Participation incentives in rank order tournaments with endogenous entry

J Econ (2008) 95:1–23DOI 10.1007/s00712-008-0017-z

Participation incentives in rank order tournamentswith endogenous entry

Timothy Mathews · Soiliou Daw Namoro

Received: 3 January 2007 / Accepted: 23 February 2008 / Published online: 5 April 2008© Springer-Verlag 2008

Abstract Rank order tournaments, in which the payment to an agent is based uponrelative observed performance, are a commonly used compensation scheme. In prac-tice, agents often compete in some (but not all) events in a set of tournaments. Thepresent study considers two mutually exclusive tournaments, in which agents them-selves decide which event to enter. An agent bases this decision upon the combinationof three distinct effects: a prize effect, a winning probability effect, and an effort costeffect. The precise impact of each of these effects is analyzed. Of particular interest isthe possibility that a field of higher quality may be attracted to the event with smallerprizes.

Keywords Tournament · Entry decision · Employee compensation

JEL Classification J33 ·M52 · C70

1 Introduction

Rank order compensation schemes provide agents with incentives to exert effortwhen effort is not easily observable by a principal. The use of such schemes hasbeen examined extensively (the pioneering works include Lazear and Rosen 1981;

T. Mathews (B)Department of Economics, Finance, and Quantitative Analysis, Kennesaw State University,1000 Chastain Rd., Kennesaw, GA 30144-5591, USAe-mail: [email protected]

S. D. NamoroDepartment of Economics, University of Pittsburgh, 4S01 W.W. Posvar Hall 230,S. Bouquet St., Pittsburgh, PA 15260, USAe-mail: [email protected]

123

2 T. Mathews, S. D. Namoro

Nalebuff and Stiglitz 1983; Green and Stokey 1983, and Mookherjee 1984). The mainfocus has been an environment in which agents compete in a single rank order tour-nament. Of particular interest are the incentives for agents to exert effort and theefficiency or inefficiency of the level of effort implemented by the principal.

In practice, agents often compete in some (but not all) events in a set of tournaments.Numerous examples can be found in professional sports such as golf, auto racing, andtennis. In each of these instances, there is a set of events, and participants choosewhich events to enter. The present study examines the entry decision by agents in amulti-tournament setting.

The participation decision of agents and the resulting field of entrants in a singletournament has been examined previously. Lazear and Rosen (1981) recognized anadverse selection problem arising when the organizer determines the field of a tour-nament by randomly choosing a pair of agents from a common pool of applicants.Agents of different types will not choose to sort themselves into applicant pools ofdifferent abilities, since: if there were a pool of high ability agents and a separate poolof low ability agents, low ability agents would prefer to enter the high ability pool.

Also focusing on the participation decision of agents in a single tournament, Ful-lerton and McAfee (1999) analyze a situation in which an arbitrary number of agentsof differing abilities wish to compete for a single prize. Efficiency often dictates thatonly the two agents of highest ability should compete in the tournament. They identifya mechanism (a slight variation of an all pay auction in which a small prize is awardedto all agents granted entry into the tournament) which allows the organizer to realizethe efficient field of entrants.

The present study distinguishes itself from the existing literature by considering amulti-tournament environment in which agents themselves choose which tournamentsto enter. One other paper addressing this issue is Friebel and Matros (2005), whichconsiders a competitive environment with homogenous workers and heterogenousfirms (differing in probability of bankruptcy). They consider a situation in which eachfirm promotes a single worker to CEO (who receives a pre-announced prize, if the firmdoes not go bankrupt), while all other workers receive nothing. Workers decide whichfirm to enter based upon the bankruptcy rates and prizes of each firm. In equilibriumworkers exert less effort in firms with higher bankruptcy rates. However, firms withhigher bankruptcy rates may offer larger prizes.

In a multi-tournament setting in which agents self select the competitive environ-ment in which they will compete, the entry decision of an agent will depend upon thecombination of three distinct effects: a “prize effect,” a “winning probability effect,”and an “effort cost effect.” The precise impact of each of these effects on the entrybehavior of agents is examined.

The prize effect refers to the fact that (with the field of entrants in an event fixed),increasing the level of prizes leads to a direct increase in the expected payoff of an agentcompeting in the event (so that competing in the event is more desirable). From here,one would intuitively expect “larger prizes” to attract entrants of “higher quality.”1

1 Empirically, Ehrenberg and Bognanno (1990) and Bronars and Oettinger (2001) analyze the tournamentparticipation decision of golfers on the PGA Tour. Both studies suggest that, in some sense, higher prizesdo attract entrants of higher ability.

123

Participation incentives in rank order tournamentswith endogenous entry 3

However, the expected payoff for an agent from entering a particular tournamentalso depends upon the identity of his rival in the tournament. The probability of winninga particular tournament for an agent clearly depends upon the quality of competitionthat he faces. An agent will have a lower probability of winning (and therefore a lowerexpected payoff) when the ability of his rival is increased. The winning probabilityeffect captures differences in the payoff of an agent due to such differences in theidentity of his rival. The optimal level of effort for an agent to exert in a particulartournament depends in part upon the quality of competition that he faces. An agentwill exert more effort (and therefore have higher effort costs and a lower expectedpayoff) when the difference in ability between himself and his rival is smaller. Theeffort cost effect captures differences in the payoff of an agent due to such differencesin the identify of his rival. Because of these two effects, the entry decision of an agentmay depend upon the entry behavior of others.

Allowing for such strategic interactions between agents, we examine the entrybehavior of heterogeneous agents (differing in their innate ability) over two mutuallyexclusive rank order tournaments (with different prize levels).2 In the model analyzedthe agents are of three different abilities: high ability, low ability, and very low ability.It may be that the unique equilibrium outcome is such that the tournament offeringlarger prizes attracts a field of lower quality entrants than the tournament offeringsmaller prizes. This counterintuitive outcome results when (relative to the prizes inthe low prize event) the prizes in the high prize event are: sufficiently large (so that forthe low ability agent the prize effect is greater in magnitude than the combination ofthe winning probability and effort cost effects, which works in the opposite directionof the prize effect), but at the same time sufficiently small (so that for the high abilityagent the prize effect is smaller in magnitude than the combination of the winningprobability and effort cost effects, which works in the opposite direction of the prizeeffect).

Amegashie and Wu (2004) obtain a result qualitatively similar to ours when exam-ining entry decisions of agents over two all pay auctions (within the context of a schooladmission process). Students of differing abilities first decide to apply to either a highor low quality school (each prefers admission to the high quality school) and thendecide how much effort to exert preparing for an entrance exam. Results from Clarkand Riis (1998) on bidder behavior in an all pay auction are applied to the choiceof effort on the entrance exam. Subsequently focusing on the application decision, itis shown that there exist equilibria in which high quality students apply to the lowquality school, while low quality students apply to the high quality school. Thus, thehigh quality school may realize a student body of lower quality (than that of the lowquality school).3

2 In some respects, the issue being considered is similar to that of “choosing the right pond” which was thefocus of Frank (1985). However, Frank pays particular attention to situations in which relative standing is ofimportance to individuals. In the setting considered here, being the highest ability agent in an environmentis not valued in and of itself, but rather only since it implies a greater likelihood of realizing a larger prize.3 In contrast to a tournament setting, in an all pay auction the allocation of the prize is deterministic (inthat the highest bidder is guaranteed to obtain the prize). Further, while Amegashie and Wu show that suchequilibria can exist, they also note that there is always an equilibrium in which the highest quality studentsapply to the high quality school.

