ) + j)gj(j) dj,

j= i = 1, 2. The assumption that the marginal productivity of effort is subject to diminishingreturns to effort implies that

e1a1

(a1, ) < e2a2 (a2, ) for a1 > a2.(A.3)

The assumption that Gi (qi | ei ) satisfies the monotone likelihood ratio property and the symme-try assumptions imply that

g1(e2(a2) e1(a1, ) + 2) < g2(e1(a1) e2(a2, ) + 1) for a1 > a2.(A.4)

It follows from (A.3) and (A.4) that

Pa1 (a1, a2, ) < Pa2 (a

2, a1, ).(A.5)

Dividing (A.1) by (A.2) and making use of (A.5) we obtain c(a1) < c(a2), which contradictsa1 > a2. The case a1 < a2 is similar.

PROOF OF PROPOSITION 2. Let a denote an arbitrary effort level that the firm can implementwhen workers are risk neutral and have accurate self-images. If workers are risk neutral and have


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beliefs of productivity given by and the firm selects a prize spread equal toy = c(a)/Pa(a, ),then the firm can implement effort level a. If workers have accurate self-images, the symmetry ofthe specialized tournament model implies that P() = 1/2, and the last term on the right-handside of (10) is zero. In this case, implementation cost is equal to C(a, ) = U+ c(a). If workershave positive self-image, P() > 1/2 and the last term on the right-hand side of (10) is negative.

Thus, C(a, ) < C(a, ), for all ( , ). PROOF OF PROPOSITION 3. Let a denote an arbitrary effort level that the firm can implement

when workers have accurate self-images. If workers have a degree of positive self-image givenby and the firm selects a utility prize spread equal to u() = c(a)/Pa(a, ), then the firmcan implement effort level a. If Pa = 0 then positive self-image has no impact on the incentivecompatibility constraint and there is no need to alter the prize spread. If Pa > 0 it follows thatu() < u() for all ( , ). If the prize spread decreases and workers are risk averse, theyface less risk. Furthermore, positive self-image relaxes the workers participation constraint.This implies that when Pa 0 the firm can implement the same effort with lower prizes for all

( , ).

PROOF OF PROPOSITION 4. Let a denote an arbitrary effort level that the firm can implementwhen workers are risk averse and have accurate self-images. If workers are risk averse andhave a degree of positive self-image given by and the firm selects a utility prize spread equalto u() = c(a)/Pa(a, ), then the firm can implement effort level a. I will now prove thatthe firm can lower implementation cost. Solving the incentive compatibility constraint and theparticipation constraint for the utility of the losing and winning prizes, we obtain

uL() = U+ c(a) P()

Pa (a, )c(a),

uW() = U+ c(a) + 1 P()Pa (a, )

c(a).

Implementation cost is given by

C(a, ) = 12

[h(uL()) + h(uW())].

The fact that P() is increasing with and the assumption that Pa(a, ) < 0 imply that

P()

Pa (a,) isincreasing with and so uL() is decreasing with . The assumption that P

()Pa (a,)1P() Pa(a, )

implies that 1P()Pa (a,)

is nonincreasing with . This in turn implies that uW() is nonincreasing with

. IfuL() is decreasing with and uW() is nonincreasing with , then C(a, ) < C(a, ) for all ( , ).

PROOF OF PROPOSITION 5. In order to prove this result I will show that the assumptions madeimply a SB() < a SB() a F B, for all < < min(, ). I will start by showing that if workershave increasing absolute risk aversion (u is concave) and hold accurate perceptions of skill, thena SB() < a F B.28


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490 SANTOS-PINTO

The second-best effort level and prizes with accurate workers, the triple (a SB(),xSB(), ySB()), are the solution to

maxa,x,y

(a,x, y) = B(a) y

s.t. Pa (a, ) [u(y + x) u(y x)] = c(a)1

2[u(y + x) + u(y x)] c(a) U.

At the optimal prize structure both constraints are binding. Since u is concave, a larger prize(increasing x but keeping y fixed) implies that each worker puts in more effort to increase theprobability of winning, that is, da/dx > 0.

The optimal reward for each worker in the first-best contract is independent of outcome,i.e., x = 0. The first-best effort level and reward, the pair (a F B, yF B), are the solution to

maxa,y (a, y) = B(a) y subject to u(y) c(a)

U. The first-order conditions to this problemare given by

u(yF B)B(a F B) = c(a F B)u(yF B) c(a F B) = U.

(A.6)

If a F B can be implemented by a tournament in the second-best setting, there must exist(x D, yD) such that

Pa(a F B, )[u(yD + x D) u(yD x D)] = c(a F B)1

2[u(yD + x D) + u(yD x D)] = u(yF B).

