The Economic Journal, Doi: 10.1111/ecoj.12473 © 2017 Royal Economic Society. Published by John Wiley & Sons, 9600 Garsington Road, Oxford OX4 2DQ, UK and 350 Main Street, Malden, MA 02148, USA.

THE PONDS DILEMMA* John Morgan, Dana Sisak and Felix V a rdy Is it better to be a big fish in a small pond or a small fish in a big pond? To find out, we study self-selection into contests. Our simple model predicts that: (i) entry into the big pond – in terms of show-up fees, number or value of prizes – is non-monotonic in ability; (ii) entry into the more meritocratic pond is likewise non-monotonic, exhibiting two interior extrema and disproportionately attracting very low ability types; and (iii) changes in rewards can produce unexpected effects, e.g. higher show-up fees may lower entry, while higher prizes or more meritocracy may lower the average ability of entrants.

Contests form an integral part of modern life; sometimes explicitly, as in innovation tournaments like the X-prize, but more often implicitly, as when individuals compete for promotions in an organisation. In a world where contests are ubiquitous, people often have to choose which contest to enter. This choice may confront them with the familiar ponds dilemma: is it better to be a big fish in a small pond or a small fish in a big pond? For instance, a golfer struggling on the PGA tour may well consider his options on the Asian tour. A freshly minted law graduate may have a choice between a ‘white shoe’ law firm in New York or a slightly less competitive and less lucrative firm in Philadelphia. A biotech start-up may have to decide whether to focus on a risky blockbuster drug or on a less risky and less profitable extension of an existing patent. Economically, the important point is that the structural properties of contests (ponds) – such as the number and value of prizes – not only affect behaviour within but also selection across contests. In this article, we present an analysis of the ponds dilemma in a simple and tractable framework. The parsimony of our modelling approach allows us to derive many testable and sometimes surprising predictions. For example: (i) entry into the big pond – i.e. the contest richer in show-up fees or in the number or value of prizes – is non-monotonic in ability; (ii) when the ponds differ in terms of meritocracy (i.e. discriminativeness), entry into the more meritocratic contest takes on two interior extrema, first reaching a minimum, then a maximum; (iii) all else equal, agents of very low ability disproportionately enter the more meritocratic contest. Nonetheless, this contest is exclusive, i.e. it attracts only a minority of the population; and (iv) changes in reward structures can produce unexpected selection effects. For instance, higher show-up fees may reduce entry, while higher prizes or more meritocracy may lower the average ability of entrants.

* Corresponding author: Felix V
ardy, International Monetary Fund, 700 19th St., Washington, DC, USA. Email: [email protected]. For valuable comments and suggestions we thank Thomas Chapman, Josse Delfgaauw, Paolo Dudine, Robert Dur, Bob Gibbons, Mitchell Hoffman, Sam Ouliaris, Sander Renes, two anonymous referees and seminar participants at Tinbergen Amsterdam, Bielefeld, Cambridge, CEU Budapest, LSE, NOVA Lisbon and Rotterdam. Dana Sisak gratefully acknowledges the financial support of the Swiss National Science Foundation through grant PBSGP1-130765. [ 1 ]

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When studying self-selection and entry, the usual first step is to postulate a fixed outside option available to all agents. We examine this scenario and show that the probability of entering the contest is strictly increasing in ability. In terms of the ponds dilemma, big fish tend to choose the big pond (i.e. they compete), while small fish tend to choose the small pond (i.e. they bow out). In light of this result, which aligns nicely with most people’s intuition, one may reinterpret the bulk of the literature on contests as pertaining to an imperfectly truncated distribution of ability types for whom participating in the contest is more profitable than a fixed outside option. This simple model of selection proves misleading when, rather than a fixed outside option, the alternative to participating in one contest is participating in another contest. When choosing between contests, agents weigh the potential rewards against the ability-dependent chances of success. The key observation is that, even though success probabilities in both contests are increasing in ability, the likelihood ratio is not. For those of extreme ability (high or low), the likelihood ratio is approximately one, since chances of success (or lack thereof) are essentially the same regardless of the pond chosen. For middling sorts, by contrast, the likelihood ratio favours the small pond. This non-monotonicity carries over to selection and, as a consequence, the common intuition for the ponds dilemma fails. Concretely, we study how a large, heterogeneous population of risk-neutral agents self-select across two mutually exclusive contests. We consider four dimensions in which the contests may differ: (i) (ii) (iii) (iv)

show-up fees; the number of (equal) prizes; the value of these prizes; and discriminativeness.

The last aspect corresponds to noisiness in performance evaluation and can be interpreted as a measure of meritocracy. While the existing literature is cognisant of the importance of discriminativeness in determining contestants’ behaviour, interpreting discriminativeness in terms of meritocracy and connecting it to entry is a contribution of our article. When agents self-select across contests, the endogenously determined ability distributions no longer correspond to simple, truncated versions of the population at large. Suppose, for instance, that the contests only differ in terms of show-up fees. Then the contest offering the higher show-up fee disproportionately attracts those of extreme ability (both high and low) and repels middling sorts. Thus, even if the underlying ability distribution is unimodal, the conditional distribution in the high show-up fee contest tends to be bimodal. Selection incentives are best understood by distinguishing between a parameter’s direct and indirect effect on payoffs. When contests differ only in terms of show-up fees, the direct effect of a higher show-up fee is to make the contest more attractive to all contestants. However, owing to the scarcity of prizes, more entry raises the performance standard required to succeed. This is the indirect effect. A higher standard is particularly costly for agents ‘on the bubble’, i.e. for middling sorts most © 2017 Royal Economic Society.

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uncertain about winning or losing. For them, the ratio of success probabilities favours the low-fee contest. Together, the direct and indirect effect cause middling sorts to be underrepresented in the high-fee contest, while extreme types are overrepresented. When contests differ in their number of (equal) prizes, only indirect effects are present. To see why, notice that the number of prizes has no direct effect on payoffs and, thus, affects behaviour only indirectly, through its effect on performance standards. Offering more prizes reduces a contest’s standard, which is most valuable for agents on the bubble but less relevant for infra-marginal types. As a result, middling sorts disproportionately enter the prize-rich contest, making abilities there more homogeneous than in the population at large. A difference in prize values has direct as well as indirect effects, both of which are type dependent. While show-up fees are equally valuable to all, higher prizes are most attractive to those anticipating to win, i.e. agents of high ability. For them, the positive direct effect of a higher prize dominates the negative indirect effect of a higher standard. As a result, high types are overrepresented in the high-prize contest. For middling sorts, it is the negative indirect effect that dominates. So they are overrepresented in the low-prize contest. Since agents of very low ability stand almost no chance of winning in either contest, they enter each contest with virtually equal probability. In other words, selection effects vanish in the lower tail. As we show, one noteworthy implication is that a contest may well raise the value of its prizes, only to see a fall in the average ability of its contestants. Perhaps the subtlest difference between contests lies in their degree of meritocracy. In our model, a decrease in meritocracy corresponds to a mean-preserving spread in the noisiness of performance evaluations. Since agents are risk-neutral, such a spread might seem immaterial. Indeed, meritocracy would not matter if measured performance and payoffs varied proportionately, as in a Roy (1951) model. In a contest, however, payoffs are a highly non-linear function of measured performance. To see the effect, notice that an increase in meritocracy reduces the chance that an agent’s performance is misevaluated. This is beneficial for high types, who worry about an evaluation that does not reflect their true ability, and detrimental to low types, who actually require a misevaluation in order to succeed. Nonetheless, selection still fails to be monotone since, at the extremes, agents care little about meritocracy. For very high types, even an adverse performance evaluation suffices to win, while for very low types, even an advantageous evaluation will not save the day. Hence, meritocracy does produce positive sorting but with waning power in the tails. We also show that, provided pecuniary motives dominate, the more meritocratic contest is exclusive; that is, it attracts only a minority of the population. Jointly, the loss of selection power in the tails and exclusivity have the counterintuitive implication that agents of very low ability disproportionately enter the more meritocratic contest. As with higher prizes, a rise in meritocracy may cause a drop in the average ability of contestants. Strictly speaking, the results and intuitions discussed so far pertain to contests differing in one dimension only. We also characterise selection when contests differ in multiple – or even all – dimensions simultaneously. In that case, show-up fees alone determine selection of very low types, while the sum of show-up fees and prize values © 2017 Royal Economic Society.

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determine the selection of very high types. Meritocracy shapes behaviour in-between these extremes, producing two interior extrema. Finally, the number of prizes affects selection only indirectly, through its effect on standards. Before proceeding, a comment on methodology is in order. In the extant literature on contests, the population of contestants is generally taken to be exogenous, while effort levels are endogenous. Initially, we focus on the polar opposite case, i.e. endogenous entry with exogenous effort. This allows us to derive crisp results with clear intuitions for all parameter values of the model. Subsequently, we show that our results carry over to environments with endogenous populations and endogenous effort, provided that structural parameters in the two contests are not too far apart. This implies that agents’ effort outlays do not differ too much depending on which contest they enter. While this approach allows us to make significant headway in analysing the ponds dilemma, it does have its limitations. For example, our model may fail to describe selfselection of law graduates between law firms in New York City and Peoria, say, because graduates expect to put in many fewer hours in Illinois than in New York. The same problem arises when analysing the choice between developing a new blockbuster drug or extending an existing patent. However, in respects other than effort, the two contests need not be close for our results to hold. For example, even if purses and prize structures of the PGA and Asian tours are very different, our results remain valid when effort is endogenous, because a golfer’s effort outlay will not depend much on which tour he chooses to enter. The article proceeds as follows. In Section 1, we introduce the baseline model with exogenous effort and endogenous selection. In Section 2, we prove existence of equilibrium. Section 3 first illustrates selection behaviour by means of a numerical example. It then proceeds with a formal analysis of selection, both for univariate as well as for multivariate differences between contests. In Section 4, we extend the model to allow for endogenous effort and show that our previous results carry over, provided the contests’ structural parameters are ‘close’. Section 5 discusses the related literature. Section 6 concludes. While intuitions for our results are provided in the main text, formal proofs have been relegated to the Appendices. There, we also study how our contest model with a continuum of players relates to the finite-player models of Tullock (1980) and Lazear and Rosen (1981). Mathematica code implementing the numerical examples is available from the authors.

1. Model Consider a unit mass of risk-neutral agents with heterogeneous abilities a 2 R: Abilities are distributed according to an atomless cumulative distribution function (CDF) G with strictly positive probability density function (PDF) g. Each agent must choose between two contests, 1 and 2. An agent of ability a entering Contest i 2 {1, 2} has measured performance yi ðaÞ; where: yi ¼ a þ e i : © 2017 Royal Economic Society.

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The random variable (RV) ei represents noise in performance measurement. Its dispersion typically differs across contests but not across agents within a contest, while its realisations are independent across contests and agents.1 We assume that the distribution of ei belongs to a location-scale family with location parameter zero and scale parameter ri [ 0: Noise ei admits a CDF F ðei =ri Þ with associated PDF ð1=ri Þf ðei =ri Þ. Density f () is assumed to be single-peaked around zero and strictly positive on R. Moreover, f is twice continuously differentiable and strictly log-concave. We interpret precision 1=ri as a measure of the meritocracy of the contest. Indeed, the greater ri , the less reliable the performance evaluation process, and, hence, the less an agent’s measured performance yi reflects his true ability a. Notice that this setup allows us to measure meritocracy by means of a single parameter, while it is still rich enough to encompass many, if not most, standard distributions.2 Sometimes we will assume that f also satisfies the following technical condition. C ONDITION 1. ðf 00 =f 0 Þ=ðf 0 =f Þ is strictly increasing in |ɛ| for ɛ 6¼ 0. Strict log-concavity of f is equivalent to ðf 00 =f 0 Þ=ðf 0 =f Þ\1: Hence, Condition 1 does not imply log-concavity, nor is it implied by it. It can perhaps best be interpreted as requiring that f becomes less log-concave when moving away from its peak. To the best of our knowledge, virtually all commonly used, strictly log-concave probability distributions satisfy this condition, including the normal, logistic, extreme value and Gumbel distributions.3 Condition 1 is only relevant to r1 6¼ r2 ; when it guarantees single-peakedness of the likelihood ratio kðaÞ  f ½ðh1  aÞ=r1 =f ½ðh2  aÞ=r2  for all h1 ; h2 2 R: Even then, its main role is to simplify exposition. For maximum clarity, we explicitly invoke the condition whenever we rely on it. Regardless of performance, an agent entering Contest i 2 {1, 2} receives a show-up fee wi  0.4 In addition, the agent earns a prize vi [ 0 iff he is among the winners of the contest. The set of winners in Contest i consists of the mass mi [ 0 of agents with the highest performance measures. Prizes are scarce overall, i.e. m1 þ m2 \1. We refer to show-up fees wi , values of prizes vi , number of prizes mi and measures of meritocracy ri as the structural parameters of the contests. Notice that the quantiles of measured performance among a continuum of agents are perfectly predictable. Hence, the 1 We have cast the model as one where performance is deterministic but noisily measured. Alternatively, one may suppose that performance itself is stochastic. The first interpretation is appropriate for settings where measurement is difficult or highly subjective. The second interpretation applies when actual performance is subject to random factors outside the control of contestants, such as in most sports. A combination of noisy performance and noisy measurement can also be accommodated. 2 In Appendix C, we study how our set-up relates to the finite-player models of Tullock (1980) and Lazear and Rosen (1981). 3 The only standard distributions we are aware of that do not satisfy Condition 1 are the Laplace, Pareto and lognormal distributions. None of these are admissible, however, because they violate strict log-concavity. One way to break Condition 1, while potentially still satisfying our other assumptions, is to have a singlepeaked density with multiple inflection points on each side of the peak. 4 Technically, negative show-up fees (i.e. entry fees) would not pose a problem. However, they do imply that expected payoffs are negative for agents of sufficiently low ability. In that case, one may want to extend the model to allow agents to earn zero by opting out entirely. While this complicates the analysis, entry into both contests remains non-monotonic in ability, ‘trailing off’ in the lower tail. (Cf. Proposition 9 in Appendix A.) Calculations are available from the authors upon request.

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condition for winning in Contest i is characterised by a deterministic performance threshold, or standard, which we denote by hi 2 ½1; 1Þ. To summarise, an agent of ability a choosing Contest i with standard hi enjoys an expected pecuniary payoff:
 hi  a pi ða; hi Þ ¼ wi þ vi F : ri Here, F  1  F denotes the decumulative distribution function (DCDF) of ei =ri . In addition to valuing money, agents also derive (potentially small) non-pecuniary payoffs from participating in each contest. These payoffs are idiosyncratic and might derive from the nature of the task required, the physical location of the contest, the personalities of the organisers and so on.5 Let d denote the difference in an agent’s non-pecuniary payoffs from participating in Contest 2 versus Contest 1. Hence, an agent non-pecuniarily prefers Contest 1 iff d ≤ 0. We assume that the realisations of d have full support on R and are i.i.d. across agents. Notice that this rules out situations where the task in one contest but not in the other is satisfying for high types, say, but much less so for low types. The distribution of d belongs to a location-scale family with location parameter and median s 2 R and scale parameter q > 0. Its CDF and associated PDF are Γ[(d  s)/q] and (1/q)c[(d  s)/q] respectively. Agents enter the contest that offers them the higher total expected payoff, which is equal to the sum of pecuniary and non-pecuniary payoffs. In case of indifference, they flip a coin. Let Hi ðaÞ denote the cumulative mass function (CMF) of endogenously determined abilities in Contest i, i.e. Hi ðaÞ is the measure of entrants into i with ability a or lower. Provided it exists, the corresponding mass density function (MDF) is denoted by hi ðaÞ, i.e. hi ðaÞ  dHi ðaÞ=da. Because we have normalised the population mass to 1, the CMFs in the two contests must add up to the CDF of abilities in the population as a whole. That is, 8a 2 R, H1 ðaÞ þ H2 ðaÞ ¼ GðaÞ. Moreover, lima!1 Hi ðaÞ must equal the fraction of the population entering Contest i, which we denote by Pri. Finally, let Gi ðaÞ denote the CDF of endogenously determined abilities in Contest i, i.e. Gi ðaÞ  Hi ðaÞ=Pri. The corresponding PDF is gi ðaÞ. To close the model, we offer a formal definition of market clearing and define equilibrium of the game as a whole. If fewer than a mass mi of agents enter Contest i, then hi ¼ 1 and all entrants receive a prize vi . In that case, we say that Contest i is uncompetitive. A contest is said to be competitive when strictly more than mi have entered. In a competitive contest, standard hi adjusts such that the mass of winners Wi – i.e. contestants whose performance exceeds the standard – equals the mass mi of prizes, i.e. hi solves:
 hi  a  F Wi ðhi Þ  dHi ðaÞ ¼ mi : ri 1 Z

1

(1)

5 In addition to realism, an advantage of including non-pecuniary payoffs is that they smooth out agents’ selection behaviour. That is, non-pecuniary payoffs make agents’ entry probabilities a continuous function of pecuniary payoffs, rather than a step function. As a consequence, entry probabilities not only indicate an agent’s preference for one contest over the other but also express the intensity of that preference.

