Team Contests with Multiple Pairwise Battles
Qiang Fuy
Jingfeng Luz
Yue Panx
April, 2012
(very preliminary)
Abstract
In this paper, we study a contest between two teams. The teams include the same number
of players. Each player of a team participates in a component battle, competing against a
matched rival from the other team. A team wins if and only if its players prevail in su¢ ciently
large number of component battles. We demonstrate in an all-pay-auction setting that the wellstudied “discouragement e¤ect”in individual multi-battle contests do not appear. Instead, the
(stochastic) outcomes of past battles do not a¤ect the stochastic outcomes of future plays. In
addition, the expected e¤ort of the contest is independent of the temporal structure of the
contest, i.e., the sequencing or clustering of component battles, or the prevailing information
disclosure and feedback policy. The analysis reveals the fundamental di¤erence in players’
strategic mindsets between individual contests and team contests. These results yield rich
implications on contest design and policy experimentation, and we show the main results and
their logic remain valid under a wide array of alternative settings.
JEL Nos: C7, D7
Key Words: Team Contests, Multi-Battle, All-Pay Auction, Collective Action, HistoryIndependence, Sequence-Independence, Temporal-Structure-Independence
We are grateful to Kai Konrad for the very inspiring and encouraging discussion. We thank Je¤rey Ely
for helpful comments and suggestions. Qiang Fu thanks Max Planck Institute of Public Finance and Tax
Law for the hospitality. A part of the project was executed when Fu visited the institute.
y
Qiang Fu: Department of Business Policy, National University of Singapore, 1 Business Link, Singapore,
117592; Tel: (65)65163775, Fax: (65)67795059, Email: [email protected].
z
Jingfeng Lu: Department of Economics, National University of Singapore, Singapore 117570. Tel :
(65)65166026. Email: [email protected].
x
Yue Pan: Department of Chemical Engineering, Email: [email protected].
1
1
Introduction
Many competitive events can be generically viewed as contests. In such contests, economic
agents expend scarce resources in order to win a limited number of prizes, while they forfeit
the resources regardless of win or loss. This type of competitive activities appears in a diverse
array of environments including political campaigns, sports, R&D races, warfare, and even
internal labor markets inside …rms.
The widespread phenomenon has spawned a wide array of formal models that is dedicated to unveiling the various strategic concerns involved in these activities. This evolving
literature has increasingly recognized that a contest often consists of more than a single
static “battle”(see Konrad, 2009). Instead, it may require rivaling parties to confront each
other on multiple fronts. One’s success cannot be accomplished in a single stroke of e¤ort,
but depends on its overall performance in a series of shots. Such multi-battle contests are
not uncommon in real world. Consider, for instance, an R&D race towards the innovation of
a …nal product (see Harris and Vickers, 1987). The development e¤ort is often an accumulative process which proceeds in multiple stages. In each stage …rms compete on a speci…c
task, i.e. the development of a component technology. A …rm attains its ultimate success
only if it achieves a su¢ cient number of advances ahead of its rival. Alternatively, to stand
out as a U.S. presidential candidate, a politician in primaries has to defeat his opponents in
the majority of state elections (see Klumpp and Polborn, 2006).
The existing studies have conventionally assumed that the grand prize of a multi-battle
contest is contested by individual contenders, and that each of them participates in all
component battles. Many contests, however, involve competitions between teams, in which
each member of a team is matched to his counterpart from the rival team on an individual
battle…eld. A team’s success requires its team members to secure su¢ cient wins in component
battles. A member of a team bene…ts from the win of the team in the overall contest, while
each member of a team decides on his contribution in his own battle. For instance, major
pharmaceutical and biotechnology companies, such as Roche and L’Oréal, have evolved into
network organizations that rely heavily on extensive webs of strategic alliances in R&D. In
a race towards a new drug, component tasks of the development project are often carried
out by a¢ liated but independent entities within each …rm’s network. Each entity retains
control over its own input on his own task. Political parties’competition in general elections
also resembles team contests. A political party wins majority status only if its candidates
prevail in a su¢ cient number of constituencies, while the success or failure of an individual
candidate to a large extent depends on his own e¤ort in raising fund and campaigning.
Alternatively, a large-scaled military operation usually includes a series of separate battles
between matched individual units, e.g., in World War II. The outcome on an individual
battle…eld depends on the maneuvers and commitments of the participating units. Such a
2
contest can be most intuitively exempli…ed by many sports events. Many team titles, such as
Davis Cup in men’s tennis, Thomas Cup in men’s badminton and Swaythling Cup in men’s
table tennis, are contested through a series of individual matches.
In team contests, one’s payo¤ depends on not only the outcome of his own battle, but
also the e¤ort contributed by his teammates. The strategic behavior of players naturally
involves coordination and collective action. They face substantially di¤erent trade-o¤s than
their counterparts in individual contests, which demands a drastically di¤erent strategic
mindsets. The literature, however, has provided little in formal modelling to shed light on
the widespread phenomenon.
In this paper, we provide a formal analysis of a multi-battle team contest. Two teams,
comprised of an equal number of players, compete for a common object. Each player of a
team competes against his matched opponent from the rival team in one component battle.
These component battles are carried out successively. Each player observes the outcomes
of previous battles and the current state of the grand contest before he sinks his outlays
on the battle…eld. A player receives a positive reward from the win of his team, which is
a public good, while he is also allowed to secure a private trophy from winning his own
battle. To provide an analogy, a newly synthesized chemical could not only contribute to
P&G’s innovation on a grooming product, but also creates private bene…t to the a¢ liated
lab that successfully develops this ingredient, i.e., royalty revenues from future alternative
uses of the chemical. Alternatively, Novak Djokovic may attach substantial importance to
his personal victory over Rafael Nadal even when he represents Serbian team in the …nal of
Davis Cup. A politician who runs for parliament membership may bene…t even more from
his individual success than from the majority status of his party.1 Matched players on each
battle…eld commonly value the private trophy. Its value, however, is speci…c to the particular
battle…eld, and could vary across di¤erent battles. Players can be heterogenous in terms of
their abilities. Their marginal e¤ort costs are drawn from possibly di¤erent distributions.
They strategically decide how much e¤ort to contribute to maximize their own expected
payo¤.
The results on team contests run in stark contrast to those from multi-battle contests
between individuals, which reveals the fundamentally di¤erent incentives induced by team
competitions as compared to individual contests. The conventional wisdom, e.g. Harris and
Vickers (1987), Klumpp and Polborn (2006), Konrad and Kovenock (2009a) and Malueg
and Yates (2010), holds that a dynamic contest demonstrates a crucial feature of “historydependence”: One’s (perhaps purely accidental) victories in early battles generate strategic
1
See Cox and Magar (1999) and Hartog and Monroe (2008) for empirical estimates of the value of a
party’s marjority status (in senate) to individual politicians or businesses. The nontrivial value of majority
status in national legislative body alludes to a team contest where team’victory bene…ts individual players.
3
momentum to him, while discourage the other, which leads the latter to give up and allows
the former to attain easy wins in subsequent confrontations. As Harris and Vickers (1987)
and Malueg and Yates (2010) show, a contest between symmetric players would evolves
into asymmetric battles, in which the leader values more winning the subsequent battles
than the laggard, and therefore places higher bids. Hence, the outcomes of early battles
distort subsequent competitions, and the outcomes of early battles predict the ultimate
winner. This “momentum e¤ect” or “discouragement e¤ect” provides a rationale to the
well observed “New Hampshire e¤ects” over the history of U.S. Presidential Primaries (see
Polborn and Klumpp, 2006). Due to the strategic importance of early battles, Polborn and
Klumpp (2006) predict that candidates invest heavily on campaigns in constituencies with
early election dates. Aldrich (1980) and Strumpf (2002) demonstrate in sequential-election
settings that the particular sequence of a given series of heterogenous battles could distort
the ultimate outcome of a multi-battle contest. The winner that stands out of a sequential
election, e.g., U.S. presidential primaries, could have been the loser if the elections in di¤erent
constituencies were ordered alternatively.
These e¤ects, however, do not arise in a team contest in which di¤erent pairs of matched
players engage in head-to-head competition in di¤erent component battles. Three main
observations from the present analysis can be highlighted as follows.
1. History-Independence We show that the outcomes of early battles do not distort future
confrontations. Throughout the contest, a player’s probability winning his battle does
not depend on the state of the contest, i.e., by how much one’s team is leading or lagging
behind the rival team, and how many additional wins each team needs to reach …nish
line. The stochastic outcome of each battle, i.e., the ex ante likelihood of each player’s
win, is purely determined by participating players’e¤ort cost characteristics.
2. Sequence-Independence Let us …x the matching of players and let each battle be contested between a …xed pair of players. The battle would yield a …xed amount of ex ante
expected overall e¤ort, which is independent of the order of the battle in the entire
sequence either. The overall expected e¤ort of the contest, as well as its stochastic
outcome, is invariable if the sequence of battles is reshu- ed.
3. Temporal-Structure-Independence (or Disclosure-Independence) The ex ante expected
e¤ort of a battle between a …xed pair of players would not vary even when the battles
of the contest, or a portion of the battles, are carried out simultaneously. So is its
stochastic outcome. To interpret it alternatively, it also predicts that the expected
e¤ort of the contest is independent of the prevailing information disclosure or feedback
policy. The expected e¤ort and stochastic outcome of a battle, as well as those of the
4
entire contest, do not vary no matter whether players observe the history of previous
battles.
