Articles in Advance, pp. 1–18ISSN 0732-2399 (print) ó ISSN 1526-548X (online) http://dx.doi.org/10.1287/mksc.2013.0798

©2013 INFORMS

Favoring the Winner or Loser in Repeated Contests

Robert RidlonKelley School of Business, Indiana University, Bloomington, Indiana 47401, rridlon@indiana.edu

Jiwoong ShinYale School of Management, Yale University, New Haven, Connecticut 06520, jiwoong.shin@yale.edu

Should a firm favor a weaker or stronger employee in a contest? Despite a widespread emphasis on rewardingthe best employees, managers continue to tolerate and even favor poor performers. Contest theory reveals

that evenly matched contests are the most intense, which implies that a contest designer can maximize eachplayer’s effort by artificially boosting the underdog’s chances. We apply this type of “handicapping” to a two-period repeated contest between employees, in which the only information available about their abilities is theirperformance in the first period. In this setting, employees are strategic and forward looking, such that theyfully anticipate the potential impact of the first-period contest result on the second-period contest and thusadjust their behaviors accordingly. The manager also incorporates these strategic behaviors of employees whendetermining an optimal handicapping policy. If employees’ abilities are sufficiently different, favoring the first-period loser in the second period increases the total effort over both periods. However, if abilities are sufficientlysimilar, we find the opposite result occurs: total effort increases the most in response to a handicapping strategyof favoring the first-period winner.

Key words : game theory; contests; handicap; incentives; ratcheting; moral hazardHistory : Received: February 10, 2012; accepted: May 14, 2013; Preyas Desai served as the editor-in-chief and

Ganesh Iyer served as associate editor for this article. Published online in Articles in Advance.

1. IntroductionGiven their limited resources, firms often face a toughdilemma on how best to motivate their employees: dothey invest in laggards or reward their top perform-ers? Still, this fundamental management question hasnot gained a consensus answer. Many experts recom-mend that managers should reward their top employ-ees and avoid coddling weaker employees with lowerstandards. For example, top sales agents often receivemore training and obtain more back-office resources(Farrell and Hakstian 2001). Indeed, $1.3 billion intraining expenses are “devoted to grooming leaders”(Kranz 2007, p. 22) and managers are often told tolook for top performers to receive specialized learn-ing opportunities (Kranz 2007). However, more than60% of employees surveyed indicated that their man-agers tolerate poor performers implying that top per-formers were not being recognized (Marchetti 2007).Furthermore, nearly two-thirds of managers claimthey spend a majority of their time dealing withand helping poor performers (Chang 2003, 2004).Such results may be strategic in the sense that favor-ing a weak performer might improve his chances ofsuccess, encouraging him to work harder. Not onlywould this improve performance by raising the skilllevel of employees, but it would also increase thegeneral competition among employees by improv-ing the performance of weak employees (Farrell and

Hakstien 2001, Chang 2004). It is therefore unclearwhat the best strategy is for the manager.To address the issue of whether a firm should favor

weaker or stronger employees, we adopt a contesttheory approach. Managers often reward employeesbased on overall evaluations over a certain period oftime, rather than on a narrowly defined sales con-test (which is a short-term temporary monetary incen-tive program for salespeople). In this study, we referto this entire evaluation time period as the contest.As such, we broadly define a contest as any compe-tition between employees, including competition forlimited support, resources, or promotion to higherranks. We refer to several reward systems firms useto motivate and encourage employees to expend theirefforts, such as monetary bonuses, promotions, ormore subtle forms of various privileges, such as thecontest prize.Contests typically involve two or more employees

competing for a single prize, and the employee whoperforms best usually wins (i.e., winner-take-all con-test). By adding this winner-take-all component, firmscan induce a significant increase in effort. However,the effect is unclear when a manager faces heteroge-neous employees who differ in their abilities. Weakeremployees often recognize their small chance of win-ning and thus have little motivation to increase theireffort (Hart et al. 1989, Murphy et al. 2004, Corsun

1

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Published online ahead of print July 26, 2013

Ridlon and Shin: Favoring the Winner or Loser in Repeated Contests2 Marketing Science, Articles in Advance, pp. 1–18, © 2013 INFORMS

et al. 2006). Consequently, stronger employees, antic-ipating less competition, also may respond with lim-ited changes to their behavior or even lower effort.In this case, the contest fails to properly motivate theemployees or meet the goals of the manager.Previous research in economics and marketing sug-

gests guidelines for contests. For example, contest the-ory suggests leveling the playing field by granting anadvantage (which is also called “handicapping” in thecontest literature) to a weaker employee, which mayincrease overall effort (Lazear and Rosen 1981, Bayeet al. 1993, Baik 1994, Liu et al. 2007). The advantageincreases the weaker employee’s chances of success,similar to giving a weaker golfer a “handicap,” whichshould make the contest more intense.1 Therefore,helping the weaker employee can make all employeescompete harder to win the contest.However, in practice, some contests actually favor

the stronger employees. For example, successful salesagents often receive more lucrative territories orproduct lines (Skiera and Albers 1998), are assignedless administrative responsibilities, obtain more back-office resources, or have more training (Farrell andHakstian 2001, Krishnamoorthy et al. 2005). Similarexamples exist in other fields. Successful researcherstend to have more grant opportunities (Che and Gale2003), winners of the regular season receive homefield advantage in a sport’s postseason, and winnersof early speed trials gain the most favorable positionin car races (Mastromarco and Runkel 2009). Similarly,by winning previous contests, well-established firmsand incumbent politicians often enjoy easier runs insubsequent contests. In most cases, such favoring ofthe winner over the loser seems to contradict the prin-ciple of maximizing effort by leveling the playingfield.We note that these existing models (for example,

Lazear and Rosen 1981, Baye et al. 1993, Baik 1994)assume that the contest designer can perfectly dis-tinguish between high- and low-ability employees.However, most managers in reality face uncertaintyabout their employees’ abilities in dynamic settings,whereby a prior round of the contest is both a sourceof profit for the firm and the source of informationabout the relative strength of the employees. There-fore, managers can base their assessment, in part, onemployees’ past performances in previous periods.Past performance clearly offers a strong, albeit noisy,signal of ability.In this article, we consider a two-period repeated

contest with uncertainty where the contest designer(i.e., manager) faces heterogeneous employees but

1 We use the paradoxical terminology of handicapping, as popu-larized by its usage in association with golf, which implies that ahigher handicap helps a player.

does not know the exact abilities of each individualemployee. The manager only receives a noisy sig-nal of employees’ abilities through first-period con-test results. In the second-period contest, the managercan assign a handicap to favor the winner or loserof the previous contest. However, in a dynamic set-ting where the two contests are linked through thehandicapping policy, such a policy might create a newincentive problem. For example, if employees antic-ipate that winning a current contest will hurt themin the future, they have less incentive to win the firstcontest.2 This “ratcheting,” by which employees mod-ify their efforts in the current period to alter incentivesin future periods, is a common problem for work-force management (Weitzman 1980, Freixas et al. 1985,Chung et al. 2011, Misra and Nair 2011).Favoring the loser of the first contest thus creates an

incentive problem of ratcheting in which both play-ers reduce their attempts to win in the first period sothey can take advantage of second-period handicap-ping. On the other hand, favoring the winner of thefirst contest would also create another type of incen-tive problem of “moral hazard,” because the winnerno longer has to work as hard in the second contest.The manager should weigh the trade-offs of these twodifferent policies (favoring the winner versus favoringthe loser) to maximize the employees’ total effort.Here, we adopt a Tullock (1980) contest model,

which uses the standard ratio-form contest suc-cess function. The model clearly demonstrates thatemployees’ incentives change over time with differenthandicapping policies (i.e., favoring weaker versusstronger employees) and fully considers the strate-gic effects of the employees (i.e., ratcheting versusmoral hazard). In particular, if abilities are sufficientlysimilar, surprisingly, favoring the first-period winnerin the second period increases the total effort overboth periods—rewarding the top performer is opti-mal. However, if abilities are sufficiently different, theopposite result holds, and total effort is maximizedby adopting a handicapping policy that favors thefirst-period loser—investing in laggards is optimal.As such, the model suggests a clear handicappingpolicy guideline for the manager when faced witha heterogenous workforce with uncertain abilities ina dynamic contest environment. We use this contestmodel and its handicapping policy as an analogy tounderstand the firm’s fundamental dilemma of whena firm should favor weaker or stronger employees tomotivate all employees.More specifically, we first find that when employees

are sufficiently different in their abilities, the standard

2 For example, Taylor and Trogdon (2002) show that low-rankingNational Basketball Association teams lose more games to ensurethey get a better draft pick for the next season. See also Stein andRapoport (2005) and Yildirim (2005).

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Ridlon and Shin: Favoring the Winner or Loser in Repeated ContestsMarketing Science, Articles in Advance, pp. 1–18, © 2013 INFORMS 3

handicapping policy of favoring the loser of the firstperiod is optimal. Even though rewarding the loserin the second period decreases the total first-periodeffort because of the ratcheting effect, the employees’strategic considerations for the second-period handi-cap overcome this effort loss in the first period. Thisis because rewarding the loser in the second periodreduces the stronger employee’s first-period incen-tives more than it reduces the weak employees’s first-period incentives. In this way, the first-period contestbecomes more equitable through the handicappingpolicy of favoring the loser. This mitigates the nega-tive impact on total first-period effort for very largeability differences.On the other hand, when employees’ abilities are

sufficiently similar, we find that favoring the winnerof the first-period contest is optimal. With little dif-ference in their abilities, the extent of advantage thewinner of the first contest receives is also small.Accordingly, the loser of the first-period contest stillhas a fair chance to win the second-period contesteven with a small disadvantage and thus experiencessufficient incentive to exert effort. Hence, the increasein effort in the first period as a result of the futurerewards for winning outweighs the effort loss in thesecond period when employees’ abilities are suffi-ciently similar.The rest of this article is organized as follows.