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Agents simultaneously and independently choose which contest to enter. A Bayesian Nash equilibrium of the game consists of a tuple f½H1 ðaÞ; H2 ðaÞ; ðh1 ; h2 Þg of CMFs Hi ðaÞ and standards hi such that: (i) conditional on Hi , standard hi clears the market for prizes in Contest i; and (ii) profit maximising entry decisions induced by ðh1 ; h2 Þ give rise to CMFs fH1 ðaÞ; H2 ðaÞg.6

2. Equilibrium We solve for equilibrium in three steps. First we show that, conditional on entry decisions characterised by ðH1 ; H2 Þ, there exist unique performance standards ðh1 ; h2 Þ that clear the market for prizes in each contest. Second, we show that standards ðh1 ; h2 Þ induce a unique pair of CMFs ðH1 ; H2 Þ. Together, these two steps define a mapping from the space of standards into itself. Finally, we show that there exists a pair ðh1 ; h2 Þ that constitutes a fixed point of the system. Notice that such a fixed point gives rise to an equilibrium f½H1 ðaÞ; H2 ðaÞ; ðh1 ; h2 Þg. 2.1. Standards Conditional on Entry Using the market-clearing condition (1), our first Lemma shows that, for a given CMF of abilities Hi , a contest’s standard hi is uniquely determined. L EMMA 1. For every Hi , i 2 {1, 2}, there exists a unique standard hi 2 ½1; 1Þ that clears the market for prizes in Contest i. From the market-clearing condition (1), it is immediate that standards are higher when prizes are scarcer, all else equal. By contrast, conditional on Hi , neither wi nor vi have any influence on hi . The reason is that, in this version of the model, ‘effort’ is equal to ability and, hence, exogenous. 2.2. Entry Conditional on Standards We now derive the unique pair of CMFs ðH1 ; H2 Þ that result from standards ðh1 ; h2 Þ. For given ðh1 ; h2 Þ, an agent of ability a enters Contest 1 iff: d  p1 ða; h1 Þ  p2 ða; h2 Þ: Hence, Pri(a), the probability that the agent enters Contest i 2 {1, 2}, is

6 Notice that the analysis remains unchanged if agents choose their contest sequentially. The reason is that, due to the atomicity of agents, ‘unilateral’ deviations do not affect the payoffs of other agents. Hence, any Bayesian Nash equilibrium of the simultaneous game corresponds to a perfect Bayesian equilibrium of the sequential game and vice versa. For the same reason, we could allow agents to switch contests upon observing the contest choices – and even performance evaluations – of other agents. What we cannot allow for is switching upon observing one’s own performance evaluation.

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 p1  p2  s Pr1ðaÞ ¼ C ¼ 1  Pr2ðaÞ: q The uniquely determined MDF hi is then given by: hi ðaÞ ¼ g ðaÞPriðaÞ;

(2)

while the associated CMF Hi ðaÞ is found by integrating hi up to a. 2.3. Fixed Point The previous steps define a function, ξ, from the space of standards [∞, ∞) 9 [∞, ∞) into itself. Specifically, each pair of standards ðh1 ; h2 Þ gives rise to a unique pair of MDFs ðh1 ; h2 Þ according to (2). In turn, each pair of MDFs ðh1 ; h2 Þ with associated CMFs ðH1 ; H2 Þ give rise to a unique pair of standards ðh1 ; h2 Þ accordingR to Lemma 1. Endowing the set of MDFs with the k  kL 1 norm – i.e. khi kL1  hi ðaÞda – it is easily verified that these mappings are continuous. Finally, notice that the function ξ is bounded from above. To see this, observe that hi takes on its largest and finite value when all agents enter Contest i. We may conclude that ξ is a continuous function on a compact space. Brouwer’s fixed-point theorem then implies that ξ has a fixed point, which we denote by ðh1 ; h2 Þ. In turn, standards ðh1 ; h2 Þ induce a pair of (internally consistent) CMFs ðH1 ; H2 Þ. Hence, equilibrium exists. Whether it is unique remains an open question. A symmetric baseline refers to a situation where the values of the structural parameters w, m, v, and r are the same in both contests and, on average, the two contests are equally attractive in non-pecuniary terms, i.e. s = 0. In that case, equilibrium is unique and takes on a particularly simple form: P ROPOSITION 1. For all structural parameters, equilibrium exists. In a symmetric baseline, equilibrium is unique, standards are the same in both contests, and 50% of every ability type enter each contest.

3. Selection In this Section, we study the selection effects of differences in structural parameters across contests. To motivate our analysis, we begin by presenting an example that illustrates the workings of the model and the selection patterns that may arise. 3.1. Example 1 Suppose that ability is standard normally distributed, differences in non-pecuniary payoffs are d N(s = 0.05, q = 0.05), and noise in performance measurement is ei Logisticð0; ri Þ, i 2 {1, 2}.7 Let ðw1 ; w2 Þ ¼ ð1:1; 1Þ, ðm1 ; m2 Þ ¼ ð0:1; 0:2Þ, 7 In anticipation of revisiting the current example in the model with endogenous effort, we assume that noise is logistic rather than normal. Normal noise does not materially change the example. However, in the endogenous-effort model, the second-order condition for optimal effort is more easily satisfied with logistic than with normal noise.

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ðv1 ; v2 Þ ¼ ð1; 1:1Þ, and ðr1 ; r2 Þ ¼ ð0:6; 1Þ. That is, Contest 1 is more meritocratic than Contest 2 and pays a 10% higher show-up fee. However, Contest 2 offers twice as many prizes and their value is 10% higher. On average, Contest 2 is somewhat more attractive in non-pecuniary terms. Case 1: For the parameter values above, equilibrium standards are ðh1 ; h2 Þ ¼ ð1:04; 1:02Þ, and the fraction of the population entering each contest is (Pr1, Pr2) = (0.24, 0.76). Figure 1(a) depicts Pr1(a), the probability of entering Contest 1 as a function of ability. The Figure shows that selection is highly non-monotonic, with two interior extrema. The resulting PDFs, gi ðaÞ, are given in Figure 1(b). Notice that the distribution of abilities in Contest 1 is bimodal. That is, Contest 1 attracts both the best and the worst. Average abilities are ðE1 ½a; E2 ½aÞ ¼ ð0:44; 0:14Þ. Hence, average ability is higher in the contest offering fewer and lower-value prizes. Case 2: Now reduce q to 0.0005. This means that there is virtually no heterogeneity in how agents perceive the two contests in non-pecuniary terms. As a result, agents with the same ability enter the same contest, and selection is essentially deterministic. This is illustrated in Figure 1(c). Figure 1(d) depicts the resulting ability distributions in the two contests. Standards are ðh1 ; h2 Þ ¼ ð1:06; 1:06Þ, (Pr1, Pr2) = (0.18, 0.82) and ðE1 ½a; E2 ½aÞ ¼ ð0:85; 0:19Þ. Case 3: Next, suppose q is very large; say, 104 . In that case, the extreme dispersion of non-pecuniary payoffs dominates all other considerations. As a result,

Pr1(a) (a)

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Fig. 1 . Example 1 in Subsection 3.1 Notes. (a) and (c) Depict the probability of entering contest 1 as a function of ability when q = 0.05 and q = 0.0005 respectively. The resulting ability distributions are given in (b) and (d). (e) and (f) depict selection and ability distributions when contest 1 is so much more attractive in pecuniary terms that contest 2 is uncompetitive. Colour figure can be viewed at wileyonlinelibrary.com. © 2017 Royal Economic Society.

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selection is virtually indistinguishable from that in a symmetric baseline, with approximately 50% of every ability type entering each contest. Case 4: Finally, reset q to 0.05 and raise v1 from 1 to 15 and w1 from 1.1 to 2. In this case, ðh1 ; h2 Þ ¼ ð1:85; 1Þ and (Pr1, Pr2) = (0.82, 0.18). This means that there are fewer entrants into Contest 2 than there are prizes. Hence, Contest 2 is uncompetitive and all entrants earn w2 þ v2 ¼ 2:1 in pecuniary payoffs. The resulting selection behaviour is depicted in Figure 1(e). Notice that the probability of entering competitive Contest 1 is strictly increasing in ability. The resulting ability distributions are shown in Figure 1(f ). Average abilities are ðE1 ½a; E2 ½aÞ ¼ ð0:31; 1:43Þ. 3.2. Analysis In Example 1, multiple forces were at play simultaneously, resulting in the rather complex selection behaviour of Figure 1. In this Section, we disentangle these forces and show that the selection properties of the example are in fact generic. Before proceeding, we may dispense with one of the model parameters by observing that s 6¼ 0 is isomorphic to a difference in show-up fees, w. To see this, observe that:  

 1 h1  a h2  a   Pr1ðaÞ ¼ C w1  w2  s þ v1 F  v2 F : q r1 r1 Since only the net of w1  w2  s figures in this expression, in the remainder of the article, we normalise s to zero and incorporate into the show-up fees any median difference in non-pecuniary payoffs across contests. 3.2.1. Uncompetitive case As illustrated in Case 4 of Example 1, when one contest is overwhelmingly more attractive than the other, the less attractive contest obtains so few entrants that it becomes uncompetitive. That is, the number of prizes, mi , exceeds the number of entrants, Pri, and all entrants win a prize. (See Appendix B for details. Also, notice that at most one contest can be uncompetitive, since the population has unit mass while m1 þ m2 \1.) The next Proposition shows that, in this case, selection into the ‘big pond’ is always monotone in ability. As a result, ability distributions in the two ponds are ordered by first-order stochastic dominance (FOSD). P ROPOSITION 2. When one contest is uncompetitive, the probability of selecting into the competitive contest is strictly increasing in ability. The ability distribution in the latter FOSDs the one in the former. When q ? 0, sorting becomes deterministic. Agents enter the competitive contest iff their ability exceeds some threshold a 2 R. To see why agents of higher ability increasingly favour the competitive contest, recall that Pr1ðaÞ ¼ Cf½p1 ðaÞ  p2 ðaÞ=qg. When Contest 2 (say) is uncompetitive, all entrants into this contest obtain w2 þ v2 in pecuniary payoffs. So the payoff difference reduces to: © 2017 Royal Economic Society.

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  h a p1 ða; h1 Þ  p2 ða; 1Þ ¼ w1  w2  v2 þ v1 F 1 : r1 Ability affects payoffs only through the chance of winning in Contest 1, F½ðh1  aÞ=r1 , which is strictly increasing in a. Therefore, Pr1(a) is also strictly increasing. When q ? 0 the effect of non-pecuniary preferences vanishes, making the sign of p1 ðaÞ  p2 ðaÞ the sole selection criterion. As a result, Pr1(a) turns into a step function and selection becomes deterministic. In a competitive contest, the abilities of winners first order stochastically dominate (FOSD) the abilities of contestants. From Proposition 2, we know that contestants’ abilities in the competitive contest FOSD those in the uncompetitive contest. By transitivity, we may then conclude that winners’ abilities in the competitive contest FOSD the abilities of ‘winners’ in the uncompetitive contest (i.e. all entrants). Finally, notice that an agent’s pecuniary payoff in the uncompetitive contest is wi þ vi , regardless of ability. Hence, the uncompetitive case is isomorphic to a model with a single contest and a fixed outside option. We may conclude that, also in that case, the probability of entering the contest is strictly increasing in ability. 3.2.2. Competitive case The competitive case, and the main focus of our analysis, occurs when the number of entrants into each contest exceeds the number of prizes on offer. The following Proposition shows that this case pertains in and around symmetric baselines. P ROPOSITION 3. Both contests are competitive in a symmetric baseline and in a neighbourhood of structural parameters around it. Moreover, this competitive region remains non-degenerate when q ? 0. In a symmetric baseline, the two contests are equally attractive in pecuniary terms. As a result, 50% of each ability type enter each contest and, since m1 ¼ m2 \1=2; both contests are competitive. Proposition 3 shows that this situation extends beyond symmetric baselines, provided the structural parameters in the two contests are not too far apart. The remainder of the article focuses on the competitive case and, if necessary, constrains the parameter space accordingly. In some cases, as when contests only differ in meritocracy, no constraints are needed. (See Lemma 11 in Appendix B.) In other cases, as when contests differ in prize values or show-up fees, the structural parameters of the two contests cannot be too far apart (see, e.g. Example 1). While we do not repeat this condition at the beginning of each formal result, it should be understood that, from hereon, both contests are assumed to be competitive in equilibrium. Notice, however, that Proposition 2 applies whenever this assumption fails. Hence, we do provide a full characterisation of selection behaviour for all parameter values. When both contests are competitive, 1 \ h1 ; h2 \ 1. Hence, the payoff difference p1  p2 is: © 2017 Royal Economic Society.

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  h a h a p1 ða; h1 Þ  p2 ða; h2 Þ ¼ w1  w2 þ v1 F 1  v2 F 2 : r1 r2

(3)

The following remark characterises selection in the tails. R EMARK 1. Show-up fees determine selection behaviour of very low types, while the sum of show-up fees and prize values determine selection of very high types. Formally:  
 w1  w2 w1 þ v1  ðw2 þ v2 Þ lim Pr1ðaÞ ¼ C : and lim Pr1ðaÞ ¼ C a!1 a!1 q q The proof of Remark 1, which is omitted, follows immediately from (3) and the fact that Pr1ðaÞ ¼ Cf½p1 ða; h1 Þ  p2 ða; h2 Þ=qg. The intuition is straight forward. In anticipation of losing in either contest, show-up fees are the sole pecuniary consideration for very low types. By contrast, in anticipation of winning, very high types only consider the sum of show-up fees and prize values. They pecuniarily prefer whichever contest offers the higher total. The derivative dðp1  p2 Þ=da can be written as:
  dðp1  p2 Þ v1 h2  a v2 =v1 ¼ f kðaÞ  : da r2 r1 r2 =r1

(4)

Recall that kðaÞ  f ½ðh1  aÞ=r1 =f ½ðh2  aÞ=r2  is the likelihood ratio of agent a just meeting the standard in each contest. Equation (4) reveals that the shape of p1  p2 , and hence of Pr1(a), crucially depends on the properties of k(a). Let k  inf a2R kðaÞ and  k  supa2R kðaÞ. The following Lemma shows that, for r1 ¼ r2 , k(a) is strictly monotone in ability and takes on values on either side of 1. This result relies on the log-concavity of f . For r1 \r2 , log-concavity implies that k(a) goes to zero in the tails. It does not pin down the number of interior extrema, however. This is where Condition 1 comes into play. Condition 1 guarantees that k(a) is single-peaked. L EMMA 2. Properties of k(a): (i) If r1 ¼ r2 then: [ [ (a) k0 ðaÞ ¼ 0 iff h1 ¼ h2 ; \ \ (b) for h1 6¼ h2 , k \ 1 \  k. (ii) If r1 \r2 then: (a) limjaj!1 kðaÞ ¼ 0 ¼ k; (b) for h1 6¼ h2 ,  k [ kðh1 Þ [ 1; (c) if f satisfies Condition 1, then k(a) is single-peaked in a 2 R. Together, Lemma 2 and (4) imply the following for selection behaviour as a function of ability. © 2017 Royal Economic Society.

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P ROPOSITION 4. Let r1 \r2 while the other structural parameters are arbitrary. If Condition 1 holds, then Pr1(a) either takes on two extrema, first a minimum and then a maximum, or is strictly decreasing. Equation (4) implies that whether Pr1(a) takes on two extrema or is strictly [  decreasing turns on ðv2 =v1 Þ=ðr2 =r1 Þ \ k. For ðv2 =v1 Þ=ðr2 =r1 Þ sufficiently close to 1, Pr1(a) has two extrema because  k [ 1. By contrast, for ðv2 =v1 Þ=ðr2 =r1 Þ 1, Pr1(a) is strictly decreasing. Characterising the exact boundary between the two cases in terms of primitives is extremely difficult, because  k depends on the standards ðh1 ; h2 Þ, which are endogenous. Proposition 4 assumes that Condition 1 holds and limits attention to the generic case where r1 6¼ r2 . When Condition 1 fails, k(a) may take on multiple interior extrema. In turn, this implies that Pr1(a) may take on up to twice the number of extrema of k(a). The knife-edge case where contests differ in multiple dimensions but are equally meritocratic is relegated to Appendices. (See Proposition 8 in Appendix A.) At the outset, we presented Example 1. It illustrated that, despite the relative simplicity of our model, selection behaviour in the ponds dilemma can be quite complex. Proposition 4 establishes that the ‘bimodal’ selection patterns of Cases 1 and 2 are, in fact, generic. Case 3’s selection profile for q ? 0 also generalises: indeed, it is easily verified that the ability space R can be partitioned into at most four intervals such that, in the limit, types belonging to the same interval enter the same contest, while types belonging to adjacent intervals enter different contests. 3.2.3. Isolating selection effects While Proposition 4 usefully describes the broad shape of the selection function, Pr1(a), in the way of intuition it has little to offer. To better understand why agents behave the way they do, we now study the selection effects of each structural parameter in isolation. That is, we analyse selection across contests that are identical in all respects save one. Together, Figure 2 and Remark 2 summarise our findings. Remark 2 is a corollary of a sequence of Propositions that have been relegated to Appendices. (See Propositions 9 to 12 in Appendix A.) R EMARK 2. Suppose contests are identical in all dimensions save one. Then selection is as follows: (i) w: Higher show-up fees attract the best and the worst, while repelling the middle; (ii) m: More prizes attract middling abilities, while not affecting selection of the best and the worst; (iii) v: Higher prizes attract the best, while not affecting selection of the worst. Middling abilities tend to be repelled; and (iv) r: Greater meritocracy attracts high types and repels low types. However, these selection effects dissipate towards the tails. One by one, we now discuss the intuitions behind these selection effects. © 2017 Royal Economic Society.