These results yield several useful implications for contest design and also practical policy
experimentation. For brevity, we discuss them only in Section 5, after we present our formal
analysis. The main logic is to be illustrated in Section 3 through a simple illustrative example.
The logic that underpins our main results surface in the simpli…ed setting, which highlight the
di¤ering strategic trade-o¤s involved in individual multi-battle contests and team contests,
respectively.
In the rest of this section, we brie‡y review the relevant literature. Section 2 sets up a
general model of multi-battle contest between teams. In Section 3, we present a simple and
stylized example. Section 4 conducts the main analysis. In Section 5, we discuss the main
implications of our results, their robustness and the a few possible extensions. Section 6
concludes this paper.
Literature
This paper primarily belongs to the literature on multi-battle contests. This strand of
literature dates back to Harris and Vickers (1987). They analyze a dynamic R&D race
where competing parties meet in a sequence of component contests successively, and one
wins a prize if and only if it has won a su¢ cient number of them. Snyder (1989), in contrast,
studies a simultaneous multi-battle contest, which models the competition for majority status
between two political parties in the parallel elections of multiple constituencies. Harris and
Vickers (1987) are among the …rst to identify the discouragement e¤ect in a race. Klumpp
and Polborn (2006), as aforementioned, focus on U.S. presidential primaries and compare
the e¢ ciency of sequential campaigning institution to that of its simultaneous counterpart.
The analysis provides a rationale for the “New Hampshire E¤ect”observed in the history of
primaries, and demonstrates the e¢ ciency e¤ect of sequential contest. Klumpp and Polborn
(2006) model each component battle as a Tullock contest.
Konrad and Kovenock (2009a) provide a complete account of sequential multi-battle
contest with each component battle being a pairwise all-pay auction, where no exogenous
noise contributes to players’winning odds in each battle. In contrast to many other studies,
the framework of Konrad and Kovenock (2009a) allows the players to be asymmetric in
various aspects, e.g., their bidding costs, the minimum numbers of battles to win for victory
in the contest, etc. Konrad and Kovenock (2010) demonstrate that the aforementioned
discouragement e¤ect can be attenuated when the two players’ bidding costs are ex ante
uncertain. Malueg and Yates (2010) show theoretically the ex post asymmetry caused by
one’s winning early battle in a best-of-three contest between two symmetric players. They
5
further provide empirical evidence from professional tennis matches.
These studies typically assume that players meet each other in disjoint battle…elds.
Kovenock, Sarangi, and Wiser (2011) introduce interdependence between component battles
by allowing for “complementarity”: A contestant wins the contest if he wins certain combination of component battles. Kovenock and Roberson (2009) study a two-stage contest. In
each stage, contestants meet each other in multiple parallel battles, and one’s win on one
battle…eld at the …rst stage grants him a head-start when they meet again on the same front.
Our paper di¤ers from these studies by studying team contest, in which each component
battle is contested by a particular pair of matched players from rival teams. The main
questions raised in this paper are analogous to those of Klumpp and Polborn (2006), i.e.
(1) How the outcomes of early battles distort those of later ones; and (2) How di¤ering
temporal structures a¤ect contestants’behaviors. The present study, however, focuses on a
di¤erent setup from Klumpp and Polborn (2006). The results thus have a di¤erent scope
of applications. For instance, our analysis could shed light on the strategic trade-o¤s and
e¢ cient design of a national general election where the candidate in each constituency plays
an active role on behalf of his political party. In contrast, Klumpp and Polborn (2006) focus
on U.S. primaries where individual contenders compete in all states. Our paper complements
that of Klumpp and Polborn (2006) in this aspect. Strumpf (2002) focus on the distortionary
e¤ect of the particular sequence of heterogenous battles on the outcome of the multi-battle
sequential contest. He shows that a player who is favored in later battles is more likely to
prevail than one who is favored in early contentions. Similar to Klumpp and Polborn (2006),
Strumpf (2002) focuses on contests between individuals, and also interpret his analysis in
the context of presidential primaries. Our setup also di¤ers from that of Strumpf (2002) in
this regard. Our setting leads to the Sequence-Independence result, which is unique in team
contest setting.
Our paper is naturally linked to the literature on group contest or contest between allied
players. This extensive literature includes Skaperdas (1998), Nitzan (1991), Esteban and Ray
(2001 and 2008), Nitzan and Ueta (2009), Münster (2007) and (2009), Konrad and Leininger
(2007), Konrad and Kovenock (2009b), among many others. These studies typically assume
that contestants in each coalition join force on a single front to produce composite output.
In contrast, our setting requires each player to carry out his part on a single front, and
matched players compete against each on a set of disjoint battles. A few notable exceptions
are more closely related to our paper, including Kovenock and Roberson (2010) and Rietzke
and Roberson (2010). These two studies consider rivalry between two allied players and one
independent player. Each of the former competes against the latter in a series of simultaneous
component battles in a Colonel Blotto games. The two studies investigate the incentives of
allied players to exchange their resources, and the subsequent strategic plays in the contests.
6
Our paper di¤er from them fundamentally in a several aspects. First, a player in their
settings …ghts a series of parallel battles (as given by the nature of Colonel Blotto games),
while one in ours does only one. Second, a player in their settings secures rent from winning
each component battle: He maximizes the sum of his private payo¤s from all his battles,
but he does not receive spillover from his ally’s win or loss. In contrast, a player in our
setting derives utility from his team’s winning the majority of battles, but may not receive
private reward from winning their own battles. Third, players in their contexts form alliance
to exchange resource, while players in our setting contribute to the teams by winning their
own battles.
Our analysis concludes that the expected e¤ort of the team contest does not dependent
on whether players learn the outcomes of previous battle. The paper can thus be linked to
the small but growing literature on communication and feedback in dynamic contests. This
strand of literature, e.g., Gershkov and Perry (2009), Aoyagi (2010), Ederer (2010), Gürtler
and Harbring (2010), and Goltsman and Mukherjee (2011), typically focuses on whether a
contest designer should reveal intermediate rankings to the contestants in a dynamic contest.
In contrast to the present paper, players in these studies compete on single tasks, and supply
continuing ‡ow of e¤orts. The winner is determined by players’accumulated output, instead
of their accumulated numbers of wins in discrete battles.
Our paper yield immediate implications on the design of election schemes. The paper is
thus marginally linked to the literature on sequential elections. Except for the few notable
exceptions such as Klumpp and Polborn (2006) and Strumpf (2002), most of these studies
concern themselves with the behavior of voters. For instance, Dekel and Piccione (2000) focus
on sophisticated voters’strategic information gathering activities. Bikhchandani, Hirshlefer
and Welch (1992) study the information cascades in primaries, and their model depict the
herding behavior of voters in later states.
2
Sequential Multi-Battle Team Contest: Setup
Two teams, indexed by i = 1; 2, compete in a contest for an indivisible object. Each team
consists of 2n + 1 players, where n is a positive integer. Players of a team i are indexed by
i(t), with t 2 f1; : : : ; 2n + 1g. The two teams compete against each other on 2n + 1 disjoint
battle…elds. Each player 1(t) of team 1 is matched to his counterpart 2(t) of the rival team
in a pairwise component battle t. A team is awarded the object if and only if its players
accumulate at least n + 1 victories in these component battles.
It should be noted that our analytical results do not depend on the ex ante symmetry
between the two teams. We discuss in Section 5 that all results extend to an alternative
setting where teams face asymmetric “…nish lines”. That is, one team can prevail by winning
7
a smaller number of component battles.
2.1
Temporal Structures
The 2n + 1 component battles are carried out successively. Before an arbitary battle t is
fought, the history of past battles, or the state of the contest, is observed by players 1(t) and
2(t). The state of the contest is summarized by a tuple (k; l), where k is the number of wins
secured by team 1 and l is that of team 2, with k + l + 1 t under a sequential contest.
The contest reaches a terminal state (k; l) once one team has accumulated exactly n + 1
victories before the other does, i.e., max(k; l) = n + 1. In this case, subsequent battles are
irrelevant for the outcome of the team contest. A component battle t is called trivial if the
contest is in a state (k; l), with max(k; l)
n + 1. However, players may still put forth
positive e¤orts if they derive private bene…t from winning their own battles, which will be
elaborated upon in more detail when we lay out the fundamentals about the payo¤ structure.
In Section 5, we present formal analysis of an alternative setting in which all or a subset
of component battles are contested simultaneously.
2.2
Payo¤s
The reward to each player arises out of two sources: team prize and private prizes. First,
one bene…ts from his team’s win. The trophy of the winning team is a public good to its
players. All players equally value the prize awarded to the team, and we normalize the
common valuation to one. Second, players reap private bene…t from the win of his own
battle, irrespective of his team’s success or failure. The winner of a component battle t
receives an individual prize t
0. The private prize purse t is speci…c to the particular
battle…eld t. It varies across di¤erent battle…elds, while it is equally valued by the two
players who vie for it.2 In the case of t > 0, each trivial battle, i.e., a battle in a state (k; l)
with max(k; l) n + 1, is no di¤erent from a standard static contest in which two players
compete for the single prize t . In the event of t = 0, each trivial battle is irrelevant as it
elicits no e¤ort.
2.3
Component Contests
In each component battle t, the two matched players simultaneously exert their e¤orts xi(t) .
Following Konrad and Kovenock (2009a and 2010), we model each component battle as an
2
For the sake of expositional e¢ ciency, we collapse the asymmetry among players into a setting of heterogenously distributed e¤ort costs. Our model can be readily adapted into a setting where players value
team victory di¤erently, and the main insights do not depend on the current modelling approach.