Section 2 describes the related literature. Section 3presents our main model of dynamic contest and itsanalysis. We provide extensions to our model in §4,and we conclude in §5.

2. Literature ReviewThe theoretical foundations of contests stem from theeconomics literature. Since Tullock (1980) publishedhis seminal contest model in which players com-pete for a single prize through the expenditure oftheir resources (sometimes called a Tullock or ratio-form contest), several models have been proposed fordifferent types of contests (Lazear and Rosen 1981,Moldovanu and Sela 2001). In particular, we followMeyer (1991, 1992) to determine how the uncertaintyabout employee types affects a manager’s handi-capping policy. Meyer (1991, 1992) analyzes a mul-tiperiod contest using a Lazear–Rosen (Lazear andRosen 1981) difference-form contest, in which suc-cess is a function of the difference in (ability-adjusted)effort levels. The Lazear–Rosen model differs frommost models in the contest literature in that (1) it doesnot follow the standard Tullock contest form (whichuses the ratio-form success function) and (2) it doesnot use a linear cost function for effort. In our model,we instead adopt the standard Tullock contest modeland attain pure-strategy equilibria where both sides

exert effort. This is not possible in difference-formasymmetric contests with nonlinear costs (Hirshleifer1989, Baik 2004).In this sense, our study extends the robustness of

Meyer’s (1992) results in the symmetric case to aTullock (1980) ratio-form contest and uncovers newresults in the asymmetric case. Meyer (1992) showsthat in a promotion setting, giving an advantage to thewinner increases overall effort (or, equivalently, mini-mizes the cost of prizes to induce the same effort). Thisresult is limted, however, to only the case of symmet-ric players, and we extend it to investigate the incen-tive problem in a two-period repeated contest betweentwo asymmetric employees (i.e., individuals with dif-ferent abilities) who strategically choose their effortsin response to the handicapping policy. Meyer (1991)analyzes the asymmetric players under uncertainty,but the incentive effect of handicapping, which is themain focus of this paper, is ignored because employeesdo not make any strategic decisions in Meyer’s (1991)model. In contrast, we focus on the incentive’s costsand benefits from the different handicapping policyunder uncertainty with asymmetric players.There is another stream of literature in contest the-

ory that considers a repeated contest with asymmet-ric information between players about their abilitiesor valuations. The players update their beliefs aboutthe abilities or valuations of their rivals based on per-formance in earlier periods. Münster (2009) allowsfor two types of valuations, and players can strate-gically choose efforts in the first-period contest toaffect beliefs in the second-period contest. Hörner andSahuguet (2007) analyze a similar setup in a repeatedauction setting. Similarly, Krähmer (2007) models aninfinitely repeated contest where players are unsureabout their relative abilities but learn about the stateover time. He finds that in some cases, the least ableagent persistently wins. However, this literature dif-fers from our paper as we consider the informationasymmetry between the firm and employees, andwe focus on the firm’s optimal handicapping policy(whether to reward the loser or winner). These papersdo not consider this issue.The model of contests (particularly, the use of

handicapping) has been extensively applied in var-ious contexts outside of sales contests. For exam-ple, Tsoulouhas et al. (2007) use a contest model tostudy the issue of employee promotion selection andshow when it is optimal to handicap insiders or out-siders for chief executive officer selection. Horskyet al. (2012) investigate the advertising agency selec-tion problem using a contest model, and Harbaughand Ridlon (2010) apply the contest model to theall-pay auction setting. In particular, Harbaugh andRidlon (2010) have a theme and structure similar tothe current model in that both investigate the issue

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Ridlon and Shin: Favoring the Winner or Loser in Repeated Contests4 Marketing Science, Articles in Advance, pp. 1–18, © 2013 INFORMS

of optimal handicapping policy between asymmet-ric players in a two-period contest setting. However,unlike Harbaugh and Ridlon (2010), who only findsupport for favoring the loser for all ability differences(i.e., the handicapping policy of rewarding the loseralways maximizes total bids in the all-pay auction set-ting), our model clearly identifies the conditions inwhich favoring the winner could be optimal. Further-more, the bid equilibrium in Harbaugh and Ridlon(2010) can only be found in mixed strategies, which isnot realistic in most contest settings. Our model over-comes this inconsistency of a mixed-strategy equi-librium and provides richer results as they relate tocompetition between asymmetric employees.3Also, the use of handicapping, or bias, broadly

applies to procurement auction settings (Laffont andTirole 1988, 1991; Celentani and Ganuza 2002; Burguetand Che 2004). In multiattribute auctions where con-tracts are awarded based on multiple attributes suchas price or quality, the auction designer uses a scoringfunction to compare bids and the bid with the high-est score wins (Engelbrecht-Wiggans et al. 2007). Inthis setting, the auction designer may bias her sub-jective evaluation of attributes or distort the relativeweights of the various attributes to favor a specificbidder. Laffont and Tirole (1991) and Celentani andGanuza (2002) investigate the issue of favoritism inthis multiattribute auction settings. Also, Laffont andTirole (1988) study whether to favor the incumbent ina regulated market and find that the auction shouldbe biased in favor of the new entrant. On the otherhand, Burguet and Che (2004) examine the optimalscoring system in multiattribute procurement auctionsand find that in the presence of corruption, handicap-ping the efficient firm is not optimal and exacerbatesthe inefficient allocation as a result of bribery. The opti-mal scoring rule for the buyer may even favor the effi-cient firm. Our paper can contribute to this scoringauction literature by identifying the conditions whenit is optimal to favor the weaker or stronger bidder ina dynamic setting.Several contests have also been examined with

company objectives other than effort maximiza-tion, such as to boost employee morale (Murphyet al. 2004), increase sales (Brown and Peterson1994), improve customer satisfaction metrics (Hauser

3 Harbaugh and Ridlon (2010) examine the auction setting wherethe expected payoff to the weaker player is zero, and therefore,there is no strategic effect of the handicapping policy on the weakerplayer in the first period (since his payoff cannot be lower thanzero). Hence, their model does not fully incorporate the tensionbetween two different incentives problems (ratcheting versus moralhazard) created by different handicapping policies, which is thefocus this current paper. Thus, their model cannot find the pure-strategy equilibrium or the conditions under which favoring thewinner can be optimal.

et al. 1994), increase accuracy in employee promotionselection (Rosen 1986, Meyer 1991, Tsoulouhas et al.2007, Ryvkin and Ortmann 2008), or identify the mostqualified bidder in the auction setting (Burguet andChe 2004, Hubbard and Paarsch 2009).In marketing, most theoretical research focuses on

the issue of optimal contest design. Kalra and Shi(2001) is among the first papers to have examinedsales contests from a game-theoretical perspective.They identify specific conditions in which the opti-mal contest design structure should include multipleprizes at varying levels to induce greater effort by allsalespeople. Krishna and Morgan (1998) further showthat winner-take-all contests are optimal when con-testants are risk neutral. Lim (2010) uses a behavioraleconomics model to demonstrate that a contest witha higher proportion of winners than losers can yieldgreater effort than one with fewer winners under cer-tain conditions. Finally, using laboratory and fieldexperiments, Chen et al. (2011) and Lim et al. (2009)show empirically that the prize structure of a salescontest indeed affects the effort of contestants.In contrast, we do not address the issue of opti-

mal structure of contests or prize amount. Instead,our focus is on finding the optimal handicapping pol-icy (whether to reward the loser or winner) takingthe common winner-take-all structures (Krishna andMorgan 1998) and prizes as given. This will giveus insight for answering management’s fundamen-tal dilemma as to whether to favor the stronger orweaker employees.4

3. ModelWe consider a model where two employees, A and B,are competing in a contest setting. These employeesare heterogeneous in their abilities, denoted by theparameters a

A> 0 and a

B> 0. Without loss of gener-

ality, we simply denote one with relatively low abil-ity as A and the other with relatively high ability asB0 Therefore, employee B is stronger and more effec-tive than the other player, employee A, in ability;aA a

B. They compete to win a single prize in a con-

test by exerting effort that increases their probabilityof winning. Note that this classification is a relativeterm for comparison between two employees. Both ofthem could have very low or very high abilities. Wealso allow the possibility that two employees have thesame ability level, which is captured by a

A= a

B.

Let eA

and eB

represent the efforts exerted byemployee A and employee B, respectively. We assumethat efforts are unobservable by the manager, whoonly identifies a winner of the contest. The abil-ity parameters directly influence the effectiveness of

4 Shin and Sudhir (2010) study another management dilemma ofwhether a firm should reward its own customers or new customersand, thus, has similar flavor with the current study.

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Ridlon and Shin: Favoring the Winner or Loser in Repeated ContestsMarketing Science, Articles in Advance, pp. 1–18, © 2013 INFORMS 5

converting effort into performance. Thus, higher abil-ity leads to higher performance, all else being equal.The probability of A winning the contest is his effortrelative to the total effort of both A and B. We specifyemployee A’s contest success probability (s

A) by fol-

lowing the standard Tullock (1980) ratio-form contestsuccess function:5

sA=

aAeA

aAeA+ a

BeB

1 (1)

where °sA/°e

A> 0 and °

2sA/°e

2A< 0 for all e

A1 such

that the extra effort exerted by player A increases hisprobability of winning at a decreasing rate. Further-more, the probability of winning decreases with hisrival’s effort whereas the ability parameter increasesthe probability of success as it becomes greater.In other words, the stronger employee A becomes, themore likely he is to win the prize.We define the relative ability as ë = a

A/a

B; here,

0 ë 1 since aA a

B. Although employees know

each other’s ability, the manager only receives infor-mation on relative ability and does not know the exactability of each employee. This is because employeesare spending more time with each other than they arewith the manager. For example, coworkers are typi-cally engaged in the same or similar tasks, and thus,they have naturally more interactions and know moreabout each other.6 Therefore, we assume that employ-ees have a greater knowledge about their coworkersthan the manager does through their personal inter-actions (Marchetti 2004).Given this basic setup, we first look at the case of a

one-period static contest before we analyze our mainmodel of two-period repeated contests. This bench-mark model serves as a baseline for our main modelanalysis.

