Economics Letters 125 (2014) 411–414

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Economics Letters

journal homepage: www.elsevier.com/locate/ecolet

The incentive effect of coarsening the competition structure in atournament✩

Sam-Ho LeeDepartment of Economics, Korea University, Republic of Korea

a r t i c l e i n f o

Article history:Received 20 August 2014Accepted 25 September 2014Available online 18 October 2014

Keywords:CoarseningTournamentIncentive effectLog-concave density

a b s t r a c t

The incentive effect of coarsening the competition structure in a tournament is studied. By coarseningthe competition structure, we mean that coarser performance measure is used while finer informationis available. Examples include letter grades or grade classes when finer numeric grades are available.Coarsening the competition structure has two countervailing incentive effects. While it reduces thelikelihood that marginal effort changes the result, the reward change will be bigger once the result ischanged. We provide a sufficient condition on the performance distribution for the reduction of workincentive by coarsening; log-concavity of the density.

© 2014 Elsevier B.V. All rights reserved.

1. Introduction

In a tournament where the reward is given by the relativerank, coarsening the performance measures can happen quitefrequently. In grading students, the letter grades such as A, B, and Care given even though numeric grades of exams are available. BoaltHall, UC Berkeley’s law school lets admission committee learn onlythe partitioned intervals LSAT score belongs to, not the exact LSATscores (Chan and Eyster, 2003).1 This paper theoretically studiesthe incentive effect of the coarseness (or fineness) of performancemeasures.

This study is especially motivated by the coarsening of gradingin Korean College Scholastic Ability Test (CSAT hereafter) in 2008.Instead of announcing the original numeric score, the Ministry ofEducation decided to provide only the nine classes the originalscore belongs to. The Ministry argued that the new system willreduce the competition to get 1–2more numeric grades. This studycan investigate the validity of this reasoning.

Coarsening the performance measure has two effects. On onehand, this will reduce the probability that the marginal effort willchange the result. As in the CSAT case, small increases in numericgrade would not change the class it belongs to. Therefore, thiseffect will reduce the work incentive, and the Ministry’s argumentcaptures this effect.

✩ The author acknowledges support from the Korean National ResearchFoundation (2014S1A5A8017336).

E-mail address: samho@korea.ac.kr.1 According to Chan and Eyster, this decision was to increase the admission of

minority students after the affirmative action is banned.

http://dx.doi.org/10.1016/j.econlet.2014.09.0280165-1765/© 2014 Elsevier B.V. All rights reserved.

On the other hand, coarseningwill increase the reward changesonce the marginal effort changes the result. Suppose that themarginal effort changes the grade class in the CSAT case. Thiswould have been a small numeric grade change before coarsening,but now it is the change in grade class. We can think of the re-ward of belonging to one grade class as the (weighted) average ofthe rewards of numeric grades in that class. Therefore, the rewardchanges one gets will be much larger after coarsening. This effectwill increase the work incentive, which is missed in the Ministry’sargument.

This paper, as a theoretical trial to address this issue, acknowl-edges the possibility that coarsening may or may not reduce theeffort incentive and provides a condition on the distribution of per-formances in a specific setting,whichwill tellwhether the coarsen-ing will increase or decrease the work incentive. In our model, theresult of the competition is determined by performance measureswhich are noisy signals of efforts (specifically, the addition of effortand error term). We show that the incentive effect of coarsening isdetermined by the distributional character of performance mea-sures. If the performance measures are informative of effort levelsin a specific sense, that is, higher effort level ismore likely aswe seehigher performance (technically called monotone likelihood ratioproperty, or MLRP), then the coarsening of the competition struc-ture, thus the less information about the effort in the final result,would reduce the effort incentive. If it is the opposite case, that is,lower effort level is more likely as we see a higher performance inthe relevant performance range, the coarsening will increase theeffort incentive.

The paper proceeds as follows. Section 1.1 will briefly discussthe related literature. Section 2 will introduce the model. Theanalysis follows in Section 3. Section 4 concludes.