8
all-pay auction. One wins the battle with certainty if he contributes a higher e¤ort than the
rival player. To the extent that both players exert the same amount of e¤orts, the winner is
picked randomly.3;4
Players can be heterogenous in terms of their competence, which is measured by the
di¤ering marginal e¤ort costs. Each player i(t) bears a constant marginal cost ci(t) > 0 for
his e¤ort entry xi(t) , with ci(t) to be distributed over a non-degenerate interval [ci(t) ; ci(t) ] 2
(0; +1) continuously and independently, according to a cumulative distribution function
Fi(t) (ci(t) ). The distribution functions, Fi(t) (ci(t) )s, and their supports are common knowledge.
The exact realization of ci(t) is known to player i(t).
The framework can ‡exibly accommodate a variety of information structures. To the
extent that the realization of ci(t) is known to player i(t) himself but not to his rival, 8i 2
f1; 2g, the component battle t is a standard incomplete-information all-pay auction (see
Moldovanu and Sela, 2001 and 2006, and Moldovanu, Sela and Shi, 2007), with (possibly) ex
ante asymmetric players. To the extent the realizations of both c1(t) and c2(t) are commonly
known, the battle t is a standard complete-information all-pay auction (see Hillman and
Riley, 1989, Baye, Kovenock and de Vries, 1996, and Konrad and kovenock, 2009a and
2010). To the extent that the realization of ci(t) is commonly known, while that of cj(t) is
learnt by player j(t) only, 8i; j 2 f1; 2g; i 6= j, the setting evolves into an all-pay auction
with one-sided asymmetric information.5 We show in the general analysis that our results
do not depend on the prevailing information structure.
2.4
Some Additional Terminologies
Let pi (k; l) denote the probability of team i’s eventual win assessed when the contest is in an
arbitrary state (k; l). Players of a team i thus expect a team payo¤ vi (k; l) = 1 pi (k; l) from
the prospect of winning the team prize, when the contest enters a state (k; l). We hereafter
name the team payo¤ vi (k; l) the continuation value of team i evaluated at a state (k; l).
Apparently, we must have v1 (k; l) = 1 if k n + 1, and v1 (k; l) = 0 if l n + 1. Similarly,
v2 (k; l) = 1 if l n + 1, and v2 (k; l) = 0 if k n + 1.
3
It should be noted that the main claims of this paper are not limited to all-pay-auction setting. We
adopt this popular form of contest technologies mainly for the sake of expositional e¢ ciency. As will be
revealed by the illustrative examples in Section 3, the logic of our main results relies little on the prevailing
winner selection mechanism.
4
Additional quali…cations are required for the zero-probability event that both players bid zero e¤orts.
Please refer to the proof of Lemma 2 for detail.
5
A similar contest is studied by Münster and Morath (2010).
9
3
Illustrative Example: Symmetric “Best-of-Three”Contest
We now present a simple “best-of-three” example to highlight the unique characteristics of
team contests. We …rst set up a benchmark contest, in which two players meet in successive
battles and one wins by winning two of them. We then present its team-contest counterpart
and compare the predictions of the two settings. Much of the logic that underpins our main
results can be revealed in the simple setting.
In each component battle, players simultaneously commit to their e¤ort outlays. For the
sake of simplicity, we assume that all players bear a unity marginal e¤ort cost, which is commonly known. Further, we assume for the moment that players do not receive intermediate
reward for winning a component battle. As mentioned in Section 2, the main logic of this
paper does not rely on speci…c contest technologies. For the moment, we only require the
winner selection mechanism of each battle to be symmetric: One is more likely to prevail if
and only if he supplies strictly more e¤ort than the other. A symmetric bidding equilibrium
is assumed to exist for symmetric players, i.e., when they are equally competent and equally
motivated. In a symmetric equilibrium, players exert the same e¤ort and each wins with a
probability 21 .
3.1
Benchmark: Individual Contest
The benchmark contest resembles the “best-of-three” contest of Malueg and Yates (2010).
Two individual players, indexed by i = 1; 2, compete for a prize of value one. In each of
the battles, players simultaneously exert their e¤ort x~i(t~) , where t~ indicates the order of the
battle.
The game can be analyzed by backward induction. Suppose that they each have won
one battle. The outcome of the subsequent battle determines the ultimate winner. The
deciding battle is apparently a symmetric match. Each player’s ex ante expected payo¤
from participating in the last battle is given by v~3 = 12 x~3 , where x~3 is the symmetric
expected equilibrium e¤ort each player exerts in the battle.
We then consider players’incentives in the second battle. When battle 2 is to be fought,
one player has won one (the …rst) battle. Without loss of generality, we assume that player
1 is the leader. Let ~ i(t~) (~
x1(t~) ; x~2(t~) ) denote a player i’s probability of winning battle t~ under
a given e¤ort pro…le (~
x1(t~) ; x~2(t~) ). He chooses x~1(2) to maximize his expected payo¤
u~1(2) = ~ 1(2) (~
x1(2) ; x~2(2) )
= v~3 +
1 + (1
x1(2) ; x~2(2) )
1(2) (~
(1
10
x1(2) ; x~2(2) ))
1(2) (~
v~3 )
x~1(2) ;
v~3
x~1(2)
(1)
By contrast, player 2, who would lose the entire contest if he loses the second battle, chooses
x~2(2) to maximize
u2(2) = 2(2) (~
x2(2) ; x~1(2) ) v~3 x~2(2) :
(2)
It is straightforward to see that player 1, as the leader, is better incentivized and becomes
a favorite in the battle: His e¤ective prize spread, i.e., the di¤erence between the payo¤ if he
wins, and that if he loses, amounts to (1 v~3 ) > 12 , while that of his rival is only v~3 < 12 . The
ex ante symmetric contest is diverted into an asymmetric path after one obtains a lead. As
a result, player 1 is more likely to win the second battle simply because he has accidentally
won the …rst.
3.2
(Sequential) Team Contests
We now demonstrate that the ex post asymmetry does not loom large in a team contest.
Two teams, each consisting of three players, compete for a prize, which is a public good
and has a value of one to all players in a team. Suppose that two teams score evenly in the
…rst two battles. Battle 3 is apparently an even one. In the symmetric equilibrium, each
player wins the deciding battle with a probability of 21 . We then look into the immediately
preceding battles. The state of the contest when battle 2 takes place can be either (1; 0) or
(0; 1). Assume without loss of generality that player 1(1) has won battle 1 on behalf of team
1.
Let i(t) (x1(t) ; x2(t) ) denote a player i’s probability of winning battle t under a given e¤ort
pro…le (x1(t) ; x2(t) ). In the second battle, player 1(2) chooses his e¤ort x1(2) to maximize his
expected payo¤
u1(2) =
1(2) (x1(2) ; x2(2) )
= v1 (1; 1) +
v1 (2; 0) + (1
1(2) (x1(2) ; x2(2) )[v1 (2; 0)
1(2) (x1(2) ; x2(2) ))
v1 (1; 1)]
x1(2) :
v1 (1; 1)
x1(2)
(3)
Player 1(2) would expect a return of exactly 1 if the team contest ends after his own battle,
i.e. v1 (2; 0) = 1. However, he expects a return, i.e., the continuation value v1 (1; 1), of exactly
1
if he loses the battle and have it proceed into a deciding battle. This fact reveals the key
2
di¤erence between the team contest and its individual counterpart. In the benchmark, player
1 expects an expected payo¤ of only 21 x~3 from losing battle 2 and being forced to …ght
battle 3. Player 1(2), however, would not invest costly input in battle 3, while he stands a
chance of 12 to win the team prize of 1. As a result, (3) can be rewritten as
u1(2) =
1
+
2
1(2) (x1(2) ; x2(2) )
1
2
x1(2) :
(4)
Player 2(2) would end up with nothing if he loses the battle, while he also expects a
return of 21 if he wins and the contest enters a state (1; 1): His teammate is expected to win
11
the even battle with a probability of 12 . That is, v2 (2; 0) = 0 and v2 (1; 1) = 21 . He then
chooses her e¤ort x2(2) to maximize her expected payo¤
1(2) (x2(2) ; x1(2) )
=
2(2) (x2(2) ; x1(2) )
=
2(2) (x2(2) ; x1(2) )
[v2 (1; 1) v2 (2; 0)]
1
x2(2) :
2
x2(2)
(5)
Functions (4) and (5) verify that player 2(2) values his win as much as his rival 1(2) does.
Battle 2 remains symmetric in spite of team 1’s lead. Each player is expected to win with a
probability of 12 . One’s initial lead does not distort subsequent battles.
3.3
Intuition
The simple example reveals the strikingly di¤erent strategic mindsets across the two environments. To illuminate its logic, let us …rst elaborate on the nature of the conventionally
held discouragement e¤ect in individual sequential contests.
For a laggard in the “best-of-three”individual contest, the bene…t of winning the second
battle is to even the score. Actual reward, however, would not accrue unless he wins the
subsequent battle. He is disincentivized by two factors. First, the uncertainty of future
battle discounts the incentive provided by the prize purse. Second, he has to continue to
sink costly e¤ort in future battle if he wins the current one. The additional outlay dissipates
his future rent, thereby attenuating his incentive to remain in contest. As evidenced by
x~3 . By contrast, the incentive of his
(2), his incentive to win battle 2 is given by 21
leading opponent to win is magni…ed by two factors. First, he would reap the actual bene…t
immediately if he wins. Second, the contest would be prolonged if he loses, which forces him
to endure another uncertain and costly battle in future. The additional outlay burns his rent
and therefore aggravates the pain from the current loss. As evidenced by our example, he
receives 1 if he wins, while he ends up with only 12 x~3 if he loses. In summary, the leader
has more to win and also more to lose, which makes him a favorite, despite their ex ante
symmetry.
