Contests with Multiple Alternative Prizes:Public-Good/Bad Prizes and Externalities

By Kyung Hwan Baik and Hanjoon Michael Jung*

Forthcoming in Journal of Mathematical Economics

Abstract We study contests in which there are multiple alternative public-good/bad prizes, and theplayers compete, by expending irreversible effort, over which prize to have awarded to them.Each prize may be a public good for some players and a public bad for the others, and theplayers expend their effort simultaneously and independently. We first prove the existence of apure-strategy Nash equilibrium of the game, then establish when the total effort level expendedfor each prize is unique across the Nash equilibria, and then summarize and highlight otherinteresting and important properties of the equilibria. Finally, we discuss the effects ofheterogeneity of valuations on the players' equilibrium effort levels and a possible extension ofthe model.

Keywords: Contest; Rent Seeking; Externalities; Public-good/bad prizes; Free riding; Existence of equilibrium; Uniqueness of the equilibrium effort levels

JEL classification: D72, H41, C72

Baik: Department of Economics, Sungkyunkwan University, Seoul 03063, South Korea (e-*

mail: khbaik@skku.edu); Jung (corresponding author): Ma Yinchu School of Economics,Tianjin University, Tianjin 300072, China (e-mail: hj.jung@tju.edu.cn). We are grateful toChris Baik, Subhasish M. Chowdhury, Amy Baik Lee, Dongryul Lee, Tim Perri, IrynaTopolyan, and two anonymous referees for their helpful comments and suggestions.

1

1. Introduction

Common are contests in which there are multiple alternative public-good/bad prizes, only

one of which will be awarded to a society of players; and the players compete, by expending

irreversible effort, over which prize to have awarded to them by a decision maker. Naturally, in1

such contests, each player has a valuation for each prize.

Examples of such contests are the ones in which there are multiple alternative industrial

policies, environmental policies, or trade policies to affect a group of firms, and the firms

compete over which policy to have adopted by the government. In these contests, some of the

firms may get benefits from the adopted policy, and others may be harmed by it. This means that

each of the policies can be viewed as a public-good/bad prize for the firms. Another example is

a contest in which there are multiple alternative economic policies to affect all the member

countries in the European Union, and the member countries compete over which economic

policy to have adopted by the Union. Yet another example is an election contest in which there

are several presidential candidates, and lobbyists or rent seekers compete, by making

contributions to the candidates' election campaign, over which candidate to have elected. No

doubt, the election result affects all the rent seekers.

Facing contests like the motivational examples above, we may well pose the following

interesting questions. For which prizes do the players expend positive effort? How many prizes

are there for which the players expend positive effort? Who expends positive effort? How many

players are there who expend positive effort? How severe is the free-rider problem? What

factors determine the effort levels expended by the players? Is there any player who expends

positive effort for more than one prize?

Accordingly, this paper models a contest involving multiple alternative public-good/bad

prizes as a strategic game, and addresses those interesting questions. It formally considers a

game in which each player's valuations for the prizes are publicly known, and the players choose

their effort levels for the prizes simultaneously and independently.

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This paper first proves the existence of a pure-strategy Nash equilibrium of the game.

Then, it identifies cases where the total effort level expended for each prize is unique across the

pure-strategy Nash equilibria.

In addition, this paper establishes the following interesting and important properties of

the Nash equilibria. First, there are at least two prizes for which the players expend positive

effort. Second, there are at least two players who expend positive effort. Third, if there are just

two prizes in total, then each player never expends positive effort for both prizes. However, if

there are more than two prizes, then some player may expend positive effort for more than one

prize. Fourth, each player expends zero effort for every prize that does not give him the highest

valuation; furthermore, he may expend zero effort for some or all of the prizes that give him the

highest valuation. Fifth, if there are just two prizes, then a player whose valuation spread that

is, the difference between his valuations for the two prizes is narrower than somebody else's

expends zero effort for both prizes and free rides; furthermore, a player whose valuation spread

is the widest may expend zero effort for both prizes. Sixth, a player with the highest valuation

for a prize (among all the players) may expend zero effort for that prize. Finally, a player with

negative valuations for all the prizes may expend positive effort for some prize or prizes.

This paper is closely related to the literature on contests with identity-dependent

externalities: See, for example, Linster (1993), Funk (1996), Jehiel et al. (1996), Esteban and

Ray (1999), Das Varma (2002), Aseff and Chade (2008), Brocas (2013), and Klose and

Kovenock (2015a, 2015b). These papers study contests in which each player's valuation for a

private-good prize depends on who is selected as the winner, and each player may have a

nonzero valuation for the prize even in case of his losing it. For example, Linster (1993)

considers -player rent-seeking contests in which each player's valuations for the single prize aren

represented by an -tuple vector, and each player's contest success function is specified by then

simplest logit-form function. Klose and Kovenock (2015b) consider -player all-pay auctions inn

which each player's valuation for the single prize may depend on the identity of the winner, so

3

that his valuations are given by an -tuple vector, and the winner is determined by the all-pay-n

auction selection rule.

The current paper differs from those papers in three ways. First, there are multiple prizes

in the current paper, only one of which is to be awarded to the players, whereas there is a single

prize in those papers. Second, each prize is a public-good/bad one in this paper, whereas the

single prize is a private-good one in those papers. Third, this paper uses a general selection

probability function (for each prize), which is different from the contest success functions (for

the players) used in those papers.

There exist many papers which study contests with a group-specific public-good

prize that is, contests in which groups of players compete to win a prize to be awarded to a

single group, and the prize is a public good only within the winning group. Examples include

Katz et al. (1990), Baik (1993, 2008), Baik and Shogren (1998), Baik et al. (2001), Epstein and

Mealem (2009), Lee (2012), Kolmar and Rommeswinkel (2013), Chowdhury et al. (2013),

Topolyan (2014), Barbieri et al. (2014), Chowdhury and Topolyan (2016a, 2016b), Barbieri and

Malueg (2016), Chowdhury et al. (2016), and Dasgupta and Neogi (2018). In these papers, the

number of groups and their sizes are exogenously given. These papers examine, among other

things, the free-rider problem and the group-size paradox.

The model in the current paper strikingly differs from the ones in those papers in three

respects. First, unlike in those papers, there are no groups (except the entire society of players)

in this paper that is, the society of players is not partitioned into groups. Second, there are

multiple alternative prizes in this paper, only one of which is to be awarded to all of the players,

whereas there is a single prize in those papers, which is to be awarded to a single group. Third,

in this paper, each prize is a public good/bad for the players precisely, it may be a public good

for some players and a public bad for the others whereas, in those papers, the single prize is a

public good only within the winning group.

The current paper is closely related to Baik (2016). He studies contests in which there

are two alternative public-good/bad prizes, and players compete over which prize to have

4

awarded to or selected for them by a decision maker. The current paper differs from Baik (2016)

in three respects. First, the current paper generalizes his model by not restricting the number of

alternative public-good/bad prizes to two. Second, the current paper formally proves the

existence of a pure-strategy Nash equilibrium in the case where there are multiple alternative

public-good/bad prizes, whereas Baik (2016) proves its existence by constructing pure-strategy

Nash equilibria in the case where there are only two public-good/bad prizes. Third, unlike Baik

(2016), the current paper identifies cases in which the total effort level expended for each prize is

unique across the pure-strategy Nash equilibria.

The rest of the paper is organized as follows. Section 2 presents a model and sets up a

simultaneous-move game. In Section 3, we prove the existence of a pure-strategy Nash

equilibrium of the game. In Section 4, we identify cases in which the total contribution or total

effort level made for each prize is unique across the pure-strategy Nash equilibria. Section 5

summarizes and highlights other interesting and important properties of the pure-strategy Nash

equilibria. Section 6 discusses the effects of heterogeneity of valuations on the players'

equilibrium effort levels and a possible extension of the model. Finally, Section 7 offers our

conclusions.

2. The model

Consider a contest in which there are alternative public-good/bad prizes, only one ofm

which will be awarded to or selected for a society of players, where 2 and 2; and then m n   

players compete, by expending effort, over which prize to have awarded to or selected for them

by a decision maker or a specified mechanism. Specifically, each prize may be a public good for

the players, it may be a public bad for the players, or it may be a public good for some players

and a public bad for the others. Whichever prize is selected, the players cannot recover their

effort expended.

Let represent the set of players, and let represent the set of prizes. Let representN M v ki

player 's valuation for prize , for each and . We assume that , where i k k M i N v R R− − −ki

5

denotes the set of all real numbers. We assume also that each player's valuations for the prizesm

are publicly known. For concise exposition, we exclude from consideration, by assuming away,

the trivial case in which every player "likes" one particular prize at least as much as every other

prize. That is, we assume that if there is prize , for , such that for all andk k M v v z M−   −ki zi

for some player , then there is some other prize , for , such that for somei N h h M v v− − hj kj

other .j N−

Let represent player 's effort level expended for prize , for each and .x i k k M i Nki − −

We assume that , where denotes the set of all nonnegative real numbers. That is,x R Rki + +−

each player is allowed to expend positive effort for any prize(s). Let represent an -tuplexi m

vector of player 's effort levels expended for the prizes, one for each prize: ( , ... ,i m xxi i´ 1

x R x i mmi im+) . Let represent the sum of effort levels that player expends for prizes 1 through ,−

so that . The cost function of player is given by ( ) for all , where x x i c c x x R ci zi i i i i + i

m

zœ

œ1œ −

represents the cost to player of expending his effort level for prizes 1 through . We assumei x mi

that the function has the properties specified in Assumption 1 below.ci

Assumption 1. We assume that c x c x for all x R where c and c ( ) 0 and ( ) 0, , w ww w wwi i i ii i i +   −

denote respectively the first and second derivatives of the function c, , .i

Let represent the sum of effort levels that players 1 through expend for prize , soX n kk

that , for each . Let ( , ... , ) . Let represent the probabilityX x k M X X R P k kj m k

n

j

m+œ ´

œ11− −X

that prize is selected, where 0 1 and 1. The probability of prize beingk P P P kŸ Ÿ œk z k

m

z

œ1

selected (or prize 's selection probability for short) depends on the players' effort levels for thek

m k P P prizes, and thus the selection probability function for prize is given by ( ). Wek kœ X

assume that the function has the properties specified in Assumption 2 below.Pk

6

Assumption 2. ( ) ( ) 0 and ( ) 0, 0.a P X P X when X` k k k zkz k

X XÎ` ` Î`2 2 Á

( ) ( ) 0 and ( ) 0 { }, 0.b P X P X for each z M k when X` k z k kzX XÎ` ` Î` − Ï2 2

( ) ( ) ( ) , { }.c P X P X for any h z M k` œ `k h k zX XÎ` Î` − Ï

( ) ( ) 1 , 0.d P m when Xk z

m

zX œ œÎ

œ1

( ) ( ) 0, 0 0.e P when X and Xk k z

m

zX œ œ

œ1

Under Assumption 2, prize 's selection probability is increasing in at a decreasingk Xk

rate, given effort levels expended for the other prizes. It is decreasing in the effort level

expended for each rival prize at a decreasing rate, . Part ( ) assumes that theceteris paribus c

marginal effect of increasing the effort level expended for each rival prize on prize 's selectionk

probability is the same across the rival prizes.2

Formally, we consider the following noncooperative simultaneous-move game. At the

beginning of the game, the players each know the valuations of all the players for the m

alternative prizes. Next, they expend their effort for the prizes simultaneously and

independently that is, player , for each , chooses his effort levels ( , ... , ) for the i i N x x− 1i mi

prizes, respectively, without knowing the other players' effort levels. Finally, one of the m

alternative prizes is selected.

Let ( ), for each , represent the expected payoff for player , given a profile of1i x xi N i−

the players' actions, where ( , ... , ). Then the payoff function for player is given byx x x´ 1 n i

v P c x1i zi z i i

m

z( ) ( ) ( ). (1)x Xœ

œ1

We assume that all of the above is common knowledge among the players. We employ

Nash equilibrium as the solution concept of the game.