123


A model is presented in Sect. 2 of two alternative tournaments, each with a fieldlimited to two entrants. The entry behavior of agents is examined in Sect. 3, focus-ing on an environment in which entry decisions are made sequentially with the agentof highest ability making the initial entry choice. Of particular interest is the coun-terintuitive possibility that a field of higher quality may enter the event with smallerprizes. Alternative entry environments are considered in Sect. 4, where it is argued thatthe resulting outcome does depend in part upon the nature of the entry process. Theresults of comparative statics exercises are discussed in Sect. 5. Section 6 concludesand discusses directions for future research.

2 Two alternative tournaments

Consider two alternative tournaments, denoted Event 1 and Event 2, each with a fieldlimited to two entrants. Let the prize structure of Event j be such that: p j is awardedto the “winner” and αp j is awarded to the “runner-up” (with 0 ≤ α < 1).4 Withoutloss of generality, let p1 = τp2 with τ ≥ 1, so that Event 1 has “larger prizes.”

Suppose there are two “strategic entrants” of differing abilities, one of “high abil-ity” (agent H ) and one of “low ability” (agent L). Assume each of these agents wishesto participate in one and only one of the events. The entry decision of agents H andL is analyzed. The primary focus is on an environment in which these agents maketheir entry decisions sequentially, with H making his choice first.5 That is, H firstchooses to enter Event 1 or Event 2. After observing the choice of H , L chooses toenter Event 1 or Event 2. The field of any event not filled by these strategic agents isfilled by “non-strategic” agents (denoted N ) of “very low ability.”6

Following the entry decisions, each tournament realizes one of four fields: {H, L},{H, N }, {L , N }, or {N , N }. Once the field of each tournament is determined, agentscompete by choosing effort levels. While the agents of type N are non-strategic withrespect to their tournament entry choice, they behave strategically when deciding howmuch effort to exert.

Turning attention to the competition within a tournament, consider an event with afirst place prize of p and a second place prize of αp. Two agents, A and B, competeby simultaneously choosing effort levels, eA ≥ 0 and eB ≥ 0. Let SAB (eA, eB) be thecontest success function for A, denoting the probability with which A outperforms B.

4 A common, positive value of α across events is observed in the prize structure offered on the PGA Tour,since of the total purse at a typical event: 18% is awarded to the first place finisher, 10% is awarded to thesecond place finisher, 6.8% is awarded to the third place finisher, on down to .2% awarded to the 70th placefinisher.5 After a detailed analysis of this entry environment in Sect. 3, the equilibria in alternate entry environmentsare discussed in Sect. 4.6 This structure is intended to capture the feature of the PGA Tour in which “Tour Members” can essen-tially enter any open events without any constraint, with the remaining slots in the field filled by “Non-TourMembers.” The assumption that agents do not wish to participate in every event is consistent with observedbehavior of PGA Tour members who each typically compete in only 20–30 weekly events per year.

123


2.1 Maintained assumptions

The following assumptions are maintained throughout.

2.1.1 Assumptions on the contest success function (ACSF)

ACSF 1: SAB (eA, eB) is homogeneous of degree zero.7

ACSF 2: SAB (e, e) = δAB ≥ 12 for all e > 0.

ACSF 3:8

∂SAB(eA, eB)

∂eA≡ S′1AB(eA, eB) > 0 ∀(eA, eB) ∈ (0,∞)2 (1)

∂2SAB(eA, eB)

∂e2A

≡ S′′11AB(eA, eB) < 0 ∀(eA, eB) ∈ (0,∞)2 (2)

∂2SAB(eA, eB)

∂e2B

≡ S′′22AB(eA, eB) > 0 ∀(eA, eB) ∈ (0,∞)2 (3)

(i) SAB(eA, 0) = 1 and (i i) SAB(0, eB) = 0 ∀(eA, eB) ∈ (0,∞)2 (4)

2.1.2 Assumptions on the cost function (ACF)

The cost of exerting effort of e is given by c(e) = keβ (the same for all agents),with k > 0 and β ≥ 1. While exerting zero effort is costless (c(0) = 0), it is costlyto exert any positive level of effort (c(e) > 0 for all e > 0). Further, effort costsare increasing in the level of effort (c′(e) = βkeβ−1 > 0) at a non-decreasing rate(c′′(e) = (β − 1)βkeβ−2 ≥ 0). Finally, for this function the elasticity of effort costsis constant, equal to εc = ec′(e)

c(e) = β ≥ 1.

2.1.3 Assumptions on the players’ abilities (APA)

Define ηAB ≡ S′1AB(1, 1) and σAB ≡ ηABβδAB

.9 If H is of higher ability than L and L isof higher ability than N , we arrive at the following logical consequences (when agentsexert equal efforts in a tournament).

APA 1: H is more likely than L to outperform N , and L is more likely to outperformN than vice versa, i.e.,

δH N > δL N > 1/2. (5)

7 Assuming the contest success function is homogeneous of degree zero implies that it is of the “ratio form”described by Hirshleifer (1989), in that the value of the function depends upon simply the ratio of eA to eB .8 For example, the function SAB (eA, eB ) = δAB ez

A(1−δAB )ez

B+δAB ezA

(with z ∈ (0, 1]) satisfies all of these

assumptions. Tournaments with contest success functions of such a logit-form have been considered byBaik (1994), Hirshleifer (1989), and Rosen (1986).9 ηAB measures the sensitivity of the contest success function for A with respect to A’s effort when bothA and B exert one unit of effort.

123


APA 2: H is more likely to outperform N than he is to outperform L , and H is morelikely to outperform L than vice versa, i.e.,

δH N > δH L > 1/2. (6)

APA 3: The closer the abilities of the agents are to one another, the more dependentis the contest outcome on effort choices. This assumption is formalized as

ηH N < ηL N and ηH N < ηH L . (7)

This final assumption is motivated by the natural idea that the winning odd-ratio inthe contest between H and N should be less sensitive to an increase in H ’s effort thanthe winning odd-ratio in the contest between L and N is to an increase in L’s effort.Under assumptions (5) and (6), Assumption APA 3 is a necessary condition for theabove notion to hold. To see this, note first that in the generic competition betweenA and B, the elasticity of the winning odd-ratio for agent A with respect to his owneffort, when both agents exert the same effort, is given by:

eA

∂(

SAB (eA,eB )1−SAB (eA,eB )

) /∂eA

SAB (eA,eB )1−SAB (eA,eB )

∣∣∣∣∣∣eA=eB=e

= S′1AB(e, e)

SAB(e, e)(1− SAB(e, e))e

=1e ηAB

δAB(1− δAB)e = ηAB

δAB(1− δAB).

This elasticity measures the increase (in percentage) of the winning odd ratio inducedby a one percent increase in A’s effort. Next, since δH N > δL N > 1/2, we have:

ηH N

δH N (1− δH N )≤ ηL N

δL N (1− δL N )⇒ ηH N < ηL N .

Note that a logit-form success function (as specified in footnote 8) can be characterizedby the fact that the elasticity of the winning odd-ratio with respect to the first argumentis constant (and equal to the parameter z). If this elasticity is also contest-independent,then restrictions (5) and (6) imply (7), and the latter is redundant.

2.2 Some consequences of the maintained assumptions

Together Assumptions ACSF 1 and ACSF 3 (1) imply ∂SAB (eA,eB )∂eB

≡ S′2AB(eA, eB) <

0 for all (eA, eB) ∈ (0,∞)2. To see this, recognize that under ACSF 1, Euler’s Homo-geneous Function Theorem implies

eA S′1AB(eA, eB)+ eB S′2AB(eA, eB) = 0. (8)

Thus, under ACSF 3 (1) we have S′2AB(eA, eB) < 0 for all (eA, eB) ∈ (0,∞)2.