The same, however, cannot be said of team contests. We …rst consider the player of the
leading team who is up for battle 2. Analogous to individual contests, the win allows him
to secure the actual prize immediately. The lead, however, allows him to slack o¤: His team
would not lose the contest immediately because of his loss and he can still receive the prize as
long as his teammate prevails. He su¤ers less from a prolonged struggle than his counterpart
in individual contests, because subsequent battle does not require his own contribution. His
stake in the current battle is thus suppressed by these e¤ects. As evidenced by (3) and (4),
a loss renders v1 (1; 1) = 12 : The third battle is an e¤ortless fair draw. He would be punished
less than his counterpart in individual contests as he free-rides on other’s e¤ort.
12
We then consider the player of the lagging team who is up for battle 2. Analogous to
his counterpart in individual contests, his valuation of the win is discounted by the future
uncertainty caused by the lag, because he receives the prize only if his team also wins battle
3. However, as he does not have to bear the cost of future battle, the rent from the win
will not be dissipated by the subsequent struggle when the contest continues. He expects a
value of v2 (1; 1) = 21 from his own win, which pays o¤ more to him than to his counterpart
in individual contests. Furthermore, the disadvantage is a double-edged sword that also
incentivizes him in a team contest. In contrast to his leading rival, the player in a lagging
team is prevented from free-riding on his teammate because the loss would end the contest
immediately. These e¤ects thus ampli…es his e¤ective “prize spread” in his battle, i.e.,
v2 (1; 1) v2 (2; 0) = 12 .
As a result, battles remain symmetric over all possible paths. We subsequently demonstrate that the result in a more generalized setting and compute the solution to the equilibrium. The logic revealed in the simple example carries over the more general setting.
4
General Analysis
We now execute our main analysis in a generalized “best-of-(2n+1)”sequential team contest.
Consider an arbitrary battle t, and suppose that the contest enters an arbitrary state (k; l)
with k; l < n + 1 and k + l + 1
t. We explore their strategic incentives in the battles.
Player 1(t) chooses his e¤ort x1(t) to maximize
u1(t) =
1(t) (x1(t) ; x2(t) )[ t
+ v1 (k + 1; l)] + (1
1(t) (x1(t) ; x2(t) )[ t
+ v1 (k + 1; l)
1(t) (x1(t) ; x2(t) ))v1 (k; l
v1 (k; l + 1)] + v1 (k; l + 1)
+ 1)
x1(t) :
x1(t)
(6)
Similarly, player 2(t)’s expected payo¤ obtains as
2(t) (x2(t) ; x1(t) )
=
2(t) (x2(t) ; x1(t) )[ t
+ v2 (k; l + 1)
v2 (k + 1; l)] + v2 (k + 1; l)
x2(t) :
(7)
In the rest of this section, we present our analysis and discuss our results in three segments.
4.1
History-Independence
We now claim that (6) and (7) again allude to a symmetric battle, in which the two players
equally value the win, i.e. v1 (k + 1; l) v1 (k; l + 1) = v2 (k; l + 1) v3 (k + 1; l), regardless
of the current state or the history of past plays. The following simple fact illuminates the
entire puzzle.
13
Lemma 1 v1 (k; l) + v2 (k; l) = 1, 8k; l 2 f1; : : : ; 2n + 1g, with k + l
2n + 1.
The simple fact of Lemma 1 does not require a formal proof. Recall that the continuation
value v1 (k; l) at a state (k; l) is simply given by v1 (k; l) = p1 (k + 1; l) 1, where p1 (k + 1; l) is
team 1’s winning likelihood assessed at a state (k; l). The continuation value to a player i(t)
is simply the payo¤ from team prize after being discounted by the probability of his team’s
eventual victory. The team prize, however, must be won by either of the two teams. The
outcome of a battle shifts the state of the contest, which changes the stochastic division of
future rent, while it does not change the overall size of the rent.
By Lemma 1, player 2(t)’s stake on the battle can be reformulated as
t
+ [v2 (k; l + 1)
=
t
+ f[1
=
t
+ [v1 (k + 1; l)
v2 (k + 1; l)]
v1 (k; l + 1)]
[1
v1 (k + 1; l)]g
v1 (k; l + 1)];
which is the exactly the same as that for player 1(t). The following obtains.
Theorem 1 Every battle t, with t 2 f1; : : : ; 2n + 1g is a common-valued all-pay auction in
which the two matched players, 1(t) and 2(t), equally value the win. In the battle t, each
player has an e¤ective prize spread of t + [v1 (k + 1; l) v1 (k; l + 1)].
Theorem 1 generalizes the insights we obtain from the “best-of-three”example. A player’s
valuation of winning his own battle depends on the state of the contest. However, the shift of
state impacts matched players symmetrically. Equally-valued battles ensue over all possible
paths.
In our model, matched players are allowed to be ex ante heterogenous, with their e¤ort
costs to be drawn from di¤erent distributions. As stated in Section 2, our setup allows for
various information structures. Each player’s e¤ort cost can be either known to himself only
or commonly known. The existing literature has provided a complete account of asymmetric all-pay auctions with complete information. In Appendix we characterize formally the
monotone equilibria of asymmetric common-valued all-pay auctions with either two-sided
asymmetric information or one-sided asymmetric information.
Let us summarize the stochastic outcome of an arbitrary two-player all-pay auction by
the tuple ( ; 1
), which indicates that player 1 wins with a probability , while the
other wins with a probability 1
. Our analysis yields the following important, although
less than surprising property, which holds under all possible information structures.
Lemma 2 Consider a two-player common-value all-pay auction in which players both value
the prize for v > 0. Its stochastic outcome in equilibrium is given by the tuple ( ; 1
), with
(c1 ; c2 ; F1 (c1 ); F2 (c2 )). Players’ expected e¤orts are linear in the prize purse v, i.e.,
E(xi ) = i v, where i
i (c1 ; c2 ; F1 (c1 ); F2 (c2 )) is a function of players’cost characteristics.
14
Proof. See Appendix.
Lemma 2 states that each player’s equilibrium winning likelihood in a common-value
all-pay auction is determined purely by players’e¤ort cost conditions, i.e., their exact e¤ort
costs, and/or their distributions, depending on the prevailing information structure. Recall
by Theorem 1 players in each battle symmetrically value their win, regardless of the state of
the contest. The probability of a player i(t)’s winning battle t, according to Lemma 2, must
depend on his and his rival’s e¤ort cost conditions only. The bidding equilibrium, as well as
the winning probability
and the expected e¤ort i (c1 ; c2 ; F1 (c1 ); F2 (c2 ))v, can be solved
for explicitly when players’cost conditions are speci…cally laid out.
Combining Theorem 1 and Lemma 2, we obtain the following main result immediately.
Theorem 2 (History-Independence) Consider an arbitrary nontrivial battle t in state (k; l)
with k; l
n, and k + l + 1
t. Player 1(t) wins with a probability t , with t
(c1(t) ; c2(t) ; F1(t) (c1(t) ); F2(t) (c2(t) )), and player 2(t) wins with the complementary probability
1
t ) does not depend on the state of the
t . The stochastic outcome of the battle ( t ; 1
contest (k; l).
By Theorem 1, the entire contest can be viewed ex ante as a series of independent draws.
The stochastic outcome of each component battle ( t ; 1 t ) is pre-determined by the pro…les
of contenders’competence, and remains invariant regardless of the actual path of the contest.
So is the stochastic outcome of the entire contest. The history of past battles does not distort
the outcomes of future battles, although players’equilibrium outlays are contingent on the
exact state of the contest.
These results are remarkably robust, and the main logic extends to much broader setting.
First, Theorem 1, which predicts the constantly symmetric valuations between matched
players, is completely independent of prevailing contest technologies. Second, the results of
history-independence would continue to hold even when players are allowed to value their
team’s win unequally. In Section 5 we discuss such an extension. The counterparts of Lemma
1 and Theorem 1 would be formulated alternatively, but the main logic leads to equivalent
predictions.
4.2
Sequence-Independence
The history-independence property allows us to compute explicitly each team’s continuation
values vi (k; l) at all possible states, and the expected e¤ort of each battle. Thanks to Lemma
1 and Theorem 1, we can focus explicitly on Team 1. For expositional e¢ ciency, we introduce
the following terminology.
In a nontrivial battle t (or a trivial battle with t > 0), a player 1(t) prevails with a
1
probability t . Let us de…ne 1 (hj m)jt+m
, with t 2 f1; : : : ; 2n + 2 mg, the probability
t
15
of players 1(t) to 1(t+m 1)’s winning exactly h of the m non-trivial battles. The probability
t+m 1
can be computed explicitly when players’ cost conditions are available for
1 (hj m)jt
1
battles t to t + m 1. We illustrate the property of 1 (hj m)jt+m
with the following
t
example.