The following features of the model are notable. First, the selection probability function

for prize is given by ( , ... , , ... , ), where represents the of effort levelsk P P X X X X sumk k k m kœ 1

7

that players 1 through expend for prize . This, together with Assumption 2, indicates thatn k

players may join forces by together expending positive effort for prize . Second, it indicatesk

also that players may compete against others to have their favorite prize awarded or selected.

Third, externalities between players may arise because the alternative prizes are public-good/bad

ones. To put this differently, players' positive effort for a prize, once the prize is selected, may3

also affect the payoffs of the players who expends zero effort (for that prize). Fourth, each

player is allowed to expend positive effort for any prize(s), and also is allowed to free ride on

others' effort. Finally, the players are not allowed to form coalitions.

The model may fit electoral competition in which each of several candidates chooses a

policy; each citizen has preferences over the policies (or the candidates), and independently

makes contributions to one (or some) of the candidates. In this electoral competition, the

citizens make strategic decisions on their contributions that is, the citizens are the

players and the policies (or the candidates) are the prizes.

Note that the payoff function for player in -player contests with identity-dependenti n

externalities is similar to function (1) (see, for example, Linster 1993, Klose and Kovenock

2015b). In such contests, player 's valuations for the single private-good prize are representedi

by an -tuple vector, ( , ... , ), where represents player 's valuation for the prize if player n v v v i j1i ni ji

wins the prize. In such contests, the probability of prize being selected, in function (1), isP kk

replaced with the probability that player wins the prize.i

Note also that, if 2, then the current contest is analytically equivalent to a contestm œ

with a group-specific public-good prize (see Baik 1993, 2008). This can be seen as follows. Let

N v v j N N N N1 1 2 1 2 1 denote the set of players such that for every , and let . (The sets,j j  − Ï´

N N v v v i N v v v1 2 1 2 1 1 2 1 2 and , are not empty.) Then, we can specify that if , and i i j i i j − œ œ

if , where , for 1, 2, represents the valuation for the prize of player in group (ori N v h j N− 2 hj hœ

group ) in the contest with a group-specific public-good prize. Using this and function (1), weh

find that, mathematically, each player in the current contest has the same objective function as

8

the corresponding player in the contest with a group-specific public-good prize (in which there

are two groups, and ).N N1 2

3. Existence of equilibrium

In this section, we establish the existence of a pure-strategy Nash equilibrium of the

game.4

Theorem 1. .There exists a pure-strategy Nash equilibrium

The proof of Theorem 1 is provided in Appendix A. To prove this theorem, we take

advantage of Theorem 3.1 in Reny (1999), which states that, given a game, if the players' action

sets are ( ) nonempty, ( ) compact, and ( ) convex, and if the players' payoff functions are ( )i ii iii iv

concave and ( ) continuous, then the game has at least one pure-strategy Nash equilibrium.v

The game under consideration satisfies only two specifically, ( ) and ( ) out of the i iii

five conditions in Theorem 3.1 in Reny (1999), so that we cannot directly apply his theorem to

prove Theorem 1 above. To resolve this (inapplicableness) problem, we organize the proof of

Theorem 1 in three steps. In Step 1, we construct a "truncated-actions game" of the original

game by placing restriction on the players' action sets. In Step 2, we show that there exists a

Nash equilibrium in such a truncated-actions game since the truncated-actions game satisfies all

the five conditions in Theorem 3.1 in Reny (1999). In Step 3, using Nash equilibria of truncated-

actions games, we show that there exists a Nash equilibrium in the original game.

As will be shown in the next section, there may exist more than one Nash equilibrium in

the game under consideration.

9

4. Uniqueness of a vector of the equilibrium effort levels for the prizes

Let ( , ... , ) (( , ... , )) denote a Nash equilibrium of the game. Letx x x* * *œ œ1 1 1n j mj* * n

jx x œ

X x k M X X m* * * *k kj m

n

jœ œ

œ11 for each , and let ( , ... , ). In this section, we examine when an -− X*

tuple vector of the equilibrium effort levels for the prizes or the total effort level expendedX*

for each prize is unique across the Nash equilibria.

We identify four cases in which the vector is unique across the Nash equilibria: twoX*

leading cases described in Theorems 2 and 3 in Section 4.2, and two additional ones described in

Theorems 4 and 5 in Section 4.3. As will be clear shortly, we impose more assumptions (or

restrictions ) in the additional cases than in the leading ones.

4.1. Preliminaries

Assumption 3 below will be assumed to hold in Theorems 2 through 5.

Assumption 3. , ( , ... , ) ( , ... , ), Consider two vectors X X and X X each having atX Xœ œ1 1m mw w w

least two positive elements If then for some prize k M we have. , X XÁ −w

` Î` Á ` Î`P X P Xk k k k( ) ( ) .X X w

Assumption 3 says that for any two different vectors of effort levels for the prizes, each

with at least two positive elements, there exists at least one prize , for , such that the first-k k M−

order partial derivative of ( ) with respect to takes different values at these vectors. NoteP Xk kX

that the function satisfying Assumption 2 may not satisfy Assumption 3, and vice versa.Pk

The following remark, whose proof is provided in Appendix B, identifies one form of the

selection probability functions (for the prizes) that satisfy Assumption 3.

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Remark 1. 3 ,The following selection probability functions satisfy Assumption : For each k M−

the selection probability function for prize k is given by P f X f X where ( ) ( ) ( ), k k k z z

m

zX œ Î

œ1

f f X and f X for all X Rk k k k +k k(0) 0, ( ) 0, ( ) 0 .œ w ww Ÿ −

The following is another assumption we will make in Theorems 2 through 5: The cost

function of player , for each , is linear: ( ) for all , where is a positivei i N c x x x R− œ −i i i i i + i) )

constant.

4.2. Two leading cases

Theorem 2 identifies the first leading case in which the vector is unique across theX*

Nash equilibria.

Theorem 2. 2, ( ) , 3 . ,Suppose that m c x x for each i N and Assumption holds Thenœ œ −i i i i)

the pair X X is unique across the Nash equilibria ( , ) .* *1 2

Theorem 2 is proved by Theorem in Baik (2016), and therefore omitted. Theorem in

Baik (2016) constructs and specifies all the pure-strategy Nash equilibria of a game which is

similar to the game considered in Theorem 2 above. It shows that there is at least one Nash

equilibrium of the game, and also that there may be multiple Nash equilibria, depending on the

players' valuations for the prizes. However, it shows that the total effort level expended for each

prize is the same across the Nash equilibria. Specifically, it establishes that the total effort level

for each prize is equal to that obtained in the unique equilibrium of the reduced game in which

only two players one with the widest positive valuation spread and one with the widest

negative valuation spread compete. 5

Theorem 2 assumes that there are two alternative public-good/bad prizes. However, in

Theorem 2, if we change the number of prizes from two to three or larger, ,m ceteris paribus

11

then the uniqueness of the vector across the Nash equilibria may not hold. The followingX*

remark illustrates this.

Remark 2. 3 2 1 0Suppose that m and n ; v v v and v v v ;œ œ œ œ œ œ œ œ11 21 32 31 12 22

c x x for i ; and P X X for k Then we have x xi i i k k zz

* *( ) 1, 2 1, 2, 3. , 1 4,œ œ Î œ œ œ Î3

111 21

œ

x x x and x ; consequently the triple X X X of the equilibrium* * * * * * *31 12 22 32 1 2 3œ œ œ œ Î0, 1 4 , ( , , )

effort levels for the prizes is not unique across the Nash equilibria.

Note that, according to Remark 1, the selection probability functions for the prizes

specified in Remark 2 satisfy Assumption 3. Remark 2 shows an example in which, when

m œ 3, there are different effort levels for some or all of the prizes across the Nash equilibria,

even if the other conditions of Theorem 2 are met: 1 4 and 1 4.X X X* * *1 2 3 œ Î œ Î

Definition 1. , , , ( For each i N we call prize h for h M his most preferred prize or his MP− −

prize for short if v v holds for all z M) .hi zi  −

The intuition behind the nonuniqueness result in Remark 2 is as follows. If a player

expends additional effort at an action profile, then he will be better off expending it for one of

his MP prizes rather than for one of his non-MP prizes, which means that, in equilibrium, no

player expends positive effort for his non-MP prizes (see Lemmas A1 and A5 in Appendix A).

In addition, if a player's valuation for a prize, say prize 1, is the same as that for another prize,

say prize 2, then his marginal payoff of additional effort for prize 1, given an action profile, is

the same as that of additional effort for prize 2 (see Lemma A2 in Appendix A). It follows

immediately from this that his expected payoff remains unchanged when changing the allocation

of his (fixed) effort level between prizes 1 and 2 (see Lemma A3 in Appendix A). In Remark 2,

player 1 has two MP prizes, prizes 1 and 2, out of three alternative ones. Thus, according to

12

Lemmas A3 and A5, it is possible to obtain multiple Nash equilibria across which player 1

allocates differently his fixed effort level between prizes 1 and 2. Indeed, we obtain multiple

Nash equilibria across which 1 4 and 1 4.x x X X* * * *11 21 1 2 œ Î œ Î

Next, Theorem 3 below identifies the second leading case in which the vector isX*

unique across the Nash equilibria. It adds an additional assumption to those in Theorem 2 in

order to remove the nonuniqueness problem illustrated in Remark 2.

Theorem 3. 3, , ( ) Suppose that m each player has just one MP prize c x x for eachœ œi i i i)

i N and Assumption holds Then the triple X X X of the equilibrium effort levels for− , 3 . , ( , , ) * * *1 2 3

the prizes is unique across the Nash equilibria.

The proof of Theorem 3 is provided in Appendix C. In Theorem 3, we further assume, as

compared to Theorem 2, that each player has just one MP prize, and hereby obtain the

uniqueness of the vector across the Nash equilibria in the case where 3.X* m œ

Note that, in Theorem 2, we do not need this additional assumption because, if a player

has two MP prizes in the case where 2, then he expends zero effort for both prizes.m œ

4.3. Two additional cases

Theorem 4 below identifies the first additional case in which the vector is uniqueX*

across the Nash equilibria. It shows that, under the assumptions of Theorem 3, the uniqueness of

the vector holds even when 4, if we limit the number of players to at most three.X* m  

Theorem 4. 4, 2 3, , ( ) Suppose that m n or each player has just one MP prize c x x for  œ i i i iœ )

each i N and Assumption holds Then the vector of the equilibrium effort levels for the , 3 . , − X*

prizes is unique across the Nash equilibria.

13

The proof of Theorem 4 is straightforward. Consider the case where 2. This case isn œ

analytically analogous to the case where 2 and 2, because each player has just one MPm nœ œ

prize and, due to Lemmas A5 and A6, the players expend positive effort only for these two

prizes in equilibrium. Hence, the case is immediately proved by Theorem 2. Next, consider the

case where 3. Since each player has just one MP prize, the proof of this case is qualitativelyn œ

the same as that of Theorem 3 for the case where 3 and 3.m nœ œ

However, in Theorem 4, the uniqueness of the vector across the Nash equilibria mayX*

not hold if we increase the number of players to four or larger, . Remark 3 belowceteris paribus

illustrates this.

Remark 3. 4 1, Suppose that m n ; v v v v v v v vœ œ œ œ œ œ œ œ œ11 22 33 44 31 41 32 42

œ œ œ œ œ Î œ œ œ œv v v v for and v v v v ; c x x13 23 14 24 21 12 43 341 0 1 2, 0 ( )% % i i iœ

for i ; and P X X for k Then we have two Nash equilibria: one 1, 2, 3, 4 1, 2, 3, 4. , œ œk k zz

œ Î4

1œ

in which x x and x for k M and i N except k i and the other in* * *ki11 22œ œ Î œ œ œ1 4 0 1, 2, − −

which x x and x for k M and i N except k i Consequently the 1 4 0 3, 4. , * * *ki33 44œ œ Î œ œ œ− −

vector X X X X of the equilibrium effort levels for the prizes is not unique across the ( , , , ) * * * *1 2 3 4

Nash equilibria.