123


Tournament level competition between A and B is analyzed in Appendix. It is shownthat in the tournament level subgame there exists a unique equilibrium with e∗A = e∗B =e∗. Thus, A outperforms B with probability δAB = SAB(e∗, e∗) = SAB(1, 1), wherethe last equality follows from ACSF 1. In equilibrium each agent has effort costs ofc(e∗) = (1−α)p ηAB

β= (1−α)pδABσAB . Thus, A and B, respectively obtain payoffs

of

A{A,B} = p(α + (1− α)δAB)− (1− α)pδABσAB = p{α + (1− α)δAB [1− σAB ]}and

B{A,B} = p(1−(1−α)δAB)− (1−α)pδABσAB = p{α + (1−α)[1−δAB(1+ σAB)]},

where p denotes the prize awarded to the winner of the contest.B{A,B} > 0 for all α ∈ [0, 1) if δAB(1+σAB) < 1, or equivalently if δAB+ ηAB

β<

1. It is shown in Appendix (see Proposition 6.1) that under Assumptions (3) and (4,i),one has δAB > ηAB and δAB + ηAB < 1. The latter inequality, along with β ≥ 1,implies B{A,B} > 0.

Further, since δAB ≥ 12 , it follows that A{A,B} ≥ B{A,B}. Thus, in such a tour-

nament the expected payoffs are such that A{A,B} ≥ B{A,B} > 0: each agent hasa positive expected payoff (so that participating in the tournament is preferable to notparticipating), and the expected payoff of the higher ability agent is greater than thatof the lower ability agent.

Note that δH L > ηH L and δL N > ηL N from Proposition 6.1 in Appendix. Theseobservations, along with assumptions (6) and (7), imply:

δH N (1− σH N ) > δH L(1− σH L) > 0 (9)

andδH N (1− σH N ) > δL N (1− σL N ) > 0. (10)

3 Composition of fields and entry decisions

Based upon the entry decisions of H and L , one of four compositions of fields acrossthe events will result. Denoting these by a, b, c, and d, we have:

Composition a: Fa1 = {H, L} and Fa

2 = {N , N }. Both strategic agents enter theevent with the larger prizes.

Composition b: Fb1 = {H, N } and Fb

2 = {L , N }. The strategic agent of higher abil-ity enters the event with the larger prizes, while the strategic agentof lower ability enters the event with the smaller prizes.

Composition c: Fc1 = {L , N } and Fc

2 = {H, N }. The strategic agent of lower abilityenters the event with the larger prizes, while the strategic agent ofhigher ability enters the event with the smaller prizes.

Composition d: Fd1 = {N , N } and Fd

2 = {H, L}. Both strategic agents enter theevent with the smaller prizes.

123


Table 1

m mH c(e∗) m

L

a τp{α+(1−α)δH L [1−σH L ]} (1−α)τpδH LσH L τp {α + (1− α)[1− δH L (1+ σH L )]}b τp{α+(1−α)δH N [1−σH N ]} (1− α)τpδH N σH N ←− FIELD 1

b FIELD 2 −→ (1− α)pδL N σL N p {α + (1− α)δL N [1− σL N ]}c p {α + (1− α)δH N [1− σH N ]} (1− α)pδH N σH N ←− FIELD 2

c FIELD 1 −→ (1− α)τpδL N σL N τp {α + (1− α)δL N [1− σL N ]}d p {α + (1− α)δH L [1− σH L ]} (1− α)pδH LσH L p {α + (1− α)[1− δH L (1+ σH L )]}

Compositions a and d will be referred to as “pooling compositions”; compositionsb and c will be referred to as “separating compositions.”

Let mi denote the expected payoff for agent i under composition m. Let p2 = p

(and therefore p1 = τp). The expected payoff and realized effort cost of each strategicagent for each composition of fields is summarized in Table 1. When interpreting theexpressions below, recall that δABσAB = ηAB

β.

To analyze entry behavior it is useful to define the following expressions:

(α) ≡ α + (1− α)δL N (1− σL N )

α + (1− α)[1− δH L(1+ σH L)] (11)

and

�(α) ≡ α + (1− α)δH N (1− σH N )

α + (1− α)δH L(1− σH L). (12)

Begin by noting that

(α) = τb

L

aL=

(1

τ

)c

L

dL

(13)

and

�(α) = τc

H

aH=

(1

τ

)b

H

dH

. (14)

It follows that (1) = 1 and (0) = δL N (1−σL N )1−δH L (1+σH L )

. Further, if

δL N (1− σL N ) > 1− δH L(1+ σH L) (15)

or equivalently

δL N − (1− δH L) >1

β(ηL N − ηH L), (16)

then ′(α) < 0 and therefore (α) > 1 for all α ∈ [0, 1).10 If (15) does not hold,then ′(α) ≥ 0 and therefore (α) ≤ 1 for all α ∈ [0, 1). Similar inspection of �(α)

10 It is straightforward to show that (15) holds (so that (α) > 1) if within each tournament the contestsuccess functions are of the logit-form (specified in footnote 8) with a common value of z.

123


reveals �(1) = 1 and �(0) = δH N (1−σH N )δH L (1−σH L )

. Further, by (9) it follows that �′(α) < 0for all α ∈ [0, 1) and �(0) > 1.

Since τ ≥ 1 we arrive at the following comparisons of the payoffs of L which holdgenerally: a

L ≥ dL and c

L ≥ bL . Comparing a

L to bL :

aL ≥ b

L ⇔ τ ≥ (α). (17)

Comparing cL to d

L :

cL ≥ d

L ⇔ τ ≥ 1

(α). (18)

By τ ≥ 1 along with (9), the payoffs of H must generally satisfy: aH ≥ d

H ,b

H ≥ cH , and b

H > dH . Comparing a

H to cH :

aH ≥ c

H ⇔ τ ≥ �(α). (19)

In terms of the equilibrium entry behavior of the agents, four distinct cases mustbe addressed (defined by the relationship of (α) to �(α)):

Case (I ): 1 ≤ �(α) ≤ (α),Case (I I ): 1 ≤ (α) < �(α),Case (I I I ): 1 < 1

(α)< �(α), or

Case (I V ): 1 ≤ �(α) ≤ 1(α)

.Equilibrium entry behavior is determined in each of these four cases, for every

τ ≥ 1. The primary focus is an environment in which the entry decisions of H and Lare made sequentially, with H making his choice first.11

An agent will decide which event to enter by comparing his expected payoff acrossthe alternative tournaments. The expected payoff from competing in any tournamentis influenced by three distinct effects: a “prize effect” (other factors fixed, the expectedpayoff of an agent is larger when tournament prizes are larger); a “winning probabilityeffect” (other factors fixed, the expected payoff of an agent is larger when his prob-ability of obtaining the first place prize is larger); and an “effort cost effect” (otherfactors fixed, the expected payoff of an agent is larger when his equilibrium level ofeffort and equilibrium effort costs are lower). In a tournament between agent A andagent B: the prize effect is dependent upon the value of τ , the winning probabilityeffect is dependent upon the value of δAB , and the effort cost effect is dependent uponthe values of ηAB and β.

Since Event 1 has larger prizes, the prize effect always makes entry into this eventmore desirable for each agent, ceteris paribus. Likewise, the directions of the winningprobability effects are straightforward to determine. For H , the winning probabilityeffect makes competition against N more desirable than competition against L (sinceδH N > δH L ); for L , the winning probability effect makes competition against N moredesirable than competition against H (since δL N > 1− δH L ).

11 The strategy choices and resulting equilibria for the alternative entry environments of “sequential entry,with L choosing first” and “simultaneous entry” are briefly discussed and summarized in the next section.