Example 1 We consider a team contest between two 13-player teams. Assume for simplicity
that each component battle is a complete-information all-pay auction, in which the realization
of each player i(t)’s marginal e¤ort cost is common knowledge. Assume for simplicity c1(t) =
1, and c2(t) = ct 1, 8 t 2 f1; : : : ; 13g. By the well-known results in contest literature, each
nontrivial component battle t has a stochastic outcome ( 2c2ct t 1 ; 2c1t ). We compute 1 (2j 3)j42 ,
the probability of team 1’s winning exactly 2 component battles among battles 2-4. Team 1
may win any of the three combination of two battles and lose the rest: (2; 3); (2; 4); (3; 4):
Hence, this probability can be computed as follows:
4
1 ( 2j 3)j2
2c3 1
1
1
2c4 1
2c2 1
[
+
]
2c2
2c3
2c4 2c3
2c4
2c3 1 2c4 1
1
+
2c2
2c3
2c4
(2c2 1) [(2c3 1) + (2c4 1)] + (2c3 1) (2c4 1)
=
:
8c2 c3 c4
Based on the notations, the following general formula can be obtained for team 1’s
continuation value v1 (k; l) evaluated at a given state (k; l). Once a contest enters a state
(k; l), with k; l < n + 1, team 1 can secure its victory by …ghting at least another n + 1 k
battles, which allows it rival team to win no additional battle.
=
Theorem 3 Suppose that a battle t is to be carried out when the contest is in a state (k; l),
with k + l + 1 t. The continuation value of team 1 in this state (k; l) can be expressed as
the follows:
8 P
t+m 1
2n (k+l)
>
kj m)
]
>
1 (n + 1
m=n+1 k [ 1(t+m)
>
>
if k; l < n + 1;
t
<
2n+1
+
(n
+
1
kj
2n
+
1
k
l)j
1
t
v1 (k; l) =
(8)
>
>
v1 (k; l) = 1
if k n + 1;
>
>
:
v1 (k; l) = 0
if l n + 1:
Proof. See Appendix.
Consider the special case of symmetric team contest, in which all battles are perfectly
symmetric. That is, the e¤ort costs of matched players are drawn from the same nondegenerate distributions in incomplete-information all-pay auction with two-sided asymmetric information, or they bear the same e¤ort costs in complete-information all-pay auction. The winner of each battle would be fairly drawn, with 1(t) = 2(t) = 21 , for all
t 2 f1; : : : ; 2n + 1g. We then obtain the following.
16
Corollary 1 In a symmetric team contest, the continuation value of team 1 when the contest
is in an arbitrary state (k; l) is expressed as the follows:
P
k+i)!
], if k; l < n + 1;
(i) v1 (k; l) = ni=0l [( 12 )n+1 k+i (n
(n k)!i!
(ii) v1 (k; l) = 1 if k n + 1;
(iii) v1 (k; l) = 0 if l n + 1.
These results allow us to compute explicitly the e¤ective prize spread of each player in a
battle t at an arbitrary state (k; l), i.e., t + [v1 (k + 1; l) v1 (k; l + 1)], with k + l + 1 t.
Theorem 4 At a state (k; l) with k + l + 1
t, the prize spread for the battle t, i.e.,
v1 (k; l + 1) is given by
t + v1 (k + 1; l)
(i) t + 1 (n kj 2n k l)j2n+1
n and min(m; l) < n;
t+1 if k; l
(ii) t + 1 if k = l = n.
(iii) t if max(k; l) n + 1.
Proof. See Appendix.
By Theorem 4, in addition to the private prize t , winning the current battle at a state
(k; l), with min(k; l) < n, would provide a marginal bene…t of v1 (k + 1; l) v1 (k; l + 1) =
kj 2n k l)j2n+1
kj 2n k l)j2n+1
1 (n
1 (n
t+1 is
t+1 to each player i(t). The expression
simply the ex ante probability of the event that player 1(t)’s teammates win exactly n k
out of the next 2n k l battles after battle t has been contested, while team 2 winning
n l out of them, conditionally on that all these battles are nontrivial. In that event, battle
t is critical to the entire contest, as its outcome would ex post break the tie and determine
the winning team.
This …nding yields interesting implications, and sheds light on the nature of a team
contest. Winning the current battle allows player 1(t) to derive additional bene…t from the
team prize if and only if this battle turns out to be a tie-breaker or it is ex post critical:
His win does not ex post contribute to the team’s ultimate victory, to the extent that his
teammates turn out to win more than n k out of them, while it does not pay o¤ (in terms
of team prize) either to the extent that they turn out to secure less than n k out of them.
This observation explains the logic of Theorem 1. A team’s lead may disincentive its players
and encourage them to free ride, as their battles are less likely to be critical, and their own
loss can be made up for by others’ wins. By contrast, a team’s lags may incentivize its
players to some extent, as their battles can be more likely to contribute to team’s win, while
they are less able to rely on the e¤orts of their teammates to remedy their own losses.
Again, the general formula allows us to obtain the following result for the special case of
symmetric team contest.
17
Corollary 2 Consider a symmetric team contest, in which each pair of matched players are
homogenous. At a state (k; l) with k + l + 1 t, the e¤ective prize spread for the battle t,
v1 (k; l + 1)], is given by
t + [v1 (k + 1; l)
(2n k l)!
1 2n k l
if k; l n and min(m; l) < n;
(i) t + ( 2 )
(n k)!(n l)!
(ii) t + 1 if k = l = n;
(iii) t if max(k; l) n + 1.
Theorem 4 and Corollary 2 allow us to derive insights into players’ e¤ort supply in
team contests. Recall by Lemma 2 that in a common-valued all-pay auction each bidder
contributes an expected e¤ort i (c1 ; c2 ; F1 (c1 ); F2 (c2 ))v. Hence, a player i(t)’s conditional
expected e¤ort in a battle t at a state (k; l), with k + l + 1 t, is simply given by
E(xi(t) (k; l)) =
i(t) (c1(t) ; c2(t) ; F1(t) (c1(t) ); F2(t) (c2(t) ))[ t
+ v1 (k + 1; l)
v1 (k; l + 1)]:
We now compute the ex ante expected e¤ort a battle t between the given pair of players
1(t) and 2(t). Let Pr( (k; l)j t), with k + l + 1 t, be the ex ante probability of the event
that the battle t takes place in a state (k; l). Further, de…ne xt x1(t) + x2(t) , which denotes
the overall e¤ort of battle t. We can then write the ex ante expected e¤ort of the battle as
the follows:
P
(9)
E(xt ) = tk=01 Pr((k; l)j t)E(xt j (k; l)):
P
Pt 1
The expected total e¤ort of the entire contest thus amounts to 2n+1
t=1
k=0 [Pr((k; l)j t)E(xt j (k; l))] .
The following thought experiment is considered. Fix the pairwise match between players. Then we reorder these battles between …xed pairs of players, and compare the resultant ex ante expected e¤orts of the contest under di¤ering sequences. De…ne t
1(t) (c1(t) ; c2(t) ; F1(t) (c1(t) ); F2(t) (c2(t) )) + 2(t) (c1(t) ; c2(t) ; F1(t) (c1(t) ); F2(t) (c2(t) )). The following
can be concluded immediately.
Theorem 5 (Sequence-Independence)
(i) A battle t between a …xed pair of players has a stochastic outcome ( t ; 1
is independent of the sequence of the entire contest.
(ii) Each battle t elicits an ex ante expected e¤ort
E(xt ) =
t t
+
t
1 (nj 2n)j t
t ),
which
;
where 1 (nj 2n)j t is the overall probability of player 1(t)’s teammates’ winning exactly n
out of the 2n battles. The ex ante expected e¤ort of a battle between a …xed pair of players
does not change when the battles are reshu- ed. So is the expected total e¤ort of the contest.
Theorem 5 states that neither the stochastic outcome of a battle between a …xed pair of
players, nor its ex ante expected e¤ort depends on the sequence of these battles. Theorem 5
18
(i) somewhat trivially re-interprets the History-Independence result. The stochastic outcome
( t;1
t ) depends on players’cost conditions only.
The general formula allows us to obtain the expected e¤ort in a symmetric contest, in
which each battle is contested by symmetric players.
Corollary 3 In a symmetric battle, each battle t elicits an ex ante expected e¤ort
E(xt ) =
t t
+
t
(2n)! 1 2n
( ) :
n!n! 2
Indeed, players’ e¤ort outlays in a battle would be contingent on the exact state of
the contest. However, the ex ante expected e¤ort, which takes into account all possible
contingencies a pair of players may face ex post, would remain independent of the order of
battles. As stated by Theorem 5, the ex ante expected e¤ort is determined entirely by the
e¤ort cost conditions of the matched players (which determine t ), the private prize of the
battle, and the cost conditions of other players (which determine 1 (nj 2n)j t ).
4.3
Temporal-Structure-Independence
In this part, we relax our assumption of sequential contest and explore the rami…cations of
alternative temporal structures. We allow component battles to be carried out (partially)
simultaneously. The entire contest, with 2n + 1 component battles, is partitioned into Z
2n + 1 clusters. Battles included in the same cluster are carried out simultaneously. Players
in each battle cannot observe the outcomes in parallel battles within the same cluster, while
they observe the history of battles contested in previous clusters. This setup accommodates
various possible temporal structures. When z = 2n + 1, the sequential setting we consider
in the baseline setting is restored. With z = 1, a perfectly simultaneous setting obtains in
which all battles are contested at the same time. We obtain the following.
Theorem 6 (Temporal-Structure-Independence) (i) Each battle t has a stochastic outcome
( t;1
t ), which is pre-determined by the distributions of their e¤ort costs, but is independent of the history of prior battles or the temporal structure of the contest.
(ii) Each battle t yields an ex ante expected overall e¤ort E(xt ) = t t + t 1 (nj 2n)j t ,
which is independent of the temporal structure.
