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 the players 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 the players expend their effort simultaneously and independently. We first prove the existence of a pure-strategy Nash equilibrium of the game, then establish when the total effort level expended for each prize is unique across the Nash equilibria, and then summarize and highlight other interesting and important properties of the equilibria. Finally, we discuss the effects of heterogeneity of valuations on the players' equilibrium effort levels and a possible extension of the 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: [email protected]); Jung (corresponding author): Ma Yinchu School of Economics, Tianjin University, Tianjin 300072, China (e-mail: [email protected]). We are grateful to Chris Baik, Subhasish M. Chowdhury, Amy Baik Lee, Dongryul Lee, Tim Perri, Iryna Topolyan, and two anonymous referees for their helpful comments and suggestions.
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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: [email protected]); Jung (corresponding author): Ma Yinchu School of Economics,Tianjin University, Tianjin 300072, China (e-mail: [email protected]). 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.
2
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.
10
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,