123


Differences in equilibrium effort and effort costs resulting from changes in theidentity of the tournament rival are captured by the parameters ηH L , ηH N , and ηL N .12

For H the winning probability effect and the effort cost effect work in the samedirection—they both make competition against N more desirable than competitionagainst L . In contrast, for L these two effects only work in the same direction whenηL N < ηH L (in which case the winning probability effect and the effort cost effectboth make competition against N more desirable than competition against H for L).If we instead have ηH L < ηL N , then for L the effort cost effect makes competitionagainst H more desirable than competition against N (that is, the effort cost effectworks in the opposite direction of the winning probability effect for L). To see whicheffect dominates we can compare the payoff for L from competing against H versuscompeting against N , across events with equal prizes (for instance, compare a

L to bL

supposing τ = 1). This is precisely a comparison of (α) to a value of 1. It followsthat for L the combination of the winning probability effect and the effort cost effect:make competition against N more desirable than competition against H when (16) issatisfied (i.e., when (α) > 1); and make competition against H more desirable thancompetition against N when (16) is violated (i.e., when (α) < 1).13

3.1 Decision by L

Analyzing the entry decisions recursively, we first focus on the choice by L . Thischoice must be analyzed in two distinct settings: situations in which H has enteredEvent 1, and situations in which H has entered Event 2.

3.1.1 Decision by L following a choice by H to enter Event 2

First consider the entry decision of L following a choice by H to enter Event 2. By(18) we know that L will enter Event 1 if and only if τ ≥ 1

(α).14 To see why L makes

this choice, note that if H has entered Event 2 then L can: compete against N in Event1 or compete against H in Event 2. When making this choice L will account for thethree distinct effects previously discussed. Recall, the prize effect makes entry intoEvent 1 more desirable.

If (16) holds (corresponding to Case (I ) or Case (I I )), then the combination of thewinning probability and effort cost effects also makes entry into Event 1 (competition

12 From Table 1 it is obvious that larger prizes lead to greater effort costs. However, the overall effect oflarger prizes is that the payoff of an agent becomes larger as prizes are increased. Thus, any difference inthe payoff of an agent across events resulting from a difference in effort costs is most relevant when theprizes across the events are relatively equal to each other (that is, when τ ≈ 1). Comparing effort costsacross events at this baseline value of τ = 1, we see that L would exert more effort against H than againstN when δH LσH L > δL N σL N or equivalently when ηL N < ηH L . From here it is clear why a comparisonof ηL N to ηH L isolates the effort cost effect.13 It is now also clear why ηL N < ηH L is sufficient to ensure that for L the combination of these twoeffects makes competition against N better than competition against H .14 Throughout the analysis it is assumed that each type of agent would choose to enter the event with largerprizes (Event 1) whenever indifferent between the two events.

123


against N ) the more desirable choice. That is, this combined effect works in the samedirection as the prize effect. In such situations, τ ≥ 1

(α)(or equivalently c

L ≥d

L ) for all τ ≥ 1. Thus (following a choice by H to enter Event 2), L will enterEvent 1.

If instead (16) does not hold (corresponding to Case (I I I ) or Case (I V )), then thecombination of the winning probability and effort cost effects makes entry into Event 2(that is, competition against H ) the more desirable choice. This combined effect nowworks in the opposite direction of the prize effect. In such situations, τ < 1

(α)(or

equivalently cL < d

L ) for relatively small values of τ , while τ ≥ 1(α)

(or equiva-

lently cL ≥ d

L ) for relatively large values of τ . That is, after H has entered Event 2,L will: enter Event 2 as well if the prizes across the events are of relatively simi-lar levels (τ sufficiently small), and enter Event 1 if the prizes across the events aresubstantially different from each other (τ sufficiently large).

3.1.2 Decision by L following a choice by H to enter Event 1

Next consider the choice of L following entry into Event 1 by H . By (17) we knowthat L will enter Event 1 if and only if τ ≥ (α). To see why this choice is madeby L , note that if H has entered Event 1 then L can: compete against H in Event1 or compete against N in Event 2. When making this choice L again considers thethree effects discussed above. The prize effect again makes entry into Event 1 moredesirable than entry into Event 2.

If (16) does not hold (corresponding to Case (I I I ) or Case (I V )), then the combina-tion of the winning probability and effort cost effects makes entry into Event 1 (compe-tition against H ) the more desirable choice. That is, this combined effect works in thesame direction as the prize effect. In such instances, τ ≥ (α) (or equivalently a

L ≥b

L ) for all τ ≥ 1. Therefore, L will enter Event 1 following a choice by H to enterEvent 1.

If instead (16) does hold (corresponding to Case (I ) or Case (I I )), then the com-bination of the winning probability and effort cost effects makes entry into Event 2(competition against N ) better than entry into Event 1. This combined effect nowworks in the opposite direction of the prize effect. Thus, τ < (α) (or equivalentlya

L < bL ) for τ sufficiently small, and τ ≥ (α) (or equivalently a

L ≥ bL ) for

τ sufficiently large. If H has entered Event 1, then L will: enter Event 2 if the prizesacross the two events are of relatively similar levels (τ sufficiently small), and enterEvent 1 if the prizes across the two events are substantially different from each other(τ sufficiently large).

3.2 Initial decision by H

Given the sequential nature of the entry decision, H can infer the subsequent choiceof L before making his own choice. From the discussion so far, the behavior of L isqualitatively different if (α) ≥ 1 or if (α) < 1.

123


3.2.1 Decision by H when (α) ≥ 1

If (α) ≥ 1 (or equivalently 1(α)≤ 1), we have either Case (I ) or Case (I I ). In

these situations H can infer that for all prize levels, L will enter Event 1 followingentry by H into Event 2. Further, if H enters Event 1, then L will: enter Event 1 ifτ ≥ (α); and enter Event 2 if instead τ < (α).

First consider τ < (α). The initial choice of H is one of competing against N inEvent 1 or competing against N in Event 2. Since Event 1 has larger prizes, enteringEvent 1 is the better choice. That is, for both Case (I ) and Case (I I ), composition bresults for τ sufficiently small.

Next consider τ ≥ (α). H can now either compete against L in Event 1 or com-pete against N in Event 2. The winning probability effect and the effort cost effect bothmake entry into Event 2 more desirable for H (since δH N > δH L and ηH N < ηH L ,respectively). Therefore, the decision by H depends upon whether the prize effect(which makes entry into Event 1 more desirable) is sufficiently strong so as to offsetthe combined impact of the other two effects. From (19) we see that this is true(so that H enters Event 1) if and only if τ ≥ �(α).

In Case (I ) we have �(α) ≤ (α). In this situation, if τ is large enough so thatτ ≥ (α), then τ ≥ �(α). That is, by the point at which the prizes in Event 1 aresufficiently large so that L does not simply “avoid competition against H ,” the prizesin Event 1 are also large enough so that H would rather compete against L for theselarge prizes as opposed to competing against N for the relatively small prizes. There-fore, in Case (I ): composition b arises for τ < (α), while composition a arises forτ ≥ (α).

In Case (I I ) we have (α) < �(α). It has already been argued that compositionb results for τ ≤ (α). However, we could now have either (α) ≤ τ < �(α)

or (α) < �(α) ≤ τ . If (α) ≤ τ < �(α), then the prizes in Event 1 are: largeenough so that L prefers to compete against H for large prizes than against N forsmall prizes, but at the same time small enough so that H prefers to compete againstN for small prizes than against L for large prizes. For τ in this intermediate range:H will enter Event 2 followed by entry into event 1 by L , leading to composition c.Finally, if (α) < �(α) ≤ τ , then the prizes in Event 1 are sufficiently large so that:not only does L prefer competition against H for large prizes over competition againstN for small prizes, but further H prefers competition against L for large prizes overcompetition against N for small prizes. For such sufficiently large τ : H will enterEvent 1 followed by entry into Event 1 by L , leading to composition a. In summary,in Case (I I ) we realize: composition b for sufficiently small values of τ ; compositionc for intermediate values of τ ; and composition a for sufficiently large values of τ .