3.1. Benchmark: One-Period Static CaseIn this benchmark case, we consider a simple one-period contest where we summarize the standardresults in the contest literature.

5 A number of studies have provided a microfoundation forthe Tullock ratio-form specification from an axiomatic approach(Skaperdas 1996, Clark and Riis 1998), which is the main reasonwhy Tullock’s ratio-form model has become a standard in contesttheory literature.6 This is consistent with sociology literature on social network andcommunity ties (Wellman et al. 1988, Podolny and Baron 1997)that supports that similarity breeds more connections (McPhersonet al. 2001). Because people generally only have significant contactswith others similar to themselves in sociodemographics such associal status or rank (Campbell et al. 1986, McPherson et al. 2001),coworkers in a similar level in the workplace hierarchy are morelikely to interact with each other and thus “know more about” eachother (Hampton and Wellman 2000).

Two employees, A and B, are competing for a pri-vately valued prize, denoted as v

Aand v

B,7 respec-

tively, and have unit costs in effort. Following thestandard Tullock (1980) ratio-form contest successfunction as stated above, employee A’s utility func-tion is written as

UA= s

A· v

AÉ e

A=

aAeA

aAeA+ a

BeB

vAÉ e

A

=ëe

A

ëeA+ e

B

vAÉ e

A0 (2)

In a similar way, employee B’s utility function is UB=

41É sA5 · v

BÉ e

B. From the first-order condition,8 it is

standard to find that the equilibrium levels of effortsfor employees A and B are, respectively,

e⇤

A=

ëvB

4vB/v

A+ë52

and e⇤

B=

ëvA

4ëvA/v

B+152

1 (3)

where °ei/°v

i> 0 and °e

i/°ë > 0 for all i 2 8A1B9,

such that equilibrium efforts increase in their own val-uation of the prize (v

i) and the relative ability ë .

We can immediately see that the ratio of effortseA/e

Bis equal to the ratio of valuations v

A/v

B. In par-

ticular, when valuations are identical (vA= v

B5, the

equilibrium levels of efforts are identical, regardlessof ability differences (i.e., e⇤

A= e

⇤

B). At the same time,

equilibrium effort levels depend on ability differencesand increase as ability differences become smaller(i.e., ë becomes greater). This captures the intuitionthat an evenly matched contest is most intense. Theseare standard Tullock contest results found in the liter-ature (e.g., Baik 1994, Nti 1999, Rosen 1986).We assume that the firm (manager) can treat

employees differently by favoring or granting anadvantage to a particular employee, which affects theoutcome of the contest. We call this artificial bias a“handicapping policy.” This handicapping policy cantake several forms in practice: for example, employeescan be assigned to different environments or prod-uct lines (Skiera and Albers 1998) or given differentamounts of training or back-office resources (Meyer1992, Krishnamoorthy et al. 2005).Let h be the handicapping policy. The handicap,

h� 0, has a multiplicative effect on the ability param-eter ë = a

A/a

Bsuch that the relative ability now

becomes hë . When h > 1, the firm favors the weakeremployee, and handicapping reduces asymmetry.

7 Throughout the paper, we maintain the assumption thatvA= v

B= v. However, as we show in our main model, the

handicapping causes the implicit valuations in the first period tovary for different employees. Therefore, the results from this asym-metric valuation case are key to understanding the mechanismunderlying our main results.8 The first-order conditions for employees A and B areëe

BvA/4ëe

A+ e

B52 = 1 and ëeAv

B/4ëe

A+ e

B52 = 1, respectively. The

second-order sufficiency condition for A (and B), °2U

A/°e

2A=

É42eBë

2/4e

B+ëe

A535v

A< 01 also holds.

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Ridlon and Shin: Favoring the Winner or Loser in Repeated Contests6 Marketing Science, Articles in Advance, pp. 1–18, © 2013 INFORMS

When h < 1, the firm favors the stronger employee,and thus, handicapping amplifies the asymmetry. Forsimplicity, we assume that heterogeneity only appearsin ability, and valuations are set to v

A= v

B= v.9

Recall that equilibrium effort levels are the samewhen valuations are identical regardless of abilityasymmetry. From (3), we express total effort as

eTotal

= eA+ e

B= 2

ëh

4ëh+ 152v0 (4)

The objective of the manager in our setting is tomaximize the total employee efforts e

Total010 Because

any asymmetry between players reduces total con-test effort, the manager has an incentive to make thecontest more equitable by giving an advantage to theweaker player. When the manager knows the abili-ties of the employees (perfect information case), theoptimal policy is to favor the weaker employee bythe exact reciprocal amount of the ability difference,h⇤ = 1/ë , equalizing the two employees’ abilities. Not

surprisingly, this maximizes total effort such that it isequal to total effort when employees are symmetric.This is a well-known result from Tullock (1980).However, in practice, the manager does not nec-

essarily have perfect information about the identitiesof the stronger and weaker employees. The managermay have a good sense of his employee pool (hence,knows the distribution of abilities) but does not knowthe exact location of each individual on the distri-bution.11 For example, in hiring new employees, thefirm has expectations of performance from strongerand weaker performers given its employee pool (thus,their ability difference ë) but cannot determine theidentity of the weaker or stronger employee. In thiscase, closing the ability gap by favoring one of theplayers might reduce total effort if it is erroneouslyapplied to the stronger player, further increasing theability disparity.In practice though, managers receive multiple noisy

signals about employees’ abilities, such as their absen-teeism, project completion, and prior performanceevaluations or previous contest results. The manager

9 Employee heterogeneity can be expressed in terms of cost of effort,prize valuation, and ability. We assume these costs and valuationsare constant between players and allow only for the ability to differ.See Baye et al. (1996) for a discussion of the equivalence of contestswith asymmetric costs, valuations, and abilities.10 Although we leave the firm’s objective in this reduced form fornow, we later formally construct a micromodel of this reduced formin which we show that the manager only needs to find the optimalh that maximizes the total employee efforts.11 For example, a company with very lucrative job positions suchas consulting or investment banking usually attracts many goodtalents and thus has a low variance and high ë , whereas manystart-up companies or low-profile manufacturing companies havea pool of employees with higher variance and lower ë0

receives such a noisy signal É about employees’ rel-ative abilities. This signal is incorrect with probabil-ity p and correct with probability 41 É p5. In otherwords, with probability p1 employee A (B5 is incor-rectly identified as stronger (weaker), and with prob-ability 41É p5, employee A (B) is correctly identifiedas the weaker (stronger) employee.

Proposition 1. When a signal É is sufficiently infor-

mative (i.e., p is sufficiently small), a handicapping pol-

icy that favors the perceived weaker employee 4h⇤> 15 is

optimal for all ability differences (ë < 1). Moreover, this

optimal handicapping policy is strictly smaller than 1/ë(h

⇤< 1/ë5.

Proof. See the appendix. ÉIn a static contest, the optimal handicapping policy

under uncertainty is to favor the perceived weakeremployee. However, this handicapping policy doesnot fully compensate the ability difference resultingfrom the uncertainty. This is different from the per-fect information case, where the handicapping pol-icy precisely equalizes the two employees’ abilities byfully compensating the exact amount of ability differ-ence (1/ë).12

3.2. Dynamic ModelIn this section,13 we extend the static benchmarkmodel to the case of a two-period repeated game andexplicitly model the source of noisy information É

as the result of a first-period contest. That is, in thistwo-period model, É is realized as a win or lose (i.e.,the endogenous outcome) of the first-period contestbetween strategic, forward-looking employees.Suppose now that the manager observes a two-

period repeated contest of prize v for each period,such that the value of the prize does not vary betweenperiods; v1 = v2 = v. By keeping the values of eachcontest the same, we can focus on the direct effectof h on the equilibrium outcome. Prior to the first-period contest, the manager commits to a handicap-ping policy h, which is observable to the employees.

12 This result is akin to that of Lazear and Rosen (1981) in adifference-form contest. As they also show that the optimal hand-icapping does not necessarily result in a fair game (i.e., equalizingthe ability difference), and the actual handicap can be less than 1/ë .Our result replicates their finding under uncertainty in a Tullockcontest. Also, the closest model to our single-period setting is thatof Dahm and Porteiro (2008), who consider the effect of a noisysignal on the efforts of agents in a Tullock contest model. However,they do not consider the optimal handicapping policy, which is thefocus of this paper. In their setting, the firm does not strategicallyset the optimal handicapping to induce higher effort; it just incor-porates this additional noisy information when it selects a betteragent.13 It is important to note that the dynamic model in this researchstudies a two-period repeated contest, not a contest that lasts overtwo periods. In other words, we study the dynamic relation of twocontests, but not a dynamic contest.

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Ridlon and Shin: Favoring the Winner or Loser in Repeated ContestsMarketing Science, Articles in Advance, pp. 1–18, © 2013 INFORMS 7

In practice, firms explicitly proclaim the way a win-ner is selected with well-defined criteria as well asa publicized benefit for the winner or the loser inthe future. In the first period, the employees com-pete in a contest, and the manager gains informationabout the employees’ relative abilities by observingwho wins or loses. This signal is still noisy because thesuccess function, or winning probability, of the first-period contest is stochastic (see Equation (1)); moreimportantly, employees may strategically adjust theirbehaviors. Using the information about who winsthe contest in the first period, the manager assesses,with some probability, who is weaker or stronger andassigns the predetermined h for the second-periodcontest. Hence, the manager treats the result of thefirst-period contest as the signal about employees’ rel-ative abilities.Here, employees are strategic and forward look-

ing, such that they fully anticipate the potentialimpact of the first-period contest result on the second-period contest, and thus they adjust their behaviorsaccordingly. The manager also considers the employ-ees’ strategic behaviors when determining an optimalhandicapping policy.