(A.7)

In general, a F B = a SB(). In order to compare the effort level at the second-best solution withthat at the first best, I consider the prize spread x D and the income yD that motivate the first-best level of effort in the second-best solution and see if the prize spread is too big or toosmall. If the sign of d/dx evaluated at (x D, yD) is negative, then the prize x D is too big andthe second-best level of effort is smaller than the first best. If the sign of d/dx evaluated at

(xD

, yD

) is positive, then the prize xD

is too small and the second-best level of effort is largerthan the first best.From the firms objective function we have that

d

dx= B(a) da

dx dy

dx.(A.8)

Differentiating the incentive compatibility and participation constraints we obtain

Pa

u(y + x)dydx + 1 u(y x)dydx 1 = c(a) dadx(A.9)


17/23


Solving (A.9) and (A.10) for dy/dx we find that

dy

dx=

c(a)c(a)

Pa S1

2u

1

2 S c(a)

c(a) Pau> 0,(A.11)

where S = u(y x) + u(y + x) and u = u(y + x) u(y x). From (A.10) we have

da

dx= 1

2c(a)

S

dy

dx+u

> 0.(A.12)

Substituting (A.12) into (A.8) we obtain

d

dx =B(a)

2c(a)S dy

dx +u

dy

dx

=

B(a)2c(a)

S 1

dy

dx+ B

(a)2c(a)

u.

Evaluating d/dx at (yD,x D) gives us

d

dx

(x D,yD)

=

B(a F B)2c(a F B)

S(x D, yD) 1

dy

dx+ B

(a F B)2c(a F B)

u

= S(x D, yD)

2u(yF B) 1 dy

dx +1

2u(yF B)u,

(A.13)

where the second equality comes from (A.6). I will now show that ifu is concave, then the signofd/dx at (yD,x D) is negative and so the second-best level of effort is less than the first best.The concavity ofu implies that u(x D, yD) < 0. We also know that dy/dx > 0. Thus, the signofd/dx at (yD,x D) is negative provided that S(x D, yD) < 2u(yF B) or

1

2[u(yD + x D) + u(yD x D)] < u(yF B).(A.14)

Since y is increasing with x we know that yD > yF B for x > 0. This together with concavity ofuimplies that

1

2[u(yD + x D) + u(yD x D)] < 1

2[u(yF B + x D) + u(yF B x D)],(A.15)

but ifu is concave we have that

1

2[u(yF B + x D) + u(yF B x D)] < u(yF B).(A.16)

Inequality (A.14) follows from (A.15) and (A.16).I will now show that ifC(a, ) < C(a, ) for all a, and Caa (a, ) > 0, then a

SB() < aSB().We know that a SB() is the solution to maxa0 B(a) C(a, ), and that a SB() is the solution


18/23

492 SANTOS-PINTO

a SB() = a F B guarantees that positive self-image is not so large as to lead the firm to choose asecond-best level of effort that is greater than the first best. Thus, under the assumptions madewe have that aSB() < a SB() a F B, for all < min(, ).

A.2. This appendix shows that if there is a weak complementarity in workers effort choices

and if workers self-image and effort levels are complements, then the firms profits are higherwith a positive self-image workforce than with an accurate workforce.

In order to prove this result, I need to provide conditions under which a workers effort isincreasing or decreasing with changes in positive self-image for fixed prizes. Worker is effortchoice problem, for a given realization of the common shock, is given by

maxaiAi

Ui (yL, ai ) + Pi (ai , aj, i ) Ui yW, aiUi yL, ai .(A.17)

Let

Ai

i , yL, yW arg max

aiAiVi

ai , aj, i , yL, yW

,

denote the set of maximizers in problem (A.17) as a function ofi , yL, and yW. For fixed prizes,the worker will never want to choose an infinite effort. So, the workers effort choice set iscompact. I also assume that Vi is order upper semicontinuous in ai . This assumption togetherwith the fact that the workers effort choice set is compact guarantees that the set of maximizersAi (i , yL, yW) is nonempty.

In order to make operational the view that higher self-image increases workers effort I usethe definition of increasing differences. This definition tells us that a function h : R2+ R has in-creasing differences in (x, ) if, for all x > x, the difference h(x, ) h(x, ) is nondecreasingin . The property of increasing differences represents the economic notion of complementar-ity.29 Athey et al. (1998) show that the set of maximizers defined by

X( ; z) arg maxxS

h (x, ) + z(x) ,

is nondecreasing in for all functions z, that is,

> implies X( ; z)S X ( ; z) ,

if and only if the function h : R2+ R has increasing differences in (x, ).30 This equivalence isused to state my first result. Define

Hi (ai , aj, i , yL, yW) Pi (ai , aj, i )

UiyW, a

iUi yL, ai .