412 S.-H. Lee / Economics Letters 125 (2014) 411–414

1.1. Related literature

The effect of a tournament competition has been analyzed sinceLazear and Rosen (1981). It is analyzed whether the tournamentstructure can reproduce the same incentive effect of the traditionalincentive schemes such as piece rate and absolute performance pay(Lazear and Rosen, 1981) or underwhat circumstance it dominatesindividual contract (Green and Stokey, 1983). The incentive effectof tournament is applied to the career concerns in the labormarketand related executive pay (Rosen, 1986 among others).

Since tournament theory is centered around its incentive effect,it is a natural question to ask how we should design a tournamentto maximize the work incentive. Moldovanu and Sela (2001, 2006)dealt with this question. In their papers, they use the frameworkof all pay auction to answer how rewards should be allocatedand whether there should be a one grand contest or some sub-contests before the final contest to maximize the effort. Thesepapers assume that rewards of each prize can be determined by thecontest designer. This paper will deal with the situation where therewards of relative ranks are exogenously given. This is relevantto LSAT or CSAT scores since the reward of entering a certain lawschool or a certain university cannot be controlled by the contestdesigner.

As the rewards of each rank are exogenously given, this paperis closely related to Moldovanu et al. (2007) and Dubey andGeanakoplos (2009). Both papers assume the specific form ofexogenous reward of relative ranks and analyze the incentive effectof categorization of ranks. This paper, on the contrary, considersthe arbitrary rewards of ranks to accommodate more generalcompetition.

2. Model

There are a continuum of agents with unit mass.2 An agent isdenoted by i ∈ [0, 1]. Each agent’s performance pi, which decidesthewinners of the competition, is a noisy signal of the effort level ei.Specifically, the performance is the sum of ei and the error term εi;

pi = ei + εi.

The performance can be changed by exerting effort with costc (e) with c ′ > 0 and c ′′ > 0. The error term εi is assumed to bei.i.d. with a probability density and distribution function g (ε) andG (ε).

Competition structure is composed of the rewards of prizesV =

vj

Jj=1 and the masses of prize winners M =

mj

Jj=1. There

are J divisions of prizes, and top m1 mass gets the first prize withreward v1, the nextm2 mass gets the second prize with reward v2,and so on. Without loss of generality,

Jj=1 mj = 1.3 The structure

of competition is CS = {V,M}. For M′=

m′

k

Kk=1 with K < J ,

we say M′ is coarser than M if some of the adjacent prizes in Mare integrated as one in M′. Then, the rewards of the prizes in anew competition V′

= {vk}Kk=1 are automatically determined. The

reward of a prize should be the weighted average of the rewardsor original prizes which were integrated.

Example 1. Suppose the original competition is {{1, 1/2, 0} ,{1/3, 1/3, 1/3}}. That is, top 1/3 mass will get the prize of reward1, middle 1/3 mass will get 1/2, and bottom 1/3 will get 0.Coarsening of this competition structure is possible in two ways.

2 Continuum of agents are an approximation of a large number of agents, whichare relevant to the interested case of CSAT.3 If there is no reward for low performers, we can assume vJ = 0.

One isV′, {2/3, 1/3}

, in which top two prizes are integrated

as one. Then V ′ should be {3/4 = 1/2 · 1 + 1/2 · 1/2, 0}. Theother is

V′′, {1/3, 2/3}

. Then V′′ should be {1, 1/4 = 1/2 · 1/2

+1/2 · 0}.

We can formally define coarsening.

Definition 1. Let the original competition structure be CS =

{V,M}, where V =vj

Jj=1 and M =

mj

Jj=1. Suppose there is

another competition structure CS ′=

V′,M′

such that V′

=

{vk}Kk=1 andM′

=m′

k

Kk=1.We sayM′ is coarser thanM and denote

M′≻ M if

(i) there are less prize divisions inM′ than M, that is J > K ,(ii) for all k′

∈ {1, . . . , K}, there exists j′ such thatj′

j=1 mj =k′k=1 m

′

k.

Competition structure CS ′ is coarser than CS, or CS ′≻ CS if

M′≻ M and V′ satisfies

j′j=1 mjvj =

k′k=1 m

′

kv′

k ifj′

j=1 mj =k′k=1 m

′

k.