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(a)

Pr1(a) (b)

1

1

1 2

1 2

w1

m1

w2

m1 a

a (c)

(d)

1

1 2

1

1 2

v1

v2

σ1

σ2

Fig. 2. Probability of Entering Contest 1 When the Contests Differ in One Dimension Only Note. Colour figure can be viewed at wileyonlinelibrary.com

Show-up fees Suppose that the two contests are identical except for their show-up fees. From Proposition 2, above, we already know that Contest 1 benefits from positive selection when w1 is so much larger than w2 that Contest 2 is uncompetitive. Selection is more nuanced in the competitive case. To see why, start from a situation where the two contests are identical and suppose that Contest 1 raises its show-up fee. This makes Contest 1 more attractive to agents of all abilities, who now enter in larger numbers. For both markets to clear, Contest 1’s equilibrium standard must rise and Contest 2’s must fall. Since the standard in Contest 1 has risen in tandem with the show-up fee, agents now face a clear trade-off: the higher show-up fee in Contest 1 (the big pond) must be weighed against the lower standard – and hence better chance of winning – in Contest 2 (the small pond). A common intuition for the ponds dilemma is that only the ablest should enter the big pond: ‘if you can’t stand the heat, stay out of the kitchen!’ While it is true that the ablest suffer little from heightened competition, so do the least able, however. For both types, a difference in standards is of little import because only extremely unlikely realisations of ei affect their almost pre-ordained success or failure. Hence, agents of extreme ability (both high and low) tend to opt for the contest with the higher show-up fee – i.e. the big pond. Not so for agents of intermediate ability, whose chances of success are noticeably hurt by a higher standard. They tend to opt for the contest with the lower show-up fee – i.e. the small pond. As a result, Pr1(a) is U-shaped. This is illustrated in Figure 2(a), which depicts the propensity to enter the big pond as a © 2017 Royal Economic Society.

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function of ability. Notice that our finding and the underlying intuition are reminiscent of Coate and Loury (1993), who show that differential hiring standards have the greatest impact on human capital acquisition of intermediate types. Because of its higher standard and greater show-up fee, we have referred to Contest 1 as the big pond. However, the following example shows that the ‘big pond’ may in fact be smaller than the ‘small pond’. That is, despite its higher show-up fee, Contest 1 may attract only a minority of agents. The driving factor is the mass of middling sorts, who are repelled by the big pond’s higher standard. The identity and size of this group depends on the number of prizes on offer and the shape of the ability distribution of the population as a whole. The upshot is that a contest may well raise its show-up fee, only to see participation decline. E XAMPLE 2. Let a N(0, 1), d N(0, 0.05), and ei Logisticð0; 1Þ. If w1 ¼ 1:1 [ 1 ¼ w2 , mi ¼ 0:4 and vi ¼ 4, i 2 {1, 2}, then Pr1 = 0.44 < 0.56 = Pr 2. Number of prizes While it is easy to see that offering more prizes increases entry, the selection effects are less clear. Who are these new entrants? Because contestants do not care about the number of prizes per se, offering more prizes only has an indirect effect, namely, a reduction in the performance standard. This is valuable regardless of ability, though more so for intermediate types, whose chances of winning improve the most. Hence, an increase in the number of prizes, mi , unambiguously raises entry of all ability types into Contest i, but especially of middling sorts. More formally, suppose that m1 [ m2 while the contests are otherwise identical. It is easily verified that limjaj!1 Pr1ðaÞ ¼ Cð0Þ ¼ 1=2. This reflects that agents of extreme ability do not care about standards. The derivative dðp1  p2 Þ=da is the same as in the show-up fee case. However, because the order of standards is reversed (i.e. h1 \h2 ), p1  p2 and Pr1(a) are inverse-U-shaped rather than U-shaped. The resulting selection pattern is depicted in Figure 2(b). Owing to its higher standard, one might consider Contest 2 to be the big pond. Notice, however, that it does not offer higher rewards as compensation. Therefore, the ‘big pond’ repels all ability types, but to differing degrees, depending on their relative chances of success in the two contests. Value of prizes The canonical ponds dilemma arises when contests differ in prize values. Naturally, higher prizes lead to an inflow of contestants and, hence, to a higher performance standard. Thus, as was the case for show-up fees, contestants face a tradeoff between payoffs and standards. However, in this case, both the costs and the benefits of entering the high-prize contest are ability dependent. While show-up fees are equally valuable to all, the expected benefit of a higher prize is proportional to an agent’s probability of winning. Therefore, all else equal, a higher prize in Contest 1 makes Pr1(a) strictly increasing in a. The cost of a higher standard continues to be greatest for intermediate types. Together, the two effects make Pr1(a) U-shaped with the right asymptote exceeding the left, i.e. lima!1 Pr1ðaÞ [ lima!1 Pr1ðaÞ.8 The resulting selection pattern is depicted in Figure 2(c). 8 To be precise: when v2 =v1 is sufficiently lopsided and k(a) is bounded, it may happen that Pr1(a) is strictly increasing rather than U-shaped. See Proposition 11 in Appendix A for details.

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As always, those of extreme ability are unaffected by the difference in standards. Yet, entry decisions differ markedly between the top and the bottom. For those at the bottom, prize differences are irrelevant because prizes are unattainable. Therefore, they perceive the two contests as equally attractive, leading to a 50–50 split. Those at the top are virtually guaranteed to win a prize in either contest. Therefore, they are much more likely to opt for high-prize Contest 1, i.e. the big pond. Still, in general, selection into the big pond fails to be monotone. As with show-up fees, the ratio of success probabilities favours the small pond for middling sorts, and this consideration tends to dominate their entry decisions. The key insight is that, regardless of whether the riches in the big pond come in the form of contingent prizes or non-contingent show-up fees, selection tends to non-monotonic in ability. (See Proposition 11 in Appendix A for details.) Meritocracy We now examine how meritocracy drives selection. For risk-neutral agents, noise in performance evaluation might seem irrelevant, because it does not affect expected performance. The flaw in this reasoning is that measurement errors have asymmetric effects, which depend on an agent’s ability relative to the standard. When an agent’s ability falls below the standard, he can only succeed if he gets a ‘lucky break’, i.e. a positive realisation of ei . When his ability exceeds the standard, he can only fail if he suffers an ‘unlucky break’, i.e. a negative realisation of ei . In the former situation, the agent seeks out noisy measurement, since therein lies his only path to success. In the latter, he avoids noisy measurement, since it constitutes his only possible undoing. Even in this case, selection is not monotone, however. To see why, recall that individuals of extreme ability – both high and low – are essentially unaffected by measurement noise, since only extremely unlikely realisations of ei can alter their almost pre-ordained success or failure. As the contests are identical in all other respects, these types enter each contest with almost equal probability. Hence, meritocracy does produce favourable selection but with waning power in the tails. This is illustrated in Figure 2(d).9 As it is more harshly revealing of true performance, we may consider the more meritocratic contest to be the big pond. Still, rewards are the same in both contests, while standards cannot be ranked. To see why either contest can have the higher standard, notice that most agents need a lucky break when prizes are scarce. This induces the bulk of the population to opt for the noisy contest and, as a result, the less meritocratic contest has the higher standard. On the other hand, when prizes are plentiful, most agents merely need to avoid an unlucky break. This induces them to opt for the more meritocratic contest, resulting in the opposite ranking of standards. Deciding which contest to enter is most complicated for types who need a lucky break in one contest but need to avoid an unlucky break in the other contest, i.e. types a 2 ½minfh1 ; h2 g; maxfh1 ; h2 g. Their predicament blunts payoff differences and makes selection less pronounced. To see why, suppose the more meritocratic contest

9 In the Figure, we have assumed that Condition 1 holds, such that k(a) is single-peaked. Otherwise, Pr1(a) may exhibit multiple extrema on either side of a~, the point, where Pr1(a) single-crosses 1/2. (See Proposition 12 in Appendix A.)

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also has the higher standard. In that case, the agent needs a lucky break in the more meritocratic contest, while he needs to avoid an unlucky break in the less meritocratic contest. Since neither contest is likely to produce the desired result, there is little to distinguish between them. Alternatively, when the more meritocratic contest has the lower standard, the agent needs a lucky break in the less meritocratic contest, while he needs to avoid an unlucky break in the more meritocratic contest. Since both contests are likely to produce the desired result, again, there is little to distinguish between them. Thus, selection is weak in this region. In Proposition 12 in Appendix A, we prove that the more meritocratic contest is exclusive. That is, provided pecuniary motives dominate, the more meritocratic contest attracts only a minority of the population. The intuition is as follows. As illustrated in Figure 2(d), agents are more likely to enter the more meritocratic than the less meritocratic contest iff their ability exceeds some threshold, a~. The threshold is such that an agent of ability a~ is equally likely to win in either contest. Within each contest, the probability of winning is strictly increasing in ability. Hence, agents who choose to enter the more meritocratic contest (tend to) have a better chance of winning than those entering the less meritocratic contest. This implies that, for a given mass of entrants, the more meritocratic contest produces more winners than the less meritocratic contest. Because the number of prizes is the same in both contests, the more meritocratic contest must attract fewer entrants. In other words, it is exclusive. Together, exclusivity and dissipation of selection power in the tails – i.e. lima!1 Pr1ðaÞ ¼ 1=2 – imply that very low types disproportionately enter the more meritocratic contest. Perversely, this means that the average ability in the more meritocratic contest may be lower than in the less meritocratic contest.10 Combining the uncovered selection effects provides an intuitive understanding of selection behaviour: Show-up fees alone determine selection of very low types, while the sum of show-up fees and prize values determine the selection of very high types. Meritocracy shapes behaviour in between these extremes, generally producing two interior extrema. Finally, the number of prizes affects selection only indirectly, through its effect on standards. 3.2.4. Stochastic ordering of abilities and limit behaviour for q → 0 Finally, we consider the implications of self-selection for the distribution of abilities across contests and study selection behaviour when non-pecuniary preferences vanish, i.e. q ? 0. From Proposition 2, we already know that, in the uncompetitive case, abilities in the competitive contest FOSD abilities in the uncompetitive contest. Hence, in line with common intuition, the ‘big pond’ attracts the best-and-the-brightest, while the ‘small pond’ is a refuge for low types. For the competitive case with w1 [ w2 , we have seen that Pr1(a) is U-shaped. This suggests that abilities in Contest 1 are more dispersed than in Contest 2. Similarly, the inverse-U-shape of Pr1(a) when m1 [ m2 suggests that, in this case, abilities in Contest 2 are more dispersed than in Contest 1. Notice, however, that the contests’ ability 10 For example, let a N(0, 1), d N(0, 0.05), and ei Logisticð0; ri Þ. If wi ¼ 1, mi ¼ 0:01, vi ¼ 1, i 2 {1, 2}, and ðr1 ; r2 Þ ¼ ð0:3; 1Þ, then E1 ½a ¼ 0:023\0:017 ¼ E2 ½a.

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distributions cannot be ranked by second-order stochastic dominance (SOSD), since neither case allows for a consistent ranking of average abilities. To remedy the situation, we introduce the concept of single-crossing dispersion (Ganuza and Penalva, 2006). D EFINITION 2. An RV with CDF J1 ðaÞ is more single-crossing (SC) dispersed than a RV with ð[Þ

CDF J2 ðaÞ iff there exists a unique a 0 2 R such that J1 ða 0 Þ ¼ J2 ða 0 Þ and, 8a \ a 0 , ð[Þ

J1 ðaÞ \ J2 ðaÞ. For example, consider two RVs drawn from normal distributions with different means and different variances. It may be verified that, regardless of the ranking of means, the distribution with the higher variance is more SC-dispersed than the distribution with the lower variance. In the next Proposition, we show that abilities of entrants in the two contests can indeed be ranked according SC dispersion, at least for small q. Furthermore, this ranking carries over to the populations of winners.11 P ROPOSITION 5. Let w1 [ w2 while the contests are otherwise identical. For small q > 0, the abilities of entrants (winners) in Contest 1 are more SC-dispersed than the abilities of entrants (winners) in Contest 2. Let m1 [ m2 and q small. Ceteris paribus, abilities of entrants (winners) in Contest 1 are less SC-dispersed than abilities of entrants (winners) in Contest 2. By attracting extreme types and repelling middling sorts, the contest with the higher show-up fee or fewer prizes attracts the more diverse talent pool. This selection pattern carries over to winners: abilities of winners are more SC-dispersed in the contest with the higher standard. For the case of show-up fees, this phenomenon is most cleanly captured in the limit when non-pecuniary considerations vanish. Such an analysis is also of independent interest, since non-pecuniary payoffs are mostly absent from the extant literature. When q ? 0, we find that entry becomes deterministic, leaving a ‘hole’ in the ability distribution of the contest offering the higher fee. That is, for w1 [ w2 , extreme types enter Contest 1 while middling sorts enter Contest 2. Notice however that, unlike entrants, winners in each contest continue to be stochastically determined. When contests differ in the number of prizes, entry remains strictly stochastic (and U-shaped) when q ? 0. This result is driven by a no-arbitrage condition which implies that, in the limit, the contests’ standards must be the same. Otherwise, all agents would choose the contest with the lower standard, which is inconsistent with it having the lower standard in the first place. When both contests have the same standard, there is no particular reason for agents with the same ability to enter the 11 For arbitrary sets of probability distributions, the concept of SC dispersion has the serious drawback that it may violate transitivity. This problem does not arise in our setting, however. To see this, fix a set of contests whose structural parameters are identical save for their show-up fees or number of prizes. Proposition 5 implies that the set can be completely ordered on the basis of SC dispersion (  SC ). Specifically, under endogenous sorting between contests (i, j), Gi ðÞ  SC Gj ðÞ iff wi  wj or mi  mj , respectively.

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same contest, even when q ? 0. However, equal standards and m1 [ m2 do imply that more contestants enter Contest 1 than Contest 2. For q ? 0, a particular ‘mixed’ entry pattern is selected among a continuum of patterns consistent with the requirements of equal standards and market clearing. Formal statements and proofs of these claims have been relegated to Appendices. (See Propositions 13 and 14 in Appendix A.) Next we turn to differences in prize values. Agents are more likely to enter the highprize contest than the low-prize contest iff their ability exceeds some threshold, a^. Hence, it would seem that high types are overrepresented in the high-prize contest and low types in the low-prize contest. However, this ignores the base rate of selection into the two contests. Relative to its population share, a type is overrepresented in Contest i iff its propensity to enter, Pri(a), is greater than the average propensity, Pri. Therefore, if v1 [ v2 , high types are indeed overrepresented in high-prize Contest 1. However, if Pr1 < 1/2, so are very low types, since they enter both contests with equal probability. The potential overrepresentation of low types in the high-prize contest implies that ability distributions in the two contests cannot be ranked by FOSD. In fact, average ability in the high-prize contest may well be lower than in the low-prize contest, as the following example illustrates. E XAMPLE 3. Suppose a N(0, 1), d N(0, 0.05), ei Logisticð0; 0:3Þ. Let v1 ¼ 1:1 [ 1 ¼ v2 , wi ¼ 1, and mi ¼ 0:05, i 2 {1, 2}. Then E1 ½a ¼ 0:023\0:020 ¼ E2 ½a. Hence, the high-prize contest attracts individuals of lower average ability. Nonetheless, for small q, it is still true that a random individual with ability greater than a^ is much more likely to enter the high-prize contest, while a random individual with ability smaller than a^ is much more likely to enter the low-prize contest. In turn, it implies that winners in the high-prize contest tend to be of higher ability than in the low-prize contest. One formalisation of this idea is to compare quantiles of entrants and winners across contests. For example, we can ask how the ability of the (lowest) 1st percentile of entrants into the high-v contest compares to the ability of the (highest) 99th percentile of entrants into the low-v contest. As we show, for small q, the former exceeds the latter with probability one. The same is true when comparing the populations of winners. In fact, Proposition 6 generalises this idea to arbitrary quantiles. P ROPOSITION 6. Let v1 [ v2 while the contests are otherwise identical. For any  [ 0 such that for all 0  q\ 0 \ p1 ; p2 \ 1, there exists a q q the following holds: with probability 1, a contestant at the p1 -th ability-quantile in Contest 1 has strictly greater ability than a contestant at the p2 -th ability-quantile in Contest 2. The same ranking applies to the abilities of winners in the two contests. When contests differ in terms of meritocracy, Pr1(a) single-crosses 1/2 from below. Hence, for small q, we can once again rank the abilities of entrants – as well as winners – in the two contests at arbitrary quantiles. We omit a formal statement and proof of this result, since it is analogous to Proposition 6. © 2017 Royal Economic Society.