Neither t nor t , nor 1 (nj 2n)j t would depend on the prevailing temporal structure
of the contest. Theorem 6 is in fact a direct extension and generalization of Theorem 2 and
5. Regardless of the temporal structure, the winner of each battle is an independent draw,
with the probability of one’s winning to be pre-determined by players’cost conditions.
By Theorem 6(ii), the team prize lures a …xed pair of players to contribute an expected
e¤ort t 1 (nj 2n)j t , where t is determined purely by the characteristics of the players.
19
The team prize is further discounted by 1 (nj 2n)j t , which, as previously stated, is the
probability of the event that player 1(t)’s teammates’winning exactly n nontrivial battles.
In other words, this is the probability of the event that battle t turns out be critical. Hence,
the ex ante expected e¤ort of the battle is independent of its order in the entire series, the
temporal structure of the contest, or the information available to the players about other
battles when committing to their outlays!
Theorem 6 further elucidates the nature of team contest and the incentive it creates. A
player’s e¤ort would ex post pay o¤ (in terms of team prize) if and only if he engages in a
deciding battle, and his e¤ort determines the win or loss of the entire contest. Consider a
battle t, which is clustered with a few other battles. As revealed by our proof, a player i(t),
when the contest is in a state (k; l), with k; l < n + 1, would respond to an e¤ective prize
spread t + 1 (n kj 2n k l)j t;>k+l . The expression 1 (n kj 2n k l; (k; l))j t;>k+l
depicts the probability of team 1’s winning exactly n k out of the 2n k l nontrivial
battles after the contest enters a state (k; l), excluding battle t itself, i.e., all the battles that
are carried out simultaneously with t or subsequent to t, with the winner of each battle t0
to be picked in an independent draw according to ( t0 ; 1
t0 ). It again refers to the event
that battle t is ex post decides the winning team. The temporal structure does not vary the
fundamental trade-o¤s faced by players.
5
Discussions and Extensions
Our results yield rich implications for the design of various competitive mechanisms. In this
part, we brie‡y discuss the implications and applications of our results. Further, our results
and the main logic would continue to hold (qualitatively) in many alternative settings. We
then discuss the robustness of our results in alternative contexts.
5.1
Implications and Applications
One straightforward application of our results is the design of team sports tournaments, e.g.,
Davis Cup (men’s tennis) and Thomas Cup (men’s badminton). In these events, players
are sorted according to their professional rankings within each team. Then each player
is matched to his counterpart from the rival team. Given the …xed matching of players
between the two teams, it leaves open the question of how to sequence the matches between
di¤erent (…xed) pairs of players. For instance, in the Davis Cup …nal, Rafael Nadal and
Novak Djokovic, the top players of Spanish and Serbian National Teams, respectively, are
set to meet. Should the match between Rafael Nadal and Novak Djokovic, the most exciting
confrontation, be scheduled for an early round or a later round? Our Sequence-Independence
result nevertheless indicates that the ex ante expected e¤ort of a match between a given pair
20
of players does not depend on its order in the sequence. Furthermore, the temporal-structureindependence result points out that the expected e¤ort in these tournaments would not vary
even if matches are to be contested (partially) simultaneously.
More importantly, the analysis allows us to address policy concerns. For instance, how
does the temporal arrangement of an election a¤ect the outcome of the election, and the
strategic behaviors of candidates? Consider, for instance, a policy designer, who attempts
to economize on campaign expenditures in political races, and one natural instrument is
to …ne-tune the timings of the elections in di¤erent constituencies. Klumpp and Polborn
(2006) explore the e¢ ciency implications of temporal arrangement in individual multi-battle
contests, i.e., U.S. presidential primaries. They show that a sequential multi-battle individual
contest, e.g. the arrangement of U.S. Primaries, leads to lesser campaign expenditures
than the arrangement of a “national election day” (a simultaneous contest), due to the
aforementioned discouragement e¤ects. Players dump more resources on early elections,
while they give up in later ones as lags accumulate.
Our analysis of team contests also addresses such policy concerns. In most of democracies,
partisan politics in national general elections resemble a simultaneous multi-battle team
contests.6 Regional candidates, who represent rival political parties in each constituency,
compete head-to-head for seats in national legislative body. Votes are cast on the election day
across all constituencies. The party that achieves majority status in legislative body forms
the government. In U.S., the election of house representatives has a simultaneous temporal
arrangement, while the election for senate has a partially sequential structure. One third
of the seats in U.S. Senate are up for election every other year, while all elections for these
seats are carried out simultaneously. Would campaign expenditures be reduced if elections
in di¤erent constituencies are carried out successively? Under a (partially) sequential race,
e.g., election for Senate in U.S., would a party’s lead distort the outcomes of subsequent
elections? Our analysis immediately sheds light on the rami…cations of the di¤ering temporal
arrangements. Our analysis suggests that the temporal structure of partisan elections in
di¤erent constituencies do not cause distortion to either their outcomes or the expected rent
dissipation rates.
Finally, our paper also sheds light on the classical question in the literature on contest
design: Whether it pays o¤ to provide intermediate feedback to contestants in a dynamic
contest? It should be noted that a (partially) simultaneous temporal structure can be equivalently interpreted as one without or with limited intermediate feedback. Under such a
feedback system, players cannot perfectly observe the history of past plays. Our analysis,
6
Empirical studies, such as Cox and Magar (1999) and Hartog and Monroe (2008), among others, have
evidenced that a party’s marjority status (in senate) generate nontrivial values to individual politicians or
businesses.
21
however, shows that feedback or disclosure policy does not impact the performance of the
contest.
5.2
Extensions
Our baseline analysis is executed in fairly general setting, and the main insights rely little
on the setup. This framework leaves open a number of possible extensions. The analysis in
our baseline setting also yields useful implications to further work. In this part, we discuss
a few possible extensions. We demonstrate the generality and versatility of the main logic
of our analysis. The discussion also allows us to ascertain and test the limit of the results in
alternative contexts.
5.2.1
Unequal “Finish Line”
One straightforward extension is to allow one team to be endowed with a headstart, such
that it can win the contest by accumulating a smaller number of victories in component
battles. Without loss of generality, we assume that team 1 is the favorite, and it wins the
…rst-past-post contest by securing k victories before team 2 wins l, with k < l. For instance,
in an R&D race towards a …nal product, one …rm may have secured patents on a few critical
component technologies, to which the other has no access.
Our analysis does not lose its bite in this context. When one team has a headstart, the
race can be viewed as a subcontest of the “best-of-(2n + 1)”contest in which one team has
accumulated a larger number of victories from prior battles. The headstart would not a¤ect
the prediction from the current model.
5.2.2
Unequally Weighed Battles
Another direct extension is to allow component battles to carry di¤erent weights for teams’
wins. To be more speci…c, let each contest include s
3 component battles, which are
i
carried out successively. A team is awarded a score yt if its player wins battle t, which can
be heterogenous across teams. A team wins the entire contest if and only if either (1) its
accumulated scores reaches a given cuto¤ before the rival team touches its …nish line, or (2)
it obtains a higher accumulated score than the rival, while either touches the …nish lines
after all s battles have been contested.
The setting allows each battle matters di¤erently in the entire contest, i.e., yt1 6= yt2 . For
instance, Normandy Invasion a¤ects the outcome of World War II more than the Battle of
Singapore. Second, it allows a given battle to contribute di¤erently to each team. A team,
by winning a battle, can move a bigger step closer to its …nish line than the other.
22
We now demonstrate that the fundamentals of our analysis continue to hold in this
context. Let the state of the contest be summarized by the tuple (~
y 1 ; y~2 ), where y~i is the
accumulated score of team i over the past battles. Let a battle t be contested in an arbitrary
state (~
y 1 ; y~2 ). Assume for simplicity that players do not receive private bene…t from winning
the battle. Again, let vi (~
y 1 ; y~2 ) be the continuation value of team i assessed when the contest
is in this state.
y 1 ; y~2 + yt2 )], while
Then player 1(t)’s e¤ective prize spread would be [v1 (~
y 1 + yt1 ; y~2 ) v1 (~
y 1 + yt1 ; y~2 )]. This alternative scoring rule does not change
player 2(t)’s is [v2 (~
y 1 ; y~2 + yt2 ) v2 (~
the reciprocity between team 1 and team 2, as alluded to by Lemma 1. Speci…cally, we will
y 1 + yt1 ; y~2 )],
y 1 ; y~2 + yt2 ) v2 (~
y 1 ; y~2 + yt2 )] = [v2 (~
continue to conclude that [v1 (~
y 1 + yt1 ; y~2 ) v1 (~
y 1 + yt1 ; y~2 ).
y 1 + yt2 ; y~2 ) = 1 v1 (~
y 1 ; y~2 + yt2 ), and v2 (~
because of v2 (~
y 1 ; y~2 + yt2 ) = 1 v1 (~
Analogous to the baseline setting, each battle would be common-valued static contest, and
its winner would be picked in an independent draw. Although winning a battle moves one
team towards the eventual victory by a di¤erent margin than it does the other, one team’s
one step further towards the …nish line implies the other team’s being pulled away from the
…nish line by a step of the same size. The symmetry highlighted by Theorem 1 persists.
In conclusion, this extension does not change the stochastic outcome of each battle, as
well as that of the entire contest. The same trade-o¤s and strategic mindset persist in the
alternative setting, although the exact formula for e¤ective prize spread and expected e¤ort
must be computed quantitatively di¤erently.