3.2.2 Decision by H when (α) < 1

If (α) < 1 (or equivalently 1(α)

> 1), we have either Case (I I I ) or Case (I V ). Hcan infer that for all prize levels, L will now always enter Event 1 following entry byH into Event 1. Further, if H were to enter Event 2, then L: would enter Event 1 ifτ ≥ 1

(α); and enter Event 2 if τ < 1

(α).

123


First consider τ < 1(α)

. The initial choice of H is one of competing against L inEvent 1 or competing against L in Event 2. Since prizes in Event 1 are larger, enteringEvent 1 is the better option. Thus, for both Case (I I I ) and Case (I V ), composition aarises for τ sufficiently small.

Next consider τ ≥ 1(α)

. H can now either compete against L in Event 1 or competeagainst N in Event 2. The winning probability and the effort cost effects both makeentry into Event 2 more desirable for H (since δH N > δH L and ηH N < ηH L , respec-tively). Therefore, the choice of H depends upon whether the prize effect (whichmakes entry into Event 1 more desirable) is strong enough to offset the combinedimpact of these other two effects. From (19), this is the case (so that H enters Event 1)if and only if τ ≥ �(α).

In Case (I V ) we have �(α) ≤ 1(α)

, so that by the point at which τ is large enough

to have τ ≥ 1(α)

we also have τ ≥ �(α). That is, if the prizes in Event 1 are largeenough so that L does not simply “follow H into Event 2,” then the prizes in Event 1are also large enough so that H prefers to compete against L for the larger prizesversus competing against N for the smaller prizes. Thus, in Case (I V ) composition aresults for all τ ≥ 1.

In Case (I I I ) we have 1(α)

< �(α). As noted, composition a results for τ < 1(α)

.It has also been argued that L will enter Event 1 following entry by H into Event 2 forτ ≥ 1

(α). However, we now can have either 1

(α)≤ τ < �(α) or 1

(α)< �(α) ≤ τ .

If 1(α)≤ τ < �(α), then the prizes in Event 1 are: large enough so that L prefers to

compete against N for large prizes than to compete against H for small prizes (recall,since (α) < 1 the combination of the winning probability and the effort cost effectsmake competition against H relatively more desirable than competition against Nfor L), but at the same time small enough so that H prefers to compete against N forsmall prizes than against L for large prizes. For τ in this intermediate range: H willenter Event 2 and L will subsequently enter Event 1, leading to composition c. Finally,if 1

(α)< �(α) ≤ τ , then the prizes in Event 1 are sufficiently large so that: not only

does L prefer to compete against N for large prizes than against H for small prizes,but further H prefers to compete against L for large prizes than against N for smallprizes. For such sufficiently large τ : H will enter Event 1 and L will subsequentlyenter Event 1, leading to composition a. To summarize, in Case (I I I ) we realize:composition a for τ sufficiently small; composition c for intermediate values of τ ;and composition a for τ sufficiently large.

3.3 Summary of equilibrium compositions

At this point it is instructive to summarize the composition that results for every valueof τ in each of the four cases. The columns labeled “H first” in Tables 2, 3, 4, 5 belowreport the resulting composition dependent upon the value of τ in relation to �(α)

and (α) or 1(α)

.15

15 The other columns relate to other entry environments, discussed in the next section.

123


Table 2Case (I ) H first L first Simultaneous

τ < �(α) b c b, c, and mixed

�(α) ≤ τ < (α) b b b

(α) ≤ τ a a a

Table 3Case (I I ) H first L first Simultaneous

τ < (α) b c b, c, and mixed

(α) ≤ τ < �(α) c c c

�(α) ≤ τ a a a

Table 4Case (I I I ) H first L first Simultaneous

τ < 1(α)

a c Mixed1

(α)≤ τ < �(α) c c c

�(α) ≤ τ a a a

Table 5Case (I V ) H first L first Simultaneous

τ < �(α) a c Mixed

�(α) ≤ τ a a a

From the columns labeled “H first” we observe: composition d never arises; com-position a (the pooling composition in which both agents enter Event 1) always ariseswhen the prizes in Event 1 are sufficiently large; and composition c (in which H entersthe event with smaller prizes, while L enters the event with larger prizes) may arise.

4 Alternate entry environments

The analysis so far has focused on a situation in which H and L sequentially decidein which tournament to compete, with H making the initial decision. However, theequilibrium composition that arises depends in part upon the nature of the entry deci-sion. That is, different outcomes may result if instead the entry decisions were madesequentially with L making the initial decision (“L first”) or if the agents made theirentry decisions simultaneously (“simultaneous”). The analyses of these alternate entryenvironments is aided by the recognition that the payoff comparisons previously madeare valid regardless of the timing of the entry decisions. Specifically, the observationsthat b

H > dH and c

L > bL are general comparisons of the payoffs of an agent

across different compositions which do not at all depend upon the mechanism bywhich the composition of fields is determined. Likewise, conditions (17), (18), and(19) are each still relevant when comparing a

L to bL , comparing c

L to dL , and

comparing aH to c

H , respectively.

123


If instead the entry decisions were made sequentially with L making the initialchoice, the analysis would proceed in a very similar fashion to the previous presenta-tion. A unique equilibrium again arises (because of the sequential nature of the entrydecisions). The resulting equilibrium composition for every τ in each of the four casesis reported in the columns labeled “L first” in Tables 2, 3, 4, 5.

If the entry choices are instead made simultaneously, there may not be a uniquepure strategy equilibrium—it could be that the unique equilibrium is in mixed strat-egies or that there are multiple equilibria. The outcome under this entry environmentis summarized in Tables 2, 3, 4, 5 in the columns labeled “simultaneous,” again forevery τ in each of the four cases.

An entry of “b, c, and mixed” indicates that there are two pure strategy equilibria(resulting in compositions b and c) as well as a third equilibrium in mixed strategies.This is the outcome when each strategic agent wants to “avoid his strategic rival.” Forexample, from the perspective of L , this requires that L prefers to compete against Nthan against H , regardless of whether competition against H would be in Event 1 orEvent 2. This occurs for relatively small τ whenever (α) > 1.

An entry of “mixed” indicates that the unique equilibrium is a mixed strategy equi-librium. This occurs when “H wants to avoid L” (in that regardless of which event Lis competing in, H wants to be in the other event) while “L wants to confront H” (inthat regardless of which event H is competing in, L wants to be in the same event).This occurs for relatively small τ whenever (α) < 1.

These qualitatively different outcomes result for τ ≈ 1 (when the magnitude of theprize effect is approximately zero) precisely because the combination of the winningprobability and the effort cost effects can work in either direction for L . Recall that forH both of these effects make competition against N more desirable than competitionagainst L . Therefore, for τ ≈ 1 H wants to “avoid L .” In contrast, for L the combi-nation of the winning probability and effort cost effects makes competition against Nmore desirable than competition against H if and only if (α) ≥ 1. Thus, for τ ≈ 1 itfollows that: L wants to “avoid H” if (α) ≥ 1, implying both separating composi-tions are stable; and L wants to “confront H” if (α) < 1, implying no combinationof pure strategies is stable.

In order to determine the impact of the nature of the entry environment on thecomposition of fields, we can compare the reported equilibria across each row inTables 2, 3, 4, 5. First observe that whenever there is a unique equilibrium in thesimultaneous move environment the same equilibrium arises if the entry decisionsare instead made sequentially (regardless of which agent makes the initial decision).Thus, for τ sufficiently large the three entry environments under consideration lead tothe same composition of fields.