The ex-post utility of the competing agents when their perfor-mances are realized is

u(p, e) =

Jj=1

vj · I{winning jth prize} − c (e) ,

where I {·} is an indicator function. An agent gets the prize of util-ity value vj as she gets jth prize. Therefore, the ex-ante expectedutility when agents choose their effort levels is

U (e) =

Jj=1

vj · Pr{winning jth prize} − c (e) .

3. Analysis

3.1. Equilibrium

The performance distribution will result from the agents’choices of the effort levels and the realizations of the error terms.The cutoffs Γ =

γj

J−1j=1 of each prize division will emerge in this

performance distribution. The agents whose performances are be-tween γj and γj−1 will get the jth prize.

With the cutoffs Γ given, each agent chooses her effort levelsto maximize the expected utility;

U(e) = v1 [1 − G (γ1 − e)] +

J−1j=2

vjG

γj−1 − e

− G

γj − e

+ vJ · G

γJ−1 − e

− c (e) . (1)

We consider a symmetric equilibrium where all agents choosethe same effort level, and let e∗ be the equilibrium effort level. Letus also define ε∗

j ≡ γj − e∗, which is the realization of the errorterm locating one’s performance at the cutoffs. Then e∗ will bedetermined by the following first order condition;J−1j=1

vj − vj+1

g

ε∗

j

= c ′

e∗

. (2)

The marginal effort will increase the agent’s performance andthis increased performance will change the prize with a certainprobability. The term g(ε∗

j ) is the probability that the marginaleffort will change the winning prize from (j + 1)th to jth and theterm

vj − vj+1

is the additional utility when this happens. The

left hand side of Eq. (2) is the marginal gain of the agent’s expectedutility when one exerts more effort. This marginal gain should thesame as the marginal cost at the optimum.

S.-H. Lee / Economics Letters 125 (2014) 411–414 413

Fig. 1. Comparison ofM and M′′ .

Given the equilibrium effort e∗, the equilibrium cutoffΓ shouldsatisfy the masses of each prize winners M in the performancedistribution. Since there are continuum of agents, the realizedperformance distribution will coincide with the ex-ante distribu-tion. Thus Γ should satisfy the following condition.

1 − Gε∗

1

= m1 (3)

Gε∗

j−1

− G

ε∗

j

= mj if j = 2, . . . , J − 1

Gε∗

J

= mJ .

The equilibriumof this competition is the pair (Γ , e∗) satisfying(2) and (3).

Lemma 1. Given the competition structure CS = {V,M}, the equilib-rium is a pair (Γ , e∗) satisfying (2) and (3).

For later use, we also note that ε∗

j satisfying (3) does not changewhen the equilibrium effort levels change. The equilibrium effortlevel shifts the performance distribution, but the shape of thedistribution is solely determined by the distribution of ε. Theequilibrium cutoffsΓ change as e∗ changes, but ε∗

j would not sincethe change in e∗ is exactly offset by the change in Γ .

3.2. Incentive effect of the coarsening

Wenowconsider another competition structure CS ′=

V′,M′

which is coarser than CS (CS ′

≻ CS). We investigate how theequilibrium effort level will change in the transition form CS to CS ′.To do that, we considerM′′ (and relevant V′′), which is the same asM except that one pair of adjacent prize divisions ofM is integratedas one; that is, m′′

l = ml + ml+1 and m′′

j = mj if j < l, = mj+1 if

j > l. Note that any coarser structureM′ is obtained by a sequenceof integrating a pair of adjacent prize divisions. If we can providea sufficient condition that work incentive increases (or decreases)as we move from M to any arbitrary M′′, that sufficient conditionwill be also valid as we move fromM to M′.

Fig. 1 shows the change from M to M′′ when the integratedadjacent prizes are in the middle of the prize structure, 2 ≤ l ≤

J − 2. All other prize divisions are the same between M and M′′

except for that lth and (l+ 1)th prize divisions inM are integratedas one division in M′′, m′′

l = ml + ml+1. Accordingly, the rewardof the integrated prize division is the weighted average of twoprize rewards, v′′

l = 1/m′′

l · (mlvl + ml+1vl+1). As explained above,ε∗

j ’s would be the same underM andM′′ though equilibrium effortmight have been changed.

We compare the effort incentives under M and M′′. Remindthat the level of equilibrium effort is determined by two factors in(2); (1) probability that marginal effort changes the prize, (2) thedifference of rewards when it is changed.