Remark 3 shows an example in which, when 4, there are different effort levels forn œ

the prizes across two Nash equilibria, even if the other conditions of Theorem 4 are met:

X X X X X X* * * * * *1 2 3 4 1 2œ œ Î œ œ œ œ1 4 and 0 in one Nash equilibrium, and 0 and

X X* *3 4œ œ Î1 4 in the other Nash equilibrium.

Remark 3, together with Theorem 5 below, implies that, in the case where 4 andm  

n   4, the nonuniqueness of the vector may arise if each player does not have the sameX*

valuation for all his non-MP prizes. Indeed, in Remark 3, each player's valuations for his non-

14

MP prizes differ. For example, player 1's valuation for prize 2 is different from that for prize 3

or 4.

Next, Theorem 5 below identifies the second additional case in which the vector isX*

unique across the Nash equilibria. It adds an additional assumption that each player has the

same valuation for all his non-MP prizes to those in Theorem 4 (or Theorem 3) in order to

remove the nonuniqueness problem illustrated in Remark 3.

Theorem 5. 4, , Suppose that m each player has just one MP prize each player is indifferent 

among all his non-MP prizes c x x for each i N and Assumption holds Then the, ( ) , 3 . ,i i i iœ −)

vector of the equilibrium effort levels for the prizes is unique across the Nash equilibria .X*

The proof of Theorem 5 is provided in Appendix D. Note that as we increase the number

of prizes, we need additional assumptions in order to establish the uniqueness of the vector X*

across the Nash equilibria. For example, in Theorem 4, we further assume, as compared to

Theorem 3, that the number of players is limited to at most 3. In Theorem 5, we further assume,

as compared to Theorem 3, that each player has the same valuation for all the prizes except his

unique MP prize.

The uniqueness result in Theorem 5 can be explained loosely as follows. Consider player

i h v who has only prize as his MP prize, and has the same valuation, , for all his non-MP prizes.i

In this case, in equilibrium, he expends zero effort for all his non-MP prizes, and may expend

positive effort for prize (see Lemmas A5 and A6). As shown in Appendix D, if he expendsh

positive effort for prize in equilibrium, then we must haveh

( ) ( ).` œ ÎP x v vh hi i hi iX* Î` )

Then, since , we must haveX xh hj

n

jœ

œ1

( ) ( ).` œ ÎP X v vh h i hi iX* Î` )

15

All this implies that, for every prize for which some player expends positive effort ink j

equilibrium, we must have

( ) ( ). (2)` œ ÎP X v vk k j kj jX* Î` )

Because 0 for prize for which every player expends zero effort in equilibrium, theX z*z œ

number of the unknowns, 's, is equal to the number of the equations from (2). Next, as shownX *k

in Appendix D, if a player is active [resp. not active] in one Nash equilibrium, then he is also

active [resp. not active] in another Nash equilibrium, if any. This indicates that there is the same

system of simultaneous equations, each from equation (2), across the Nash equilibria. Therefore,

as shown in Appendix D, the vector which satisfies the equality conditions from (2) is uniqueX*

across the Nash equilibria.

The uniqueness of the vector across the Nash equilibria implies that the effort levelX*

expended for each prize is the same across the Nash equilibria. This in turn implies that the

probability of each prize being selected (or each prize's selection probability) is the same across

the Nash equilibria.

Szidarovszky and Okuguchi (1997) and Cornes and Hartley (2005) establish the

uniqueness of Nash equilibrium in contests in which individual players compete to win a private-

good prize. Theorem 5 may cover their uniqueness results as special cases. This can be seen as

follows. Suppose that , and that player , for , has just one MP prize, say prize form n i i N iœ −

i M i N− −, which is different from those of the other players. Then, we can specify that, for

and for , if , and 0 otherwise, where represents player 's (positive)k M v v k i v v i− ki i ki iœ œ œ

valuation for the prize in their papers. Using this and function (1), we find that, mathematically,

each player in this case has the same objective function as the corresponding player in their

papers.

Finally, Baik (1993, 2008) establishes the uniqueness of Nash equilibrium in contests in

which groups of individuals compete to win a group-specific public-good prize. Theorem 5 may

also cover his uniqueness result as a special case. This can be seen as follows. Suppose that

16

player , for , has just one MP prize, say prize for . Let denote the set of playersi i N h h M N− − h

whose MP prize is prize . Suppose that the number of such sets and their sizes are the same ash

the number of groups and their sizes in his papers, respectively. Then, for , we can specifyh M−

that, for and for , if and , and 0 otherwise, where i N k M v v k h i N v v− − −ki hj h ki hjœ œ œ

represents the (positive) valuation for the prize of player in group (or group ) in his papers.j N hh

Using this and function (1), we find that, mathematically, each player in this case has the same

objective function as the corresponding player in his papers.

5. Properties of the Nash equilibria

In this section, we summarize and highlight other interesting and important properties of

the Nash equilibria of the game, which are obtained in the course of the analysis in Sections 3

and 4.

5.1. Prizes with positive effort and those with zero effort in equilibrium

For which prizes do players expend positive effort in equilibrium? For which prizes do

players expend zero effort in equilibrium? The following properties give answers to these

questions.

First, it follows from Lemmas A4 and A6 in Appendix A that there are at least two prizes

for which the players expend positive effort. This can be explained as follows. If there were to

be no prize for which the players expend positive effort, then some player would have an

incentive to expend a positive effort level of for one of his MP prizes and increase his expected%

payoff. This implies that there is no Nash equilibrium in which no prize has positive effort.

Next, if there were to be just one prize for which the players expend positive effort, then those

players who expend positive effort would have an incentive to decrease their effort for that prize

and increase their expected payoffs. This implies that there is no Nash equilibrium in which just

one prize has positive effort.

17

Second, each of the prizes for which the players expend positive effort must be some

player's MP prize. This can be explained by Lemma A1 in Appendix A. Lemma A1 says that, at

an action profile, if a player expends additional effort, then he will be better off expending it for

a prize for which he has a higher valuation (rather than for a prize for which he has a lower

valuation). It implies that, in equilibrium, no player expends positive effort for his non-MP

prizes (see Lemmas A5 and A6 in Appendix A).

Third, if 2, then each player never expends positive effort for both prizes. This canm œ

be explained as follows. If a player has the same valuation for both prizes, then he expends zero

effort for them because which prize to be selected for the society does not matter to him. If a

player has different valuations for the prizes, then he never expends positive effort for his non-

MP prize (see Lemmas A5 and A6 in Appendix A). However, Lemmas A5 and A6 imply that

some player may expend positive effort for more than one prize, if 3 and the player hasm  

multiple MP prizes. In fact, this is shown and explained in Remark 2.

Finally, it is possible that every player expends zero effort for some player's MP prize. It

is also possible that more than one player expends positive effort for a prize.

5.2. Active players and free riders in equilibrium

Who does expend positive effort in equilibrium? Who does expend zero effort and free

ride in equilibrium? The following properties give answers to these questions.

First, it follows from Lemmas A4 and A6 in Appendix A that there are at least two

players who expend positive effort. This can be explained as follows. If there were to be no

player who expends positive effort, then some player would have an incentive to expend a

positive effort level of for one of his MP prizes and increase his expected payoff. This implies%

that there is no Nash equilibrium in which no player expends positive effort for any prize. Next,

if there were to be just one player who expends positive effort, then he would have an incentive

to decrease his effort and increase his expected payoff. This implies that there is no Nash

equilibrium in which just one player expends positive effort.

18

Second, if a player has the same valuation for all the prizes, then he expends zero effort

for the prizes, regardless of the valuation, and free rides. This is simply because which prize to

be selected for the society does not matter to him.

Third, a player may expend positive effort for a prize only if that prize is one of his MP

prizes. Put differently, each player expends zero effort for his non-MP prizes; furthermore, he

may expend zero effort for some or all of his MP prizes (see Lemmas A5 and A6 in Appendix

A). This can be explained by the fact that, at an action profile, if a player expends additional

effort, then he will be better off expending it for a prize for which he has a higher valuation (see

Lemma A1 in Appendix A).

Fourth, suppose that there are two alternative public-good/bad prizes, and that the other

conditions in Theorem 2 are satisfied. Consider player , for , who has only prize , fori i N h−

h M− , as his MP prize. Then, similarly to the derivation of expression (D2) in Appendix D, we

can write one of his first-order conditions as follows:

( ) ( ), (3)` Ÿ ÎP x v vh hi i hi kiX* Î` )

for { }.k M h− Ï

Since , the left-hand side of expression (3) has the same value, ( ) ,X x P Xh hj h h

n

jœ `

œ1X* Î`

for those players who have only prize as their MP prize. It follows then that only the playersh

with the lowest value for the right-hand side of expression (3) (among those players who have

only prize as their MP prize) may expend positive effort for prize in equilibrium. To put thish h

differently, player whose marginal gross payoff, ( ) ( ) , equals his marginalj v v P xhj kj h hj Î`` X*

cost, in the Nash equilibrium may expend positive effort for prize . On the other hand, a)j, h

player whose marginal gross payoff is less than his marginal cost in the Nash equilibrium that

is, one who expects the total effort level for prize in the Nash equilibrium to be large enoughh

from his perspective expends zero effort for both prizes and free rides on others' effort.

19

This result is also established in Baik (2016). He shows that, given 1 for all ,)i œ i N−

player whose valuation spread, defined as ( ), is the widest may expend positive effortj v vhj kj

for prize ; player whose valuation spread is narrower than somebody else's expends zero efforth t

for both prizes and free rides on others' effort. Of course, player with theif there is just one

widest valuation spread, only the player expends positive effort for prize .then h

This result indicates that externalities between players may arise. Such externalities (if

any) arise because the alternative prizes are public-good/bad ones, and the selection probability

function for prize is given by ( , ... , , ... , ), where represents the ofk P P X X X X sumk k k m kœ 1

effort levels that players 1 through expend for prize .n k

Fifth, suppose that there are more than two alternative public-good/bad prizes, and that

the other conditions in Theorem 5 are satisfied. Consider player who has only prize as hisi h

MP prize, and has the same valuation, , for all his non-MP prizes. Then, we have expressionvi

(D2) in Appendix D as one of his first-order conditions:

( ) ( ). (4)` Ÿ ÎP x v vh hi i hi iX* Î` )

Using this expression, we obtain the following property of the Nash equilibria, similar to

the one in the preceding paragraphs. Among those players who have only prize as their MPh

prize, only the players with the lowest value for the right-hand side of expression (4) or,

equivalently, player whose marginal gross payoff, ( ) ( ) , equals his marginalj v v P xhj j h hj Î`` X*

cost, in the Nash equilibrium may expend positive effort for prize , while the rest expend)j, h

zero effort for every prize in and free ride on others' effort. M

Note that the valuation spreads, ( ) for each , and the marginal costs, for each ,v v i ihi i i )

matter to identify the players who expend positive effort for prize . Note also that theh

equilibrium total effort level for prize is independent of the number of players who have onlyh

prize as their MP prize, these players' valuations for each prize, the sum of their valuations forh

prize , and their valuation spreads, ( ) for each , unless changes in these come with ah v v ihi i

change in the lowest of the values for the right-hand side of expression (4).6

20

Sixth, a player with the highest valuation for prize (among all the players), for ,k k M−

may expend zero effort for that prize; furthermore, even a player with the highest valuations

across all the prizes expend zero effort for every prize and free ride., if any, may

Finally, a player with negative valuations for all the prizes in this case, every prize is a

(public) bad for him may expend positive effort for some prize or prizes in order to have his

best prize public-bad awarded to or selected for the society of the players.