3.2.1. Second Period. We start by considering thesecond period. Instead of receiving an exogenous sig-nal É with uncertainty p, the manager receives theendogenous signal from the first-period contest asa win or loss, with probability equal to the first-period success function. As we see from the bench-mark, the second-period equilibrium efforts of bothemployees are identical since v

A= v

B= v. However,

given the handicapping policy h, the exact level ofeffort depends on which employee won the first-period contest.The impact of handicapping policy h on the abil-

ity gap is either ëh if the handicapping policy offavoring the loser is correctly applied to the weakeremployee or ë/h if it is incorrectly applied to thestronger employee. For example, when ë = a

A/a

B=

12 ,

employee A must exert twice as much effort asemployee B to equal his chance of winning the con-test without any handicapping policy. If a handicap-ping policy of h= 1/ë = 2 is correctly applied to theweaker employee A (i.e., hë = ha

A/a

B= 1), A only

has to equal B’s effort to achieve an equal chance ofwinning. However, A would have to exert four timesas much effort as employee B if the handicap wereincorrectly applied to B (i.e., ë/h= a

A/ha

B=

14 ).

Hence, if the weaker employee A wins the contestin the first period, he is believed (incorrectly) to be thestronger employee, and the handicapping policy offavoring the loser is erroneously applied to the truly

stronger employee. In this case, we can find the equi-librium second-period efforts from Equation (3). Theequilibrium effort in the second period will be

ef= e

f

A= e

f

B=

ëh

4ë +h52v0 (5)

Again, both employees exert the same amount ofeffort ef in equilibrium since the valuation for the con-test is the same (i.e., v

A= v

B= v). The superscript f

represents their “false” identity.But if employee A loses the contest in the first

period, he will be correctly identified as the weakeremployee and receive a handicap of h. Equilibriumlevels of effort for both employees in the secondperiod are hence

et= e

t

A= e

t

B=

ëh

41+ëh52v1 (6)

where the superscript t represents their “true”identity.To calculate the second-period success function,

we insert these equilibrium efforts into Equa-tion (1). Thus, the second-period success functions foremployee A are conditional on the outcome of thefirst-period contest. If A loses (L) in the first period(and thus, equilibrium effort is e

t = 4ëh/41+ëh525v,

the contest success function for employee A is

sA óL = Pr4A wins óA loses first-period contest5

=ëh

1+ëh0 (7)

If he wins (W ) in the first period (and thus, equilib-rium effort is e

f = 4ëh/4ë +h525v, the contest success

function for employee A is

sA óW = Pr4A wins óA wins first-period contest5

=ë

ë +h0 (8)

Employee A’s and employee B’s second-periodexpected utilities are, respectively,

E4UA125= s

A· 64s

A óW 5vÉ ef

2 7+ 41É sA5

· 64sA óL5vÉ e

t

27

and E4UB125= 41É s

A5 · 641É s

A óW 5vÉ ef

2 7

+ sA· 641É s

A óL5vÉ et

271

(9)

where sA

is the probability that employee A winsin the first-period contest (s

A= ëe

A/4ëe

A+ e

B5 from

Equation (1)).

3.2.2. First Period. Now, we turn to the first-period utility to understand the employees’ optimaleffort decisions. Since employees are strategic andforward looking, they incorporate the fact that their

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Ridlon and Shin: Favoring the Winner or Loser in Repeated Contests8 Marketing Science, Articles in Advance, pp. 1–18, © 2013 INFORMS

first-period efforts affect not only their current first-period utility but also their future second-period util-ity. Thus, employee A chooses an effort level in thefirst period that maximizes his total expected utility:

E4UA5 = s

AvÉ e

A+ Ñ

�sA· 64s

A óW 5vÉ ef

2 7

+ 41É sA564s

A óL5vÉ et

27 1 (10)

where Ñ is the discount factor, which we normalize to1 for simplicity. The first-order condition is14

°sA

°eA

v+°s

A

°eA

44sA óW 5vÉ e

f

2 5É°s

A

°eA

44sA óL5vÉ e

t

25= 1

,°s

A

°eA

óA= 1 ,

ëeB

4ëeA+ e

B52ó

A= 11 (11)

where

óA= v+ 864s

A óW 5vÉ ef

2 7É 64sA óL5vÉ e

t

279

= v+ v

✓ë

ë +h

◆2

É

✓ëh

1+ëh

◆2�0

Here, óAcaptures employee A’s implicit value of win-

ning the first-period contest. The first term (v) rep-resents the utility from the first-period contest, andthe second 64s

A óW 5vÉef

2 7 and third 64sA óL5vÉe

t

27 termsrepresent the extra future value (or cost) of winningthe first-period contest, depending on ë and h.Similarly, the first-order condition for maximizing

employee B’s total utility function is

°41É sA5

°eB

óB= 1 ,

ëeA

4ëeA+ e

B52ó

B= 11 (12)

where

óB= v+ 6841É s

A óL5vÉ et

29É 841É sA óW 5vÉ e

f

2 97

= v+ v

✓1

1+ëh

◆2

É

✓h

ë +h

◆2�0

Here, óBrepresents employee B’s implicit value of

winning the first-period contest.From Equations (11) and (12), we find that first-

period equilibrium efforts are

e⇤

A=

ëóB

4óB/ó

A+ë52

and

e⇤

B=

ëóA

4ëóA/ó

B+ 152

0

(13)

As a result, the probability of employee A winningthe first-period contest is simply

sA=

ëóA

ëóA+ó

B

0 (14)

14 It is easy to check that the second-order condition, °2UA/°e

2A< 0,

is still satisfied.

Note that the implicit valuations in the first period,ó

Aand ó

B, are independent of first-period efforts

(eA1 e

B). That is, they are treated as exogenous prizes

just as in the static, single-period contest case. How-ever, a change in handicapping policy h (whether tofavor the winner or loser of the first-period contest aswell as how much benefit or favor to give) affects theeffort levels of both employees in the first period byeither increasing or decreasing the implicit value ofwinning in the first period (ó

A1ó

B). This is the strate-

gic effect of h on the first-period efforts.The following proposition shows how differently a

handicapping policy affects the implicit value of win-ning in the first period for different employees, whichultimately affects their effort levels.

Proposition 2. The value of winning the first-period

contest decreases in handicapping policy h 4°óA/°h < 0,

°óB/°h< 0). Moreover,

(a) under a handicapping policy of favoring the loser

(i.e., when h > 1), the value of winning the first-period

contest is such that óB<ó

A< v;

(b) under a handicapping policy of favoring the winner

(i.e., when h < 1), the value of winning the first-period

contest is such that óB>ó

A> v; and

(c) without a handicapping policy (i.e., h= 1), the valueof winning the first-period contest converges to the static

single-period case for both employees: óA=ó

B= v.

Proof. See the appendix. ÉProposition 2 suggests that when h becomes larger

(i.e., the manager favors the loser of the first periodmore), the value of winning in the first perioddecreases for both employees (°ó

A/°h < 0 and °ó

B/

°h< 0) since the loser can benefit in the second periodfrom the handicapping policy h. This is the strategiceffect of h due to the dynamic relationship betweentwo-period contests. What is more interesting andsurprising is that the value for the stronger employeeis greater or smaller than that of the weaker employeedepending on the handicapping policy (i.e., whetherh < 1 or h > 1). Figure 1 illustrates this relationshipbetween the handicapping policy (h) and the value ofwinning the first-period contest (when ë = 005).When the firm favors the loser (i.e., h > 1), it

reduces the value of winning the first-period contestbecause of the future punishment for current suc-cess: ó

A< v and ó

B< v. As a result, both employ-

ees may modify their efforts by holding back in thefirst period. This dynamic arises from the ratcheteffect identified in previous literature (Weitzman 1980,Freixas et al. 1985). Moreover, this decrease in value isgreater for the stronger employee who is more likelyto win the current contest, and thus, ó

B< ó

A< v for

h> 10On the other hand, when the firm favors the win-

ner (h< 1), the value of winning the first-period con-test is higher than the static single-period case as a

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Figure 1 The Value of Winning the First-Period Contest

10

Value

h

v

Favoring the loserFavoring the winner

!B

!A

result of the future rewards for the current success:ó

A> v, ó

B> v for h < 1. More importantly, the value

of winning for the stronger employee is greater thanthat of the weaker employee when h < 1. This isbecause the stronger employee B benefits more fromthe increased asymmetry in the second-period con-test, whereas the weaker employee A is still morelikely to lose even with an advantage since the hand-icapping does not fully compensate the ability dif-ference as a result of the uncertainty (see Proposi-tion 1). Hence, employee B would value winningthe first-period contest more than employee A would(ó

A<ó

B).

The implicit value of winning varies for eachemployee under different handicapping policies andthus affects their effort levels differently. From Equa-tion (13), it is clear that the direct effect of its ownimplicit value of winning is to increase the first-periodeffort: °e

A/°w

A> 0 and °e

B/°w

B> 0. The higher the

stake, the more effort employees exert.However, changes in rival’s efforts are more

ambiguous. The indirect effect of the implicit valueof winning for employee A 4B5 on the effort ofemployee B 4A5 is not necessarily monotonic:

°eA

°óB

=ëó

2A4ëó

AÉó

B5

4ëóA+ó

B53

ø 0 , ë ø óB

óA

1

°eB

°óA

=ëó

2B4ó

BÉëó

A5

4ëóA+ó

B53

∑ 0 , ë ø óB

óA

0

(15)

The following proposition summarizes these indi-rect effects of the implicit value of winning on theeffort of the other competitor with different handicap-ping policies, which is the key factor that drives ourmain results of optimal handicapping choice in thenext section.