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3.3. Grading on the Curve Rather than fixing the number of prizes, contest organisers sometimes fix the fraction of contestants who win a prize. For example, a student may have to be in the top 20% of his class to get an A or, as was the case under Jack Welch’s ‘Rank-andYank’ system at General Electric, the bottom 10% of the workforce is fired. We now analyse the ponds dilemma under this alternative scenario, which we refer to as ‘grading on the curve’. Formally, suppose that fractions l1 ; l2 2 ð0; 1Þ of top performers in Contests 1 and 2 win a prize. Then, for i 2 {1, 2}, the market clearing conditions become:
 hi  a  F Wi ðhi Þ ¼ dHi ðaÞ ¼ mi ¼ li Pr i: ri 1 Z

1

For the rest, the model remains unchanged. First notice that, as before, equilibrium is characterised by a tuple of mutually consistent CMFs and performance standards f½H1 ðaÞ; H2 ðaÞ; ðh1 ; h2 Þg. Lemma 7 in the Appendix establishes that such an equilibrium indeed exists. Next notice that both contests are always competitive, since 0 \ l1 ; l2 \ 1 and q > 0. Finally, observe that the payoff difference p1 ða; h1 Þ  p2 ða; h2 Þ is the same as in the baseline model and given by (3). Hence, for given standards ðh1 ; h2 Þ, Pr1(a) remains unchanged. It is easily verified that Proposition 4 – which characterised the shape of the selection function Pr1(a) for contests that differ in meritocracy – did not depend on the values of ðh1 ; h2 Þ or their ranking. Hence, the Proposition carries over to the model with grading on the curve. The intuition is that standards play no role in the tails of the ability distribution and that the effect of differences in meritocracy, which shape behaviour in between these extremes, does not depend on standards either. The latter observation also explains why Panel (d) in Figure 2 remains unchanged when we move from the baseline model to grading on the curve. Unlike in Panel (d), the shape of Pr1(a) in Panels (a, b), and (c) does depend on the rankings of standards, which in the baseline model are unambiguous. Indeed, Panel (b) is inverse-U -shaped precisely because h1 \h2 when m1 [ m2 , while Panels (a) and (c) are U-shaped because h1 [ h2 . If these rankings are carried over to the model with grading on the curve, so would the shapes of the selection curve (with l1 ; l2 replacing m1 ; m2 ). However, this is not the case. In fact, even in a symmetric baseline, standards cannot be ranked when grading is on the curve. We show this by means of an example. E XAMPLE 4. Suppose grading is on the curve. Let a N(0, 1), d N(0, 0.05) and ei Logisticð0; 1Þ, while li ¼ 0:7 and wi ¼ 1 ¼ vi , i 2 {1, 2}. This symmetric baseline has the following equilibria: (i) h1 ¼ h2 ¼ 1:02, Pr1(a) = 1/2, and E1 ½a ¼ 0 ¼ E2 ½a. (ii) ðh1 ; h2 Þ ¼ ð1:02; 0:62Þ, (Pr1, Pr2) = (0.9994, 0.0006) and ðE1 ½a; E2 ½aÞ ¼ ð0:001; 1:51Þ. Because h1 \h2 , Pr1(a) is inverse-U-shaped. (iii) ðh1 ; h2 Þ ¼ ð0:62; 1:02Þ, (Pr1, Pr2) = (0.0006, 0.9994) and ðE1 ½a; E2 ½aÞ ¼ ð1:51; 0:001Þ. Because h1 [ h2 , Pr1(a) is U-shaped. © 2017 Royal Economic Society.

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Example 4 illustrates that equilibrium standards can differ across ex ante identical contests. To understand why this may happen, notice that the mass of entrants does not affect competitiveness when grading is on the curve. Rather, competitiveness and standards are determined by the average ability of those who do enter. When l is large, it is the low-ability types who are uncertain about winning or losing. They are disproportionately attracted to the contest which happens to have the lower standard. This reduces the average ability of agents in the low-standard contest and raises it in the high-standard contest, further lowering standards in the former and raising standards in the latter. It constitutes a self-reinforcing process that can give rise to (very) asymmetric equilibria. The implicit function theorem implies that, also in a neighbourhood of a symmetric baseline, standards cannot be ranked . Hence, when grading is on the curve and the fraction of winners is high, Panels (a), (b) and (c) may be U-shaped or inverse-Ushaped, depending on which contest happens to have the higher standard. To conclude, when contests differ in meritocracy, the selection pattern under grading on the curve is qualitatively the same as is in the baseline model. When the two contests are equally meritocratic, selection patterns may differ across equilibria and can deviate from the patterns found in the baseline.

4. Endogenous Effort In the baseline model of Section 1, ‘effort’ was exogenous and equal to ability. Yet, in practice, agents’ effort levels may vary with the structural parameters of the contest. Moreover, anticipated effort may play a role in deciding which contest to enter. Therefore, we now add endogenous effort back into the model. As before, a unit mass of agents chooses between two contests, 1 and 2. The determination of success and failure in each contest is analogous to the earlier model. However, measured performance now depends on endogenous effort rather than exogenous ability. Specifically, measured performance of an agent who exerts effort x 2 [∞, ∞) in Contest i is: yi ¼ x þ ei :

(5)

Our assumptions on ei are the same as before.12 For an agent of ability a 2 R, the cost of exerting effort x 2 [∞, ∞) is given by c(x, a). In addition to continuity and differentiability (twice), we impose the following, fairly standard properties on the cost function: for all a 2 R: (i) (ii) (iii) (iv)

c(∞, a) = 0; @cðx; aÞ=@xjx¼1 ¼ 0 and @c(x,a)/@x > 0 for x 2 R; @ 2 cðx; aÞ=ð@xÞ2 is strictly positive and bounded away from zero; and @c(x,a)/@a < 0 and @ 2 cðx; aÞ=ð@a@xÞ\0 for x 2 R.

12 To avoid negative effort, one may interpret x as the log of ‘true’ effort X 2 [0, ∞). Measured performance of an agent who exerts effort X 2 [0, ∞) in Contest i is Yi ¼ X  Ei , where noise Ei 2 ð0; 1Þ. Taking logs, we get back yi ¼ x þ ei , where yi ¼ ln Yi 2 ½1; 1Þ and ei ¼ ln Ei 2 R.

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An agent of ability a who exerts effort x in Contest i with standard hi enjoys an expected pecuniary payoff:
 hi  x pi ðx; a; hi Þ ¼ wi þ vi F  cðx; aÞ: ri

(6)

Non-pecuniary payoffs, d, are the same as before, and total payoffs continue to be the sum of pecuniary and non-pecuniary payoffs. Agents simultaneously and independently choose which contest to enter and how much effort to exert.13 If fewer than mi enter Contest i, then hi ¼ 1 ¼ xi ða; 1Þ and all entrants receive a prize vi . Otherwise, for a given CMF Hi ðaÞ, effort schedule xi ða; hi Þ and performance standard hi constitute an equilibrium of Contest i if: (i) xi ða; hi Þ is optimal for every a 2 R; (ii) hi is such that the mass of winners, Wi , equals the mass of prizes, mi . Hence, an equilibrium ½xi ða; hi Þ; hi  of Contest i satisfies: xi ða; hi Þ 2 arg sup pi ðx; a; hi Þ; and x  Z 1   hi  xi ða; hi Þ   F Wi ðhi Þ ¼ dHi ðaÞ ¼ mi : ri 1 A Bayesian Nash equilibrium of the full game consists of a tuple: f½H1 ðaÞ; H2 ðaÞ; ½x1 ða; h1 Þ; h1 ; ½x2 ða; h2 Þ; h2 g of CMFs Hi ðaÞ and contest equilibria ½xi ða; hi Þ; hi , i 2 {1, 2}, such that if Hi assigns positive mass density to type a in Contest i, then this type cannot gain by switching contests. 4.1. Equilibrium We solve for equilibrium as before, save for the additional consideration of effort optimisation. First, we characterise the optimal-effort schedule xi ða; hi Þ conditional on standard hi . Second, for each contest, we determine the market-clearing standard hi conditional on CMF Hi . Third, we derive agents’ entry decisions and resulting CMFs ðH1 ; H2 Þ conditional on standards ðh1 ; h2 Þ. Together, these three steps define a mapping from the space of performance standards into itself. Finally, we show that there exist standards ðh1 ; h2 Þ that constitute a fixed point of the system. These standards gives rise to an equilibrium f½H1 ðaÞ; H2 ðaÞ; ½x1 ða; h1 Þ; h1 ; ½x2 ða; h2 Þ; h2 g. We begin by characterising the optimal effort profile, x  ða; hÞ, conditional on standard h. (We suppress subscript i in the remainder of this Section, because it plays no role.) Differentiating (6) with respect to x yields the following first-order condition (FOC) for optimal effort:

13 Again, the analysis remains unchanged if agents move sequentially or if they can switch contests and adjust their effort upon observing others’ entry and effort choices. As before, the argument relies on the atomicity of individuals and, hence, the absence of aggregate uncertainty.

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v hx f r r

 

23

@cðx; aÞ ¼ 0: @x

The second-order condition (SOC) for the FOC to characterise a maximum can be written as: f0



  hx hx @ 2 cðx; aÞ @cðx; aÞ [ 0: f þr r r @x ð@xÞ2

Notice that the SOC is always satisfied when x ≥ h. When x < h, performance measurement must be sufficiently noisy or the log of marginal costs must increase sufficiently fast. For the remainder of the analysis we assume that the SOC is satisfied. Because the marginal cost of effort is strictly decreasing in a, the optimal-effort schedule is strictly increasing. All else equal, effort is increasing in v as well, because a higher prize raises the marginal benefit of effort. The effect of an exogenous rise in standards critically depends on whether an agent needs a lucky break or needs to avoid an unlucky break. When the agent needs to avoid an unlucky break, increasing the standard raises his effort. To see this, notice that a higher h narrows the ‘gap’ |x  h|. Since the density of ɛ is single-peaked around zero, this narrowing raises the marginal benefit of effort. Hence, optimal effort increases. By contrast, when an agent needs a lucky break, a higher standard widens the gap between effort and standard. Again owing to the single-peakedness of f , the marginal benefit of effort falls and so does optimal effort. Interestingly, effort is not uniformly increasing in meritocracy either. To see why, notice that a fall in r lifts the peak of f and thins the tails. This raises the marginal benefit of effort for agents operating close to the standard but reduces it for those operating farther away. Naturally, optimal effort follows suit. Put differently, a rise in meritocracy discourages low types, encourages intermediate types and makes high types complacent. Returning to the two-contest environment, the remainder of the equilibrium derivation proceeds along the same lines as in the exogenous-effort model. This allows us to conclude that Propositions 1, 2 and 3 carry over to the model with endogenous effort. In particular: (i) equilibrium exists; (ii) it is unique in a symmetric baseline, with 50% of every ability type entering each contest; (iii) when q ? 0, both contests remain competitive in a neighbourhood of a symmetric baseline; and (iv) when one contest is uncompetitive, selection into the competitive contest is strictly increasing in ability. (For a formal proof of these claims, see Proposition 15 in Appendices.) 4.2. Selection Around a Symmetric Baseline We now revisit the sorting effects of differences in structural parameters. For tractability reasons, we focus on a neighbourhood of structural parameters around a © 2017 Royal Economic Society.

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symmetric baseline. That is, the two contests cannot be ‘too different’. In that case, all our previous findings carry over. Formally: P ROPOSITION 7. In a neighbourhood of structural parameters around a symmetric baseline, selection patterns in the endogenous-effort model are the same as in the exogenous-effort model of Section 1. That is, mutatis mutandis, Remark 2 and Propositions 4 to 6 continue to hold. The proof of Proposition 7 relies on the envelope theorem. In the model of Section 1, ‘effort’ was fixed at a regardless of the values of structural parameters. By contrast, effort is now parameter dependent. Indeed, even a marginal change in one or more parameters has a first-order effect on optimal effort. However, by the envelope theorem, this effort adjustment only has a second-order effect on payoffs. Hence, when studying Pr1(a) in a neighbourhood of a symmetric baseline, we can ignore changes in xi ðaÞ and pretend that effort is exogenous. In fact, this argument holds around any parameter point – not merely around a symmetric baseline. However, in a symmetric baseline, every type’s effort level is the same across contests. Therefore, the cost of effort differences out of p1  p2 , which makes the endogenous-effort model locally isomorphic to the exogenous-effort model. This allows us to reinterpret an individual’s effort at a symmetric baseline as his type and apply all the arguments and machinery of the exogenous-effort model. Proposition 7 raises the question as to how close the two contests have to be for the selection pattern in the endogenous-effort model to be similar to that in the exogenous-effort model. To get a feel, we reanalyse Example 1 with endogenous effort. E XAMPLE 5. Suppose a N(0, 1), d N(s = 0.05, q = 0.05), and ei Logisticð0; ri Þ, i 2 {1, 2}. Let ðw1 ; w2 Þ ¼ ð1:1; 1Þ, ðm1 ; m2 Þ ¼ ð0:1; 0:2Þ, ðv1 ; v2 Þ ¼ ð1; 1:1Þ, ðr1 ; r2 Þ ¼ x a ð0:6; 1Þ, and cðx; aÞ ¼ ðe e  e x  1Þ=ðe e  1Þ. Hence, probability distributions and parameter values are as in Example 1, while the cost function is chosen such that it satisfies our assumptions. Specifically, for r not too small, the SOC is satisfied.14 (i) The effort schedules in the two contests are shown in Figure 3(a), while Pr1(a) is depicted in Figure 3(b). The PDFs of abilities in the two contests are given in Figure 3(c). Standards are ðh1 ; h2 Þ ¼ ð0:40; 0:36Þ. The fraction of the population entering each contest is (Pr1, Pr2) = (0.25, 0.75), while average abilities are ðE1 ½a; E2 ½aÞ ¼ ð0:32; 0:11Þ. (ii) When q is reduced to 0.0005, effort, selection, and ability densities are as in Figure 1(d), (e), and (f ), respectively. Standards are ðh1 ; h2 Þ ¼ ð0:43; 0:42Þ, while (Pr1, Pr2) = (0.20, 0.80) and ðE1 ½a; E2 ½aÞ ¼ ð0:49; 0:13Þ.

14 Our cost function is rather complicated and ugly. A simpler cost function, such as cðx; aÞ ¼ e bx =e a , a > 0, b > 1, would not materially affect our findings. However, it would complicate the exposition. The reason is that, for sufficiently low ability types, cðx; aÞ ¼ e bx =e a may violate the SOC. (Alternatively, the FOC may not even have a solution.) As a result, xi ðaÞ becomes discontinuous. Specifically, below a certain ability, contestants ‘drop out’ and choose x = ∞. This discontinuity is immaterial, however, because all the relevant integrals such as Wi and Hi remain continuous. Therefore, equilibrium continues to exist, and the discontinuity of xi ðaÞ has only minor effects on standards and selection. Details are available from the authors upon request.

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(a)

(b) x*1 (a) 5

–4

–2

x*2 (a)

2

(c) Pr1 (a) 1.0 0.9

4

–10 –4 –2 (e)

–2

0.1 0

0.2 2

4

ρ = 0.05

–4

–2

2

2

4

2

4

(f)

1.0 0.9

5 –4

0.4

0.5

–5

(d)

g1(a) g2(a)

g(a)

Pr1

1.2 0.8

4

0.5

–5

0.4

–10 –4

0.1 –2 0 2 4 ρ = 0.0005

–4 –2

0

Fig. 3. Optimal Effort, Selection, and the Resulting Ability Distributions in the Two Contests of Example 5, for q = 0.05 (Top) and q = 0.0005 (Bottom) Note. Colour figure can be viewed at wileyonlinelibrary.com

The similarities between Figures 1 and 3 are quite striking. Indeed, the selection function and the resulting endogenous ability distributions are almost indistinguishable.15 Further simulations with different sets of parameters suggest that, even when the visual likeness with the exogenous-effort model is lost, Pr1(a)’s characteristic ‘bimodal’ shape is maintained. This provides some comfort that the results and intuitions of the exogenous-effort model are reasonably robust and, for most intents and purposes, carry over to the endogenous-effort model. 4.3. Effort Across Contests Agents of the same ability who enter the same contest exert the same effort. However, across contests, their effort levels will generally differ, i.e. x1 ða; h1 Þ 6¼ x2 ða; h2 Þ. This raises the question which contest induces agents to work harder and how the prospect of hard work is related to entry. Intuitively, the relationship between effort and entry could go either way. Since effort is costly, one might argue that agents should avoid contests that make them work hard. Conversely, if agents work hard, the rewards must be high, which should be attractive.

15 It is easy to destroy any visual likeness by changing the cost function’s dependence on a. For example, if x cðx; aÞ ¼ ðe e  e x  1Þ=a, then the horizontal axes in Figure 3 are stretched out by a factor exp [exp (a)]. However, provided multiplicative separability between effort and ability is maintained, any change in the cost function’s dependence on a simply corresponds to a relabeling of types. In that sense, our initial choice was without loss of generality.

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Our next Lemma shows that effort and entry are, in fact, unrelated. Rather, it is the change in the probability of entry in a neighbourhood around an agent – i.e. the slope of Pr1(a) – that ‘determines’ which contest elicits the greater effort. Specifically: L EMMA 3. [

x1 ðaÞ ¼ x2 ðaÞ () \

dPr 1ðaÞ [ ¼ 0: \ da

Notice that Lemma 3 holds globally, i.e. not just in a neighbourhood of a symmetric baseline. The intuition for the Lemma is as follows. The envelope theorem implies that the rise in payoffs associated with a small rise in ability is equal to the cost savings from exerting the same level of effort at higher ability. Due to the sub-modularity of costs in x and a, i.e. @ 2 cðx; aÞ=ð@a@xÞ\0, these cost savings are increasing in the initial level of effort. Hence, when p1 ðaÞ increases faster in a than p2 ðaÞ, it must be that x1 ðaÞ [ x2 ðaÞ, and vice versa. Finally, recall that dPr1(a)/da takes on the same sign as dðp1  p2 Þ=da. The sign of the slope of Pr1(a) is therefore a sufficient statistic for ranking an agent’s effort across contests. From Propositions 4 and 7 we know that, in a neighbourhood of a symmetric baseline, the slope of Pr1(a) is determined by meritocracy alone. In combination with Lemma 3, this allows us to rank agents’ efforts across contests. Let a0 denote the lower, and a1 the higher value of a where kðaÞ ¼ ðv2 =v1 Þ=ðr2 =r1 Þ.16 Then: C OROLLARY 1. Agents of intermediate ability work harder in the more meritocratic contest, while agents of extreme ability work harder in the less meritocratic contest. Formally, suppose Condition 1 holds and r1 \r2 . In a neighbourhood of a symmetric baseline: x1 ðaÞ [ x2 ðaÞ iff a0 \a\a1 : In subsection 4.1, we observed that, in a single-contest, an increase in meritocracy discourages low types, encourages intermediate types and makes high types complacent. Corollary 1 can be viewed as an across-contest analogue. However, in one important respect the result in Corollary 1 is stronger: it compares contests with (somewhat) different w, m, v, and r, and shows that, qualitatively, effort comparisons only depend on the difference in r. On the other hand, the single-contest result is more robust, as it extends to global comparisons of r.