LEMMA A.1. Ai (i , yL, yW) is nondecreasing in i if and only if Hi has increasing differences

in (ai , i ).

PROOF. An application of Theorem 2.3 in Athey et al. (1998).

29 Ifh is a benefit function and x > x, then the incremental benefit of increasing x to x is h(x, ) h(x, ). Ifhhas increasing differences in (x, ), then the incremental benefit from increasing x to x when = is higher than thei l b fi f i i h f


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Lemma A.1 states that if a workers self-image and effort are complements, then the higheris self-image the higher is a workers set of optimal effort choices. When self-image and effortare complements, an increase in self-image leads to an increase in effort, since the increase inthe perceived incremental probability of winning the tournament times the utility prize spreadis higher when self-image is higher.

We are interested in finding necessary and sufficient conditions on the structure of the effortchoice subgame that, together with the conditions found on the workers individual effort choiceproblems, allow us to know how the set of pure-strategy Nash equilibria effort levels changeswith positive self-image.

Milgrom and Roberts (1990) show that if the game is a supermodular game, then it hasa pure-strategy Nash equilibrium. Furthermore, they show that is a supermodular gamewhere the payoff functions are parameterized by ; then it is possible to provide compara-tive static results that link a change in with a change in the smallest and largest of Nashequilibrium of . I make use of these results to prove existence of equilibrium in the work-ers effort choice subgame and to state comparative static results relating the workers degreeof positive self-image to the smallest and the largest Nash equilibrium of the effort choicesubgame.

Let e(1, 2, yL, yW) = {{1, 2}, (Ai , Vi , i {1, 2}),} denote the simultaneous effort choice

subgame for levels of positive self-image (1, 2) and for prize structure (yL, yW). According toMilgrom and Roberts (1990), e is a supermodular subgame if (i) A

i is a compact interval in R,(ii) Vi (ai , aj) is order upper semicontinuous in ai for fixed aj and order continuous in aj forfixed ai , and Vi (ai , aj) has a finite upper bound, (iii) Vi has increasing differences in ai , and (iv)Vi has increasing differences in (ai , aj).

We see that e satisfies condition (i) since, for any finite prize structure, it is never optimalfor the workers to choose an infinite amount of effort. e also satisfies the first requirement ofcondition (ii) since we have assumed before that Vi is order upper semicontinuous in ai , i

=1, 2.

Condition (iii) is satisfied trivially since workers choice variables are scalars. So, for e to be asupermodular subgame we need to assume that it also satisfies condition (iv) and the second andthird requirements in condition (ii). The next result guarantees the existence of a pure-strategyNash equilibrium in e by imposing the remaining conditions that make it a supermodularsubgame.

LEMMA A.2. If Hi has increasing differences in (ai , aj), j= i, i = 1, 2, Vi is order continuousin aj for fixed ai , j= i, i = 1, 2, and Vi has a finite upper bound, then e(1, 2, yL, yW) has apure-strategy Nash equilibrium.

PROOF. The assumption that Hi has increasing differences in (ai , aj), j= i, i = 1, 2, impliesthat V

i

has increasing differences in (ai

, aj

), j= i, i = 1, 2, since the interaction between a1

anda2 in the workers interim perceived payoff functions is only through Hi . The assumption thatVi is continuous in aj and has a finite upper bound together with the fact that Vi has increasingdifferences in (ai , aj), j = i, i = 1, 2, imply that all the conditions required for e(1, 2) tobe a supermodular game are satisfied. But then, by Theorem 5 in Milgrom and Roberts (1990),e(

1, 2) has a pure-strategy Nash equilibrium.

The assumption that Hi has increasing differences in (ai , aj), j= i, i = 1, 2, imposes themissing structure in the e that, together with the two other assumptions, allows us to use theorder-theoretic approach to state this existence result. This assumption restricts the type ofinteraction between the workers choice variables by forcing a1 and a2 to be weak complements.

That is, I restrict attention to effort choice subgames where a workers increase in effort makesit more desirable for his opponent to increase effort too.

The assumptions that guarantee that condition (ii) is verified rule out the possibility that


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494 SANTOS-PINTO

satisfy condition (ii) and there is no pure-strategy equilibrium but there exists a mixed strategyequilibrium.31

LEMMA A.3. If e(1, 2, yL, yW) is a supermodular subgame and H

i has increasing dif-

ferences in (ai , i ), i = 1, 2, then the smallest and the largest pure-strategy Nash equilibria ofe(

1, 2, yL, yW) are nondecreasing functions of (1, 2).

PROOF. An application of Theorem 6 in Milgrom and Roberts (1990).

Lemma A.3 states that if there is a weak complementarity in the workers effort choices andif workers self-image and effort levels are complements, then the higher is the workers degreeof positive self-image the higher will be the smallest and the largest Nash equilibria effort levelsofe.