5.2.3
Asymmetric Valuations
In our baseline analysis, we assume that all players equally value the team prize. We now
relax this assumption. Assume that each player i(t)’s valuation of the team prize ! i(t) is
a random draw according to a continuous distribution function Fi(t) (! i(t) ). We continue to
allow for all possible information structures, such that the realization of ! i(t) may or may
not be known to player j(t). We normalize each player’s marginal e¤ort cost to one.
We assume …rst that players do not derive private bene…t from winning their own battles.
Suppose that the contest is currently in a state (k; l) with k; l < n+1. We continue to denote
by pi (k; l) a team i’s winning likelihood assessed when the contest is in a state (k; l). Then
player 1(t) maximizes
u1(t) =
1(t) (x1(t) ; x2(t) )p1 (k
= p1 (k; l + 1)! 1(t) +
+ 1; l)! 1(t) + (1
1(t) (x1(t) ; x2(t) )[p1 (k
1(t) (x1(t) ; x2(t) ))p1 (k; l
+ 1; l)
+ 1)! 1(t)
p1 (k; l + 1)]! 1(t)
x1(t) :
p2 (k + 1; l)]! 2(t)
x2(t) :
x1(t)
Similarly, player 2(t)’s expected payo¤ obtains as
u2(t) = p2 (k + 1; l)! 2(t) +
2(t) (x1(t) ; x2(t) )[p2 (k; l
23
+ 1)
(10)
A similar argument to Lemma 1 can be invoked here. Because p2 (k; l+1) = 1 p1 (k; l+1)
and p2 (k + 1; l) = 1 p1 (k + 1; l), the e¤ective prize spread for player 2(t) is rewritten as
[p2 (k; l + 1)
= f[1
p2 (k + 1; l)]! 2(t)
p1 (k; l + 1)]
= [p1 (k + 1; l)
[1
p1 (k + 1; l)]g ! 2(t)
p1 (k; l + 1)]! 2(t) :
Hence, we can conclude that in an arbitrary battle t, a player i(t) have an e¤ective prize
spread [p1 (k + 1; l) p1 (k; l + 1)]! i(t) . The coe¢ cient [p1 (k + 1; l) p1 (k; l + 1)] is determined
by the state of the contest, but it is symmetric across matched players. Hence, the change
in the state of the contest would scale up or down matched players’valuations of the win
! 1(t)
, remains constant
by the same factor. The ratio of players’e¤ective prize spread, i.e., !2(t)
regardless. Standard argument in contest or auction literature allows us to conclude that each
battle has a stochastic outcome ( t ; 1
t ), which is pre-determined by the characteristics
of players’prize valuations. It does not depend on the history of past battles, and the winner
of each battle is independently drawn. The entire set of results in our baseline analysis will
be preserved in the alternative setting.
However, history of plays would cause distortion to the extent that players derive private
bene…t from winning their own battles. Let i(t) be a player i(t)’s private reward for winning
the battle, regardless of his team’s win or less. In this case, a player i(t)’s e¤ective prize
spread would become i(t) + [p1 (k + 1; l) p1 (k; l + 1)]! i(t) . The expected payo¤ from team
prize is still scaled up or down by the same factor, i.e., [p1 (k + 1; l) p1 (k; l + 1)], across
the two players. The change in state, however, does not a¤ect the private prize i(t) . In this
case, the stochastic outcome of a battle would depend on the state of the contest when the
battle is being carried out, as it a¤ects the ratio of the two players’e¤ective prize spread,
1(t) +[p1 (k+1;l) p1 (k;l+1)]! 1(t)
i.e., 2(t)
. The winner of each battle would not be independently drawn.
+[p1 (k+1;l) p1 (k;l+1)]! 2(t)
However, it should be noted that the simultaneous ‡uctuation in expected team prize
still o¤sets the externality between battles to varying degree, depending on the particulars of
the contest. The distortion caused by history of plays tends to be less signi…cant than that
in a contest between individuals. The results would be qualitatively more robust especially
when the value of private bene…t is relatively small. A complete analysis of this extension
goes beyond the scope of this study. However, it is worth serious research e¤ort and will be
attempted by the authors in future work.
5.2.4
Non-deterministic Winner Selection Mechanisms
The baseline analysis focuses on a contest with deterministic winner selection mechanism,
i.e., one wins with certainty if his e¤ort exceeds that of his rival. In this part, we consider
24
an extension in which the winner selection mechanism includes noisy factors, such that the
player who contributes more e¤ort may not win.
We begin the discussion with the most popularly adopted Tullock contest success function
(see, for example, Klumpp and Polborn, 2006). Under this setting, a player i(t) has a winning
probability
t
xr1(t)
;
(11)
(x
;
x
)
=
2(t)
1(t) 1(t)
xr1(t) + xr2(t)
with rt 2 (0; 1). When rt approaches in…nitely, the Tullock contest converges into an all-pay
auction. This contest success function is axiomatized by Skaperdas 1996 and Clark and Riis
1998. Fu and Lu (2012) paves it a micro foundation in a noisy-ranking approach. Assume
for tractability that the realizations of players’marginal e¤ort costs are commonly known.7
The result stated in Theorem 1 does not depend on the prevailing winner selection mechanism. Each battle continues to be a common-value one-shot contest. With Tullock contest
technology, a player’s winning probability purely depends on the ratio between players’marginal cost. This result is well known for contests with small r, in which a pure-strategy
equilibrium exists. For brevity, we omit the proof for this statement in a setting with large
r, but it is available from the authors upon request. As a result, the entire set of results we
obtain from the all-pay auction setting would continue to hold.
To put it more generally, our results remain robust when the winner selection mechanism
is homogenous of degree zero, i.e., pi(t) (x1(t) ; x2(t) ) = pi(t) ( x1(t) ; x2(t) ), with > 0. Both
Tullock contest, and its limiting case of all-pay auction, demonstrate this property.
However, additional caution is needed when the winner selection mechanism does not satisfy this requirement. For instance, let a player i(t)’s winning probability take the following
form:
f (x1(t) )
p1(t) (x1(t) ; x2(t) ) =
;
(12)
f (x1(t) ) + f (x2(t) )
with f ( ) to be continuous and strictly increasing. Assume that f ( ) is concave, such that
a pure-strategy equilibrium exists. In a common-value one-shot contest with a prize v, the
equilibrium is determined by
f (x2(t) ) f (x1(t) )
c1
= 0
=
;
0
f (x2(t) ) f (x1(t) )
c2
f 0 (xi(t) )f (xj(t) )
v = ci ; 8i; j 2 f1; 2g; i 6= j:
[f (x1(t) ) + f (x2(t) )]2
7
It should be noted that Tullock contest model has limited tractability when the success function becomes
excessively elastic, i.e., when r is large. Due to the technical di¢ culty, the analysis on Tullock contest
remains incomplete. Alcade and Dahm (2010) and Wang (2010) demonstrate the existence of all-pay auction
equilibrium in Tullock contest with large r under complete-information settings. The equilibrium behavior
of a Tullock contest with large r and incomplete information has yet to be characterized in the literature.
25
f (x
)
1(t)
The ultimate winning probability f (x1(t) )+f
is independent of v if and only if f (x) = axr ,
(x2(t) )
with a > 0 (see Clark and Riis, 1996). To the extent that this property is absent, externalities
among battles arise. In that case, the stochastic outcome of a battle depends on the exact
value of the e¤ective prize spread, which is apparently state-dependent.
However, recall again that Theorem 1 holds regardless of the prevailing contest technologies. The balancing force it exercises remains in place to varying degree. It would continue
to partially o¤set the externality between battles, and counteract (partially) the distortion
caused by history of plays.
Furthermore, it is worthwhile to be pointed out that our results in symmetric contests
would not be a¤ected by winner selection mechanism. Consider a symmetric contest in
which all battles are contested between equally competent players. Suppose that the contest
technology of each battle allows for a symmetric equilibrium, which, for instance, is feasible
under ratio-form contest success function (16) with regularly-behaved production function
f ( ). In this case, the winner of each battle would be picked in an independent and fair draw,
with each to win with a probability of exactly 12 . The distortion caused by history of plays
on the outcomes of the contest is again fully o¤set. To put it di¤erently, it can be inferred
that the history-independence principle we establish in all-pay-auction setting tends to be
qualitatively more robust in a context where rival players are more closely matched in terms
of their competence.
Despite the independence in terms of winner selection mechanisms, history of plays would
still a¤ect the e¤ort supply in the contest. It remains curious how battles between …xed pairs
of players should be ordered in a sequence, under various contest technologies. The problem
is technically challenging, while a full solution is expected to generate rich implications on
contest design. These extensions will be explored by the authors in future e¤orts.
6
Concluding Remarks
In this paper, we study multi-battle-contest between two teams. The contest is comprised of a
series of pairwise battle fought between matched players from rival teams. We demonstrate in
an all-pay-auction setting that the well-studied “discouragement e¤ect”in individual multibattle contests do not appear. The history of plays in the sequential contest does not cause
distortion to the stochastic outcomes of future battles. In addition, the expected e¤ort of the
contest is independent of the temporal structure of the contest, i.e., the sequencing or clustering of component battles. These results yield rich implications for policy experimentation
and contest design.
The present paper contributes to the burgeoning literature on multi-battle contests by
introducing a setting that involves collective actions. Team competitions, as stated in In-
26
troduction, are pervasive in real world, while it has been rarely studied in way of formal
modelling. Our analysis unveils the fundamental di¤erence in players’ strategic mindsets
between individual contests and team contests.