6. Discussion

6.1. Heterogeneity of valuations

We have seen that the players' valuations for the prizes affect important properties or

aspects of the Nash equilibria of the game, such as the number of the equilibria, the uniqueness

of the vector across the equilibria, the effort level expended for each prize, and the effortX*

level expended by each player.

In particular, we have seen that, in equilibrium, each player expends zero effort for his

non-MP prizes, but may expend positive effort for some or all of his MP prizes (see Lemmas A5

and A6 in Appendix A). Accordingly, we conclude easily that, as far as each player's MP prizes

remain the same, his equilibrium effort levels for his non-MP prizes are independent of the

degree of heterogeneity of valuations, across the players or across the prizes. Indeed, they

remain unchanged at zero. However, each player's equilibrium effort levels for his MP prizes

may not be independent of the degree of heterogeneity of valuations, even in the case where

every player's MP prizes remain the same. This is because each player's equilibrium effort levels

for his MP prizes, as explained in Section 5.2, depend on the relevant players' valuation spreads

(see expressions (3) and (4)).

Remark 4 below illustrates of heterogeneity of valuations (both the effects across the

players and across the prizes) on the players' equilibrium effort levels.

21

Remark 4. 2 , , , , 0Suppose that m n ; v v v and v for ;œ œ œ œ œ œ 11 21 12 22α α α α α

c x x for i ; and P X X X for k Then there exists a unique Nashi i i k k( ) 1, 2 ( ) 1, 2. , œ œ Îœ œ1 2

equilibrium in which x x and x x Consequently as the * * * *11 22 21 12œ œ Î œ œα 2 0. , heterogeneity

parameter increases each player's equilibrium effort level for his non-MP prize remainsα ,

unchanged at zero but his equilibrium effort level for his MP prize , increases in proportion to

the parameter .α

In Remark 4, each player's valuation spread between his MP prize and non-MP prize is

2 . Hence, as the parameter increases, each player's valuation spread increases,α αheterogeneity

which in turn gives each player an incentive to expend more effort for his MP prize.

Consequently, as the degree of heterogeneity of valuations increases, each player's equilibrium

effort level for his MP prize increases.

6.2. A possible extension

The results of the model are obtained at a considerable level of generality, for example

without specific restrictions on the form of the selection probability functions for the prizes.

Nevertheless, there exist only "sincere-voting" equilibria in which the players never expend

positive effort for their non-MP prizes (see Lemmas A5 and A6 in Appendix A). In this

subsection, we consider an extended (or modified) model which yields a "strategic-voting"

equilibrium in which one of the players expends positive effort only for his non-MP prize.

The sincere-voting behavior in an equilibrium of the model comes from Assumptions 1

and 2. This can be explained as follows. Assumption 1 ensures that player i's marginal cost of

expending additional effort is the same across the prizes. Assumption 2 ensures that player i's

marginal revenue from expending additional effort is greater when he expends it for a higher-

valuation prize than when he does for a lower-valuation prize. These two facts then imply that

player i's marginal payoff, which is the difference between his marginal revenue and his

22

marginal cost, from expending additional effort is greater when he expends it for his MP prize

than when he does for his non-MP prize, which leads to the sincere-voting behavior.

Accordingly, we may in principle obtain a strategic-voting equilibrium by extending (or

modifying) either assumption or both. Remark 5 below obtains a strategic-voting equilibrium by

extending Assumption 1.

Remark 5. 3 2 2, 1, 0, 0, Suppose that m and n ; v v v v v andœ œ œ œ œ œ œ11 21 31 12 22

v ; and P X X X X for k Suppose that player 's cost function is32 1 2 3œ œ4 ( ) 1, 2, 3. 1k kœ Î

C x x x for and player 's cost function is C x there1 1 11 21 31 2 2 27( ) 6 2 ( ) . x xœ Î œ" "   Then,

exists a unique Nash equilibrium in which x x and* *21 32

2 2 2œ œ 4 (4 ) , 16 (4 ) , " " " "Î Î

x x x x Consequently player expends positive effort only for prize* * * *11 31 12 22œ œ œ œ 0. , 1 2,

which is not his MP prize.

In Remark 5, we have that ` œ ` œ `C x C x C x1 11 1 21 1 11Î` Î` Î Î`1 and 1 , and hence that "

Á Î``C x1 21. This means that player 1's cost function specified in Remark 5 does not satisfy

Assumption 1. It is easy to see that the selection probability functions satisfy Assumption 2.

Remark 5 shows that player 1 expends positive effort for one of his non-MP prizes, prize

2. The logic behind this result is as follows. Player 1's cost function does not satisfy

Assumption 1, which implies that his marginal cost of expending additional effort may not be the

same across the prizes. Indeed, it becomes smallest when he expends additional effort for prize

2. Thus, when , his marginal payoff from expending additional effort for prize 2 is greater"   6

than that for prize 1 or that for prize 3. As a consequence, player 1 expends positive effort only

for prize 2, in equilibrium, which is not his MP prize.

7. Conclusions

We have studied contests in which there are multiple alternative public-good/bad prizes,

only one of which will be awarded to a society of players; and the players compete, by

23

expending irreversible effort, over which prize to have awarded to or selected for them by a

decision maker. Specifically, each prize may be a public good for some players and a public bad

for the others.

Formally, we have considered the following game. At the beginning of the game, the n

players each know the valuations of all the players for the alternative prizes. Next, theym

choose their effort levels for the prizes simultaneously and independently. Finally, one of the m

alternative prizes is selected.

We have first proved the existence of a Nash equilibrium of the game. Then, we have

identified four cases two leading cases and two additional ones in which the vector of X *

the equilibrium effort levels for the prizes is unique across the Nash equilibria. Next, we have

summarized and highlighted other interesting and important properties of the Nash equilibria.

Finally, we have discussed the effects of heterogeneity of valuations on the players' equilibrium

effort levels and a possible extension of the model.

This paper has assumed that every player's valuations for the prizes are publicly known.

A possible extension of this paper would be to consider a model in which some or all of the

players' valuations for the prizes are imperfectly known to the players. In this paper, we have

assumed that only one of the multiple alternative public-good/bad prizes is awarded to the

players. It would be interesting to consider a model in which more than one of them is awarded

to the players. In this paper, we have assumed that the players do not sabotage each other. It

would be interesting to consider a model in which players are allowed to sabotage each other.

We leave these extensions and/or modifications for future research.

24

Footnotes

1. In general, we define a contest as a situation in which players or groups of players

compete by expending irreversible effort to win a prize. Examples are rent-seeking contests,

election campaigns, environmental conflicts, litigation, patent contests, all-pay auctions, sporting

contests, etc. In the literature on the theory of contests, seminal papers include Tullock (1980),

Rosen (1986), Dixit (1987), and Hillman and Riley (1989); other important works include Nitzan

(1991), Baik and Shogren (1992), Baye et al. (1993, 1996), Clark and Riis (1998), Moldovanu

and Sela (2001), Epstein and Nitzan (2007), Congleton et al. (2008), Siegel (2009), Konrad

(2009), and Vojnovic (2015).w

2. Contest success functions that satisfy all the parts of Assumption 2 are extensively used

in the literature on the theory of contests. See, for example, Tullock (1980), Hillman and Riley

(1989), Katz et al. (1990), Nitzan (1991), Epstein and Nitzan (2007), Epstein and Mealem

(2009), Konrad (2009), Kolmar and Rommeswinkel (2013), Dasgupta and Neogi (2018), and

Baik and Jung (2019).

3. For discussions of different types of externalities arising in contests, see Konrad (2009).

4. Throughout the paper, we restrict our attention to only pure-strategy Nash equilibria of

the game. Thus, henceforth, we may call them simply Nash equilibria of the game.

5. Baik (2016) defines player 's between prizes 1 and 2 as ( ), fori valuation spread v v1 2i i

each .i N−

6. These values for the right-hand side of expression (4) are computed for the players who

have only prize as their MP prize.h

7. Note that each player's cost function is defined on his set of actions, which is notR3+, and

on the set of the sum of his effort levels for the three prizes.

25

Appendix A: Proof of Theorem 1

In this appendix, we first state and prove several lemmas, which serve as preliminaries to

Theorem 1, and then prove Theorem 1.

Lemma A1. Consider an action profile at which X and X hold for, ( , ... , ), 0 0 x x xœ 1 n g h

g h M with g h Suppose that for i N we have v v for k t M Then there exists a, . , , , . − Á − −ki ti

positive real number such that x x ( ) ( ) .$ `   `1 1 $i ki i tix xÎ` Î`

Proof. Given the action profile , player 's marginal payoff from increasing isx i x ki

x v P x c x x` ` `1i ki zi z ki i i ki

m

z( ) ( ( ) ) ( ) , (A1)x XÎ` œ Î` Î`

œ1

and that from increasing isxti

x v P x c x x` ` `1i ti zi z ti i i ti

m

z( ) ( ( ) ) ( ) . (A2)x XÎ` œ Î` Î`

œ1

Let ( ) ( ) . Then, to prove Lemma A1, we should show thatK x x´ ` `1 1i ki i tix xÎ` Î`

K 0.

Using expressions (A1) and (A2), we obtain

( ( ) ( ) ) ( ( ) ( ) ). (A3)K v P x P x v P x P xœ Î` Î` Î` Î`ki k ki k ti ti t ti t ki` ` ` `X X X X

Note that the following facts lead to expression (A3). First, given , the marginal cost ofxi

increasing is equal to that of increasing : ( ) ( ) . Second, ( )x x c x x c x x P xki ti i i ki i i ti z ki` ` `Î` œ Î` Î`X

œ `P x z M k t cz ti( ) holds for any { , }, due to part ( ) of Assumption 2.X Î` − Ï

Next, due to the fact that ( ) 1, we obtain ( ) ( ) . This m m m

z z zz z ki z ti

œ œ œ1 1 1P P x P xX X Xœ ` œ `Î` Î`

equation then reduces to ( ) ( ) ( ) ( ) because` ` œ ` `P x P x P x P xk ki k ti t ti t kiX X X XÎ` Î` Î` Î`

26

` œ `P x P x z M k tz ki z ti( ) ( ) holds for any { , }. Now we can rewrite expression (A3)X XÎ` Î` − Ï

as

( ) ( ( ) ( ) ). (A4)K v v P x P xœ Î` Î`ki ti k ki k ti` `X X

Finally, the following facts related to the right-hand side of expression (A4) lead us to

conclude that 0. First, by assumption, we have . Second, given the assumption thatK v v ki ti

z k

z k kiÁ

X P x a ` 0, we have ( ) 0 due to part ( ) of Assumption 2. Third, we have:X Î`

` ` œ œP x X b P x Xk ti k k ti k( ) 0 if 0 (due to part ( ) of Assumption 2), and ( ) 0 if 0X XÎ` Î`

(due to part ( ) of Assumption 2).e

Lemma A2. Suppose that for i N we have v v for k t M with k t Then given an, , , . , − − Áki tiœ

action profile we have x x, ( , ... , ), ( ) ( ) .x x x x xœ ` œ `1 n i ki i ti1 1Î` Î`

Proof. The proof of Lemma A2 is similar to that of Lemma A1, and therefore omitted. Lemma

A2 comes immediately from the following fact about the two components of function (1). Given

the action profile , we have: ( ( ) ) ( ( ) ) and ( )x X X m m

z zzi z ki zi z ti i i ki

œ œ1 1v P x v P x c x x` œ ` `Î` Î` Î` œ

`c x xi i ti( ) .Î`

In Lemma A3 below, we let ( , ... , ) and ( , ... , ) be two action profilesx x x x x xœ œ1 1n nw w w

which are different only in that, for , we have and with i N x x x x x x x x− Á Á œ ki ti ki tiki kiti tiw w w w

for , with . Then Lemma A3 is immediate from Lemma A2. k t M k t− Á

Lemma A3. Suppose that for i N we have v v for k t M with k t Then player i's, , , . − − Áki tiœ

expected payoff at the action profile is the same as that at the action profile : ( ) ( ).x x x xw w1 1i iœ