5. Related Literature The ponds dilemma has stimulated interest since antiquity and, as attested by the many books and articles written about it, continues to do so. Caesar’s claim that he would ‘rather be first in this village than second in Rome’ (Plutarchus, n.d.), is perhaps the earliest literary expression of the dilemma. It is also the title of a recent paper on the topic by Damiano et al. (2010). The actual phrase ‘big fish in a small pond’ is of 16

Because  k [ 1, a0 and a1 always exist in a neighbourhood of a symmetric baseline.

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American origin. It appears to have been coined by the Galveston Daily News in 1881. The popular book by Frank (1985), entitled Choosing the Right Pond, builds on and expands the metaphor. And in his recent best-seller David and Goliath, Gladwell (2013) continues the discussion, arguing in favour of being a big fish in a small pond, rather than a small fish in a big pond. Generally, the technical literature on the ponds dilemma is of recent vintage. It includes papers by Azmat and M€ oller (2009, 2012), Damiano et al. (2010, 2012), Leuven et al. (2010, 2011) and Konrad and Kovenock (2012). An important exception is the seminal paper by Lazear and Rosen (1981) on rank-order tournaments. Even though their main focus is on competition in a single contest, Lazear and Rosen do show that self-selection sorts workers inefficiently across contests. In Leuven et al. (2010), abilities are binary and success is determined by a parametric contest success function (Tullock, 1980). Their main finding is that highability individuals are not necessarily attracted by higher prizes. This is consistent with our findings in so far as, also in our model, the attractiveness of higher prizes is non-monotonic in ability. Meritocracy, show-up fees and number of prizes do not feature in their analysis. Leuven et al. (2011) conduct a field experiment to disentangle the selection and incentive effects of contests. They find that selection effects dominate. Azmat and M€ oller (2009) study how competing contests should be structured in order to maximise participation. Their main finding – for which they find empirical support in professional road running – is that the more discriminatory the contest, the more prizes should be offered. Unlike in our model, contestants in Azmat and M€ oller (2009) are identical in ability and the contest success function is parametric. In Azmat and M€ oller (2012), abilities are binary. The authors show that the fraction of highability agents choosing the more competitive, high-prize contest is a decreasing function of their population share. Data on entry into marathons support their finding. Also studying entry into contests, Konrad and Kovenock (2012) show that mixing in the entry stage can lead to coordination failure in entry decisions. This coordination failure shelters rents, even among homogenous contestants. Damiano et al. (2010, 2012) study sorting across organisations, focusing on pecking order and peer effects. In their 2010 paper, individuals only care about the average ability of their peers and their own place in the pecking order. The authors show that high and low types self-segregate, while middling sorts are present in both organisations. In their 2012 paper, individuals still care about the average ability of their peers but money is now a consideration as well. Competing organisations try to maximise the average ability of their workforce. The authors show that, while both organisations attract some high-ability types, equilibrium is asymmetric. Moreover, the ‘low-ability organisation’ offers a steeper wage schedule than the ‘high-ability organisation’. Compared to the extant literature, our modelling innovations allow us to analyse the ponds dilemma in considerable generality. We do not restrict the distribution of abilities and allow for a relatively broad class of noise distributions. Simultaneous differences in discriminativeness, show-up fees and the number and value of prizes are also an original contribution of the current article. Our analysis uncovers an interplay between the direct and indirect effects of differences in structural parameters. Jointly, these effects explain selection behaviour across contests. © 2017 Royal Economic Society.

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A small but growing literature considers self-selection into alternative remuneration schemes. Lazear (2000) studies output per worker in a firm that changes from fixed wages to piece rates. He finds that as much as 50% of the resulting increase in productivity comes from positive selection, while the other half can be attributed to an increase in the productivity of existing workers. In the auction literature, Moldovanu et al. (2008) consider quantity competition between two auction sites, while McAfee (1993), Peters and Severinov (1997) and Burguet and Sakovics (1999) study competition by means of reserve prices. One of the workhorses of the labour literature is the Roy model (see Roy, 1951; as well as Borjas, 1987; Heckman and Honor
e, 1990; and Heckman and Taber, 2008). As in our model, agents in the Roy model self-select into the sector that provides them with the highest expected payoff. However, an important difference is that, in the Roy model, abilities are sector-specific. This makes that entry decisions are driven by comparative advantage. Depending on the variances and correlation of an agent’s abilities in the two sectors, either sector may benefit from positive selection. In our model, comparative advantage plays no role, because an agent’s ability is the same in both contests. On the other hand, we allow for general-equilibrium effects that are absent from the Roy model. Specifically, additional entry into a contest negatively affects the expected payoffs of agents already there. In turn, this may induce these agents to reconsider their own choice of contest. Similarly, changes in effort in one contest affect equilibrium entry and effort in both contests. Finally, our article is related to the literature on Hotelling’s ‘linear city’ model and its many variants. (See d’Aspremont et al., 1979, for a correction to the original analysis by Hotelling, 1929.) In these games of positioning and competition, equilibrium entails positioning strategies that mitigate competition. In the case of the ponds dilemma, mitigation would suggest that individuals of similar ability split up across contests in order to soften competition down the line. Perhaps surprisingly, we have shown that the opposite occurs in our model: individuals of similar ability (tend to) enter the same contest. This is a consequence of our focusing on large contests, which preclude ‘market-impact effects’. That is, the presence or absence of a single agent has no measurable effect on the competitiveness of a contest. Hence, the dyadic nature of competition in small contests, which is most intense between agents of the same ability, is lost and replaced by an anonymous battle against a seemingly fixed standard. The result is homophily rather than mitigation because, if one agent pecuniarily strictly prefers a particular pond, so too do all other agents of similar ability. Thus, selfselection in the ponds dilemma induces sorting rather than splitting.

6. Conclusion In this article, we analyse various versions of the ponds dilemma – that is, the question whether it is better to be a big fish in a small pond or a small fish in a big pond. A common intuition is that bigger fish (i.e. those of higher ability) are more likely to choose the big pond (the contest with the greater rewards and more intense competition). Unless one of the ponds is entirely uncompetitive, we show that this intuition is incorrect. The key insight is that the likelihood ratio of success across contests varies non-monotonically with ability, because extreme types can safely ignore © 2017 Royal Economic Society.

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differences in competitiveness while middling sorts cannot. This non-monotonicity carries over to selection such that, in large regions of the ability distribution, bigger fish are less likely to choose the big pond. Changes in reward structures can have unexpected selection effects. For instance, offering a higher show-up fee makes the distribution of contestants bimodal, since such a policy attracts the extremes while repelling the middle. Even a seemingly straightforward increase in prize values yields non-obvious selection patterns, owing to the competitive changes wrought. While higher prizes do attract high types, they also drive out middling sorts and have little effect on the selection of low types. As a result, a contest may well raise the value of its prizes, only to see the average ability of contestants fall. A different kind of trade-off arises when contests differ in meritocracy (i.e. discriminativeness). Here, agents must compare the benefit of a ‘lucky break’ in measured performance against the cost of an ‘unlucky break’. We obtain the intuitive result that higher types are overrepresented in the more meritocratic contest, while lower types are underrepresented. However, selection effects attenuate toward the tails, because extreme types find meritocracy almost irrelevant to their choice of contest. This has the striking implication that very low types disproportionately enter the more meritocratic contest. Our model is quite general in a number of respects. We impose essentially no restrictions on the distribution of abilities. We allow for both pecuniary and nonpecuniary preferences, encompassing the ‘neoclassical’ case where the latter are vanishingly small. And, apart from log-concavity and the restriction to location-scale families, we make few assumptions about the distribution of noise in performance measurement. Probably the most important limitation of our model is the restriction to prizes of equal value within each contest. Unfortunately, we do not see an easy way to relax this assumption in a tractable manner. In his lecture notes on the Roy model, Autor (2003) observes that ‘self-selection points to the existence of equilibrium relationships that should be observed in ecological data, and these can be tested without an instrument. In fact, there are some natural sciences that proceed almost entirely without experimentation – for example, astrophysics. How do they do it? Models predict non-obvious relationships in data. These implications can be verified or refuted by data, and this evidence strengthens or overturns the hypotheses. Many economists seem to have forgotten this methodology’. We believe that some of the predictions of our model constitute such non-obvious relationships. We look forward to them being verified – or perhaps refuted – by the data.

Appendix A. Proofs A.1. Equilibrium Proof of Lemma 1. If Hi is such that Pri  mi , then all individuals win a prize and hi ¼ 1. If Pri [ mi , standard hi solves:
 hi  a F dHi ðaÞ ¼ mi : ri 1

Z Wi ðhi Þ ¼ © 2017 Royal Economic Society.

1

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i.e. it equalises the mass of individuals achieving or exceeding the standard, Wi ðhi Þ, to the mass mi of promotion opportunities. To see that hi exists and is unique, notice that: (i) Wi ðhi Þ is continuous and strictly decreasing in hi ; (ii) Wi ðhi Þ ! Pri [ mi when hi ! 1; and (iii) Wi ðhi Þ ! 0 \mi when hi ! 1. Proof of Proposition 1. For arbitrary structural parameters, existence of equilibrium was proved in the main text. For the case of symmetric baselines, we first show that equilibrium standards must be the same across contests. Suppose not. Because s = 0, strictly more than 50% of each ability type enter the contest with the lower standard; say, Contest 1. This implies that the mass of winners in Contest 1 is strictly greater than in Contest 2. Since m1 ¼ m2 ¼ m, this is inconsistent with equilibrium. Finally, as standards are identical across contests and s = 0, 50% of each ability type enter each contest. Applying Lemma 1, we know that this selection pattern uniquely determines the (identical) standards in the two contests. Hence, equilibrium is unique.

A.2. Uncompetitive Case Proof of Proposition 2. Assume, without loss of generality, that Contest 1 is competitive and Contest 2 is uncompetitive. Then h1 [ h2 ¼ 1, such that:
  h a p1 ðaÞ  p2 ðaÞ ¼ w1 þ F 1 v1  w 2  v2 : r1 This payoff difference is strictly increasing in a. By monotonicity of Γ, the same is true for Pr1ðaÞ ¼ Cf½p1 ðaÞ  p2 ðaÞ=qg. To prove FOSD, observe that G1 ðaÞ\G2 ðaÞ iff: Z a Z a Pr1ðaÞg ðaÞda ½1  Pr1ðaÞg ðaÞda 1 \ 1 : Pr1 1  Pr1 This is equivalent to: Z

a

1

½Pr1ðaÞ  Pr1 g ðaÞda\0:

Finally, notice that the last inequality indeed holds for all a, because Pr1(a) is strictly increasing in a and the LHS converges to zero when a ? ∞. We now turn to the second part of the Proposition. Let ~ h1 denote the limit value of the standard in Contest 1 as q ? 0. First, we prove that this limit indeed exists. Suppose to the contrary that, as q ? 0, there exists a sequence of equilibrium thresholds that does not converge. Then there are at least two convergent subsequences, A, B, with differing limit values ~ hA1 , ~hB , such that, wlog, ~hA [ ~hB . The resulting entry pattern in the limit of subsequence A is: 1 1 1

Pr1ðaÞ !

8 > > 0 > > < > 1=2 > > > : 1


 A 1 w2  w1 þ v2 ~  if a\h1  r1 F v1
 w  w1 þ v2 ; 2 A 1 if a¼~ h1  r1 F v1 otherwise

(A.1)

and similarly for sequence B. Since ~hA1 [ ~hB1 , the set of entering types in the limit of sequence A is a strict subset of the set of entering types in the limit of sequence B. Because of this subset © 2017 Royal Economic Society.

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~hA 1

31

[ ~hB1 ,

property and the fact that the mass of winners under A is strictly smaller than under B. Hence, market clearing must be violated in at least one of these cases. Therefore, all subsequences converge to the same standard, ~h1 , and entry is as in (A.1) with ~ hA1 ¼ ~ h1 . An  ~ analogous argument shows that h1 is, in fact, unique.

A.3. Competitive Case Recall that interior equilibrium standards ðh1 ; h2 Þ are characterised by the market clearing conditions:  Z 1
h1  a  F dH1 ðaÞ ¼m1 ; r1 1  Z 1
h2  a F dH2 ðaÞ ¼m2 : r2 1 Denote the left-hand side of this system by Sðh1 ; h2 Þ and denote the first and second component of Sðh1 ; h2 Þ by S1 and S2 , respectively. The following two Lemmas are used in the proof of Proposition 3 below. L EMMA 4. In a symmetric baseline, the Jacobian of Sðh1 ; h2 Þ is non-singular for generic values of q. Proof. When evaluated at a symmetric 2 @S1 6 @h1 det6 4 @S2 @h1

baseline, we have to show that: 3 @S1 @h2 7 7 6 0: ¼ @S2 5 @h2



ðh1 ;h2 Þ¼ðh ;h Þ

Notice that: @S1 ¼ @h1

Z

1

@h1 ða; h1 ; h2 Þ da þ F1 @h1 1

Z

1

1

1 f1 h1 ða; h1 ; h2 Þda; r1

where Fi and fi are short for F½ðhi  aÞ=ri  and f ½ðhi  aÞ=ri . Similarly: Z 1 @S1 @h1 ða; h1 ; h2 Þ F1 ¼ da: @h2 @h2 1 Recall that h1 ða; h1 ; h2 Þ ¼ g ðaÞC½ðp1  p2 Þ=q. Differentiating h1 with respect to h1 , we find: @h1 ða; h1 ; h2 Þ 1 @p1 ða; h1 Þ v1 ¼ g ðaÞ c ¼ g ðaÞcf1 : @h1 q @h1 qr1 Here, c is short for c½ðp1  p2 Þ=q. At a symmetric baseline this reduces to: @h1 ða; h1 ; h2 Þ @h2 ða; h1 ; h2 Þ  v ¼ ¼ g ðaÞcð0Þf ; @h1 qr @h2 ðh1 ;h2 Þ¼ðh ;h Þ ðh1 ;h2 Þ¼ðh ;h Þ where f   f ½ðh  aÞ=r. Similarly, @h1 ða; h1 ; h2 Þ @h2 ða; h1 ; h2 Þ  v ¼ g ðaÞcð0Þf : ¼ @h2 qr @h1 ðh1 ;h2 Þ¼ðh ;h Þ ðh1 ;h2 Þ¼ðh ;h Þ

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Let @S1 =@hi  @S1 =@hi jðh1 ;h2 Þ ¼ ðh ;h Þ . into @S1 =@h1 and @S2 =@h1 we find: @S1 ¼ @h1

Z

Substituting the expressions for @h1 =@h1 and @h1 =@h2

1

1

Cð0Þ  F cð0Þ

 v 1  @S2 f g ðaÞda ¼  ; @h2 q r

and @S2 ¼ @h1

Z

1

1

v1  @S1 F cð0Þ f g ðaÞda ¼  : @h2 qr

Therefore: @S1 ¼ Cð0Þ @h1

Z

1

1  @S1 @S1 f g ðaÞda   ¼ N   ; @h @h r 1 2 2

where Ξ > 0. Due to symmetry, the determinant of the Jacobian of S then simplifies to: 2

@S1 6 @h1 det6 4 @S2 @h1

3 @S1

 @S1 @S1 @S1 @S1 @h2 7 7 : ¼ þ  @S2 5 @h1 @h2 @h1 @h2 @h2 ðh1 ;h2 Þ¼ðh ;h Þ

(A.2)

Since @S1 =@h1 ¼ N  @S1 =@h2 and Ξ > 0, the first factor in (A.2) is strictly positive at the baseline. Thus, it remains to show that the second factor, @S1 =@h1  @S1 =@h2 , is non-zero. Substituting and simplifying yields the required condition:  Z 1 v 1    Cð0Þ  2F cð0Þ f g ðaÞda 6¼ 0: q r 1 Solving for q we get: Z

1

F f  g ðaÞda cð0Þ 1 Z 1 q 6¼ 2v : Cð0Þ f  g ðaÞda 1



Notice that h does not depend on q since, in a symmetric baseline, selection is 50–50 irrespective of q. Hence, the RHS is a strictly positive constant. Therefore, generically, the Jacobian is non-singular at the baseline. L EMMA 5. Fix some r1 ; r2 [ 0, m1 ; m2 [ 0, and m1 þ m2 \1. If the contests’ show-up fees w and prizes v are sufficiently close together, a measure of individuals strictly greater than mi enter Contest i 2 {1, 2} when q → 0. Proof. Suppose to the contrary that, no matter how small jw1  w2 j and jv1  v2 j, when q ? 0, fewer than mi individuals enter Contest i. In that case, Prj [ 1  mi individuals enter Contest j 6¼ i. Moreover, since 1  mi [ mj , it follows that Contest j is competitive. This configuration yields standards hi ¼ 1 and 1 [ hj  hj [  1. Here, hj is a lower bound on hj that is reached when only the lowest-ability ð1  mi Þ-quantile of individuals enter Contest j. Because hi ¼ 1, all individuals entering Contest i win a prize with certainty. The pecuniary payoff of entering this contest is therefore wi þ vi . The pecuniary payoff of entering Contest j is wj þ F½ðhj  aÞ=rj vj . Hence, when q ? 0, an ability type a enters Contest j iff: © 2017 Royal Economic Society.