Proposition 3. (a) Under a handicapping policy of

favoring the loser (h> 1), where óB<ó

A< v,

°eA

°óB

< 0 and°e

B

°óA

> 0

if employee abilities are very different 4ë <óB/ó

A51

°eA

°óB

> 0 and °eB/°ó

A< 0

otherwise 4ë �óB/ó

A50

(b) Under a handicapping policy of favoring the winner

(h 1), where óB>ó

A> v,

°eA

°óB

0 and°e

B

°óA

� 00

Proof. See the appendix. ÉA handicapping policy of favoring the loser obvi-

ously reduces both players’ valuations but in differ-ent amounts. When abilities are sufficiently different(ë < ó

B/ó

A), a decline in ó

Acauses employee B to

lower his effort (°eB/°ó

A> 0) since he is playing an

even weaker opponent. However, employee A low-ers his effort when the value to employee B increases(°e

A/°ó

B< 0 ). This result stems from the fact that the

increased effort from the stronger player as a result ofthe increase in the valuation of winning ó

Bmakes the

first-period contest even more asymmetric such thatweaker player (employee A) has very little chanceof winning. In this case, the weaker player simplyreduces his effort cost in the first period.On the other hand, when employees’ abilities are

similar (ë � óB/ó

A), a change in h has the op-

posite effect. Employee A surprisingly lowers hiseffort further when the value to employee B

declines (°eA/°ó

B> 0). Likewise, when ó

Aincreases,

employee B increases his effort. In other words, theweaker employee has a fair chance of winning thecontest when employees’ abilities are similar and thustries to match his rival in the contest. However, thestronger employee, B, now finds it optimal to reducehis effort cost in the first period and tries to bene-fit from the handicapping policy of favoring the loserin the second period, which makes the contest evenmore asymmetric.In addition, a handicapping policy of favoring the

winner (h < 1) increases both players’ valuations forwinning the first-period contest. An increase in ó

A

causes employee B to raise his effort (°eB/°ó

A> 0)

to match the competitor in the contest. However,employee A lowers his effort when the value toemployee B increases (°e

A/°ó

B< 0 ). This is very sim-

ilar to the case of when employees are very differentunder a handicapping policy of favoring the loser: theincreased asymmetry in the first period as a result of

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the increase in óBcauses employee A to have very

little chance of winning. Hence, employee A finds itoptimal to reduce his effort cost, even for all abilitydifferences (°e

A/°ó

B 0 and °e

B/°ó

A� 0).

This indirect effect of handicapping policy on therival’s effort is the key factor that mitigates thefirst-period effort losses from favoring the loser andexplains how a handicapping policy of favoring theloser can increase total effort if the contest is asym-metric enough. To what degree these effects impactoverall effort will be shown in the next section.

3.3. Optimal Handicapping Policy: Favoring theLoser or Winner

In the previous section, we showed that favoring thewinner (h < 1) has the direct effect of increasing thevaluation of winning the first period (ó

A1ó

B), lead-

ing to higher total efforts in the first period. But italso has a separate and unique indirect effect on eachemployee’s effort (°e

A/°ó

B< 01 °e

B/°ó

A> 0). Further-

more, favoring the winner may increase the abilitygap in the second period, lowering total efforts in thesecond period. The manager should consider all threeeffects when she chooses the optimal handicappingpolicy.Let e

Total1 and e

Total2 be the total efforts of the first

period and second period, respectively. Given a hand-icapping policy h and ability difference ë1 the totalexpected equilibrium effort for both periods is

E6eTotal

4h1ë57

= E6eTotal1 4h1ë5+ e

Total2 4h1ë57

= eA1 + e

B1 + 6sA· 2 · ef2 + 41É s

A5 · 2 · et271

where

eA14óA

1óB1ë5=

ëóB

4óB/ó

A+ë52

1

eB14óA

1óB1ë5=

ëóA

4ëóA/ó

B+ 152

1

óA4h1ë5= v

1+

✓ë

ë +h

◆2

É

✓ëh

1+ëh

◆2�

= v · óA4h1ë ó v= 151

óB4h1ë5= v

1+

✓1

1+ëh

◆2

É

✓h

ë +h

◆2�

= v · óB4h1ë ó v= 151

ef

2 4h1ë5=v4ëh5

4ë +h521 e

t

24h1ë5=v4ëh5

41+ëh521

and sA=

ëóA

ëóA+ó

B

0 (16)

Recall that since employees’ valuations are identi-cal in the second period, their equilibrium efforts inthat period are also the same; if employee A wins in

the first period (with probability sA), the equilibrium

effort is ef

2 for both employees, whereas if he loses inthe first period (with probability 1É s

A), it is e

t

2.The objective of the manager is to maximize the

firm’s profit, which is a function of total employeeeffort è

M= ñ · eTotal ÉC4v5, where ñ is the parameter

that captures the firm’s product efficiency and C4v5 isthe cost for prize v.Here, the contest prize clearly affects the equi-

librium effort levels; i.e., more precisely, eTotal =

eTotal

4h1v1ë5. Thus, when the manager designs thecontest in practice, she should optimize not onlythe handicapping policy h but also the contest prizeamount v for a given ë . However, all ó

Aand ó

Bare

linear functions of v (i.e., óA4h1v1ë5 = v · ó

A4h1ë ó

v = 15 and óB4h1v1ë5 = v · ó

B4h1ë ó v = 15, where

óA4h1ë ó v = 15 and ó

B4h1ë ó v = 15 are ó

A, ó

Bwhen

v = 1). Thus, the first-period efforts eA1 and e

B1are also linear functions of v (i.e., e

A14óA1ó

B1ë5 =

4ëóB5/44ó

B/ó

A5 + ë5

2 = 4vëóB5/44ó

B/ó

A5 + ë5

2

and eB14óA

1óB1ë5 = 4ëó

A5/4ë4ó

A/ó

B5+ 152 =

4vëóA5/4ë4ó

A/ó

B+ 152). Moreover, s

A(which is

equal to 4ëóA5/4ëó

A+ó

B5 = 4ëó

A5/4ëó

A+ ó

B5) is

independent of v, and therefore, the second-periodexpected efforts (s

A· 2 · ef2 + 41É s

A5 · 2 · et2) are again

a linear function of v. Therefore, eTotal

4h1v1ë5 =

v · eTotal4h1ë5.Now, the firm’s profit function can be rewritten as

èM

= v · ñ · eTotal4h1ë5 É C4v5. This implies that thechoice of optimal handicapping policy h is indepen-dent of the contest prize v. Hence, to maximize profit,the manager only needs to find the optimal h thatmaximizes the total employee efforts for any given v.In other words, for any given contest prize, our resultof a handicapping policy is always optimal, and thusthe objective of the manager reverts to finding theoptimal h irrespective of the value of v.15 However,it is important to note that, in general, the optimalcontracting will be a function of both the handicap-ping h and the prize value v0 Only when the prizevalue v is symmetric does the problem degenerate toone with optimizing only the handicapping h. Oth-erwise, the firm should optimally choose both thehandicapping h and the optimal size of the prize v

simultaneously.Also, we note that h affects employees’ implicit val-

uations in the first period through óAand ó

B. Given a

manager’s handicapping policy in the second period,employees choose their efforts in the first period, fullyanticipating the consequence of their choices in thesecond period.

15 The optimal prize v clearly affects the incentives of the agents bysteepening or moderating the efforts. The optimal v can be foundas C 0

4v⇤5= e

T , and thus, it is independent of h and depends only onthe specification of cost function C4v50 For example, if we imposea convex cost of C4v5= v

2/2, then v

⇤ = eT .

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Proposition 4. For any given v > 0, when employee

abilities are very different (ë is sufficiently small), a hand-

icapping policy of favoring the loser (h> 1) maximizes the

expected total effort. Otherwise, when employee abilities are

similar (ë is sufficiently large), a handicapping policy of

favoring the winner (h< 1) maximizes total effort.

Proof. See the appendix. ÉThere is a fundamental trade-off between the effort

levels across two periods when the manager employsa handicapping policy. On the one hand, a handi-capping policy of favoring the loser (h > 1) alwaysincreases effort in the second period because it levelsthe playing field and encourages the weaker player.Yet favoring the loser reduces the incentives of play-ers to win the first-period contest because of thefuture punishment for the winner, tempering thegains from the second period.On the other hand, a handicapping policy of favor-

ing the winner (h < 1) always increases effort in thefirst period because of the additional future rewardfor the winner in the second period. However, thisreduces effort in the second period because the win-ner of the first contest would no longer need to workas hard in the second period, tempering the gainsfrom the first period.Overall, the manager must balance these trade-

offs when choosing a handicapping strategy. Whenemployees are very different in their abilities (i.e.,small ë), the handicapping policy of favoring theloser (h > 1) can intensify competition betweenemployees in the second period, and this benefitof increased effort outweighs the loss of effort inthe first period. Both employees reduce effort inthe first period to take advantage of future bene-fit, but employees do not race to the bottom as aresult of the strategic indirect effect of the weakeremployee A (°e

A/°ó

B< 0 from Proposition 3). The

value to stronger employee B decreases significantly(i.e., ó

B<ó

A< v from Proposition 2), thus lowering

his first-period effort. This has an indirect effect on theweaker employee’s effort because he now has a higherchance to win the first-period contest against a morerestrained stronger employee and, thus, increases hiseffort. This mitigates the loss in total first-period effortfor the manager. This only occurs when employeesare very different in their abilities (i.e., small ë), andin this case, favoring the loser (h > 1) maximizesexpected total effort.In contrast, when employee abilities are very simi-

lar (i.e., large ë), the cost in lost effort associated withthe handicapping policy of favoring the loser (h> 1) isgreater. Again, the value for winning the first-periodcontest decreases for both employees A and B, reduc-ing their efforts. Unlike the case when ë is small (oremployees are very different), the indirect strategic

effect of the reduced implicit value causes the weakeremployee to further reduce his effort (°e