27

In the game under consideration, given effort levels of all the other players, player , fori

each , faces the following maximization problem:i N−

x x Maximize ( ), ... , 1i mii1 x

such that 0 for each .x k Mki   −

Let ( , ... , ) denote an action profile at which the first-order conditions for maximizingx x xb b b´ 1 n

1j i mib b

i( ), for any , are satisfied, where ( , ... , ) for each . Then, for each x xj N x x i N i N− − −b ´ 1

and , we have:k M−

( ) 0 for 0 (A5)` 1i kibkixb Î` œx x

or

( ) 0 for 0. (A6)` Ÿ1i kibkixb Î` œx x

Let denote the set of action profiles at which the first-order conditions for maximizingZ

1j( ), for any , are satisfied. Lemma A4 identifies action profiles which are excluded fromx j N−

the set .Z

Lemma A4. x( ) a Let be the action profile at which each player expends zero effort for every0

prize Then we have that Z b Let be an action profile at which X. . ( ) ( , ... , ) 0x x x x0  œ w w w1 n k

w

holds for some prize k M and X holds for any t M with t k Then we have that− − Áwt œ 0 .

x x x xw  œ Z c Let be an action profile at which x holds for some player i N. ( ) ( , ... , ) 0 1 n i −

and x holds for any j N with j i Then we have that Zj œ Â0 . .− Á x

Proof. ( ) Consider player , for , whose valuation for prize , for some , is at leasta i i N k k M− −

as high as that for every other prize, and is greater than that for some other prize , for : Inh h M−

terms of symbols, for all and .v v z M v vki zi ki hi  −

28

Given zero effort levels of the other players, if player expends zero effort for everyi

prize, then his expected payoff is

v P c1i zi z i

m

z( ) (0, ... , 0) (0).x0 œ

œ1

On the other hand, if he expends a positive effort level of only for prize , then his expected% k

payoff is

v c1i ki i( ) ( ).x+ œ %

Clearly, under Assumption 1 and parts ( ) and ( ) of Assumption 2, for any sufficiently small ,a e %

there exists a positive real number such that ( ) ( ) , so that we have$ 1 1 $i ix x+   0

( ( ) ( )) 0. This implies that the first-order condition, ( ) 0,1 1 1i i i kix x x+ Î   Î ` Ÿ0 0% $ % Î`x

does not hold (see expression (A6)).

( ) At the action profile , we have a player, say player , who expends positive effort forb ixw

prize . His effort level for prize is denoted by . Thus his expected payoff at isk k xwki x w

v c x1i ki i ki( ) ( ).xw œ w

Now consider an action profile, denoted by , which is the same as the action profile with thex x ww w

exception that player 's effort level for prize is now . Assume that 0. Then,i k x x xww w wwki ki ki

under Assumption 1 and parts ( ) and ( ) of Assumption 2, we havea e

v c x1 1i ki i iki( ) ( ) ( ),x xww wœ ww

so that we have

( ) 0.lim limx x x xx x x x

c x c xi kiww w ww w

ww w ww w

ww w

ki ki ki ki

i i

ki ki ki ki

i iki ki

Ä Ä

( ) ( )

( ( ) ( ))1 1x xww w

œ œ c xw w

This implies that the first-order condition, ( ) 0, does not hold (see expression (A5)).`1i kixw Î` œx

29

( ) First, we know from part ( ) that if 0 holds for some prize and 0c b x k M xki ti œ−

holds for any with , then .t M t k Z− Á x Â

Next, consider the case where 0 and 0 hold, and , for , withx x v v g h Mgi hi gi hi   −

g h t t M x v vÁ −. In this case, there exists a prize, say prize for , such that 0 and forti ti ki Ÿ

any with 0. We have thenk M x− ki

x v P x c x x` ` `1i ti zi z ti i i ti

m

z( ) ( ( ) ) ( )x XÎ` œ Î` Î`

œ1

( ( ) ) ( ) (A7)Ÿ ` `mz

ti z ti i i tiœ1

v P x c x xX Î` Î`

Note that the following two facts lead to inequality (A7). First, if 0 for , thenx k Mki −

v v P x b x s Mki ti k ti si  ` œ and ( ) 0 due to part ( ) of Assumption 2. Second, if 0 for , thenX Î` −

` œ ` œP x e P xs ti z ti

m

z( ) 0 due to part ( ) of Assumption 2. Next, since ( ) 0 holds,X XÎ` Î`

œ1

inequality (A7) is reduced to

( ) ( ) .` Ÿ `1i ti i i tix Î` Î`x c x x

Due to Assumption 1, the right-hand side of this inequality is negative, which implies that

` Â1i ti( ) 0. This leads to the fact that (see expression (A5)).x xÎ`x Z

Lemma A5 characterizes action profiles in the set .Z

Lemma A5. x x xConsider an action profile in the set Z Suppose that for i N, ( , ... , ), . , b b bœ −1 n

and k M we have that v v for all z M and v v for t M with k t Then, we have , . −   − − Áki zi ki ti

that x and xb bki ti  œ0 0.

Proof. Lemma A1, together with parts ( ) and ( ) of Lemma A4, yields thata b

` `1 1i ki i ti( ) ( ) . This, together with expressions (A5) and (A6), implies thatx xb bÎ` Î`x x

`   œ1i tib bki ti( ) 0. It then follows from expressions (A5) and (A6) that 0 and 0.xb Î`x x x

30

Lemma A6. An action profile is a Nash equilibrium if and only if it is in the set Z.

Proof. First the "forwards" proof is simple. Any action profile which is a Nash equilibrium

satisfies the first-order conditions for maximizing ( ), for any , and thus it belongs to the1j x j N−

set .Z

Conversely, we need to prove that any constitutes a Nash equilibrium. Let ( ,x xb − Z wi

xbi) denote the action profile at which every player except player , for , , chooses hisj i i j N−

action as specified by , whereas player chooses the action . To achieve this "backwards"x x xb bj ii w

proof, we have to show that, for every player , ( ) ( , ) for every action of playeri 1 1i i i i ix x x xb b  w w

i.

Fix player , for . For expositional simplicity, assume without loss of generality thati i N−

his valuation for prize 1 is at least as high as that for every other prize: for all .v v z M1i zi  −

First, using Lemmas A4 and A5, we obtain that 0 holds for some player withx j Nbj −

j i x x z M v vÁ − , and that 0 and 0 for any with , which are used below in thisb bi zi i zi1 1  œ

proof.

Next, we show below that, given player s action , there exists player 's action, i' ix xwi i

H

´   œ( , ... , ), such that ( ) ( , ) ( , ) and ( ) 0 for any withx x i ii x z MH H1i mi i i i i zii i

H 1 1x x x xH b b

w −

v v1i zi .

( ) Suppose that for some . For expositional simplicity, we assumea v v k M1i ki −

without loss of generality that 2. We show that, given the action , there exists an action k œ x xw wwi i

of player such that ( ) ( , ) ( , ), ( ) ( , ) 0 with 0, ori i ii x x1 1 1i i i ii i i i i i ix x x x x xww w wwb b b

ww  ` Î` œ1 1

` Ÿ1i ii i i i zi zi( , ) 0 with 0, ( ) 0, and ( ) for any {1, 2}.x xww b

ww ww ww wÎ` œ œ œ − Ïx x iii x iv x x z M1 1 2

Let ( , , ... , ) be an action of player such that ( , ) 0 withx x x1 1i i mi i ii i i´ `x x x i x1

1 2 1w

w 1 b Î` œ

x x x x i1 1 11 1 11i i ii ii i ` Ÿ0, or ( , ) 0 with 0. Then maximizes player 's expected payoff1 x x1 b

Î` œ

subject to the nonnegativity constraint, given his effort levels ( , ... , ) for the other 1x x mw2i mi

w

prizes and the list of the other players' actions, so that we have ( , ) ( , ).x x x x xb b b i i i i ii i1 11   w

31

Note that is uniquely defined as a finite real number because the function is strictlyx11i i1

concave in and its first-order partial derivative is negative for some large .x x1 1i i

Here we have two cases: Either 0 or 0. If 0, then player 's action isx x x iw w w2 2 2i i i iœ œ x1

exactly the action , which we try to find. On the other hand, if 0, then we take thexwwi ixw2

following steps to show the existence of player 's action .i xwwi

Step x 1. Let a real number be defined as12i

inf { : (( , , , ... , ), ) 0 for all ( , ]}.x x x x x x x x x x1 0 1 02 12 22 2 23 2i ii ii i i ii mi i iœ ` −1 w w

w xb Î`

Then, in the interval ( , ], player 's expected payoff decreases in his effort level ; it isx x i x02 2 2i i i

w

maximized at his effort level ; and we have (( , , , ... , ), ) ( , ). Notex x x x x1 1 12 1 2 3i i ii ii mi i i i1 1w

w x x xb b  w

that, because ( ) ( , ) 0 and ( ) ( ) is continuous in , the real numberi x ii x x` `1 1i i i i ii ix x x1 b Î` Î`2 2 2

x x x i1 12 2 2i i iis well-defined and we always have . Part ( ) comes from the following facts. First, w

if the players expend zero effort levels for every prize except prize 2 at the action profile ( ,x1i

x x xb b

w wi i i i ii i), then as shown in the proof of part ( ) of Lemma A4, we have ( , ) ( )b x c x` œ1 1 Î` 2 2

0. Second, if the players expend positive effort for at least two distinct prizes at the action

profile ( , ), then Lemma A1, together with ( , ) 0, leads to ( ,x x x x x1 1 1i i i i ii i i

b b ` Ÿ `1 1Î`x1

xbi i) 0.Î`x2

Step x x x x i . We iterate Step 1 to find an action ( , , , ... , ) of player such that∞ 11 2 3i i i mi

∞ ww

x x x x x∞ ∞ w2 2

11 3i ii ii i mi i i iœ  0 and (( , , , ... , ), ) ( , ).1 1w

x x xb bw

Step L i x x x x . Lastly, we find player 's action, ( , , , ... , ), such that ( ,x x∞ ∞i i´ ∞ ∞ w

1 2 3i i i mi iw 1

x x xb b

wi i i ii i i) ( , ) and maximizes player 's expected payoff given his effort levels ( , , ...  1 w x i x x∞ ∞

1 2 3

, ) for the other 1 prizes and the list of the other players' actions. Now this action x mwmi i x xb

i∞

of player is exactly the action , which we try to find.i xwwi

( ) Next, suppose that for some {1, 2}. Using steps similar to thoseb v v k M1i ki − Ï

above, we can show that, given the action , there exists player 's action, ( , ... , ),x xw wwwi i i mii x x´ www ww

1w

32

such that ( ) ( , ) ( , ), ( ) ( , ) 0 with 0, or ( ,i ii x x1 1 1 1i i i i ii i i i i i ix x x x x x xwww www wwwb b bi

www  ` `∞ Î` œ1 1

xb

www www www www wi i i zi zii ki) 0 with 0, ( ) 0 and 0, and ( ) for any {1, 2, }.Î` œ œ œ œ − Ïx x iii x x iv x x z M k1 1 2Ÿ

( ) Eventually, we can show that, given the action , there exists player 's action, c ix xwi i

H

´   œ( , ... , ), such that ( ) ( , ) ( , ) and ( ) 0 for any withx x i ii x z MH H1i mi i i i i zii i

H 1 1x x x xH b b

w −

v v1i zi .