THE PONDS DILEMMA

wj þ F

hj  a rj

33

! vj  wi þ vi :

This is equivalent to: a  hj  rj F 1


 w i  w j þ vi  vj : vj

It follows that, for q ? 0, Prj equals: 
 
 wi  wj þ vi  vj wi  wj þ vi  vj \1  G h1  rj F 1 : 1  G hj  rj F 1 vj vj

(A.3)

Finally, notice that when wi  wj þ vi  vj \vj F f½h1  G 1 ðmi Þ=rj g, the RHS of (A.3) is strictly smaller than 1  mi . This contradicts the notion that, no matter how small jw1  w2 j and jv1  v2 j, Prj [ 1  mi . Proof of Proposition 3. From Proposition 1 we know that, in a symmetric baseline, 50% of every ability type enter each contest. Hence, Pr1 = Pr2 = 1/2. Because m1 ¼ m2 ¼ m\1=2, we may conclude that both contests are competitive. From Lemma 4 we know that the Jacobian of S is non-singular at a symmetric baseline. Hence, at such a point, we may apply the implicit function theorem (IFT) to the market-clearing conditions Sðh1 ; h2 Þ ¼ ½m; mT . The IFT implies that equilibrium standards ðh1 ; h2 Þ remain finite in a neighbourhood of structural parameters around a symmetric baseline. Thus, both contests are competitive. Finally, Lemma 5 implies that the competitive region remains non-degenerate when q ? 0. Let fi  f ½ðhi  aÞ=ri  and let fi 0 denote the derivative of fi with respect to its argument gi ðaÞ  ðhi  aÞ=ri . Proof of Lemma 2. i(a). Differentiating k(a) with respect to a we obtain: k0 ðaÞ ¼

f10 f2 þ f20 f1 rðf2 Þ2

;

which takes the sign of the numerator. From log-concavity of f we know that f 0 ðÞ=f ðÞ is strictly decreasing. Hence, for h1 [ h2 : f10 =f1 \f20 =f2 : This implies that k0 ðaÞ [ 0. For h1 \h2 the argument is analogous. The result for h1 ¼ h2 is trivial. i(b). If h1 [ h2 , then we know from Part (i(a)) that k0 ðaÞ [ 0. Hence: k ¼ lim kðaÞ\kðh2 Þ\1\kðh1 Þ\ lim kðaÞ ¼  k; a!1

a!1

where the second and third inequalities follow from single-peakedness of f. For h1 \h2 , the argument is analogous. © 2017 Royal Economic Society.

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ii(a). Notice that limjaj!1 kðaÞ ¼ 0 is equivalent to: lim log

jaj!1

f ½g1 ðaÞ ¼ lim log f ½g1 ðaÞ  log f ½g2 ðaÞ ¼ 1: f ½g2 ðaÞ jaj!1

Now consider the two cases: 1. g1 ðaÞ  g2 ðaÞ: Then, by concavity of log f : log f1  log f2  f20 =f2  ½g1 ðaÞ  g2 ðaÞ:

(A.4)

Hence, it suffices to show that the RHS goes to ∞ when |a| ? ∞. First notice that lima!1 g1 ðaÞ  g2 ðaÞ ¼ 1 and lima!1 g1 ðaÞ  g2 ðaÞ ¼ 1, because r1 \r2 by assumption. Next notice that, by continuity of f , limjaj!1 f ¼ 0, such that limjaj!1 log f ¼ 1. Hence, there must be an argument g0 and a b > 0 such that d log f =dgjg ¼ g0 \  b. By strict concavity of log f , it follows that d log f =dg \ d log f =dgjg¼g0 \  b for all g [ g0 . Hence, d log f =dg ¼ f 0 =f is strictly negative and bounded away from zero for g ? ∞. Similarly, there must be an argument g00 and a b > 0 such that d log f =dgjg¼g0 [ b. By strict concavity of log f , d log f /dg > b for all g\g00 . 0 Hence, d log f =dg ¼ f 0 =f is strictly positive and bounded away fromzero when g ? ∞. It then follows that the RHS of (A.4) must go to minus infinity when |a| ? ∞. 2. g1 ðaÞ  g2 ðaÞ: Then, by concavity of log f : log f2  log f1  f20 =f2  ½g2 ðaÞ  g1 ðaÞ; This is equivalent to: log f1  log f2  f20 =f2  ½g1 ðaÞ  g2 ðaÞ: An analogous argument as in 1. now establishes the required limit inequality. ii(b). For h1 6¼ h2 , k [ kðh1 Þ [ 1 follows immediately from single-peakedness of f around zero and r1 \r2 . ii(c). Single-peakedness of k(a) follows from Lemma 6 given below. Proof of Single-peakedness of k(a):17 L EMMA 6. k(a) is single-peaked Proof. Let gi ðaÞ ¼ ðhi  aÞ=ri and, without loss of generality, let r1 \r2 . We have: k0 ðaÞ ¼

f1 ðr2 f10 =f1 þ r1 f20 =f2 Þ: r1 r2 f2

As a tends to ∞, g1 ðaÞ [ g2 ðaÞ [ 0, which implies f10 =f1 \ f20 =f2 \ 0 by the fact that f peaks at 0 and is log-concave. Therefore, k0 ðaÞ [ 0 for a sufficiently small. As a tends to ∞, g1 ðaÞ \ g2 ðaÞ \ 0, which implies f10 =f1 [ f20 =f2 [ 0. Therefore, k0 ðaÞ\0 for a sufficiently large. It follows that there exists a  such that k0 ða  Þ ¼ 0. Moreover, at such a point a  :

17 We thank an anonymous referee for suggesting the following shorter formulation for the proof of single-peakedness.

© 2017 Royal Economic Society.

THE PONDS DILEMMA

k00 ða  Þ ¼

f2 f100 =r21  f1 f200 =r22 ðf2 Þ2

35

:

Using the fact that r1 =r2 ¼ ðf10 =f1 Þ=ðf20 =f2 Þ at the point a  , we obtain that k00 ða  Þ has the same sign as: f100 =f10 f200 =f20  0 : f10 =f1 f2 =f2 Furthermore, since 1 [ r1 =r2 ¼ ðf10 =f1 Þ=ðf20 =f2 Þ, we have 0 [ g1 ða  Þ [ g2 ða  Þ for fi 0 =fi [ 0; i ¼ 1; 2 and g2 ða  Þ [ g1 ða  Þ [ 0 for fi 0 =fi \0; i ¼ 1; 2, by the log-concavity of f . Finally, Condition 1 then implies that k00 ða  Þ\0 in both cases. This proves that k0 ðaÞ is single-crossing from above, which implies that k(a) is single-peaked. Proof of Proposition 4. Recall that dðp1  p2 Þ=da ¼ f ½ðh2  aÞ=r2 ½kðaÞ  ðv2 =v1 Þ=ðr2 =r1 Þ ðv1 =r1 Þ. From Lemma 2 part (ii) we know that k(a) is single-peaked and converges to zero in the tails. Hence, if ðv2 =v1 Þ=ðr2 =r1 Þ\k, then dðp1  p2 Þ=da is U-shaped, crossing the x-axis twice, first from below and then from above. In turn, this implies that p1 ðaÞ  p2 ðaÞ and Pr1(a) take on two extrema, first a minimum and then a maximum. If ðv2 =v1 Þ=ðr2 =r1 Þ [  k, then dðp1  p2 Þ=da\0. Hence, p1 ðaÞ  p2 ðaÞ and Pr1(a) are strictly decreasing in ability. The next Proposition, omitted from the main text, deals with the case where contests differ in multiple dimensions but are equally meritocratic. P ROPOSITION 8. Let r1 ¼ r2 while other structural parameters are arbitrary. Then Pr1(a) is either single-peaked or monotone in ability. Proof. To prove Proposition, we show that: (i) If v2 =v1 2 ðk; kÞ, then Pr1(a) is single-peaked, taking on a minimum iff h1 [ h2 ; (ii) If v2 =v1 62 ðk; kÞ, then Pr1(a) is strictly monotone; it is increasing iff v2 =v1 \k: (i) Recall that dðp1  p2 Þ=da ¼ ð1=rÞf ½ðh2  aÞ=r½kðaÞv1  v2  while, from part (i) [ [ of Lemma 2, we know that k0 ðaÞ ¼ 0 iff h1 ¼ h2 . Hence, if v2 =v1 2 ðk;  kÞ, then \ \   p1 ðaÞ  p2 ðaÞ is single-peaked, taking on a minimum iff h1 [ h2 . Finally, by monotonicity of Pr1(a) in p1 ðaÞ  p2 ðaÞ, Pr1(a) inherits these properties. (ii) If v2 =v1 \k, then dðp1  p2 Þ=da ¼ ð1=rÞf ½ðh2  aÞ=r½kðaÞv1  v2  [ 0 for all a. If v2 =v1 [ k, then dðp1  p2 Þ=da\0 for all a. By monotonicity of Γ, the same holds for Pr1(a).

A.4. Isolating the Selection Effects Proof of Remark 2. The remark is a corollary of Propositions 9–12, below. P ROPOSITION 9. A higher show-up fee disproportionately attracts the best and the worst, while repelling the middle. Formally, let w1 [ w2 while the contests are otherwise identical. In equilibrium: (i) (ii) (iii) (iv)

Pr1(a) is U-shaped in ability; 1=2 \ limjaj!1 Pr1ðaÞ ¼ C½ðw1  w2 Þ=q \ 1; h1 [ h2 ; and Pr1 can be greater or smaller than 1/2.

Proof. First we prove part (iii), namely, that h1 [ h2 . Suppose, by contradiction, that h1  h2 . In that case, Contest 1 is pecuniarily strictly more attractive to all agents. Hence, strictly more © 2017 Royal Economic Society.

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than 50% of every ability type enter this contest. In combination with h1  h2 , this means that there are strictly more winners in Contest 1 than in Contest 2. However, this is inconsistent with equilibrium because m1 ¼ m2 . For h1 [ h2 , Lemma 2 part (i) implies that k(a) is strictly increasing in a and k \ 1 \  k. From (4) it then follows that dðp1  p2 Þ=da single-crosses zero from below. Hence, p1  p2 and Pr1(a) are U-shaped in a. This proves part (i). Finally, part (ii) is proved in the text, while part (iv) is proved by example: Suppose a N (0, 1), d N(0, 0.05), and ei Logisticð0; 1Þ. Let w1 ¼ 1:1 [ 1 ¼ w2 . If mi ¼ 0:1 and vi ¼ 1, i 2 {1, 2}, then Pr1 = 0.71 > 0.29 = Pr2. However, if mi ¼ 0:4 and vi ¼ 4, i 2 {1, 2}, then Pr1 = 0.44 < 0.56 = Pr2. Hence, Pr1 may take on values on either side of 1/2. P ROPOSITION 10. Offering more prizes attracts all types but disproportionately those of middling ability. Formally, let m1 [ m2 while the contests are otherwise identical. In equilibrium: (i) (ii) (iii) (iv)

Pr1(a) is inverse-U-shaped; limjaj!1 Pr1ðaÞ ¼ 1=2; h1 \h2 ; and ∀a, Pr1(a) > 1/2.

Proof. The pecuniary payoff difference is:  

  h a h a  F 2 v: p1  p2 ¼ F 1 r r Hence, limjaj!1 Pr1ðaÞ ¼ Cð0Þ ¼ 1=2. To prove that h1 \h2 , suppose by contradiction that h1  h2 . In that case, at least 50% of every ability type enter Contest 2. As a result, the number of winners in Contest 2 is greater than the number of winners in Contest 1. This is inconsistent with equilibrium because, by assumption, m1 [ m2 . Because h1 \h2 while the contests are otherwise identical in all payoff relevant dimensions, we have that Pr1(a) > 1/2 for all a. For h1 \h2 we know from Lemma 2 part (i) that k(a) is strictly decreasing, taking on values on either side of 1. Equation (4) then implies that p1  p2 is inverse-U-shaped in a. By monotonicity of Γ, the same holds for Pr1(a). The next Proposition deals with differences in prize values, v. For r1 ¼ r2 , we know from Lemma 2 that k(a) is strictly monotone and takes on values on either side of 1. When contests only differed in show-up fees or number of prizes, this was enough to establish single-peakedness of the payoff difference. (See Propositions 9 and 10, above.) Here, this is no longer the case. Inspection of (4) reveals that the sign of dðp1  p2 Þ=da also depends on the prize ratio v2 =v1 and whether k(a) is bounded. For r1 ¼ r2 , we say that k(a) is bounded if k > 0 and  k\1. For example, the logistic distribution falls into this category, since its likelihood ratio runs from e 1=r to e 1=r . We say that k(a) is unbounded if k = 0 and k ¼ 1 for r1 ¼ r2 . The normal distribution is a case in point.18 As we now show, monotone selection requires that v2 =v1 62 ðk;  kÞ – i.e. k(a) is bounded and the prize ratio is sufficiently lopsided. Alternatively, when v2 =v1 2 ðk;  kÞ – i.e. k(a) is unbounded or the prize ratio is sufficiently close to 1 – then dðp1  p2 Þ=da changes sign exactly once, which makes Pr1(a) single-peaked. P ROPOSITION 11. Higher prizes most strongly attract the best-and-the-brightest, while not affecting entry decisions of the worst. 18 For ease of exposition, our definitions ignore the ‘semi-bounded’ cases, where k > 0 and k ¼ 1 or vice versa. For example, the extreme value distribution fall into this category. These cases are handled like the bounded or the unbounded case, depending on whether v2 =v1 is smaller or greater than 1.

© 2017 Royal Economic Society.

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37

Formally, let vi [ vj while the contests are otherwise identical. In equilibrium: (i) hi [ hj . (ii) lima!1 Pr1ðaÞ ¼ 1=2 and lima!1 Pr1ðaÞ ¼ C½ðv1  v2 Þ=q.  then: (iii) If v2 =v1 2 ðk; kÞ (a) (b) (c) (d)

ð\Þ

ð\Þ

9^ a 2 R such that PriðaÞ [ 1=2 iff a [ a^; Pri(a) is U-shaped on ð1; a^Þ; Pri(a) is strictly increasing on ½^ a ; 1Þ; and Pri can be greater or smaller than 1/2.

(iv) If v2 =v1 62 ðk; kÞ then, 8a 2 R, Pri(a) > 1/2 and strictly increasing. Proof. (i) The proof is analogous to that of Proposition 9 part (iii). (ii) Trivial. (iii) Suppose, without loss of generality, that v1 [ v2 . Part (i) then implies that h1 [ h2 . In turn, we may apply Lemma 2 part (i) to conclude that k0 ðaÞ [ 0. Next, recall that dðp1  p2 Þ=da ¼ 1=rf ½ðh2  aÞ=r½kðaÞv1  v2 . Hence, for v2 =v1 2 ðk;  kÞ, dðp1  p2 Þ=da single-crosses zero from below, which makes p1  p2 and Pr1(a) U-shaped in a. Now recall from part (ii) that lima!1 Pr1ðaÞ ¼ 1=2 and lima!1 Pr1ðaÞ ¼ C½ðv1  v2 Þ=q [ 1=2, where the inequality follows from v1 [ v2 . Combining the U-shapedness of Pr1(a) with these limit values implies parts (i) and (ii). Part (iii) is proved by example. Let a N(0, 1), d N(0, 0.05), ei Logistic ð0; 0:6Þ, mi ¼ 0:1, wi ¼ 1, i 2 {1, 2} and v2 ¼ 1. If v1 ¼ 2, then ðh1 ; h2 Þ ¼ ð1:54; 0:91Þ and Pr1 = 0.44 < 0.56 = Pr2. If v1 ¼ 5, then ðh1 ; h2 Þ ¼ ð1:73; 0:47Þ and Pr1 = 0.51 > 0.41 = Pr2. (iv) From Lemma 2 part (i) we know that k < 1. Hence, if v2 =v1 \k, then v1 [ v2 . By part (i), h1 [ h2 . By the expression for dðp1  p2 Þ=da above, if v2 =v1 \k, then p1  p2 and Pr1(a) are strictly increasing in a. An analogous proof holds for v2 =v1 [  k [ 1. Finally, to see that Pri(a) > 1/2, combine part (ii) with the observation that Pr1(a) is strictly increasing in a. P ROPOSITION 12. Meritocracy attracts high types and repels low types. However, these selection effects dissipate toward the tails. The majority of the population enters the less meritocratic contest. Formally, let r1 \r2 while the contests are otherwise identical. In equilibrium: (i) (ii) (iii) (iv) (v)

ð\Þ

ð\Þ

Pr1ðaÞ [ 1=2 iff a [ a~  ðr2 h1  r1 h2 Þ=ðr2  r1 Þ; limjaj!1 Pr1ðaÞ ¼ 1=2; For small q, Pr1 < 1/2; Either contest may have the higher standard; and If Condition 1 holds, Pr1(a) is single-peaked on either side of a~.

Proof. (i) From (3) it follows that p1  p2 single-crosses zero from below at a~  ðr2 h1  r1 h2 Þ= ðr2  r1 Þ. This implies the claim. (ii) From (3) it also follows that limjaj!1 p1  p2 ¼ 0. This implies the claim. (iii) From part (i) we know that Pr1(a) single-crosses 1/2 from below at a~. This implies that, in the limit for q ? 0, agents enter Contest 1 iff a [ a~ . (A more careful proof of this claim is analogous to the proof of the limit result in Proposition 6, below, and hence omitted.) Therefore, limq!0 Pr1 ¼ 1  Gð~ a Þ and limq!0 Pr2 ¼ Gð~ a Þ. Now suppose by contradiction that Pr1 ¼ 1  Gð~ a Þ  1=2 for q ? 0. Then: © 2017 Royal Economic Society.