32 I use Lemma A.3 to characterize the impact of worker positive self-image on the firmsprofits when self-image and effort are complements.

THEOREM A.1. If e(1, 2, yL, yW) is a supermodular game and H

i has increasing differences

in (ai , i ), i = 1, 2, then the firms profits are higher in T(1, 2) than in T(1, 2), with i >i , i

=1, 2.

PROOF. We know from Lemma A.3 that if workers self-image and effort are comple-ments the smallest and the largest pure-strategy Nash equilibria ofe(

1, 2, yL, yW) are largerthan the smallest and the largest pure-strategy Nash equilibria of e(

1, 2, yL, yW). Further-more,the workers positiveself-image relaxesthe workers participation constraints. This impliesthatthefirmcanimplementthesameactionswithlowerprizesorobtainmoreoutputforthesameprizes. One way or the other the firms profits are higher in T(1, 2) than in T(1, 2).

A.3. This appendix shows that if output is exponentially or normally distributed, then thereis a nonmonotonic relation between effort and self-image but the firm can still be better off with

a positive self-image workforce.

PROPOSITION A.1. If workers are risk averse, the output of worker i has the exponential distri-bution with mean ai and worker i perceives his output to have the exponential distribution with

mean ai, then the firms profits are higher in T() than in T(1), for all1 < .

PROOF. In this case, P(Qii > qj) = exp{qj/ai }, and

P(ai , aj, ) = P(Qi i > Qj) = 1aj

+0

exp

qj

ai+ qj

aj

dqj = a

i

aj + ai .(A.18)

The cross partial of P(ai , aj, ) with respect to ai and is

Pai (ai , aj, ) = a

j(aj ai )(aj + ai )3 .

The sign of Pai is positive when aj/ai > and negative when aj/ai < . Now, let a denote

an arbitrary effort level that the firm can implement when workers are risk averse and have

31 In order to see this consider the extreme case where the distribution of the idiosyncratic shocks is degenerate, that

is, the outcome of the tournament is completely deterministic. In this case, as Nalebuff and Stiglitz (1983) point out,

each worker can assure that he wins the tournament by increasing his effort slightly above that of his opponent. Butthen, beyond some critical effort level, it is better to shirk and be certain to receive the losing prize than incurring in a

very high disutility of effort and capturing the winning prize. Although there exists no pure-strategy Nash equilibrium,

N l b ff d S i li (1983) h h i h i d l h i i d N h ilib i


21/23


accurate self-images. The assumption of symmetry and (A.18) imply that P() = /(1 + ) andPai (a, ) = /[a(1 + )2]. Thus, the utility of the losing and winning prizes is

uL() = U+ c(a) (1 + )ac (a) ,(A.19)

uW() = U+ c(a) +1 +

ac (a).(A.20)

The utility prize spread that implements effort level a is

u() = (1 + )2

ac(a).(A.21)

If workers are accurate, then = 1 and uL(1) = U+ c(a) 2ac(a), uW(1) = U+ c(a) +2ac (a), and u(1) = 4ac(a). If workers have positive self-image, then (A.19), (A.20), and(A.21) imply that the firm is able to implement effort level a by increasing the prize spread butsimultaneously reducing the winners and the losers prizes. This implies that C(a, ) < C(a, 1)for all > 1.

PROPOSITION A.2. If workers are risk averse, the output of worker i has the normal distributionN(ai , 2) , and worker i perceives his output to have the normal distribution N(ai , 2) ; thenthe firms profits are higher in T(, a) than in T(1, a), for all (, a) such that 1 < < and

a <

2

(1) .

PROOF. In this case, worker is unconditional perceived probability of winning the tournamentis given by

P(ai , aj, ) = P(Qii > Qj) = P(ai + i > aj + j)= P(ai aj > i j) = (ai aj),

where () is the distribution function of a normal random variable with mean 0 and variance22. The cross partial of P(ai , aj, ) with respect to ai and is

Pai (ai , aj, ) = 1

2

1 ai aj

22ai

exp

1

42(aiaj)2

.(A.22)

We see from (A.22) that the sign of Pai is positive when 22 > (ai aj)ai and negative

when 22 < (ai aj)ai .33 Now, let a denote an arbitrary effort level that the firm can imple-ment when workers are risk averse and have accurate self-images. The assumption of symmetry,(A.22), and that < imply that Pai > 0 at the symmetric equilibrium. It follows from Propo-sition 3 that the firm will reduce the mean utility prize and lower the prize spread when it wishesto implement a and the workforce has positive self-image. This implies that C(a, ) < C(a, 1),

for all (, a) such that 1 < < and a < 2(1)

.


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496 SANTOS-PINTO

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