Finally, to complete the proof, we only need to show that ( ) ( , ). Let real1 1i i i ix x xb b  H

numbers and be defined as and , respectively. Then, becauseb b x xH Hœ œ m m

z z

bzi zi

œ1 1œ

H

x x z M v v bbzi zi i zi iœ œH 0 for any with , we obtain, using Lemma A3, that (( , 0, ... , 0),− 1 1

x x x x xb b b b i i i ii i i) ( ) and (( , 0, ... , 0), ) ( , ). Using Lemma A3, we obtain also thatœ œ1 1 1H H

1 1i ii mi ib b bi i(( , 0, ... , 0), ) (( , , ... , ), ) for any small positive real number ; ifb x x x % % %x xb b

œ 1 2

b z M v v x b x œ œ0, then for some prize with and 0, (( , 0, ... , 0), ) (( , ...− 1 1i zi i ib bzi i i1 1% xb

, , ... , ), ) for 0 . Next, because maximizes player 's expected payoff,x x x x ib b b bzi mi i zi i % %xb

Ÿ 1

we have ( ) (( , , ... , ), ). Similarly, we have ( ) (( , ... , ,1 1 1 1i i i ib b b b bi i imi i zix x xb b b x x x x x1 2 1 % %

... , ), ). These inequalities, together with the preceding equalities, yield (( , 0, ... , 0),x bbmi i ixb

1

x x x xb b b b i i i ii i i) (( , 0, ... , 0), ) and (( , 0, ... , 0), ) (( , 0, ... , 0), ). Next, using 1 1 1b b b % %

these inequalities and the definition of the partial derivative, we obtain that (( , 0, ... , 0),`1i b

x xb bii i i i i) 0 with 0, or (( , 0, ... , 0), ) 0 with 0. Since the function Î` Î` œx b b x b1 1œ ` Ÿ1 1

is strictly concave in , this implies that maximizes player 's expected payoff given his effortx b i1i

levels (0, ... , 0) for the other 1 prizes and the list of the other players' actions. This inm xbi

turn implies that (( , 0, ... , 0), ) (( , 0, ... , 0), ). Using this inequality and the1 1i ii ib x xb b   H

equalities obtained above that (( , 0, ... , 0), ) ( ) and (( , 0, ... , 0), ) ( ,1 1 1 1i i i ii i ib x x x xb b b œ œH H

x x x xb b b i i ii i), we obtain that ( ) ( , ).1 1  H

Proof of Theorem 1. To prove this theorem, it suffices, due to Lemma A6, to show that the set

Z is not empty. In other words, it suffices to show that there exists an action profile at which the

first-order conditions for maximizing ( ), for any , are satisfied.1j x j N−

33

We organize the proof in three steps. In Step 1, we construct a "truncated-actions game"

of the original game by placing restriction on the players' sets of actions. In Step 2, we show that

there exists a Nash equilibrium in such a truncated-actions game. In Step 3, using Nash

equilibria of truncated-actions games, we show that, in the original game, there exists an action

profile at which the first-order conditions for maximizing ( ), for any , are satisfied.1j x j N−

Step truncated-actions game 1. Define an as a game which is the same as the one in%

Section 2, with the exception that each player's effort levels for the prizes are now limited as

follows: For each , we assume that [ , ], where prize is one of his MP prizes (seei N x B k− ki − %

Definition 1) and 0 , and that {0} for any with . We call prize player −% B x z M z k kzi − Á

i's target MP prize.

Step i i N 2. In an truncated-actions game, player 's set of actions, for each , is% −

nonempty, compact, and convex; and his payoff function is concave in . Also, an truncated-xki %

actions game is - , the term introduced by Reny (1999), because the payoffbetter reply secure

functions for the players are all continuous. Accordingly, in an truncated-actions game, there%

exists a Nash equilibrium, which is verified by Theorem 3.1 in Reny (1999).

Step t 3. Consider now an truncated-actions game, for 1, 2, ... , where, for each%t œ

i N x B k x z M z k− − Á, we have [ , ] for his target MP prize , and {0} for any with .ki zit− −%

Note that denotes raised to the power of . We assume that is less than unity. We assume% % %t t

also that the upper bound is sufficiently large that each player expends an effort level less thanB

B kfor his target MP prize in equilibrium. Then, there exist Nash equilibria in these %t

truncated-actions games, and they all belong to the set [0, ] {0} . Since the set [0,B n n m‚ ( 1)

B] {0} is compact and every sequence in a compact set has a convergent subsequence,n n m‚ ( 1)

there exists a limit of a subsequence of those equilibria. Without loss of generality, we assume

that a sequence of action profiles { } is a subsequence of Nash equilibria of the truncated-xt ∞œt

t1 %

actions games and has a limit of the action profile . We claim and prove below that this limitxp

x xp satisfies the first-order conditions for maximizing ( ), for any , in the original game.1j j N−

34

First, we prove that the action profile satisfies, in the original game, the first-orderxp

condition for maximizing ( ), for each , over the effort level expended for his target1i kix i N x −

MP prize . If 0, then we have ( ) 0 for sufficiently large because { } is ak x x tpki i ki t `1 x xt tÎ` œ ∞

œ1

sequence of Nash equilibria of the truncated-actions games. This leads to%t

( ) ( ) 0. (A8)limt

i ki i kiÄ∞

` œ ` œ1 1x xt pÎ` Î`x x

Consider the other case where 0. Since the action profile , for 1, 2, ... , is a Nashx tpki œ œxt

equilibrium of the truncated-actions game, and thus satisfies the first-order conditions, by%t

mimicking the proof of part ( ) of Lemma A4, it is straightforward to show that the equilibriuma

total effort level, , is strictly greater than for sufficiently large . The total effort level,n

j

t tj

œ1x n t%

n

j

pj

œ1x , is also positive because the action profile is the limit of a sequence of Nash equilibriaxp

of the truncated-actions games (see the proof of part ( ) of Lemma A4). Using these facts, we%t a

obtain that the first-order partial derivative ( ) converges to ( ) . This,` `1 1i ki i kix xt pÎ` Î`x x

together with ( ) 0 for sufficiently large , leads to` Ÿ1i kixt Î`x t

( ) ( ) 0. (A9)limt

i ki i kiÄ∞

` œ ` Ÿ1 1x xt pÎ` Î`x x

Second, we prove that the action profile satisfies, in the original game, the first-orderxp

condition for maximizing ( ), for each , over the effort level expended for his other1i hix i N x −

MP prize . Using Lemma A2 and expressions (A8) and (A9), it is straightforward to obtain thath

x xphi i hiœ ` Ÿ0 and ( ) 0.1 xp Î`

Finally, we prove that the action profile satisfies, in the original game, the first-orderxp

condition for maximizing ( ), for each , over the effort level expended for his non-MP1i six i N x −

prize . Recall that the action profile is the limit of a sequence of Nash equilibria of the s xp %t

truncated-actions games. Using this fact and mimicking the proofs of parts ( ) and ( ) ofa b

Lemma A4, it is straightforward to show that the players expend positive effort for at least two

35

prizes at the action profile . Then, using Lemma A1 and expressions (A8) and (A9), we obtainxp

that 0 and ( ) 0.x xpsi i siœ ` Ÿ1 xp Î`

Appendix B: Proof of Remark 1

Consider two vectors, ( , ... , ) and ( , ... , ), each having at least twoX Xœ œX X X X1 1m mw w w

positive elements. We prove Remark 1 by showing that if, for every prize , we havek M−

` Î` œ ` Î` œP X P Xk k k k( ) ( ) , then must hold.X X X Xw w

Suppose that, for every prize , we have ( ) ( ) . Then, fork M P X P X− ` Î` œ ` Î`k k k kX X w

every prize , we havek M−

f X f X f X f X f X f X f X f Xw w w w w w

œ œ œ œk k k kk z z k k z z z k z

m m m m

z z z zz z( )( ( ) ( )) ( ( )) ( )( ( ) ( )) ( ( )) .

1 1 1 1

2 2 Î œ Î

This expression can be rewritten as

f X f X f X f X f X f X f X f Xw w w w w w

œ œ œ œk k k kk z z k k z k z z z

m m m m

z z z zz z( )( ( ) ( )) ( )( ( ) ( )) ( ( )) ( ( )) .

1 1 1 1

2 2 Î œ Î

Let ( ) ( ). Then, since 0, we haveα α´ Î m m

z zz z z z

œ œ

w

1 1f X f X

f X f X f X f X f X f Xw w w w w

œ œk k k kk z z k k z k

m m

z zz( )( ( ) ( )) ( )( ( ) ( )) 1. (B1)

1 1 Î œα α α

First, we show that (B1) holds for every prize only if 1. Suppose on thek M− α œ

contrary that 1. In this case, for any , if ( ) ( ), then we haveα α  k M f X f X− k k k kw

f X f X f X f X f X f Xw w w w w

œ œk k k kk z z k k z k

m m

z zz( ) ( ) and ( ) ( ) ( ) ( ), so that the left-hand side of (B1) Ÿ α α α

1 1

is less than unity. On the other hand, if ( ) ( ), then since ( ) ( ), wef X f X f X f Xk k k z z zk

m m

z zz œα αw w

œ œ

1 1

have ( ) ( ) for some prize with . This, together with the same argument asf X f X t M t kt t t t α w − Á

above, leads to the conclusion that, for prize , the left-hand side of (B1) is less than unity.t

36

Hence, if 1, then (B1) does not hold for every prize . Next, suppose that 1.α α k M−

Similarly, we obtain again that (B1) does not hold for every prize .k M−

Next, we show that ( ) ( ) for every prize . Suppose on the contrary thatf X f X k Mk k k kœ w −

f X f X t M f X f Xt t t tt t t t( ) ( ) for some prize . In this case, we have ( ) ( ) and Ÿw w w w−

m m

z zz z t t z tz t

œ œ

w w

1 1f X f X f X f X( ) ( ) ( ) ( ) because we have 1 or, equivalently, œα

m m

z zz z z z

œ œ

w

1 1f X f X t( ) ( ). This leads to the conclusion that, for prize , the left-hand side of (B1) isœ

less than unity. On the other hand, suppose that ( ) ( ) for some prize . In thisf X f X t Mt t t t w −

case, we have ( ) ( ) for some prize with because we havef X f X s M s ts s s s w − Á

m m

z zz z z z

œ œ

w

1 1f X f X( ) ( ). This, together with the same argument as above, leads to the conclusionœ

that, for prize , the left-hand side of (B1) is less than unity. Hence, if ( ) ( ) for somes f X f Xt t t tÁ w

prize , then (B1) does not hold for every prize .t M k M− −

Finally, because, for every prize , the function is strictly increasing in andk M f R− k +

f X f Xk k k k( ) ( ), we obtain . This completes the proof.œ w X Xœ w

Appendix C: Proof of Theorem 3

To prove Theorem 3, we need the following lemma.

Lemma C1. 3, 3, , ( ) Suppose that m n each player has just one MP prize and c x x forœ   i i i iœ )

each i N If there exists a Nash equilibrium x x x with X x for . , (( , , )) , 0 − xN œ œ œN N N n N Nj j j j k kj

n

j1 2 3 1

1œ

œ

some prize k then there exists no Nash equilibrium in which a player or players expend positive,

effort for prize k .

37

Proof. Without loss of generality, we assume that there exists a Nash equilibrium, (( ,xN œ xNj1

x x X X XN N n N N Nj j j2 3 3 1 21, )) , with 0. Then, due to Lemmas A4 and A6, we have 0 and 0, andœ œ

at least two players are active that is, they expend positive effort in this equilibrium. This,

together with Lemmas A5 and A6, implies that each active player has either prize 1 or prize 2

(but not both) as his MP prize, at least one of the active players has prize 1 as his MP prize, and

at least one of the active players has prize 2 as his MP prize. Then, without loss of generality,

we assume that players 1 and 2 are active in the equilibrium, player 1 has prize 1 as his MP

prize, and player 2 has prize 2 as his MP prize.