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  1
 h  a~ h a h  a~ Pr1 F 1 m¼ g ðaÞda [ F 1 Pr1 ¼ F 2 r r r a~
   Z a~
 h  a~ h a Pr2 [ g ðaÞda = m,  F 2 F 2 r r 1 Z

where we have used that the chance of winning is strictly increasing in a and, at a~, the same across contests. Contradiction. Hence, in the limit for q ? 0, Pr1 < 1/2. By continuity of Pri in q, the exclusivity of Contest 1 extends to a neighbourhood of q around zero. (iv) We establish the result by example. Let a N(0, 1), d Nð0; q2 Þ, ei Nð0; r2i Þ, v = w = 1, ðr1 ; r2 Þ ¼ ð0:5; 1Þ, and q = 0.1. If m = 0.2, then h1 ¼ 0:51 [ 0:40 ¼ h2 . If m = 0.1, then h1 ¼ 1:15\1:21 ¼ h2 . Hence, h1 and h2 cannot be ranked. (v) This claim follows from the expression for dðp1  p2 Þ=da in (4), single-peakedness of k(a) and limjaj!1 kðaÞ ¼ 0 proved in Lemma 2 part (ii), and the fact that k [ 1 [ r1 =r2 .

A.5. Stochastic Ordering and Limit Behaviour Recall that we have constrained the parameter space such that both contests are competitive in equilibrium. For the limit results in Propositions 13, 14 and 6, we require that the entire converging (sub)sequence of equilibria is competitive for q ? 0. As before, this amounts to the assumption that structural parameters are not too far apart. P ROPOSITION 13. Let w1 [ w2 while the contests are otherwise identical. When q ? 0, selection becomes deterministic. Extreme ability types enter Contest 1, while middling sorts enter Contest 2. Formally, for any convergent (sub)sequence of equilibria, there exist a pair fa; ag 2 R2 , a\ a , such that:  0 if a\a\ a lim Pr1ðaÞ ¼ : 1 otherwise q!0 Proof. For q ? 0, consider a convergent (sub)sequence of equilibria with limit standards ðh1 ; h2 Þ . First, we show that when q ? 0, p1 ðaÞ  p2 ðaÞ remains U-shaped. To see this, notice that the arguments in the proof of Proposition 9 establishing this result for fixed q > 0 continue to hold without modification when q ? 0. U-shapedness of p2 ðaÞ  p1 ðaÞ implies that, in pecuniary terms, almost all ability types have strict preferences over contests. Because pecuniary payoffs determine entry decisions when q ? 0, this means that individuals of the same ability choose the same contest – namely, the one that strictly maximises their pecuniary payoffs. Because limjaj!1 p1 ðaÞ  p2 ðaÞ ¼ w1  w2 [ 0, when q ? 0, extreme ability types enter Contest 1. Let a 0 denote the point where p2 ðaÞ  p1 ðaÞ takes on its minimum. By assumption, w1 and w2 are sufficiently close together such that both contests are competitive. Therefore, it must be that p1 ða 0 Þ  p2 ða 0 Þ\0 for q ? 0. Once more using single-peakedness of p2 ðaÞ  p1 ðaÞ, we may conclude that, for a convergent (sub)sequence of equilibria, there exist 1 \ a \ a \ 1 such that:  0 if a\a\a : lim Pr1 ¼ 1 otherwise q!0 P ROPOSITION 14. Let m1 [ m2 while the contests are otherwise identical. When q ? 0, selection remains strictly stochastic. Standards in the two contests converge, while Pr1(a) remains inverse U-shaped. © 2017 Royal Economic Society.

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limq!0 h1

Formally, for any convergent (sub)sequence of equilibria, ¼ 
  1 v h a \ lim Pr1ðaÞ ¼ C c f \1; 2 q!0 r r

limq!0 h2



¼ h , and

where c > 0 is a constant. Proof. For q ? 0, consider a convergent (sub)sequence of equilibria with limit standards ðh1 ; h2 Þ. First notice that, in the limit, h1 ¼ h2 ¼ h . Otherwise, when q ? 0, all individuals enter the contest with the lower performance standard. This is inconsistent with both contests being competitive. Assuming that Pr1(a) converges when q ? 0, the propensity to enter Contest 1 in the limit is:      9 8 h ðqÞ  a > h ðqÞ  a > > >  F 2 F 1 < = r r lim Pr1ðaÞ ¼ C v lim : q!0 q!0 > > q > > : ; Applying l’H^ opital’s rule, we get:

  dh2 dh1 v h  a lim  : lim Pr1ðaÞ ¼ C f q!0 q!0 dq r r dq 

(A.5)

It remains to prove that dh2 =dq  dh1 =dq ! c, 0 < c < ∞, when q ? 0. First, by contradiction, suppose that there exist multiple convergence points c1 ; c2 , where, wlog, c1 \c2 . From (A.5) it then follows that, in the limit with convergence point c2 , a larger fraction of every ability type enters Contest 1 than in the limit with convergence point c1 . In turn, this means that there are strictly more winner under c2 than under c1 . Hence, market clearing is violated either for c1 or c2 . We may conclude that, in fact, c1 ¼ c2 . Next, suppose that c ≤ 0 or c = ∞. If c ≤ 0 then, in the limit, less than 50% of each ability type enter Contest 1. Because m1 [ m2 , this would imply that h1 \h2 , contradicting our conclusion above that h1 ¼ h2 ¼ h . If c = ∞ then, in the limit, almost everybody enters Contest 1, contradicting our result above that both contests are competitive. Proof of Proposition 5. For w1 [ w2 , we prove the result for entrants. For m1 [ m2 , we prove it for winners. The omitted proofs are analogous. w1 [ w2 ; entrants. Observe that: Z Pr2 G1 ðaÞ  G2 ðaÞ ¼

a 1

Z Pr1ðaÞg ðaÞda  Pr1

a

1

½1  Pr1ðaÞg ðaÞda

Pr1 Pr2

;

and d½G1 ðaÞ  G2 ðaÞ Pr2  Pr1ðaÞ  Pr1  ½1  Pr1ðaÞ ¼ g ðaÞ: da Pr1 Pr2 From Proposition 13 we know that limq!0 Pr1ðaÞ ¼ 0 if a \ a \ a , and 1 otherwise. This implies: 8 1 > > g ðaÞ\0 if a\a\ a d½G1 ðaÞ  G2 ðaÞ < Pr2 ¼ lim q!0 > da > : 1 g ðaÞ [ 0 otherwise. Pr1 © 2017 Royal Economic Society.

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Hence, limq!0 d½G1 ðaÞ  G2 ðaÞ=da changes sign exactly twice. Because G1 and G2 are CDFs, both run from 0 to 1. Together, these observations imply that G1 single-crosses G2 from above. By continuity, the same holds for q > 0 but small. Therefore, abilities in Contest 1 are more SC-dispersed than in Contest 2. m1 [ m2 ; winners. The CDF of winners’ abilities in Contest i, Ki ðaÞ, is given by:
 Z 1 a  hi  a F PriðaÞg ðaÞda: Ki ðaÞ ¼ mi 1 ri Hence:

and


  
   h a h a Pr1ðaÞ  m1 F 2 ½1  Pr1ðaÞ g ðaÞda m2 F 1 r r ; K1 ðaÞ  K2 ðaÞ ¼ 1 m1 m2 Z

a


 
  h a h a Pr1ðaÞ  m1 F 2 ½1  Pr1ðaÞ m2 F 1 d½K1 ðaÞ  K2 ðaÞ r r ¼ g ðaÞ: da m1 m2

From Proposition 14 we know that limq!0 Pr1ðaÞ ¼ Cfcðv=rÞf ½ðh  aÞ=rg. This implies:
  
  h a v h  a ðm1 þ m2 ÞC c f  m1 F d½K1 ðaÞ  K2 ðaÞ r r r ¼ g ðaÞ; lim q!0 da m1 m2 which takes the sign of:


 v h  a ðm1 þ m2 ÞC c f  m1 : r r

Notice that this expression is strictly negative for large |a|. This follows from m1 [ m2 and Γ(0) = 1/2. Hence, limq!0 d½K1 ðaÞ  K2 ðaÞ=da changes sign either zero or an even number of times. Because f () is single-peaked, the derivative changes sign at most twice. Because K1 and K2 are CDFs, both run from 0 to 1. Furthermore, K1 6¼ K2 . Hence, limq!0 d½K1 ðaÞ  K2 ðaÞ=da changes sign at least once. Together, these observations imply that K1 single-crosses K2 from below. By continuity, the same holds for q > 0 but small. Therefore, winners’ abilities in Contest 1 are less SC-dispersed than in Contest 2. Proof of Proposition 6. First we prove that selection becomes deterministic in the limit for q ? 0. That is, agents enter high-v Contest 1 iff their ability exceeds a threshold level, a^. Suppose that v2 =v1 2 ðk; kÞ. For vi and vj sufficiently close, such that both contests are competitive in the limit, consider a converging (sub)sequence of equilibria with limit standards ðh1 ; h2 Þ 2 ð1; 1Þ2 . Because h1 ; h2 [  1, the same arguments as in the proof of part (iii) of Proposition 11 imply that, for q ? 0, there continues to exist an a^ 2 R such that ð\Þ

ð\Þ

ð\Þ

ð2Þ

p1 ðaÞ  p2 ðaÞ [ 0 iff a [ a^. Hence, in the limit, (almost) all a [ a^ enter Contest 1 . In turn, this also implies that, in the limit, almost all winners in Contest 1 are of greater ability than all winners in Contest 2. When v2 =v1 62 ½k; k, it is never the case that both contests are competitive in the limit. To see this, suppose by contradiction that both contests remain competitive. In that case, part (iv) of Proposition 11 continues to hold and, therefore, pi ðaÞ  pj ðaÞ [ 0 for all a. Hence, everybody enters the high-prize contest when q ? 0. Contradiction. Next we prove the quantile rankings of entrants and winners for small but positive q. © 2017 Royal Economic Society.

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For q ? 0, consider a converging (sub)sequence of equilibria with limit standards ðh1 ; h2 Þ 2 ½1; 1Þ2 . For a given value of q, let ai ðpi ; qÞ denote the ability of an individual in the pi -th quantile of Contest i. (Formally, ai ðpi ; qÞ  Gi1 ðpi ; qÞ.) Above, we have shown that sorting is ‘perfect’ in the limit, i.e. individuals choose to enter the contest with the higher prize iff their ability exceeds a^. Hence, limq!0 a1 ðp1 ; qÞ [ a^ [ limq!0 a2 ðp2 ; qÞ. By continuity, a1 ðp1 ; qÞ [ a^ [ a2 ðp2 ; qÞ continues to hold for q sufficiently small. Finally, the argument for the quantile ranking of winners is analogous. This proves the claim.

A.6. Grading on the Curve L EMMA 7. Suppose grading is on the curve. For all structural parameters, equilibrium exists and both contests are competitive. Proof. To see that both contests are competitive, recall that 0 \ l1 ; l2 \ 1 and q > 0. Hence, by construction and irrespective of how few contestants have entered a particular contest, prizes are scarce in both contests. This proves the result. To prove existence, fix some CMF Hi , i 2 {1, 2} and observe that, under grading on the curve, standard hi clears the market for prizes in Contest i iff:  Z 1
hi  a Wi ðhi Þ ¼ dHi ðaÞ ¼ mi ¼ li Pri: F ri 1 For Hi given, the mass of prizes li Pri is fixed. Hence, Lemma 1 carries over to grading on the curve. That is, for every CMF Hi , i 2 {1, 2}, there exists a unique standard hi 2 ½1; 1Þ that clears the market. (In fact, hi 2 ð1; 1Þ, because contests are always competitive.) Next, fix some standards ðh1 ; h2 Þ and observe that the payoff difference p1 ða; h1 Þ  p2 ða; h2 Þ is the same as in the baseline model. Similarly, Pr1ðaÞ ¼ Cf½p1 ða; h1 Þ  p2 ða; h2 Þ=qg and hi ðaÞ ¼ g ðaÞ PriðaÞ, i 2 {1, 2}. The previous steps define a function, ξ, from the space of standards [∞, ∞) × [∞, ∞) into itself. From here, the proof of existence proceeds as in the baseline model.

A.7. Endogenous Effort The following Lemma is a useful building block in proving that, in the model with endogenous effort, each Hi induces a uniquely determined hi . L EMMA 8. Properties of x(a, h): (i) dx(a, h)/dh is bounded strictly below 1. Formally, there exists a f > 0 such that dx(a, h)/dh < 1  f for all x 2 R. (ii) For all ‘a’, limh!1 h  xða; hÞ ¼ 1 and limh!1 h  xða; hÞ ¼ 1. Proof. (i) Implicitly differentiating the FOC for optimal effort we get: dxða; hÞ ¼ dh

vf 0 þ

vf 0 : @ 2 cðx; aÞ ð@xÞ2

The SOC for a maximum guarantees that the denominator of this expression is positive. The result then follows from the f 0 being bounded and @ 2 cðx; aÞ=ð@xÞ2 being bounded away from zero. (ii) The FOC and single-peakedness of f around zero imply that: @c  f ð0Þv; @x © 2017 Royal Economic Society.

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and, therefore:
x

@c @x

1 ½f ð0Þv;

where ð@c=@xÞ1 ðÞ denotes the inverse of @c/@x with respect to x. Hence, x is bounded and limh!1 h  xða; hÞ ¼ 1. Next notice that: d dxða; hÞ ½h  xða; hÞ ¼ 1  : dh dh Part (i) implies that d[h  x(a, h)]/dh is strictly positive and bounded away from zero. Hence, limh!1 h  xða; hÞ ¼ 1. Lemma 8 allows us to show that standards are unique. Formally: L EMMA 9. In a contest with endogenous effort, there exists a unique equilibrium standard hi for every Hi . Proof. If Hi is such that Hi ð1Þ  mi , then all individuals win a prize and hi ¼ 1. If Hi ð1Þ [ mi , then the equilibrium standard hi solves:   hi  xi ða; hi Þ dHi ðaÞ ¼ mi : F ri 1

Z Wi ðhi Þ ¼

1

(A.6)

An implication of Lemma 8 part (ii) is that Wi ðhi Þ ! Hi ð1Þ [ mi when hi ! 1, and Wi ðhi Þ ! 0\mi when hi ! 1. Continuity of Wi ðhÞ in hi and the intermediate value theorem then imply that there exists a hi such that (A.6) holds. To prove uniqueness, it suffices to show that Wi ðhi Þ is strictly decreasing in hi . By Lemma 8 part (i), dxi =dhi \1. Hence:     Z 1  Z 1 d hi  xi ða; hi Þ 1 hi  xi ða; hi Þ dxi ða; hi Þ  F dHi ðaÞ ¼  1 dHi ðaÞ\0: f dhi 1 ri ri dhi 1 ri We are now in a position to prove: P ROPOSITION 15. An equilibrium exists in the selection model with endogenous effort. In a symmetric baseline, the equilibrium is unique. Both contests are competitive and have the same standard. 50% of every ability type enter each contest. When q ? 0, both contests remain competitive in a neighbourhood of a symmetric baseline. When one contest is uncompetitive, the probability of selecting into the competitive contest is strictly increasing in ability. Proof. From Lemma 9 we know that there exists a unique equilibrium standard hi for every Hi . All the other steps in the proof of Proposition 15 are identical to those in the exogenouseffort model. Proof of Proposition 7. The result follows from the fact that, in a neighbourhood of a symmetric baseline, we may reinterpret an individual’s endogenous equilibrium effort as his exogenous ability type, and treat the problem as one of pure selection without effort. This transformation is justified as follows: (i) In a symmetric baseline, equilibrium effort of each ability type is the same across contests. Since effort is strictly increasing in ability, this implies that there exists a unique mapping from effort to ability and vice versa. © 2017 Royal Economic Society.

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(ii) Since effort is the same in both contests, the cost of effort differences out when calculating the payoff difference across contests in a symmetric baseline. (iii) The envelope theorem implies that, in a neighbourhood of structural parameters around a symmetric baseline, we may ignore changes in equilibrium effort when calculating payoff differences across contests. Hence, in such a neighbourhood, we can reinterpret (endogenous) equilibrium effort in the symmetric baseline as a new (but still exogenous) ability type. Together, these observations imply that, in a neighbourhood of a symmetric baseline, the model with endogenous effort is isomorphic to one with exogenous ability/effort types. Hence, all selection results carry over. Proof of Lemma 3. First notice that dPr1(a)/da takes on the same sign as dðp1  p2 Þ=da. Next observe that, by the envelope theorem: Z x  ðaÞ 2 1 dðp1  p2 Þ @c½x1 ðaÞ; a @c½x2 ðaÞ; a d cðx; aÞ  dx: ¼ ¼ @a dxda da @a x2 ðaÞ Finally, recall that d2 cðx; aÞ=ðdxdaÞ is strictly negative by assumption. Hence, dðp1  p2 Þ=da takes on the same sign as x1 ðaÞ  x2 ðaÞ.