A/°ó

B>

0 from Proposition 3), instigating a true “race tothe bottom.” Note that this indirect effect is greaterthan the indirect strategic effect on employee B sinceó

B<ó

A< v (i.e., ó°e

A/°ó

Bó> ó°e

B/°ó

Aó). Furthermore,

because the abilities are similar, any advantage gainedfrom the handicapping policy would only marginallyimprove effort in the second period. Therefore, whenemployee abilities are very similar, a handicappingpolicy in favor of the loser (h> 1) cannot be advanta-geous for the manager.A handicapping policy in favor of the winner

(h< 1), on the other hand, motivates employees tocompete more intensely in the first period, outweigh-ing the costs of lackluster performance in the sec-ond period. In other words, the potential incentiveproblem of moral hazard from favoring the winneris not severe when players are very similar in theirabilities. Even with a small advantage to the winner,the loser still has sufficient incentive to exert effortin the second-period contest. In anticipation of sucheffort, the stronger player still responds to the con-test with sufficiently high effort level in the secondperiod. Hence, when the ability difference is small(i.e., ë is sufficiently large), the cost of effort lossin the second period is more than compensated bythe increased effort in the first period—favoring thewinner (h< 1) maximizes the expected total effort.For example, when employees are identical in ability,ë = 1, the total expected effort is 43h + 15v/4h+ 152,which is maximized by a handicapping policy offavoring the winner h=

13 .

This is in stark contrast to the static case, wherefavoring the perceived weaker player (h > 1) isoptimal for all ability differences. In contrast, in adynamic setting, the manager sometimes maximizesthe total effort by favoring the perceived strongerplayer who wins the first-period contest. By increas-ing the incentives in the first-period contest throughthe future rewards for the first-period contest suc-cess, the manager can maximize the total efforts fromboth employees. This only arises from the dynamicincentives created by the handicapping policy, andhence, in the static case, we could not find the sit-uation where favoring the perceived stronger playeroptimal.We illustrate the relationship between the relative

ability ë and the optimal handicapping policy h inFigure 2(a), when v = 1. This clearly demonstratesthat favoring the winner (h < 1) is beneficial to themanager’s attempt to raise effort when employees aresimilar in their abilities (in this particular example,when ë > 0036). Favoring the loser (h > 1) is benefi-cial only when employees are sufficiently different intheir abilities (when ë < 0036).

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Ridlon and Shin: Favoring the Winner or Loser in Repeated Contests12 Marketing Science, Articles in Advance, pp. 1–18, © 2013 INFORMS

Figure 2 Optimal Handicapping Policy and Expected Total Effort for v = 1

1 0.36 100.360

1Favoringthe winner

h < 1

Favoringthe winner h* < 1

Favoringthe loser

h > 1

Favoringthe loser h* > 1

(a) Effort-maximizing handicap

1 With optimal handicap, h*

Without handicap, h = 1

eTotal

12

(b) Maximum efforth

!!

Next, we investigate the relationship between therelative ability ë and the expected total effort fromboth periods E4eTotal5 under the optimal handicappingpolicy h

⇤. As Figure 2(b) shows, the expected totaleffort E4eTotal5 under the optimal policy h

⇤ is alwaysgreater than the expected total effort without anyhandicapping policy (i.e., h= 1 for all ë).To better understand the underlying forces behind

this result, we decompose the total effort into indi-vidual effort by period, as shown in Figure 3. First,we note that there is a single crossover between thefirst- and second-period effort at the point where theoptimal handicapping policy is h = 1 (in this partic-ular case, ë = 0036). A handicapping policy of h = 1is equivalent to the case of a no handicapping policy

Figure 3 Effort by Period and Employee Type as a Function of Ability Differences for h= h⇤

10.36

(a) First-period effort levels (b) Second-period effort levels

0.360 1

h < 1 h < 1h > 1 h > 1

0

1

e e

eB, 1

eB, 2 = eA, 2

eA, 1

e1Total(h = 1) e2

Total(h = 1)

414

!!

(or static contest case). Hence, effort levels from boththe first and the second periods are identical. Sec-ond, Figure 3(b) also illustrates that the efforts of bothemployees are equal in the second period because theexogenous value for the contest prize (v) is the samefor both.When employees’ abilities are sufficiently differ-

ent (ë < 0036), a handicapping policy of favoring theloser h > 1 clearly raises the second-period effort byboth employees (see Figure 3(b)). Moreover, the first-period effort by the weaker employee A is greaterthan that of the stronger employee B (see Figure 3(a)).This is because the potential effort loss in the firstperiod is mitigated by the indirect effect identifiedin Proposition 3. The decreased value of winning

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Ridlon and Shin: Favoring the Winner or Loser in Repeated ContestsMarketing Science, Articles in Advance, pp. 1–18, © 2013 INFORMS 13

the first period, or “loser’s bonus,” has a strategiceffect on employee B to reduce his effort. This reduc-tion in effort from a restrained employee B causesemployee A to increase his effort in the first period.In other words, the change in the total first-periodeffort is ambiguous and small. This effect is offsetby the overwhelming increase in effort in the secondperiod.On the other hand, when employees’ abilities are

quite similar (ë > 0036), favoring the winner increasesthe first-period effort by raising the value of winningin the first period for both employees (see Figure 3(a)).Although the strategic effect from an increased val-uation of winning in the first period is greater forstronger employee B (see Proposition 2), the indirecteffect from the weaker player is also increased effort.These effects are higher than the effort reduction fromincreased asymmetry in the second-period contest.

4. Extension: Commitment vs.Flexibility

In this extension, we relax two key assumptions inour main model: (1) the manager credibly commits toa handicapping policy prior to the first-period con-test,16 and (2) the employees have a greater knowl-edge about their coworkers than the manager. Thoughthese assumptions seem reasonable (since firms tendto operate in an environment with enforceable con-tracts with its employees, clients, and suppliers, andemployees have naturally more ineractions with eachother), an interesting issue about whether the firmprefers to commit to a handicapping policy arises ifthe manager expects to receive more relevant infor-mation regarding the identities of employees (who isthe weaker employee, who is the stronger one) dur-ing the first period.17 In these circumstances, elimi-nating all ex post possibilities through commitment

16 This is a critical assumption because our main result of the“favoring the winner” case in Proposition 4 only arises with com-mitment. Without commitment, the manager only maximizes thesecond-period efforts and thus always favors the loser since favor-ing the weaker is optimal under uncertainty for a static case (seeProposition 1).17 In this case, one may consider expanding the contract space andcondition the handicapping policy on the total effort. There areseveral papers in contest theory literature that allow agents toupdate their beliefs about the abilities of their rivals based on per-formance in earlier periods (Hörner and Sahuguet 2007, Krahmer2007, Münster 2009). However, these papers only consider strate-gic interactions between employees and ignore the issue of firm’sstrategic choice of handicapping policy. In contrast, we considernot only strategic interactions between employees but also strategicinteractions between the firm and employees (i.e., optimal handi-capping policy). This extra layer of complexity makes it intractableto allow conditioning the handicapping policy on the first-periodeffort.

seems unlikely to be a good idea, and the managermay value the flexibility to act on this information.To address this issue, we compare our main model

of commitment with the following benchmark sce-nario of no commitment. In this benchmark caseof no commitment, we also relax the assumptionthat employees have superior information to themanager—the manager perfectly learns the abilitiesof both employees at the end of the first period.This setup serves to “stack the deck” against findingcommitment-related benefit because it assumes thebest possible situation for the no-commitment (flexi-bility) case; that is, the information that the managercan act on at the beginning of the second periodis now perfect information. In reality, the informa-tion is far from perfect, which should lower thepotential benefit from the flexibility. Hence, with this“perfect information” under the no-commitment case,one may wonder whether the manager still prefers tocommit to a handicapping policy.We first note that under the no-commitment sit-

uation, the manager chooses a policy h only afterthe first period. If the manager knows employees’identities perfectly, she can achieve the first-best out-come in the second period by favoring the weakeremployee by the exact amount of the ability dif-ference, h= 1/ë , yielding v/2 in total second-periodeffort (Tullock 1980). Since employees fully anticipatethat the manager will perfectly know their identitiesin the second period, there is no strategic componentof effort in the first period.18 Thus, the first period inthe repeated contest reverts to the static contest resultwhere total effort is equal to 2ëv/4ë + 152. With per-fect information, the total effort from both players inboth periods is

eTotal

=2ëv

4ë + 152+

v

2=

v41+ 6ë +ë25

241+ë521 (17)

where eTotal denotes the effort level under no commit-

ment, which simplifies to v when ë = 1.We can easily see that the total effort under

commitment is lower than that of no commitment(eTotal < e

Total) over the range of asymmetry wherefavoring the loser is optimal (i.e., when employeesare very different or ë is sufficiently low), since the

18 Here, we are simply imposing the best possible scenario for theno-commitment case by assuming that the manager receives per-fect information. However, in practice, the issue of whether orhow the manager can acquire the perfect information is important.For example, employees may strategically conceal their abilities toavoid ratcheting. Once one has a micromodel about the interac-tions between the manager and employees during the contest, thenone can analyze the dynamics of information revelation. But this isbeyond the scope of the current research. We thank an anonymousreviewer for highlighting this important issue.