Let ( , , ). Then, since the first-order conditions for maximizing ( ), forX xN œ X X XN N Nj1 2 3 1

each , are satisfied in the Nash equilibrium , we have (see expressions (A5) and (A6) andj N− xN

Lemmas A5 and A6):

v P x v P x v P x 11 1 11 21 2 11 31 3 11 1( ( ) ) ( ( ) ) ( ( ) ) 0,` ` ` œX X XN N NÎ` Î` Î` )

v P x v P x v P x 12 1 22 22 2 22 32 3 22 2( ( ) ) ( ( ) ) ( ( ) ) 0,` ` ` œX X XN N NÎ` Î` Î` )

and

v P x v P x v P x 13 1 33 23 2 33 33 3 33 3( ( ) ) ( ( ) ) ( ( ) ) 0.` ` ` ŸX X XN N NÎ` Î` Î` )

Due to the fact that ( ) ( ) ( ) 1, we have that ( ) ( )P P P P x P x1 2 3 1 2X X X X X œ `` Î` Î`N Nii ii

` œ œ œP x i P3 3( ) 0 for each 1, 2, 3. We have also that ( ) 0, and thus thatX XN NÎ` ii

` œ œP x P x3 11 3 22( ) ( ) 0. Hence, these selected first-order conditions, which willX XN NÎ` ` Î`

be used below, can be rewritten as

v v P x ( )( ( ) ) 0, (C1)21 11 2 11 1 Î` ` œX N )

v v P x ( )( ( ) ) 0, (C2)12 22 1 22 2 Î` ` œX N )

and

v v P x v v P x ( )( ( ) ) ( )( ( ) ) 0. (C3)13 33 1 33 23 33 2 33 3 Î` Î` ` ` ŸX XN N )

38

Now, we have two cases to consider. First, consider the case where none of the players

in the contest have prize 3 as their MP prizes. In this case, it is immediate from Lemmas A5 and

A6 that there exists no pure-strategy Nash equilibrium in which a player or players expend

positive effort for prize 3.

Next, consider the case where some player, say player 3, has only prize 3 as his MP

prize. (Recall that player 1 has prize 1 as his MP prize and player 2 has prize 2 as his MP prize.)

It first follows from Lemmas A5 and A6 that each player with prize 1 or prize 2 as his MP prize

expends zero effort for prize 3 in any pure-strategy Nash equilibrium. Then, to complete the

proof in this second case, it suffices to show that player 3 expends zero effort for prize 3 in any

pure-strategy Nash equilibrium. Suppose on the contrary that there exists a Nash equilibrium,

xw œ œ(( , , )) , in which player 3 expends positive effort for prize 3. Let forx x x X xw w w w w

œ1 2 3 1

1j j j z zj

nj

n

jœ

each 1, 2, 3, and let ( , , ). Then, since the first-order conditions for maximizing z X X Xœ œX w w w w

1 2 3

1j( ), for each , are satisfied in the Nash equilibrium , we have:x xj N− w

v P x v P x v P x 11 1 11 21 2 11 31 3 11 1( ( ) ) ( ( ) ) ( ( ) ) 0,` ` ` ŸX X Xw w wÎ` Î` Î` )

v P x v P x v P x 12 1 22 22 2 22 32 3 22 2( ( ) ) ( ( ) ) ( ( ) ) 0,` ` ` ŸX X Xw w wÎ` Î` Î` )

and

v P x v P x v P x 13 1 33 23 2 33 33 3 33 3( ( ) ) ( ( ) ) ( ( ) ) 0.` ` ` œX X Xw w wÎ` Î` Î` )

Since we have that ( ) ( ) ( ) 1, and thus that ( ) ( )P P P P x P x1 2 3 1 2X X X X X œ `` Î` Î`w wii ii

` œ œP x i3( ) 0 for each 1, 2, 3, these selected first-order conditions can be rewritten asX w Î` ii

v v P x v v P x ( )( ( ) ) ( )( ( ) ) 0, (C4)21 11 2 11 31 11 3 11 1 Î` Î` ` ` ŸX Xw w )

v v P x v v P x ( )( ( ) ) ( )( ( ) ) 0, (C5)12 22 1 22 32 22 3 22 2 Î` Î` ` ` ŸX Xw w )

and

v v P x v v P x ( )( ( ) ) ( )( ( ) ) 0. (C6)13 33 1 33 23 33 2 33 3 Î` Î` ` ` œX Xw w )

39

The second term in the left-hand side of (C4) is positive because player 1 has only prize 1 as his

MP prize so that we have and and, under our hypothesis that 0, we v v v v X21 11 31 11 3 w

have ( ) 0 due to part ( ) of Assumption 2. Using these facts, (C1), and (C4), we` P x b3 11X w Î`

obtain

P x P x` `2 11 2 11( ) ( ) . (C7)X Xw Î` Î`N

Similarly, using (C2) and (C5), we obtain

P x P x` `1 22 1 22( ) ( ) . (C8)X Xw Î` Î`N

Under our hypothesis that 0, we have that ( ) in (C7) and ( ) in (C8)X P x P xw3 2 11 1 22 ` `X Xw wÎ` Î`

are nonpositive due to parts ( ) and ( ) of Assumption 2. Also, we have thatb e

` œ ` ` œ `P x P x P x P x c1 22 1 33 2 11 2 33( ) ( ) and ( ) ( ) due to part ( ) of Assumption 2.† † † †Î` Î` Î` Î`

Finally, since player 3 has only prize 3 as his MP prize, we have and . Thesev v v v13 33 23 33

facts, together with (C7) and (C8), imply

v v P x v v P x v v P x( )( ( ) ) ( )( ( ) ) ( )( ( ) )13 33 1 33 23 33 2 33 13 33 1 33 Î` Î` Î`` ` `X X Xw w N

( )( ( ) ). `v v P x23 33 2 33 Î`X N

This expression, together with (C3) and (C6), yields the following contradiction:

v v P x v v P x)3 13 33 1 33 23 33 2 33œ ` ` ( )( ( ) ) ( )( ( ) ) Î` Î`X Xw w

( )( ( ) ) ( )( ( ) ) . ` ` Ÿv v P x v v P x 13 33 1 33 23 33 2 33 3 Î` Î`X XN N )

Therefore, in this second case too, there exists no pure-strategy Nash equilibrium in which a

player or players expend positive effort for prize 3.

Lemma C1 implies that, under the conditions stated herein, if there is an equilibrium in

which the players expend positive effort for all the three prizes, then there is no equilibrium in

which they expend positive effort only for one or two prizes.

40

Proof of Theorem 3. ( ) Consider the case where 2. This case is analytically analogous toa n œ

the case where 2 and 2, because one of the three prizes is not any player's MP prizem nœ œ

and, due to Lemmas A5 and A6, both players expend zero effort for the prize in equilibrium.

Hence, the case is immediately proved by using Theorem 2.

( ) Next, consider the case where 3. Let (( , , )) be a Nashb n x x x  œxN N N N nj j j j1 2 3 1œ

equilibrium (see Theorem 1). Let for each 1, 2, 3, and let ( , , ).X x z X X XN N N N Nz zj

n

jœ œ œ

œ11 2 3X N

Then, due to Lemmas A4 and A6, we have 0 and 0, for some , 1, 2, 3 withX X h tN Nh t œ

h tÁ , and at least two players are active that is, they expend positive effort in this

equilibrium. This, together with Lemmas A5 and A6, implies that at least one of the active

players has only prize as his MP prize, and at least one of the active players has only prize ash t

his MP prize. Then, without loss of generality, we assume that players 1 and 2 are active in the

equilibrium, player 1 has only prize 1 as his MP prize, and player 2 has only prize 2 as his MP

prize. Note that this assumption, together with Lemmas A5 and A6, implies that 0 andX N1

X N2 0.

Now, we have two cases to consider: one where 0, and the other where 0.X XN N3 3œ

In the first case where 0, it follows from Lemma C1 that there exists no Nash equilibriumX N3 œ

in which a player or players expend positive effort for prize 3. This implies that this first case is

analytically analogous to the case where 2 and 3. Hence, the proof of Theorem 3 inm nœ  

this first case is immediately done by using Theorem 2.

Next, consider the second case where 0. If is the only Nash equilibrium, thenX N3 xN

the proof is trivial. Suppose that it is not the only one. Let (( , , )) be any otherxw œ x x xw w w1 2 3 1j j j

njœ

Nash equilibrium. Let for each 1, 2, 3, and let ( , , ). Note that,X x z X X Xw w w w w

œz zj

n

jœ œ œ

11 2 3X w

since 0 for all 1, 2, 3, we have 0 for all due to Lemma C1. Then, to proveX k X kNk k œ w

Theorem 3 in this second case, it suffices to show that ( ) ( ) for each` Î` ` Î`P X P Xk k k kX XN œ w

k œ 1, 2, 3, because Assumption 3 holds by hypothesis. Without loss of generality, we assume

that player 3 expends positive effort for prize 3 in the Nash equilibrium , which implies, due toxw

41

Lemmas A5 and A6, that he has prize 3 as his MP prize. (Note that players 1 and 2 expend zero

effort for prize 3 in the Nash equilibrium because none of them have prize 3 as their MPxw

prizes.) Then, we have the following expressions:

v v P x v v P x ( )( ( ) ) ( )( ( ) ) 0, (C9)21 11 2 11 31 11 3 11 1 Î` Î` ` ` ŸX Xw w )

v v P x v v P x ( )( ( ) ) ( )( ( ) ) 0, (C10)12 22 1 22 32 22 3 22 2 Î` Î` ` ` ŸX Xw w )

and

v v P x v v P x ( )( ( ) ) ( )( ( ) ) 0. (C11)13 33 1 33 23 33 2 33 3 Î` Î` ` ` œX Xw w )

(Note that these expressions are selected among the first-order conditions for maximizing

1j( ), for each , which are satisfied in the Nash equilibrium . We use the fact thatx xj N− w

` Î` Î` Î`P x P x P x i1 2 3( ) ( ) ( ) 0 for each 1, 2, 3. For detailed derivations,X X Xw w wii ii ii ` ` œ œ

see the proof of Lemma C1.)