For relatively small values of τ (those for which there is a mixed strategy equilibriumof the simultaneous move game), the outcome is qualitatively different for (α) > 1versus (α) < 1. If (α) > 1 then for such τ we have b

H > cH > a

H > dH and

cL > b

L > aL > d

L , giving us an “anti-coordination game” in that both strategicagents agree that either separating composition is better than either pooling compo-sition. In a sequential environment there is a first mover advantage in that the agentchoosing first is able to guarantee that we realize his preferred separating composition.

123


If (α) < 1 then for such small τ we still have bH > c

H > aH > d

H but nowhave a

L > dL > c

L > bL . The agents are now playing a game of “cat and mouse”

in that H prefers either separating composition over either pooling composition, whileL prefers either pooling composition over either separating composition. In a sequen-tial environment there is now a second mover advantage. The agent going secondessentially chooses which type of composition will arise (either separating or pooling),while the agent going first essentially chooses which composition of this type arises.

For each of the three entry environments considered, we still have the following tworesults: composition d is never realized (as a pure strategy equilibrium); and compo-sition a is realized for τ sufficiently large. The intuition for the latter of these insightsis clear—the prizes in Event 1 can always be made “so large” so that the prize effectdominates the combination of the winning probability and effort cost effects, makingentry into Event 1 a dominant choice for each agent. In conclusion, we see that for(α) > 1 we essentially realize a separating composition for τ sufficiently small anda pooling composition for τ sufficiently large. However, for (α) < 1 we still realizea pooling composition for τ sufficiently large, but may realize either a pooling compo-sition or a separating composition (with the resulting type of composition dependingin part upon the entry environment) for smaller values of τ . Finally, the realizationof composition c is not an artificial construct of focusing on sequential entry with Hchoosing first. Rather, inspection of Tables 2, 3, 4, 5 reveals that composition c is infact less likely to result for “H first.”

5 Comparative statics

In order to see how the composition depends upon the parameters of the model, wewill focus on the sequential entry environment in which H makes his entry choicebefore L (“H first”). The results of comparative statics exercises are discussed forchanges in p1, p2, δH L , δH N , δL N , ηH L , ηH N , and ηL N individually, with all otherfactors fixed.

5.1 Changes in prize levels

It is straightforward to see how the outcome depends upon p1 or p2. Since τ = p1p2

,increasing p1 or decreasing p2 increases τ (and decreasing p1 or increasing p2decreases τ ). As changes in p1 or p2 do not influence �(α) or (α), we need not beconcerned with switching between Cases (I ), (I I ), (I I I ), and (I V ) as prize levelsare changed. Therefore, whichever case is relevant, we simply need to read down thecolumn for “H first” to see how the composition changes as τ increases.

Conducting this exercise in Case (I I ) or Case (I I I ) (the cases in which composi-tion c may arise), we see that changes in the prize structure can lead to unanticipatedchanges in the composition of fields. Note that Event 1 attracts a field of: {H, L}under composition a; {H, N } under composition b; and {L , N } under compositionc. In terms of the quality of the entrants, {H, L} is the strongest and {L , N } is theweakest of these fields.

123


First consider 1 ≤ (α) < �(α) (Case (I I )) and suppose initially τ < (α) (sothat composition b results and Event 1 attracts entrants of {H, N }). If Event 1 increasestheir prizes to a level for which (α) ≤ τ < �(α), composition c results and Event 1attracts entrants of {L , N }. As a result, Event 1 attracts a lower quality field afteroffering larger prizes.16

An examination of the entry decisions of H and L reveal why such unanticipatedchanges could result. When τ < (α) < �(α), the prizes in Event 1 are sufficientlysmall so that both H and L prefer to compete against N for small prizes as opposedto competing against their strategic rival for large prizes. However, when the prizesare changed so that (α) ≤ τ < �(α), the prizes in Event 1 are: now large enoughso that L prefers to compete against H for large prizes over competing against N forsmall prizes, but still small enough so that H prefers to compete against N for smallprizes over competing against L for large prizes.

Now consider 1 < 1(α)

< �(α) (Case (I I I )) and suppose initially τ < 1(α)

(sothat composition a results and Event 1 has entrants of {H, L}). If Event 1 increases theirprizes so that 1

(α)≤ τ < �(α), then composition c arises and Event 1 has entrants

of {L , N }. As a result, Event 1 again attracts a lower quality field after offering largerprizes.17

An examination of the entry decisions of H and L again reveals why this unex-pected change occurs. First recall that since (α) < 1 it follows that for all prize levelsL prefers to compete against H for larger prizes as opposed to against N for smallerprizes (since for L the combination of the winning probability and effort cost effectsmakes competition against H more desirable than competition against N ). Further, forτ < 1

(α)the prizes in Event 1 are small enough so that L prefers to compete against

H for small prizes than against N for large prizes. Thus, for such small τ , H cannotavoid competing against L . Therefore, composition a results since for H competingagainst L for large prizes is better than competing against L for small prizes. However,when the prizes are changed so that 1

(α)≤ τ < �(α), then the prizes in Event 1 are

now: large enough so that L prefers to compete against N for large prizes as opposedto competing against H for small prizes, but at the same time small enough so that Hprefers to compete against N for small prizes as opposed to competing against L forlarge prizes.

5.2 Changes in winning probabilities

First note that as δH N is increased: �(α) increases; but (α) and 1(α)

are not changed.Thus, if the winning probability for H when competing against N is larger, we are

16 Similarly, in this case Event 2 could possibly attract a higher quality field by offering smaller prizes.Start by supposing τ < (α) and consider a decrease in p2 resulting in (α) ≤ τ < �(α) (a changetaking us from composition b to composition c). After decreasing their prize level, Event 2 attracts a fieldof {H, N } instead of {L , N }.17 Again, in this case Event 2 could attract a field of higher quality by offering smaller prizes. Start bysupposing τ < 1

(α)and consider a decrease in p2 so that 1

(α)< τ < �(α) (a change taking us from

composition a to composition c). After decreasing their prizes, Event 2 attracts a field of {H, N } instead of{N , N }.

123


more likely to have Case (I I ) instead of Case (I ) and more likely to have Case (I I I )instead of Case (I V ). As a result, composition c is more likely to arise and compositiona is less likely to arise.

Next note that as δL N is increased: �(α) is not changed; (α) increases; and 1(α)

decreases. To see the impact of this increase in the winning probability for L whencompeting against N , start by noting that if Case (I ) is relevant this increase makes usrealize composition b instead of composition a for some τ . If Case (I I ) is relevant,this change gives us composition b instead of composition c for a range of τ , and maytake us from Case (I I ) to Case (I ). If we initially have Case (I I I ), this increase inδL N makes us realize composition c instead of composition a for certain τ , and maytake us from Case (I I I ) to Case (I I ). Finally, starting at Case (I V ), this change doesnot alter the resulting composition until the point at which we move from Case (I V )

to Case (I I I ). In total this change makes a separating composition more likely and apooling composition less likely.

Finally note that an increase in δH L will: decrease �(α); increase (α); anddecrease 1

(α). By Condition (9): δH L < δH N − 1

βηH N + 1

βηH L . As δH L approaches

this upper limit: �(α)→ 1 and (α)→ α+(1−α)δL N (1−σL N )

α+(1−α){(1−δH N )− 1

β (2ηH L−ηH N )} . If (α)>1,

then this limiting value of (α) is also greater than 1. However, if (α) < 1, thenthis limiting value of (α) is greater than 1 if and only if δH N − (1 − δL N ) >1β

[(ηL N − ηH L)+ (ηH N − ηH L)] (a condition which may or may not be true).Considering such an increase in the winning probability for H when competing

against L , we see that when Case (I ) is relevant this change makes us realize com-position b instead of composition a for some τ . If we initially have Case (I I ), thischange makes us realize composition b instead of composition c for some values of τ

and makes us realize composition a instead of composition c for some other values ofτ . Further, such a change eventually takes us from Case (I I ) to Case (I ). Thus, when(α) > 1, an increase in δH L makes composition c less likely to arise.