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Ridlon and Shin: Favoring the Winner or Loser in Repeated Contests14 Marketing Science, Articles in Advance, pp. 1–18, © 2013 INFORMS

second-period effort is always higher under no com-mitment (eTotal2 < e

Total2 ), and the first-period effort is

lower than in the static case under this handicappingpolicy (eTotal1 < e

Total1 since ó

i v for all i = 8A1B9; see

Proposition 2). Hence, the manager values flexibil-ity more than commitment when employees are verydifferent.However, we find that even if the manager can

benefit from flexibility by obtaining the perfectinformation at the end of first period, she prefers tocommit to a handicapping policy of favoring the win-ner based on the first-period contest result when ë

is sufficiently large or employees are quite similar.19In spite of having perfect information about employ-ees, the manager can be better off by forgoing thisinformation and committing to the handicapping pol-icy of favoring the winner when employees are quitesimilar. In this case, the employees’ future considera-tion raises the first-period effort because of the handi-capping policy of favoring the winner. This increase inthe first-period effort can more than compensate theeffort loss from the inefficiency caused by using noisyinformation based on the first-period contest result(instead of using perfect information about employeetypes). For example, in the symmetric case of ë = 1,the optimal handicapping is h

⇤ =13 , and the total

effort is eTotal = 9

8v > v= eTotal.

This result highlights the value of a dynamic natureof handicapping policy; by dividing the contest intoseparate but mutually dependent contests, a managercan increase total effort above that achieved from per-fect information by committing to a handicappingpolicy. Therefore, the effect of a handicapping pol-icy is not merely shifting employee’s efforts betweentwo periods, but in fact, it increases the total effortlevel above the perfect information case as a resultof employees’ strategic behaviors arising from thedynamic relationship between two contests through ahandicapping policy.

5. ConclusionManagers constantly face the problem of motivatinga heterogeneous workforce. In particular, the issueof whether managers should invest in laggards orreward their top performers to motivate heteroge-neous employees still does has not come a consensusanswer. In this research, we use a Tullock contestmodel and handicapping policy to address this issue.We show that by dividing the contest into sepa-rate but mutually dependent contests, a manager can

19 The issue of commitment and flexibility is investigated in manydifferent settings, and similar results are reported (for example,Amador et al. 2006, Zeithammer 2007). It has been omitted here forbrevity, but the detailed proof of this result is available from theauthors upon request.

increase total effort by committing to a certain hand-icapping policy.A conflict arises, however, between favoring a

poor performer and rewarding a top performer in adynamic setting. Favoring the loser increases effortin the second period at the expense of reducing eachemployee’s incentive to win the first period becauseof the future punishment for the winner; this can beseen as the “ratchet effect.” Also, favoring the win-ner increases effort in the first period because of thefuture reward for the winner but would also createanother type of incentive problem—the “moral haz-ard,” where the winner no longer has to work as hard.The manager should weigh the trade-offs of these twodifferent policies (favoring the winner versus favoringthe loser) to maximize the employees’ total effort.We find that if abilities are sufficiently similar,

favoring the winner in the second period increasesthe total effort over both periods—rewarding the topperformer is optimal. However, if abilities are suffi-ciently different, the opposite result holds, and thetotal effort is maximized by adopting a handicappingpolicy that favors the loser—investing in laggards isoptimal. As such, the model suggests a clear handi-capping policy guideline for the manager when facedwith a heterogeneous workforce with uncertain abil-ities in a dynamic contest environment. Moreover,handicapping is common practice in various settings:from sports events (for example, golf and sailing)to social systems (such as affirmative actions).20 Thisstudy can broadly apply to those various settings andprovide useful insight about what types of systemscan help participants in those settings to put in theirbest efforts and increase the efficiency of the system.Also, we show that a manager can increase

total effort above the level achieved from perfectinformation by committing to a handicapping pol-icy. This arises from employees’ strategic behaviorsbecause of the dynamic relationship between two con-tests through handicapping policy. Furthermore, thisimplies that even if the manager knew the ability ofeach employee, a handicapping policy of favoring thewinner has a greater incentive effect than evening thecontest by favoring the loser. This result uniquely con-tributes to the existing contest literature concerningmaximizing effort and handicapping.There are many other factors that affect the effec-

tiveness of a handicapping policy such as fairnessor prize structure. For instance, handicapping theweaker employee may cause the stronger employeeto feel that the contest has been unfairly alteredsuch that the latter’s work appears underappreciated,

20 We thank an anonymous reviewer for encouraging us to thinkabout the broader implications of handicapping and suggestingthese examples.

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Ridlon and Shin: Favoring the Winner or Loser in Repeated ContestsMarketing Science, Articles in Advance, pp. 1–18, © 2013 INFORMS 15

which may eventually lead to disenchantment for thestronger player. This point is worth examining froma goal theory perspective, as Murphy et al. (2004) do.Our paper complements this goal theory perspectiveand contributes to understanding why favoring theloser could lower the effort from a stronger employee.Yet the proposed model sometimes seems to favor theMatthew effect, by which winners only win becausethey have won in the past, not because of their supe-rior ability (Merton 1968). We believe that our modelcan offer another explanation for the Matthew effect.Also, we have only examined a two-period re-

peated contest. A natural extension would involve amultiperiod contest; it would be interesting to exam-ine whether our main results can hold beyond twoperiods. Unlike the two period case where the hand-icapping policy is employed only once at the begin-ning of the second period, the manager can employdifferent handicapping policies for each period, andthus, the optimal handicapping profile would involvea series of different handicapping policies over entireperiods, which is beyond the scope of our currentwork. We leave it for future research to explore thisissue.Finally, we only examine the two-employee case;

other research has examined a much larger pool ofparticipants, in which case the optimal proportion ofwinners in a contest can be an important issue (Lim2010). However, if managers only have to distinguishbetween a pair of leaders, it might not be practical toinclude more than two agents in a competition, say,for the same client account. Nevertheless, an appli-cation to the multiemployee case would broaden theimplications of our study.

AcknowledgmentsThe authors are grateful to the editor, the associate editor,and two anonymous reviewers for detailed feedback thatgreatly improved the paper. They thank Kyung Hwan Baik,Michael Baye, Timothy Derdenger, Eunkyu Lee, Noah Lim,JohnMaxwell, Subrata Sen, K. Sudhir, Birger Wernerfelt, andespecially Rick Harbaugh for their helpful comments. Also,they thank seminar participants at the Quantitative Market-ing and Economics Conference at Chicago, the North EastMarketing Consortium Conference at Harvard, CarnegieMellon University, Indiana University, Sungkyunkwan Uni-versity, the University of Rochester, the University of Texasat Austin, Washington University in St. Louis, and Yale Uni-versity for helpful feedback.

Appendix

Proof of Proposition 1. Given a handicapping pol-icy h and uncertainty p1 we can calculate the equilibriumefforts under handicapping from Equation (3). With prob-ability 1 É p, employee A is (correctly) identified as theweaker employee and receives a handicap of h. Equilib-rium levels of effort for both employees then are e

t = et

A

= et

B= 4ëh/41+ëh5

25v. Since vA = vB = v, both employees

exert the same amount of effort et in equilibrium, where

the superscript t represents their true identity. But withprobability p, employee A is identified (incorrectly) as thestronger employee, and the handicap is erroneously appliedto the truly stronger employee B. The equilibrium levels ofeffort for both employees are e

f = ef

A= e

f

B= 4ëh/4ë +h5

25v,

where the superscript f represents their false identity.For a given ë 2 60117 and p 2 601 1

2 5, the manager solvesthe following maximization problem:

maxh>0

E4eTotal

5 = p

✓2v

✓ë/h

41+ë/h52

◆◆

+ 41É p5

✓2v

✓ëh

41+ëh52

◆◆0

The first-order condition with respect to h yields

°E4eTotal

5

°h= 2ëv

✓41É p541Éhë5

41+ëh53É

p4hÉë5

4h+ë53

◆0

It can be easily seen that 6°E4eTotal5/°h7h=1 = 2ëv4441É 2p5 ·41 É ë55/41+ë5

35 � 0 (equality holds when ë = 1), and

°E4eTotal

5/°h < 0 for all h � 41/ë54> 150 By continuity,there must be at least one point h

⇤ 2 6111/ë5 at which6°E4e

Total5/°h7h=h⇤ = 0.

Moreover, we can show that °E4eTotal

5/°h > 0 for allh< 1.

1. When h ë4< 151 it is trivially satisfied.2. When ë < h < 11 we show that 41 É p541 É hë5/

41+ëh53> p4hÉë5/4h+ë5

3 by rearranging the inequality

41Ép541Éhë5

41+ëh53>

p4hÉë5

4h+ë53

,

✓1Ép

p

◆✓h+ë

1+ëh

◆3✓1Éhë

hÉë

◆>1

, p<4h+ë5

341Éhë5

41Éë256h41+h25+h41+h25ë2É41É6h2+h45ë7= p0

In the second inequality, as p # 01 the (left-hand side) goesto infinity, and thus, the inequality is always satisfied. Moreprecisely, we can find the condition such that when p < p,°E4e

Total5/°h> 0 for all h< 1.

In summary, when p < p, °E4eTotal5/°h > 0 for all h 1,and °E4e

Total5/°h < 0 for all h � 41/ë54> 15. Therefore,

E4eTotal

5 attains its maximum at some point h⇤ 2 6111/ë5,

where 6°E4eTotal

5/°h7h=h⇤ = 0 by the mean value theorem.Hence, the optimal handicapping policy is strictly smallerthan 1/ë .

Proof of Proposition 2. First, it is obvious that óA =

óB = v when h= 1. Next, note that óA = v+ë2v/4ë +h5

2 É

4ëh52v/41+ëh5

2 and óB = v+41/41+ëh552vÉ4h/4ë+h55

2v.