On one hand, coarsening the competition structure will reducethe probability thatmarginal effortwould change the prizes.Whilethe marginal effort would change the prize from (l + 1)th to lthwith probability g

ε∗

l

under M, that possibility disappears as

two prizes are integrated as one under M′′. Therefore, the workincentive would decrease by

(vl − vl+1) gε∗

l

. (4)

On the other hand, coarsening the competition structure wouldincrease the difference of rewardswhen the prize is changed. Sincethe reward of the integrated prize v′′

l is larger than vl+1, thereis larger reward difference with the prize right below, v′′

l+1 =

vl+2 than in M. Likewise, since v′′

l is smaller than vl, there islarger reward difference with the prize right above, v′′

l−1 = vl−1.Therefore, work incentive would increase byv′′

l − vl+1g

ε∗

l+1

+

vl − v′′

l

g

ε∗

l−1

= (vl − vl+1)

ml

m′′

lg

ε∗

l+1

+

ml+1

m′′

lg

ε∗

l−1

, (5)

where the second line is the simplification using v′′

l = 1/m′′

l ·

(mlvl + ml+1vl+1).The total effect on effort incentive of the coarsening is depen-

dent on the relative size of (4) and (5). If we subtract (4) from (5),we get the net incentive effect of the coarsening:

(vl − vl+1)

ml

m′′

lg

ε∗

l+1

+

ml+1

m′′

lg

ε∗

l−1

− g

ε∗

l

= (vl − vl+1)

ml · ml+1

m′′

l

g

ε∗

l−1

− g

ε∗

l

G

ε∗

l−1

− G

ε∗

l

−

gε∗

l

− g

ε∗

l+1

G

ε∗

l

− G

ε∗

l+1

(6)

wherewe useml = Gε∗

l−1

−G

ε∗

l

andml+1 = G

ε∗

l

−G

ε∗

l+1

.

Note that the sign of total effect is solely determined by thedistribution of ε. Since the term vl − vl+1 is factored out, the signof the net effect (6) is only dependent on the weight with which itis divided and the density of each cut point.4 Since this weight isrepresented by the distribution functionG (ε) from the equilibriumcondition (3), the sign of net effect is dependent only on thedistribution of ε, not rewards v. We can provide the sufficientcondition on g (ε) which guarantees that the coarsening reducesthe work incentive.

To provide the condition, we first define log-concavity (and log-convexity) of the density function and some of its properties.

Definition 2 (Log-Concavity). A function g is log-concave (log-convex) on the interval (a, b) if ln g is a concave (convex) functionon (a, b), or g ′ (x) /g (x) is monotone decreasing (increasing) on(a, b).

Remark 1. 1. Regardless of the distribution of ε, p is informativeabout e in the sense of first order stochastic domination (FOSD).That is, e′

+ε first order stochastically dominates e+ε if e′ > e.2. If ε has a log-concave density function, p = e + ε is more

informative about e in that it satisfies Monotone LikelihoodRatio Property (MLRP): if you observe higher p, e is more likelyto be higher. That is,

gp′

− e′

g(p′ − e)≥

gp − e′

g (p − e)

if p′ > p and e′ > e.

4 The size of the net effect, of course, is dependent on the prize difference,vl − vl+1 .

414 S.-H. Lee / Economics Letters 125 (2014) 411–414

The log-concavity of density function is used inmany economicapplications because of its MLRP implication, and there are manycommon distributions with log-concave density.5 Some propertiesof log-concave functions are listed for this paper’s use.

Lemma 2. (i) If a function g is log-concave on (a, b), then

gx′′

− g

x′

G (x′′) − G (x′)

<g

x′

− g (x)

G (x′) − G (x)for all x < x′ < x′′

∈ (a, b) .