We have also the following expressions, which are satisfied in the Nash equilibrium xN

(see the proof of Lemma C1):

v v P x v v P x ( )( ( ) ) ( )( ( ) ) 0, (C12)21 11 2 11 31 11 3 11 1 Î` Î` ` ` œX XN N )

v v P x v v P x ( )( ( ) ) ( )( ( ) ) 0, (C13)12 22 1 22 32 22 3 22 2 Î` Î` ` ` œX XN N )

and

v v P x v v P x ( )( ( ) ) ( )( ( ) ) 0. (C14)13 33 1 33 23 33 2 33 3 Î` Î` ` ` ŸX XN N )

To show that ( ) ( ) for each 1, 2, 3, we first show that` Î` ` Î`P X P X kk k k kX XN œ œw

` Î` ` Î` ` Î` ` Î`P x P x P x P x3 11 3 11 3 11 3 11( ) ( ) . Suppose on the contrary that ( ) ( ) .X X X XN Nœ w w

In this case, using (C9) and (C12), we obtain

P x P x` `2 11 2 11( ) ( ) . (C15)X XN Î` Î`w

42

For this derivation, note that we have and because player 1 has prize 1 as hisv v v v 21 11 31 11

MP prize, and ( ) 0 and ( ) 0 because we have 0 and 0` ` P x P x X X3 11 3 11 3 3X Xw Î` Î`N w N

(see part ( ) of Assumption 2). Similarly, using (C10) and (C13), we obtainb

P x P x` `1 22 1 22( ) ( ) . (C16)X XN Î` Î`w

Since we have 0 for all 1, 2, 3, we have that ( ) in (C15) and ( )X k P x P xwk œ ` `2 11 1 22X Xw wÎ` Î`

in (C16) are negative due to part ( ) of Assumption 2. Also, we have that ( )b P x` 1 22† Î`

œ ` ` œ `P x P x P x c1 33 2 11 2 33( ) and ( ) ( ) due to part ( ) of Assumption 2. Finally,† † †Î` Î` Î`

since player 3 has prize 3 as his MP prize, we have and . These facts,v v v v13 33 23 33

together with (C15) and (C16), imply

v v P x v v P x v v P x( )( ( ) ) ( )( ( ) ) ( )( ( ) )13 33 1 33 23 33 2 33 13 33 1 33 Î` Î` Î`` ` `X X XN N w

( )( ( ) ). `v v P x23 33 2 33 Î`X w

This expression, together with (C11) and (C14), yields the following contradiction:

v v P x v v P x)3 13 33 1 33 23 33 2 33  Î` Î` ( )( ( ) ) ( )( ( ) )` `X XN N

( )( ( ) ) ( )( ( ) ) . ` ` œ v v P x v v P x13 33 1 33 23 33 2 33 3 Î` Î`X Xw w )

On the other hand, suppose that ( ) ( ) . In this case too, the argument` Î` ` Î`P x P x3 11 3 11X XN w

similar to the above leads to a contradiction. Hence, ( ) ( ) must hold.` Î` ` Î`P x P x3 11 3 11X XN œ w

Similarly, we can show that ( ) ( ) and ( )` Î` ` Î` ` Î`P x P x P x1 22 1 22 2 33X X XN Nœ œw

` Î`P x2 33( ) .X w

Finally, due to part ( ) of Assumption 2, ( ) ( ) implies thatc P x P x` Î` ` Î`3 11 3 11X XN œ w

` Î` ` Î` ` Î` ` Î` ` Î`P x P x P x P x P x3 22 3 22 1 22 1 22 1 33( ) ( ) ; ( ) ( ) implies that ( )X X X X XN N Nœ œw w

œ œ` Î` ` Î` ` Î` ` Î`P x P x P x P x1 33 2 33 2 33 2 11( ) ; and ( ) ( ) implies that ( )X X X Xw wN N

œ œ ` Î` ` Î`P x P P P P x2 11 1 2 3 1( ) . Due to the fact that ( ) ( ) ( ) 1, we have ( )X X X X Xw Nii

` ` œ ` ` œP x P x P x P x P x i2 3 1 2 3( ) ( ) ( ) ( ) ( ) for each 1,X X X X XN NÎ` Î` ` Î` Î` Î`ii ii ii ii iiw w w

2, 3. Using these facts, we obtain that ( ) ( ) for each 1, 2, 3.` Î` ` Î`P X P X kk k k kX XN œ œw

43

Appendix D: Proof of Theorem 5

Suppose that player , for , has just one MP prize, say prize for , and isi i N h h M− −

indifferent between all the prizes except his MP prize: and for any { }v v v v k M hhi ki ki i − Ïœ

and for some . Let represent a vector of the equilibrium effort levels corresponding tov Ri − X*

the Nash equilibrium : ( , ... , ). Then, since the first-order conditions forx X* * œ X X* *m1

maximizing ( ), for each , are satisfied in the Nash equilibrium , we have (see1j x xj N− *

expressions (A5) and (A6) and Lemmas A5 and A6):

v P x v P x 1 1i hi mi m hi i( ( ) ) ( ( ) ) 0. (D1)` ` ŸX X* *Î` â Î` )

Due to the fact that ( ) 1, we have also ( ) 0. Using this, we can rewrite m m

z zz z hi

œ œ1 1P P xX Xœ ` œ* Î`

player 's selected first-order condition (D1) asi

( ) ( ). (D2)` Ÿ ÎP x v vh hi i hi iX* Î` )

In equilibrium, expression (D2) holds for every player whose MP prize is prize .h

Because the left-hand side of expression (D2) has the same value for these players, it follows

that, among these players, only the players with the lowest value for the right-hand side may

expend positive effort for prize , while the rest expend zero effort for every prize in (seeh M

Lemmas A5 and A6). Henceforth, for concise exposition, we assume that there is a unique

player who has the lowest value for the right-hand side of expression (D2). (The proof can be

completed without this assumption, but it would be cumbersome.)

Next, suppose that there are prizes, each of which is some player's MP prize, andmw

( ) prizes, each of which is nobody's MP prize, where 2 min{ , }. Let m m m m n M Ÿ Ÿw w w

represent the set of the players' MP prizes: {1, ... , }. For each , there is a uniqueM m h Mw wœ w −

player, among the players whose MP prize is prize , who has the lowest value for the right-handh

side of expression (D2). We isolate these players, and let represent the set of the isolatedm N w w

44

m N m jw w players: {1, ... , }. Then, without loss of generality, we assume that player , for eachw œ

j N j− w, has prize as his MP prize.

Note that, in equilibrium, player , for each , may expend positive effort for prize ,j j N j− w

but expends zero effort for every prize except prize . Note also that, in equilibrium, the playersj

who do not belong to the set expend zero effort for every prize in .N Mw

Now, define ( ) ( ) ( ) ( ) . Then, for anyP P X P X P XX X X X´ Î` Î` â Î`` ` `1 2 2 3 1m

vector that has at least two positive elements, ( ) is well defined and has a nonpositive valueX XP

(see parts ( ) and ( ) of Assumption 2). We take two steps to complete the proof. In Step 1, web e

show that the value ( ) of the funtion is unique across the Nash equilibria of the game. InP PX*

Step 2, using Assumption 3 and the fact proved in Step 1, we show that the vector is uniqueX*

across the Nash equilibria.

Step 1. Let and represent the vectors of the equilibrium effort levelsX Xw ww

corresponding to the Nash equilibria, and , respectively. Then, due to Lemmas A4 and A6,x xw ww

X X w ww and each have at least two positive elements. Suppose, by way of contradiction, that we

have ( ) ( ). Assume that ( ) ( ) , where 0. The following expressionP P P PX X X Xw ww w wwÁ œ % %

holds for player , for , whose MP prize is prize (see expressions (D2)):j j N j − w

( ) ( ),` Ÿ ÎP x v vj jj j jj jX w Î` )

where is player 's valuation for every prize except his MP prize. Due to the fact thatv jj

m m

z zz z j

œ œ1 1P P X( ) 1, we have ( ) 0, which yieldsX Xœ ` œw Î`

( ) ( ) ( ) . (D3)` œ ` œ `P x P x P Xj jj z jj z jz j z j

X X Xw w wÎ` Î` Î` Á Á

Using expression (D3) and the fact that ( ) ( ) ( )P P X P XX X Xw w wœ ` ` 1 2 2 3Î` Î` â

`P X cm( ) , which can be rewritten, due to part ( ) of Assumption 2, asX w Î` 1

P P X P X t M s M t( ) ( ) ( ) for any and any { }, we obtainX X Xw w wœ ` `z t

z t t sÁ

Î` Î` − − Ï

45

( ) ( ) ( ) ( ) , (D4)` œ ` œ `P x P X P P xj jj z j j kjz j

X X X Xw w w wÎ` Î` Î`Á

for any { }. Next, Using expressions (D2) and (D4), we obtaink M j− Ïw

( ) ( ) ( ) ( ), (D5)` œ ` Ÿ ÎP x P P x v vj jj j kj j jj jX X Xw w wÎ` Î` )

for any { }.k M j− Ïw

We derive two observations from inequality (D5). First, if ( ) decreases by 0 andP X w %

player , for , is active that is, he expends positive effort for prize before and afterj j N j− w

this decrease, then ( ) must decrease by as much as in order to satisfy inequality`P xj kjX w Î` %

(D5) with equality. Note that both ( ) and ( ) have nonpositive values. Second, ifP P xX Xw w` j kjÎ`

player is active in the Nash equilibrium , so that we have ( ) 0 andj P xx Xw w` j kjÎ`

Î` P P x v v( ) ( ) ( ), then he must be also active in the Nash equilibrium .X X xw w ww ` œ Îj kj j jj j)

That is, we must have ( ) 0 and ( ) ( ) ( ). Note` ` œ ÎP x P P x v vj kj j kj j jj jX X Xww ww wwÎ` Î` )

that otherwise we would have ( ) ( ), which contradicts that is a Nash P v vX xww ww Î)j jj j

equilibrium of the game. We obtain this last strict inequality using the following facts. First, we

have ( ) ( ) ( ) and ( ) 0. Second, if player is not active in Î` P P x v v P jX X Xw w w ` œ Î j kj j jj j)

the Nash equilibrium , so that 0, then due to part ( ) of Assumption 2, we havexww X ewwj œ

` œ ` œ P X P x P Pj k j kj( ) 0, and thus we have ( ) 0. Third, we have ( ) ( ).X X X Xww ww ww wÎ` Î`

Next, using Lemmas A4 and A6, we know that at least two players in are active in theN w

Nash equilibrium , each expending positive effort for a different prize in . Using the twoxw M w

observations derived in the preceding paragraph, we obtain: If player , for , is active in thej j N− w

Nash equilibrium , so that 0, then we have ( ) ( ) for anyx X Xw w wwX P X P Xwj j k j k ` œ ` Î` Î` %

k M j j j N− Ï −w w{ }. In addition, we have the following facts. If player , for , is not active in the

Nash equilibrium , so that 0, then due to part ( ) of Assumption 2, we havexw X ewj œ

` œP X k M j b ej k( ) 0 for any { }. Due to parts ( ) and ( ) of Assumption 2, we haveX w Î` − Ï

` ŸP X k M zz k( ) 0 for any { }.X ww Î` − Ï

46

Consequently, using these facts, we have

P P X P X P X( ) 2 ( ) ( ) ( ) 2X X X Xw w w w œ ` ` ` % %1 2 2 3 1Î` Î` â Î`m

( ) ( ) ( )  ` ` `P X P X P X1 2 2 3 1X X Xww ww wwÎ` Î` â Î`m

( ),œ P X ww

which contradicts the hypothesis that ( ) ( ) . This contradiction proves that theP PX Xw ww œ %

value ( ) of the funtion is unique across the Nash equilibria.P PX*

Step 2. Let represent a vector of the equilibrium effort levels corresponding to theX*

Nash equilibrium . Due to Lemmas A4 and A6, has at least two positive elements. Then,x X* *

for each , we have (see expression (D5)):j N− w

( ) ( ) ( ) ( ), (D6)` œ ` Ÿ ÎP x P P x v vj jj j kj j jj jX X X* * *Î` Î` )

for any { }.k M j− Ïw

The value of the right-hand side of inequality (D6) is unique across the Nash equilibria.

We know from Step 1 that the value of ( ) is unique across the Nash equilibria. The value ofP X*

`P x jj kj( ) is unique across the Nash equilibria: It is equal to 0 if player is not active acrossX* Î`

the equilibria, and is equal to ( ) ( ) if he is active across the equilibria. Note that,P v vX* Î)j jj j

as explained above, if a player is active [resp. not active] in one equilibrium, then he is also

active [resp. not active] in another equilibrium, if any. These facts imply that ( ) , for`P xj jjX* Î`

each , is unique across the Nash equilibria.j M− w

Next, in any Nash equilibrium, we have 0 for every . Thus we haveX z M M*z œ − Ï w

` œP X k M z ez k( ) 0 for any { }, due to part ( ) of Assumption 2. In addition, similarly toX* Î` − Ï

the derivation of expression (D6), we obtain ( ) ( ) ( ) for any ` œ `P X P P Xz z z kX X X* * *Î` Î`

k M z P X z M M− Ï Î` − Ï{ }. These imply that ( ) , for each , is unique across the Nash` z zX* w

equilibria.

47

Hence, we have that ( ) , for each , is unique across the Nash equilibria.`P X k Mk kX* Î` −

This, together with Assumption 3, leads to the conclusion that the vector is unique across theX*

Nash equilibria.

48

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