If Case (I V ) is relevant, an increase in δH L may make us transition to Case (I I I ).If Case (I I I ) is relevant, this change would make us realize composition c instead ofcomposition a for some values of τ and realize composition a instead of compositionc (that is, the opposite change) for some other values of τ . Further, an increase in δH L

could make us transition from Case (I I I ) to either Case (I I ) or Case (I V ).

5.3 Changes in effort costs

First, observe that as ηH N is increased: �(α) decreases; but (α) and 1(α)

are notchanged. An increase in ηH N increases the effort costs for H of competing againstN . Such an increase will not alter the outcome at all in Cases (I ) or (I V ). However,if we initially have Case (I I ), this increase makes us realize composition a insteadof composition c for some τ , and may take us from Case (I I ) to Case (I ). Simi-larly, under Case (I I I ), an increase in ηH N makes us realize composition a insteadof composition c for some τ , and may take us from Case (I I I ) to Case (I V ). Intotal, an increase in ηH N makes composition c less likely and composition a morelikely.

123


Second, note that an increase in ηL N will: not change �(α); but will decrease(α) and increase 1

(α). If Case (I ) is relevant, this increase in effort costs for

L from competing against N makes us realize composition a instead of composi-tion b for some τ , and eventually takes us from Case (I ) to Case (I I ). Startingin Case (I I ), this increase makes composition c arise instead of composition b forsome τ , and may take us from Case (I I ) to Case (I I I ). If we initially have Case(I I I ), this change makes composition a arise instead of composition c for cer-tain τ , and eventually makes Case (I V ) relevant. Within Case (I V ), an increasein ηL N does not alter the realized composition. In summary, an increase in ηL N

makes a pooling composition more likely and a separating composition lesslikely.

Finally, consider an increase in ηH L . As the cost of exerting effort is increased inthe competition between H and L: �(α) increases; while (α) increases and 1

(α)decreases. Further, it can be shown that (α) > �(α) is sufficient to guarantee∂(α)∂ηH L

>∂�(α)∂ηH L

. Therefore, if (α) > �(α), then an increase in ηH L widens the gapbetween (α) and �(α). From here, if Case (I ) is initially relevant, an increase inηH L will: make us realize composition b instead of composition c for some τ , butcannot take us from Case (I ) to Case (I I ). If instead (α) < �(α), then we stillhave ∂(α)

∂ηH L>

∂�(α)∂ηH L

for (α) sufficiently close in value to �(α). Thus, if Case (I I )is relevant, an increase in ηH L will: give rise to composition b instead of compo-sition c for certain τ , give rise to composition c instead of composition a for othervalues of τ , and may take us from Case (I I ) to Case (I ). Starting at Case (I I I ), anincrease in ηH L : will widen (in both directions) the intermediate range of τ for whichcomposition c arises instead of composition a, and may take us from Case (I I I ) toCase (I I ). Finally, within Case (I V ) an increase in ηH L will not directly alter thecomposition of fields, but may take us to Case (I I I ). In summary, as ηH L increases,a separating composition is realized more often and a pooling composition is realizedless often.

6 Conclusion and future research

In many tournament environments, agents compete in some (but not all) tournamentsin a set of events. The decision by an agent regarding which competitive environmentto enter should depend upon three distinct effects: a “prize effect,” a “winning proba-bility effect,” and an “effort cost effect.” The precise impact of each of these effects andthe resulting equilibrium entry behavior of agents were determined. Intuitively, onewould expect that (for a particular tournament) “larger prizes” would attract entrantsof “higher quality.” However, it is shown that a field of higher quality may be attractedto an event offering smaller prizes. This counterintuitive outcome arose when (relativeto the prizes in Event 2) the prizes in Event 1 were: sufficiently large (so that for L theprize effect is greater in magnitude than the combination of the winning probabilityand effort cost effects, which works in the opposite direction of the prize effect), but atthe same time sufficiently small (so that for H the prize effect is smaller in magnitudethan the combination of the winning probability and effort cost effects, which worksin the opposite direction of the prize effect).

123


Empirically, Ehrenberg and Bognanno (1990) and Bronars and Oettinger (2001)have provided some preliminary insights into the entry decisions of professional golf-ers. Bronars and Oettinger observe that an increase in total prizes at only tournamentk will lead more “unconstrained” golfers (those who can enter any event during thecurrent season) to enter tournament k. Ehrenberg and Bognanno find evidence tosuggest that “exempt” golfers (those who are assured entrance into all events duringthe following year, regardless of performance during the current year) are more likelythan “non-exempt” golfers to enter tournaments with larger prizes.18 In the end, bothstudies suggest that in some sense “higher prizes” do attract participants of “higherquality.” However, neither study makes a fine enough distinction among the typesof participants to address the questions raised here. Specifically, neither makes botha clear distinction between constrained and unconstrained participants, along with afurther distinction regarding the quality of the unconstrained participants. Ultimately,an experimental analysis should be conducted, to determine whether or not “smallerprizes” may possibly attract entrants of “higher quality.”

Acknowledgments We would like to thank Pradeep Dubey, Alexander Matros, and Drew Fudenberg,for some helpful comments and suggestions. Previous versions of this paper were presented at the 2005Stony Brook International Conference on Game Theory, the Spring 2006 Midwest Economic Theory Meet-ings, and seminars at California State University-Northridge, California State University-Long Beach, andKennesaw State University.

Appendix

A1. Computation of the equilibrium outcome and payoffs

The tournament level competition between A and B (as described in Sect. 2) is ana-lyzed. Since the “winner” receives p and the “runner-up” receives αp, the payoffof A is A = αp + (1 − α)pSAB (eA, eB) − c(eA) and the payoff of B is B =p − (1− α)pSAB (eA, eB)− c(eB). From here:

∂A

∂eA= (1− α)pS′1AB(eA, eB)− c′(eA)

and∂B

∂eB= −(1− α)pS′2AB(eA, eB)− c′(eB)

⇒∂2A

∂e2A

= (1− α)pS′′11AB(eA, eB)− c′′(eA) ≤ 0

and∂2B

∂e2B

= −(1− α)pS′′22AB(eA, eB)− c′′(eB) ≤ 0.

18 The “non-exempt” players considered by Ehrenberg and Bognanno may be either “constrained” or“unconstrained” as defined by Bronars and Oettinger.

123


Since SAB(eA, 0) = 1 for all eA > 0 and SAB(0, eB) = 0 for all eB > 0, therecannot be a pure strategy equilibrium in which either agent exerts zero effort.19 To seethis, consider the choice by agent i if his rival (agent j) chooses e j = 0. Suppose ichooses ei = ε > 0. Since any ei > 0 results in agent i outperforming agent j withprobability one, agent i would be better off choosing any ei ∈ (0, ε). Since this istrue for any ε > 0, it follows that there is no pure strategy equilibrium characterizedby e j = 0 and ei > 0. Further, a choice of zero effort by both agents cannot be anequilibrium. To see this, consider the choice of effort by the low ability agent (B) whenthe high ability agent (A) exerts zero effort. If SAB(0, 0) ∈ (0, 1], agent B realizes agreater expected payoff from eB > 0 so long as c(eB) < (1− α)pSAB(0, 0). This isclearly true for eB sufficiently close to zero. Likewise, if instead SAB(0, 0) = 0, thenagent A is better off exerting a small amount of effort (as opposed to zero effort) solong as c(eA) < (1−α)p. For eA sufficiently close to zero this condition holds. Thus,there are no pure strategy equilibria characterized by zero effort by either agent.20

From here, we should attempt to identify an interior solution, in which each agentexerts positive effort.