The inequality is opposite if g is log-convex.(ii) If a density function g is log-concave on (a, b), its distribution

function G and reliability function G = 1 − G are also log-concave. (Bagnoli and Bergstrom, 2005)

Proof. As g ′(t)g(t) >

g ′(x′)g(x′) or g ′ (t) g

x′

> g ′

x′

g (t) for all t <

x′, x′

x g ′ (t) gx′

dt >

x′

x g ′x′

g (t) dt . Therefore, g(x′)−g(x)

G(x′)−G(x)>

g ′(x′)g(x′) . Likewise, g ′(x′)

g(x′) >g(x′′)−g(x′)G(x′′)−G(x′)

. Combining the two will givethe desired result. For the proof of (ii), see Bagnoli and Bergstrom(2005). �

We are ready to provide the sufficient condition on g (ε) ofthe reduction (or increase) of work incentive by coarsening. From(6), the sign of net effect is determined by that of the expression,gε∗l−1

−g(ε∗

l )

Gε∗l−1

−G(ε∗

l )−

g(ε∗l )−g

ε∗l+1

G(ε∗

l )−Gε∗l+1

. By Lemma 2, this expression is

negative (positive) as ε∗

l−1 > ε∗

l > ε∗

l+1 if g (ε) is log-concave (log-convex). Therefore, when the coarsening happens in the middleof prize hierarchy, it will decrease (increase) the work incentiveif the error term has the log-concave (log-convex) density functionin that range.

If the density function is log-concave and therefore perfor-mance p is informative about effort in MLRP sense, the finer com-petition structure, where performance difference is recognized asoften as possible, will increase thework incentive. On the contrary,if the higher performance is less likely with higher effort in therelevant range, the coarser competition structure, which does notrecognize the performance difference, will increase the work in-centive.

We cando a similar analysis and calculate the net effect as abovewhen the integrated prizes are in the end, i.e. l = 1 or l = J − 1.The total effect of those cases is

(v1 − v2)m1

g

ε∗

2

1 − G

ε∗

2

−g

ε∗

1

1 − G

ε∗

1

when l = 1

vJ−1 − vJ

mJ

g

ε∗

J−2

G

ε∗

J−2

−g

ε∗

J−1

G

ε∗

J−1

when l = J − 1.

At the end of prize hierarchy the work incentive of the coarsenedcompetition structure tends to be smaller. Consider coarseninghappens in the high end, l = 1. The part of reward difference,which would have been attributed to the reward difference withhigher prize if it is in the middle, would go away since there is

5 For the list of commondistributionswith log-concave density function and theirproperties, see Bagnoli and Bergstrom (2005).

no higher prize. Therefore, coarsening is more likely to reduce thework incentive.

However, the log-concavity of g (ε) is still a sufficient conditionfor the decrease in work incentive since the log-concavity of thedensity function g implies the log-concavity of distribution andreliability functions G and 1 − G by Lemma 2.6

As noted before, the change fromM toM′ is the result of sequen-tial change of integrating a pair of adjacent prize divisions. Thus thesufficient condition stated above is also valid in comparison of Mand any coarser competition structureM′.

Proposition 1. Coarsening the competition structure will decreasethe effort if the density function of error term g is log-concave inthe relevant range. It will increase the effort if the density functionis log-convex in the relevant range and the distribution function G(or reliability function 1 − G) is log-convex if the coarsening happensat the end of hierarchy.7

4. Conclusion

In this paper, we investigated how coarsening the competitionstructure affects the work incentive of participants. If we assumethat agents’ performance by which the prizes are determined isthe sum of the effort and the error term, we can provide the suf-ficient condition on the observable performance distribution thatthe coarsening reduces the work incentive. Note that we do notneed any information on rewards since the sufficient condition isonly relevant to the performance distribution. This sufficient con-dition is log-concavity of the density function; that is, if the per-formance is informative about the effort level in MLRP sense, thecoarsening of competition structure, by making the performancedifference not recognized in the relevant range will reduce thework incentive.

We should admit that the empirical application of this paper’sframework is quite limited and not directly relevant to the mainmotivation; coarsening of Korean CSAT. This is pursued in thecompanion paper, Han et al. (2014).

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6 The log-convexity of g (ε), on the contrary, does not guarantee the oppositeconclusion since it does not imply log-convexity of G and 1 − G.7 Note that the distribution function and the reliability function cannot be log-

convex for the whole range at the same time. While g must be increasing if G is tobe log-convex, g must be decreasing if 1 − G is to be log-convex. Therefore, thisresult does not conflict with the fact that the complete coarsening (the same prizefor all the competitors) will always reduce the work incentive.