Appendix B. Competitive Versus Uncompetitive Contests In this Appendix, we derive some results regarding the (un)competitiveness of contests. We begin by identifying situations where only one of the contests is competitive. For w and v, competitiveness turns on the intuitive condition that the difference between contests should not be too large. For example, if Contest 1 offers an enormous show-up fee relative to Contest 2, Contest 1 will attract so many entrants that Contest 2 becomes uncompetitive. The other two parameters, m and r, do not (necessarily) have this property. In the case of discriminativeness, this is intuitive: there is no reason for one or the other contest to become uncompetitive solely due to a difference in discriminativeness. For a difference in the number of prizes, we show that both contests remain competitive provided pecuniary factors dominate. The intuition is as follows. If contestants mainly care about money, small differences in standards are sufficient to induce large differences in entry across contests. Therefore, the endogenous adjustment of standards in response to a difference in the number of prizes suffices to maintain competitiveness of both contests. The following Lemmas formalise these intuitions by stipulating conditions such that one of the contests becomes uncompetitive in the case of w and v, or both contests remain competitive in the case of r and m. For the result regarding w and v, the ‘all else equal’ condition is not needed, i.e. parameters in the two contests can be arbitrary. Formally: L EMMA 10. Fix all structural parameters save w1 (v1 ). For w1 (v1 ) sufficiently large, Contest 2 is uncompetitive. Proof. Notice that:  
 
  h a 1 h a w1  w2 þ v1 F 1 lim C  v2 F 2 g ðaÞda r2 q r1 1 v1 !1 ( " ! # ) Z 1 ^ h1  a 1 lim C  v2 g ðaÞda ¼ 1  1 ¼ 0; 1  w1  w2 þ v1 F v !1 q r1 1 1 Z

lim Pr2 ¼1 

v1 !1

1

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where ^h1 denotes the (finite) market clearing threshold if everybody entered Contest 1. Hence, for v1 sufficiently large, Pr2\m2 , such that Contest 2 is uncompetitive. The argument for w1 is analogous. For differences in r and m, the next two results provide conditions that ensure that both contests remain competitive. Obviously, such results cannot be obtained for arbitrary parameter values since, for sufficiently large differences in w or v, one contest is uncompetitive. Accordingly, we restrict attention to the ‘all else equal’ case. Formally: L EMMA 11. Suppose the two contests are identical save for their level of meritocracy. For all ðr1 ; r2 Þ 2 ð0; 1Þ2 , both contests are competitive. Proof. Suppose not. Then the standard in the uncompetitive contest is ∞. Hence, all agents monetarily prefer this contest. As a result, more than 50% of each ability type enter. However, this is inconsistent with this contest being uncompetitive because 2m < 1 ⇔ m < 1/2. Contradiction. To guarantee that both contests are competitive when there is an imbalance in the number of prizes across contests, pecuniary considerations must come sufficiently to the fore. The reason is that, when non-pecuniary motives dominate, each contest attracts roughly half of most ability types. Formally: L EMMA 12. Suppose the two contests are identical save for the number of prizes. For any ðm1 ; m2 Þ such that m1 þ m2 \1, both contests are competitive if q is sufficiently small. Proof. Suppose by contradiction that Contest 2, say, is uncompetitive for q sufficiently small. Then, h2 ¼ 1, while h1 [  1, because prizes are scarce in the aggregate. Since w2 ¼ w1 and v2 ¼ v1 , Pr2 becomes: Z Pr2 ¼ 1 

1

1

C

 
   v  h1  a  1 g ðaÞda: F q r1

(B.1)

Provided that limq!0 h1 ðqÞ [  1, (B.1) implies that limq!0 Pr2 ! 1 [ m2 . Notice, however, that this contradicts Contest 2 being uncompetitive for small q. To finish the proof, it remains to show that limq!0 h1 ðqÞ [ 1. Suppose to the contrary that limq!0 h1 ðqÞ ¼ 1. Let W ðqÞ ¼ W1 ðqÞ þ W2 ðqÞ, i.e. W(q) is the sum total of winners in both contests for a given value of q. Since limq!0 h1 ðqÞ ¼ 1 and Contest 2 is uncompetitive by assumption, we have limq!0 W ðqÞ ¼ 1 [ m1 þ m2 . But this contradicts the notion that h1 ðqÞ is an equilibrium standard. Hence, limq!0 h1 ðqÞ [ 1.

Appendix C. Connection to Tullock (1980) and Lazear and Rosen (1981) In this Appendix, we study how our contest model with a continuum of players relates to the finite-player models of Tullock (1980) and Lazear and Rosen (LR, 1981). In Tullock (1980), winning probabilities are determined by a reduced-form ‘contest success function’ (CSF), while LR (1981) rely on an actual comparison of measured performances to identify the winner(s) of the contest. To illustrate the difference, suppose there are n agents j 2 {1, . . ., n} with abilities a ¼ fa1 ; . . .; an g, aj 2 R. In an LR-style contest, which—following their terminology – we will from now on refer to as a tournament, measured performances y ¼ fy1 ; . . .; yn g are given by yj ¼ aj þ ej , where noise ej is i.i.d. and location-scale (0, r). (As in the baseline model of Section 1, we assume that ‘effort’ is exogenous and equal to ability. This simplifies the exposition but is not essential.) With a single prize v, agent j wins the tournament with probability

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e  e  ai  aj j i  ; 8i 6¼ j : r r By contrast, in a Tullock-style lottery contest, player j’s winning probability is given by the CSF Rn ! ð0; 1Þ: wðaj Þr PrTC ½winðaj Þ ¼ Pn (C.1) r ; i¼1 wðai Þ PrLR ½winðaj Þ ¼ Pr

which maps the profile of abilities, a, into a winning probability for a player with ability aj . Here, wðÞ : R ! Rþ is some strictly positive, strictly increasing, and differentiable function, while r > 0 is a measure of the discriminativeness of the contest. In a 2-player setting, Fu and Lu (2012) show that a Tullock contest is isomorphic to an LR tournament with extreme value type I noise (EVTI). To see why, recall that the difference of two EVTI random variables is logistically distributed. Hence: a1 a2

e  e a2  a1  e r 1 2 PrLR ½winða1 Þ ¼ Pr  ¼ a1 a2 r r 1þe r a1

¼

er e

a2 a1 rþ r

e

¼

wða10 Þr ¼ PrTC ½winða10 Þ: þ wða20 Þr

wða10 Þr

Here, r = 1/r, while aj0  w1 ðe aj Þ, which constitutes a strictly monotone relabelling of ability. For a single prize, the isomorphism between a Tullock contest and an LR-EVTI tournament easily extends to n players, n > 2. With multiple prizes, the relationship becomes more complicated. Fu and Lu (2012) show that for k < n prizes of weakly decreasing value v1  . . .  vk  0, a single LR-EVTI tournament is equivalent to a sequence of nested Tullock contests: In each round 1 to k, one winner is chosen by means of a standard Tullock CSF. The winner receives the highest remaining prize and exits. The process ends when all prizes have been allocated. We now restrict attention to the special case where the k prizes are of equal value. Assume that players’ abilities a ¼ fa1 ; . . .; an g are i.i.d. according to the PDF g(a), a 2 R. Then, in an LR tournament, measured performances y ¼ fy1 ; . . .; yn g are i.i.d. according to the convolution of  a and ɛ. Its DCDF, BðÞ, is given by:  ¼ BðyÞ

Z

y  a  F g ðaÞda: r 1 1

(C.2)

Agent j wins a prize iff his performance, yj , exceeds the k-th order statistic, yðkÞ , of the performance draws of the n  1 other agents. Hence: PrLR ½winðaj Þ ¼ Pr ½ej [ yðkÞ  aj : Because yðkÞ is stochastic, PrLR ½winðaj Þ is hard to calculate for intermediate values of n and k. However, if n becomes large while the fraction of winners m  k/n < 1 remains constant, then: Pr yðkÞ ! B1 ðmÞ:

In turn, this implies that:  1  B ðmÞ  a : PrLR ½winðaÞ ! Pr ½ej [ B1 ðmÞ  a ¼ F r

(C.3)

Notice that the resulting LR tournament is equivalent to our baseline ‘contest’ model with h ¼ B1 ðmÞ. © 2017 Royal Economic Society.

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We now try to express PrLR ½winðaÞ in (C.3) as a contest success function. With a unit mass of contestants and a fraction m 2 (0, 1) of equal prizes, a CSF is a mapping of ability profiles into a winning probability for a player of ability a. An ability profile is given by a PDF g(). If the DCDF  BðyÞ in (C.2) is invertible in closed form, then substituting B1 ðmÞ into (C.3) yields a CSF representation of PrLR ½winðaÞ. Unfortunately, closed-form solutions for B1 ðmÞ may not exist. For example, if noise is EVTI then: PrLR ½winðaÞ ¼ 1 

1  ; a  B1 ðmÞ exp exp r

R1   is not invertible in closed and BðyÞ ¼ 1 f1  1= exp exp½ðy  aÞ=rgg ðaÞda. Because this BðyÞ form, an LR-EVTI tournament with a continuum of contestants does not have a CSF representation – and, hence, as n ? ∞, neither does the limit of nested Tullock contests with k = m 9 n prizes. We now define (the analogue of) a Tullock CSF for the continuum: with a unit mass of contestants and a mass m 2 (0, 1) of equal prizes, a Tullock CSF is given by:19 wr ðaÞ

PrTC ½winðaÞ  m Z

wr ðaÞg ðaÞda

:

a2A

Here, w() has the same properties as in (C.1) above. To ensure that PrTC ½winðaÞ  1 for all a 2 A R, we also require that m,w(), and A R satisfy: wr ðsup AÞ 

1 m

Z wr ðaÞg ðaÞda: a2A

Notice that the Tullock CSF clears the market for prizes for all admissible m and w(). Also, notice that we can always relabel abilities according to a 0  ln wðaÞ, which transforms the Tullock CSF into its ‘canonical form’: e ra

PrCTC ½winða 0 Þ ¼ m Z

0

0

a0 2A0

e r a cða0 Þda0

;

where cða 0 Þ is the PDF of a 0 . In the remainder, we assume that this transformation has taken place and revert to using a and g() to denote ability and its PDF. We now show that a Tullock contest in canonical form is equivalent to an LR tournament with exponential noise. P ROPOSITION 16. An LR tournament with a unit mass of contestants and 0 < m < 1 prizes is isomorphic to a canonical Tullock contest iff noise is exponential. Formally: PrLR ½winðaÞ ¼ PrCTC ½winðaÞ () where z ¼ r ln½ð1=mÞ

19

R a2A

e Expð1Þ; r ¼ 1=r ; h ¼ z; r

e ða=rÞ g ðaÞda.

We thank an anonymous referee for suggesting this formulation.

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Proof. We seek necessary and sufficient conditions such that:
  h a e ra F : ¼mZ r e r a g ðaÞda a2A

Observe that: 2 mR

3

6 1 e 1 ¼ exp6 41=r  r ln Z r a g ðaÞda e a2A ra


 7 a 7e 1=r ¼ exp  z  a ; 5 a 1=r e 1=r g ðaÞda m

a2A

R

where z  ð1=r Þ lnfð1=mÞ a2A exp½a=ð1=r Þg ðaÞdag. Hence, an LR tournament gives rise to a canonical Tullock contest iff: (i) (ii) (iii)

FðÞ is the DCDF of the Exponential distribution; r = 1/r; and h ¼ z.

Finally, notice that h ¼ z indeed clears the market for prizes. Because the exponential distribution is not strictly log-concave, the selection results in the main text do not apply to Tullock contests. We now proceed to fill this gap. Consider a unit mass of agents self-selecting into one of two Tullock contests. As in the main text, Contest i = 1, 2 offers a show-up fee wi [ 0 and mi prizes of value vi [ 0, with 0 \ mi \ 1 and m1 þ m2 \1. First, observe that the results for the uncompetitive case remain unchanged. That is, selection into the competitive contest is strictly increasing in ability. Next, consider the competitive case. Conditional on mass density hi ðaÞ of contestants having entered Contest i, agent a’s probability of winning is: PrTC ½wini ðaÞ ¼

wri ðaÞ : hi ðaÞ da wri ðaÞ Pri a2A

mi Z Pri

Here, mi =Pri\1, since both contests are competitive. P ROPOSITION 17. Selection across two competitive Tullock contests is as follows: 1. If r1 ¼ r2 , then Pr1(a) is monotone. ð\Þ

2. If r1 [ r2 , then Pr1(a) is (inverse-)U-shaped. Proof. For an agent of ability a, the payoff difference between entering Contest 1 and 2 is: p1 ðaÞ  p2 ðaÞ ¼ w1  w2 þ v1 PrTC ½win2 ðaÞ  v2 PrTC ½win2 ðaÞ: Differentiating with respect to a, we find that d½p1 ðaÞ  p2 ðaÞ=da takes on the same sign as: Z wr1 ðaÞh1 ðaÞda v2 m2 r2 a2A Z wr2 r1 ðaÞ: 1 v1 m1 r1 r2 w ðaÞh2 ðaÞda a2A ð\Þ

Hence, if r1 ¼ r2 , then Pr1ðaÞ ¼ Cf½p1 ðaÞ  p2 ðaÞ=qg is monotone. If r1 [ r2 , then Pr1(a) is (inverse-)U-shaped. © 2017 Royal Economic Society.

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To understand what drives the selection, recall from the main text that dPr 1(a)/da takes on the same sign as kðaÞ  ðv2 =v1 Þ=ðr2 =r1 Þ , where kðaÞ ¼ f ½ðh1  aÞ=r1 =f ½ðh2  aÞ=r2  (see (4)). Notice that, for exponential noise and r1 ¼ r2 , k(a) is independent of a, rather than monotone. As a consequence, Pr1(a) is monotone rather than single-peaked. Next, when r1 6¼ r2 , log-linearity implies that k(a) is monotone in a instead of single-peaked. As a consequence, Pr1(a) is single-peaked rather than exhibiting two extrema. Finally, in light of Proposition 17, one may wonder about selection in situations where noise is log-convex. It is readily characterised: log-convex noise reverses the results in the main text. Specifically, the selection curves in Figure 2 are transformed into their mirror images with respect to the horizontal line Pr = 1/2. In this case, higher show-up fees attract intermediate abilities and deter the extremes, while more prizes attract the extremes and deter intermediate abilities. Perhaps most strikingly, higher prize values and greater meritocracy now produce negative selection.

University of California Erasmus University Rotterdam and Tinbergen Institute International Monetary Fund Submitted: 23 April 2016 Accepted: 26 October 2016

References Autor, D. (2003). ‘Lecture note: self-selection – The Roy model’, Available at: http://economics.mit. edu/files/551 (last accessed: 4 November 2016). Azmat, G. and M€ oller, M. (2009). ‘Competition among contests’, RAND Journal of Economics, vol. 40(4), pp. 743–68. Azmat, G. and M€ oller, M. (2012). ‘The distribution of talent across contests’, Barcelona GSE Working Paper, January 2012. Borjas, G. (1987). ‘Self-selection and the earnings of immigrants’, American Economic Review, vol. 77(4), pp. 531–53. Burguet, R. and Sakovics, J. (1999). ‘Imperfect competition in auction designs’, International Economic Review, vol. 40(1), pp. 231–47. Coate, S. and Loury, G. (1993). ‘Will affirmative action eliminate negative stereotypes?’, American Economic Review, vol. 83(5), pp. 1220–40. Damiano, E., Li, H. and Suen, W. (2010). ‘First in village or second in Rome’, International Economic Review, vol. 51(1), pp. 263–88. Damiano, E., Li, H. and Suen, W. (2012). ‘Competing for talents’, Journal of Economic Theory, vol. 147(6), pp. 2190–219. d’Aspremont, C. Jaskold Gabszewicz, J. and Thisse, J.-F. (1979). ‘‘On hotelling’s “stability in competition”’, Econometrica, vol. 47(5), pp. 1145–50. Frank, R. (1985). Choosing the Right Pond: Human Behavior and the Quest for Status, New York: Oxford University Press. Fu, Q. and Lu, J. (2012). ‘Micro foundations of multi-prize lottery contests: a perspective of noisy performance ranking’, Social Choice and Welfare, vol. 38(3), pp. 497–517. Ganuza, J.-J. and Penalva, J.S. (2006). ‘On information and competition in private value auctions’, mimeo, available at: http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1001885 (last accessed: 4 November 2016). Gladwell, M. (2013). David and Goliath: Underdogs, Misfits, and the Art of Battling Giants, New York: Little, Brown, and Company. Heckman, J. and Honor
e, B. (1990). ‘The empirical content of the Roy model’, Econometrica, vol. 58(5), pp. 1121–49. Heckman, J. and Taber, C. (2008). ‘Roy model’, in (S. Durlauf and L. Blume, eds.), The New Palgrave Dictionary of Economics, 2nd edition. London, UK: Palgrave Macmillan. Hotelling, H. (1929). ‘Stability in competition’, ECONOMIC JOURNAL, vol. 39(153), pp. 41–57. Konrad, K. and Kovenock, D. (2012). ‘The lifeboat problem’, European Economic Review, vol. 56(3), pp. 552– 59. Lazear, E. (2000). ‘Performance pay and productivity’, American Economic Review, vol. 90(5), pp. 1346–61. © 2017 Royal Economic Society.

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49

Lazear, E. and Rosen, S. (1981). ‘Rank-order tournaments as optimal labor contracts’, Journal of Political Economy, vol. 89(5), pp. 841–64. Leuven, E., Oosterbeek, H. and van der Klaauw, B. (2010). ‘Splitting tournaments’, IZA Discussion Paper 5186. Leuven, E., Oosterbeek, H., Sonnemans, J. and der Klaauw, B. (2011). ‘Incentives versus sorting in tournaments: evidence from a field experiment’, Journal of Labor Economics, vol. 29(3), pp. 637–58. McAfee, R. (1993). ‘Mechanism design by competing sellers’, Econometrica, vol. 61(6), pp. 1281–312. Moldovanu, B., Sela, A. and Shi, X. (2008). ‘Competing auctions with endogenous quantities’, Journal of Economic Theory, vol. 141(1), pp. 1–27. Peters, M. and Severinov, S. (1997). ‘Competition among sellers who offer auctions instead of prices’, Journal of Economic Theory, vol. 75(1), pp. 114–79. Plutarchus, L.M. (n.d.). ‘Life of Caesar’, Available at: http://classics.mit.edu/Plutarch/caesar.html (last accessed: 4 November 2016). Roy, A. (1951). ‘Some thoughts on the distribution of earnings’, Oxford Economic Papers, vol. 3(2), pp. 135–46. Tullock, G. (1980). Towards a Theory of the Rent-seeking Society, Northampton, MA: Edward Elgar Publishing.

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