At an interior solution, eA and eB should be such that ∂A∂eA= 0 and ∂B

∂eB= 0

simultaneously. This requires − S′1AB (eA,eB )

S′2AB (eA,eB )= c′(eA)

c′(eB ). Recall that since SAB (eA, eB)

is homogeneous of degree zero condition (8) must hold, which can be expressed as

− S′1AB (eA,eB )

S′2AB (eA,eB )= eB

eA. As a result, in order for ∂A

∂eA= 0 and ∂B

∂eB= 0 simultaneously, it

must be that eBeA= c′(eA)

c′(eB )or equivalently eAc′(eA) = eBc′(eB). Since ec′(e) = βkeβ

is increasing in e, this requires eA = eB .∂A∂eA= (1 − α)p ∂SAB (eA,eB )

∂eA− c′(eA) can now be examined, restricting attention

to eA = eB = e. Since SAB (eA, eB) is homogenous of degree zero, S′1AB(eA, eB) ishomogeneous of degree−1, implying S′1AB(e, e) will: tend to∞ as e→ 0, tend to zeroas e→∞, and strictly decrease as e is increased. Recall that c′(e) is positive, finite,

and non-decreasing. Together, these insights imply ∂A∂eA

∣∣∣(eA,eB )=(e,e)

is: decreasing in

e, positive for e sufficiently small, and negative for e sufficiently large. Thus, there is

19 Assumption ACSF 1 (i.e., the assumption of homogeneity of degree zero) is sufficient but not neces-sary to ensure that there is not an equilibrium in which either agent exerts zero effort. This assumptionimplies that SAB (eA, eB ) is discontinuous at (eA, eB ) = (0, 0). Indeed, for any eA > 0 and eB > 0, byACSF 1: SAB (eA, eB ) = SAB (rAB , 1), where rAB = eA

eB(i.e., the value of the function depends upon

only the ratio of eA to eB , as noted in footnote 7). Continuity of SAB (eA, eB ) at (0, 0) would requireSAB (rAB , 1) = SAB (0, 0) for all rAB ∈ [0,∞], which could only be true if SAB (eA, eB ) were a constant(i.e., if SAB (eA, eB ) = SAB (0, 0) for all eA ≥ 0 and eB ≥ 0). Thus, discontinuity of SAB (eA, eB ) at(0, 0) is an immediate consequence of assuming homogeneity of degree zero.20 As stated in footnote 19, assumption ACSF 1 (which implies SAB (eA, eB ) is discontinuous at (eA, eB ) =(0, 0)) is sufficient but not necessary to guarantee that there is not an equilibrium in which either agentexerts zero effort. To see what could happen if SAB (eA, eB ) were continuous at (eA, eB ) = (0, 0): relaxassumption ACSF 1 (i.e., the assumption of homogeneity of degree zero), maintain assumptions ACSF 2,ACSF 3 (1) − (3), and ACF, and now assume S′2AB (eA, eB ) < 0 (which was previously a consequenceof ACSF 1 together with ACSF 3 (1)). If β > 1 (so that c′(0) = 0), then A would exert a positive level ofeffort if S′1AB (0, eB ) > 0 for all eB ≥ 0 and B would exert a positive level of effort if S′2AB (eA, 0) < 0for all eA ≥ 0. If β = 1 (so that c′(e) = k for all e ≥ 0), then each agent would exert zero effort ifS′1AB (0, 0) ≤ k

(1−α)p and −S′2AB (0, 0) ≤ k(1−α)p .

123


a unique e∗ ∈ (0,∞) for which ∂A∂eA

∣∣∣(eA,eB )=(e∗,e∗)

= 0 and ∂B∂eB

∣∣∣(eA,eB )=(e∗,e∗)

= 0

simultaneously. This unique equilibrium effort level, e∗A = e∗B = e∗ > 0, must satisfy

S′1AB(e∗, e∗)c′(e∗)

= 1

(1− α)p. (20)

With e∗A = e∗B = e∗, in equilibrium SAB(e∗, e∗) = SAB(1, 1) = δAB .Since S′1AB is homogeneous of degree −1, condition (20) can be written as

S′1AB(e∗, e∗)c′(e∗)

= S′1AB(1, 1)

e∗c′(e∗)= 1

(1− α)p

⇔ e∗c′(e∗) = (1− α)pηAB,

where ηAB ≡ S′1AB(1, 1).

For c(e) = keβ we have noted that εc = ec′(e)c(e) = β. Thus, the equilibrium condition

above can be expressed asc(e∗)β = (1− α)pηAB, (21)

implying that the equilibrium effort cost of each agent is c(e∗) = (1− α)p ηABβ

.

A2. A useful proposition

Proposition 6.1 Any homogeneous-of-degree zero contest success function SAB sat-isfies the inequalities (i) 0 < ηAB < δAB and (i i) ηAB + δAB < 1.

Proof The proof relies on viewing the contest success function alternatively as astrictly concave function of its first argument or a convex function of its second argu-ment.

The inequality in (i) follows from the well known characterization of concave func-tions as lying below their tangent lines at any point:

SAB(x, 1) ≤ SAB(1, 1)+ S′1AB(1, 1)(x − 1) ∀x ∈ (0,∞).

Since the above holds with strict inequality for x �= 1, the desired result follows byletting x tend toward zero.

Using the reverse inequality which characterizes convex functions,

SAB(1, x) ≥ SAB(1, 1)+ S′2AB(1, 1)(x − 1) ∀x ∈ (0,∞).

Since the above holds with strict inequality for x �= 1, inequality (ii) is obtained by let-ting x tend toward zero (and noting that SAB(1, 0) = 1 and S′2AB(1, 1) = −S′1AB(1, 1)

(homogeneity of degree zero)). ��

123


References

Amegashie JA, Wu X (2004) Self selection in competing all-pay auctions. Department of Economics,University of Guelph, Working paper

Baik KH (1994) Effort levels in contests with two asymmetric players. South Econ J 61(2):367–378Bronars SG, Oettinger GS (2001) Effort, risk-taking, and participation in tournaments: evidence from

professional golf. Department of Economics, University of Texas, Working paperClark DJ, Riis C (1998) Competition over more than one prize. Am Econ Rev 88(1):276–289Ehrenberg RG, Bognanno ML (1990) Do tournaments have incentive effects? J Polit Econ 98(6):1307–

1324Frank RH (1985) Choosing the right pond. Oxford University Press, OxfordFriebel G, Matros A (2005) A note on CEO compensation, elimination tournaments and bankruptcy risk.

Econ Gov 6(2):105–111Fullerton RL, McAfee RP (1999) Auctioning entry into tournaments. J Polit Econ 107(3):573–605Green JR, Stokey NL (1983) A comparison of tournaments and contracts. J Polit Econ 91(3):349–364Hirshleifer J (1989) Conflict and rent-seeking success functions: ratio vs difference models of relative

success. Public Choice 63(2):101–112Lazear EP, Rosen S (1981) Rank-order tournaments as optimum labor contracts. J Polit Econ 89(5):841–

864Mookherjee D (1984) Optimal incentive schemes with many agents. Rev Econ Studies 51(3):433–446Nalebuff BJ, Stiglitz JE (1983) Prizes and incentives: towards a general theory of compensation and com-

petition. Bell J Econ 14(1):21–43Rosen S (1986) Prizes and incentives in elimination tournaments. Am Econ Rev 76(4):701–715

123

Participation incentives in rank order tournaments with endogenous entry

Documents

Transcript of Participation incentives in rank order tournaments with endogenous entry