Hence,

óBÉóA =

✓1

1+ëh

◆2

É

✓h

ë+h

◆2

Éë

2

4ë+h52+

4ëh52

41+ëh52

�v

=

1+4ëh5

2

41+ëh52É

ë2+h

2

4ë+h52

�v

=2vëh41Éh

2541Éë

25

41+ëh524ë+h520 (18)

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Ridlon and Shin: Favoring the Winner or Loser in Repeated Contests16 Marketing Science, Articles in Advance, pp. 1–18, © 2013 INFORMS

Therefore, óB É óA > 0 if and only if h < 1 (note thatë < 15.

Moreover, óA = v + ë2v/4ë +h5

2 É 4ëh52v/41+ëh5

2>

v if and only if h < 1, because ë2v/4ë +h5

2<

4ëh52v/41+ëh5

2 () 1 < h. The results in the propositionfollow.

Proof of Proposition 3. From the first-order condi-tion, °eA/°óB = ëó

2A4ëóAÉóB5/4ëóA +óB5

3 and °eB/°óA =

Éëó2B4ëóA ÉóB5/4ëóA +óB5

3.1. Under a handicapping policy of favoring the

loser, óB < óA < v. Hence, if ë < óB/óA, then°eA/°óB = ëó

2A4ëóA É óB5/4ëóA +óB5

3< 0 and °eB/°óA =

Éëó2B4ëóA É óB5/4ëóA +óB5

3> 0. Otherwise (i.e., ë �

óB/óA), °eA/°óB > 0 and °eB/°óA < 0.2. Under a handicapping policy of favoring the win-

ner, óB > óA > v. Thus, °eA/°óB = ëó2A4ëóA ÉóB5/4ëóA +

óB53< 0 and °eB/°óA = ëó

2B4óB ÉëóA5/4ëóA +óB5

3> 0.

3. Without a handicapping policy, óA = óB = v. Thus,°eA/°óB = 0 and °eB/°óA = 0.

Proof of Proposition 4. First, we show that there existsa unique equilibrium of efforts by both employees for allë and h. We have already established that the first- andsecond-order conditions are satisfied, yielding the existenceof an equilibrium. Therefore, to prove uniqueness, we exam-ine the Hessian of UA and UB :

H =

2

6664

°2UA

°e2A

°2UB

°eA°eB

°2UA

°eA°eB

°2UB

°e2B

3

77751 (19)

which simplifies to

H =

2

6664

°2sA

°e2A

vA É°2sA

°eA°eB

vB

°2sA

°eB°eA

vA É°2sA

°e2B

vB

3

77751 (20)

since valuations are exogenous to efforts. If H is negativedefinite for all eA and eB , then the equilibrium is unique(Rosen 1965). This claim is easy to verify since the first com-ponent is negative by assumption and the determinant,

óH ó=É°2sA

°e2A

°2sA

°e2B

vAvB É°2sA

°eA°eB

°2sA

°eA°eB

vAvB > 01 (21)

is positive definite, provided that

É°2sA

°e2A

°2sA

°e2B

>°2sA

°eA°eB

°2sA

°eA°eB

0 (22)

We have already established existence, so we can evaluateEquation (22) for each success function:

• From Equation (7), when Pr4A wins second period ó

A loses first period5= sA óL = ëheA/4ëheA + eB51

óH ó> 01 becauseh2ë

2

4eB + eAhë54 > 00

• From Equation (8), when Pr4A wins second period ó

A wins first period5= sA óW = 4eA4ë/h55/4eA4ë/h5 + eB51

óH ó> 01 becauseh2ë

2

4eAë + eBh54 > 00

• From Equation (1), when Pr4A wins first period5 =

sA = ëeA/4ëeA + eB51

óH ó> 01 becauseë

2

4eB + eAë54 > 00

Next, we show that the optimal handicapping policy isdecreasing in ë . From Equation (16), we know that for agiven ë , total effort is

E6eTotal

4h1ë57= eA1 + eB1 + 6sA · 2 · ef2 + 41É sA5 · 2 · et

271

where

eB14óA1óB1ë5=ëóB

44óB/óA5+ë521

eB14óA1óB1ë5=ëóA

4ë4óA/óB5+ 1521 sA =

ëóA

ëóA +óB

1

óA4h1ë5= v+ v

✓ë

ë +h

◆2

É

✓ëh

1+ëh

◆2�1

óB4h1ë5= v+ v

✓1

1+ëh

◆2

É

✓h

ë +h

◆2�1

ef

2 4h1ë5=v4ëh5

4ë +h521 and e

t

24h1ë5=v4ëh5

41+ëh520 (23)

At equilibrium h = h⇤1 the first-order condition with

respect to h satisfies

°E6eTotal

7

°h

����h=h⇤

=°E6e

Total7

°óAB

dóA

dh+

°E6eTotal

7

°óB

dóB

dh+

dE6eTotal

7

dh

=

°eA1

°óA

+°eB1

°óA

+ 24ef2 É et

25°sA

°óA

�dóA

dh

+

°eA1

°óB

+°eB1

°óB

+ 24ef2 É et

25°sA

°óB

�dóB

dh

+ 2sAde

f

2

dh+ 241É sA5

det

2

dh= 01

where

°eA1

°óA

=2ëóAó

2B

4ëóA +óB53 > 01

°eB1

°óA

=ëó

2B4óB ÉëóA5

4ëóA +óB53 > 01

ef

2 É et

2 � 0 (when 41Éh2541Éë

25� 051

°sA

°óB

=ëóB

4ëóA +óB52 > 01

°óA

°h=É2ë2

✓1

4h+ë53+

h

41+ëh53

◆< 01

°óB

°h=É2ë

✓h

4h+ë53+

141+ëh53

◆< 01

°eA1

°óB

=Éëó

2A4óB ÉëóA5

4ëóA +óB53 1

°eB1

°óB

=2ë2

ó2AóB

4ëóA +óB53 > 01

°sA

°óA

=ÉëóA

4ëóA +óB52 < 01

def

2

dh=

ë4ë Éh5

4h+ë531 and

det

2

dh=

ë41Éhë5

41+hë530 (24)

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Ridlon and Shin: Favoring the Winner or Loser in Repeated ContestsMarketing Science, Articles in Advance, pp. 1–18, © 2013 INFORMS 17

By substituting these results, the first-order condition ofEquation (24) simplifies to

°E6eTotal

7

°h

=É2ë22ëóAó

2B+ëó

2B4óBÉëóA5

4ëóA+óB53 +

41Éë5241Éh5

2ëóB

4ëóA+óB52

�

·

✓41+ëh5

3+h4h+ë53

4h+ë5341+ëh53

◆

É2ëÉëó

2A4óBÉëóA5+2ë2

ó2AóB

4ëóA+óB53 É

ëóA41Éë5241Éh5

2

4ëóA+óB52

�

·

✓h41+ëh5

3+4h+ë53

4h+ë5341+ëh53

◆

+2ë

ëóA+óB

✓41+ëh5

3+4h+ë53

4h+ë5341+ëh53

◆=0

, Éë⇥2ëóAó

2B+ëó

2B4óBÉëóA5

+41Éë5241Éh5

2ëóB4ëóA+óB5

⇤441+ëh5

3+h4h+ë5

35

+6ëó2A4óBÉëóA5É2ë2

ó2AóB

+41Éë5241Éh5

2ëóA4ëóA+óB574h41+ëh5

3+4h+ë5

35

+41+ëh53+4h+ë5

34ëóA+óB5

2=01

where óA4h1ë5= v+ v64ë/4ë + h552 É 4ëh5/41+ëh55

27 and

óB4h1ë5= v+ v641/41+ëh552 É 4h/4ë +h55

27.

Using this, we can further simplify the first-order condi-tion and define the implicit function F 4h1ë5:

°E6eTotal

7

°h= 0

, F 4h1ë5=⇥ë4X

2Y

2Éh

2Y

2+X

2543X2

Y2Éh

2Y

2+X

2

ÉëX2Y

2Éë41Éh

2542Éë541+ 2hë +h

255

+ë41Éh2541Éë5

24X

2Y

2Éh

2Y

2+X

25

· 441+ë5X2Y

2Éh

2Y

2+X

2Éë

34Y

2ÉX

255⇤

⇥ 4Y3+hX

35+

⇥4X

2Y

2Éh

2Y

2+Y

2542ëX2

Y2

Éh2Y

2+X

2+ 4X

2Y

2Éë

241Éh

2542Éë5

· 41+ 2hë +h2555+ 4X

2Y

2Éë

2Y

2+ë

2X

25

· 441Éë5X2Y

2Éh

2Y

2+X

2Éë

34Y

2ÉX

255⇤

⇥ 4hY3+X

35+X

3Y

2ÉY

2= 01

where X = h+ë and Y = 1+ëh0

We differentiate F 4h1ë5 with respect to h1ë . After a fewalgebraic steps, we can see that dF /dh < 0 for all h > 0and ë 2 60117, and dF /dë > 0 if h > 0 and ë ë , where ë

solves the equation 4dF /dë54h1 ë5= 0. By the implicit func-tion theorem, dh/dë = É4°F /°ë5/4°F /°h5 < 00 All we needto show is that when ë is large such that ë > ë1 it is thecase that h⇤

< 10 We prove it by contradiction. Suppose thatë is sufficiently large and the optimal handicapping policyis h⇤ � 10 Then, it must be the case that 4°E6eTotal7/°h54h= 15> 00 However, we can easily see that

°E6eTotal

4h= 157°h

= 241+ 5ë É 5ë3É 33ë3

É 37ë4É 9ë5

+ 5ë6+ë

75< 01 if ë > ë ⇡ 00380

This contradicts the assumption. Thus, when ë > ë1 it isthe case that h⇤

< 10In particular, when ë = 11 E6e

Total4h1ë = 157= v41+ 3h5/

41+h52> 1 from Equation (16). Total effort is therefore max-

imized at h = 1/3.

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