Our analysis is one of the early steps towards a complete understanding of this intriguing
phenomenon. As discussed in Section 5, the framework leaves open a tremendous room for
many extensions. These variations would introduce a wide array of new elements that enrich
the strategic interactions in the game and uncover other forces involved in the trade-o¤s
players face in team competitions. They all deserve to be explored more comprehensively.
As we have demonstrated, the main logic of our baseline analysis extends beyond the current
setting, and it. Our technical results, as implied by our discussion on extensions, would also
be useful in analyzing team contests in alternative settings.
In addition to them, the framework can be extended in many other directions. In our
baseline analysis, we assume that players’ e¤ort costs are independently distributed. One
interesting but technically challenging extension is to allow players’cost distributions involve
team-speci…c characteristics, i.e., the costs of players from the same teams have a¢ liated
distributions. For instance, players of a tennis team often join training camps before major
tournaments. The joint training program a¤ects the performance of all players in a team.
In that case, the team contest would involve strategic information updating as players of
later rounds would infer the competence levels of their rivals by observing the past plays.
Alternatively, a player’s performance and incentives can be perturbed by various behavioral
factors. For instance, a player may use his teammates’performance as a reference to set his
own expectations: He can be pressurized when teammates perform well, or can be relieved of
pressure otherwise. Team contests provide an interesting context in which we can introduce
such peer e¤ects into formal modelling. These extensions, among many others, would give
rise to substantially more extensive strategic interaction, and broaden the scope of this
framework. They deserve to be studied seriously, and will be attempted by the authors in
future. In addition, this paper generates a number of testable hypotheses. The theoretical
predictions have yet to be tested either in …eld or in laboratory to test their limit and
to identify other factors that could be involved in the strategic trade-o¤s of players. An
experimental study is in progress to shed light on economic agents’behavior in multi-battle
team contest.
Appendix
Proof of Lemma 2
Proof. We consider common-valued all-pay auction. Players both value the prize for v.
27
Altogether, the analysis is conducted in three cases.
Case 1: Complete-Information All-Pay Auction
Proof. We …rst consider an all-pay auction in which players’marginal costs are commonly
known. Contestant 1 enjoys a rent of v cc12v , while contesant 2 simply ends up with zero
expected payo¤. The equilibrium can be characterized by the following equations:
F2 (x1 )v
c1 x1 = v
F1 (x2 )v
c2 x2 = 0:
c1 v
;
c2
Hence, their equilibrium mixed strategies are F2 (x2 ) = (1 cc21 ) + c1vx2 ; and F1 (x1 ) =
v
R cv2 c2 x2 c1
c1 c2 x22 c2
c 2 x1
. In the equilibrium, contestant 2 wins with a probability 0 v
dx2 = 2v2
=
v
v
c1 c2 ( cv )2
0
c1
:
2c2
=
Apparently, the expected e¤ort is also linear in v, and the coe¢ cienct is a
function of (c1 ; c2 )
2
2v 2
Case 2: Incomplete-Information All-Pay Auction with Two-Sided Asymmetric
Information.
Proof. Bidder i0 s marginal bidding cost is ci follows distribution Fi (ci ) on support [ci ; ci ]
where ci > 0. The density is denoted by fi (ci ) 2 [f ; f ] where f > 0. Consider the following
di¤erential equation system
0
1 (b)
=
0
2 (b)
=
2 (b)
;
vf1 ( 1 (b))
1 (b)
;
vf2 ( 2 (b))
(13)
(14)
with boundary conditions i (0) = ci ; i = 1; 2. i (b); i = 1; 2 can be pinned down starting
from b0 = 0. As the slopes are known from (13) and (14) at b0 = 0, i (b) can be pinned
at a higher b1 :Then the slopes at b1 can be pinned down, and we can move to a higher b2 .
Clearly, i ( ); i = 1; 2 determined by (13) and (14) strictly decrease.
If there exists a common b > 0 such that i (b) = ci ; i = 1; 2, we de…ne ci = ci , i = 1; 2.
Otherwise suppose i ( ) …rst reaches ci at some b > 0, while j (b) > cj . Let ci = ci , and
de…ne j as the weaker bidder. We show there exists cj 2 (cj ; cj ) and common b > 0 such
that i (b) = ci ; i = 1; 2 where i ( ); i = 1; 2 are solutions of (13) and (14) with boundary
conditions i (0) = ci ; i = 1; 2.
Extend the support of fj (cj ) to allow cj < cj , and de…ne fj (cj ) = fj (cj ), 8cj < cj .
8^
cj 2 [cj ; cj ], consider i ( ); i = 1; 2 determined by (13) and (14) with boundary conditions
^j . De…ne b(^
cj ) such that i (b) = ci and de…ne (^
cj ) = j (b(^
cj )).
i (0) = ci = ci ;and j (0) = c
28
Clearly, b(^
cj ) is continuous, and (^
cj ) is also continuous. In addition, when c^j = cj , we have
(cj ) > cj . When c^j = cj , we have (cj ) < cj . Therefore, there must exist c^j 2 (cj ; cj ) such
that (^
cj ) = j (b(^
cj )) = cj . De…ne cj = c^j and b = b(cj ). By de…nition, we have (13) and
(14) with boundary conditions i (0) = ci ; i = 1; 2 admit solution of decreasing functions
i ( ); i = 1; 2 such that i (b ) = ci ; i = 1; 2.
We are now ready to characterize the bidding equilibrium. In a winner-take-all all pay
auction with two bidders who are privately endowed with information about their bidding
costs, we derive the bidding equilibrium. We assume a tie-breaking rule which favors the
stronger bidder i.
Bidder i values the prize by v; i = 1; 2. Bidder i0 s marginal bidding cost is ci follows
distribution Fi (ci ) on support [ci ; ci ] where ci > 0. The density is denoted by fi (ci ) 2 [f ; f ]
where f > 0.
We look for (weakly) monotonic pure strategy bidding equilibrium denoted by bi (ci ); i =
1; 2;which are continuous and only potentially have mass point at zero. Clearly, bi (ci ); i = 1; 2
must share a common upper bound b. Assume bi (ci ) = 0; i = 1; 2 when ci ci 2 [ci ; ci ], and
bi (ci ); i = 1; 2 strictly decrease when ci ci . ci are de…ned as above. For stronger bidder i,
de…ne 'i (b) bi 1 (b) 2 [ci ; ci ] = [ci ; ci ]; where b 2 [0; b].
i (bi ; ci )
= v Pr(bj (cj )
bi )
ci bi ; 8ci 2 [ci ; ci ]; 8bi
0:
(15)
For stronger bidder i of type ci 2 [ci ; c^i ] chooses bi to maximize his expected payo¤:
max
bi 0
i (bi ; ci )
= v Pr(bj (cj )
c i bi
bj 1 (bi ))
c i bi
Fj (bj 1 (bi ))]
ci bi :
= v Pr(cj
= v[1
bi )
The …rst order condition is given by:
dbj 1 (bi )
vfj (bj (bi ))
dbi
1
For weak bidder j, de…ne 'j (b)
j (bj ; cj )
bj 1 (b) 2 [cj ; cj ]; where b 2 [0; b].
cj bj ; 8cj 2 [cj ; cj ]; 8bj
= v Pr(bi (ci ) < bj )
For weak bidder j of type cj 2 [cj ; cj ] chooses bj
max
bj 0
(16)
ci = 0:
j (bj ; cj )
0 to maximize his expected payo¤:
= v Pr(bi (ci ) < bj )
c j bj
= v Pr(ci < bi 1 (bj ))
c j bj
Fi (bi 1 (bj ))]
c j bj :
= v[1
29
0:
The …rst order condition is given by:
vfi (bi 1 (bj ))
dbi 1 (bj )
dbj
cj = 0:
(17)
Solving bi (ci ); i = 1; 2 is equivalent to solving for 'i (b); i = 1; 2. (16) and (17) can be
rewritten as
'j (b)
; i; j = 1; 2; i 6= j;
(18)
'0i (b) =
vfi ('i (b))
with boundary conditions 'i (0) = ci ; i = 1; 2:Clearly, i ( ); i = 1; 2 determined by (13) and
(14) with boundary conditions i (0) = ci = ci ;and j (0) = cj must satisfy (18). In addition,
i (b ) = ci ; i = 1; 2. Let bi (ci ) 2 [0; b ]; ci 2 [ci ; ci ]; i = 1; 2 denote the inverse of i ( ). For
the weaker bidder j, we de…ne bj (cj ) = 0; 8cj 2 [cj ; cj ]. Note for the weaker bidder j; if he is
type cj , then his optimal bid is zero which renders a zero payo¤ for him. This means that for
a type cj > cj , the optimal bid must be zero. The reasons are as follows. If type cj > cj has
an optimal bid that di¤ers from zero and renders positive expected payo¤, then type cj can
adopt the same bid and earn a positive expected payo¤. Let i (ci ) = bi (ci ; v = 1); i = 1; 2.
We now establish that bi (ci ; v) = i (ci ) v; i = 1; 2 are bidding equilibrium for a general prize
v.
We claim that equilibrium bidding strategy takes the form bi (ci ; v) = i (ci ) v, where
i (ci ) = bi (ci ; v = 1).
For this purpose, it su¢ ces to show that for given v = v; 8 > 0; i = 1; 2; we have that
~
~bi (ci ) = bi (ci ) are the new equilibrium. De…ne inverse functions '
~ i (~b) ~bi 1 (~b) = 'i ( b ) 2
[ci ; ci ]; ; i = 1; 2 on [0; b]. We need to show that they satisfy
'
~ 0i (~b) =
'
~ j (~b)
; i; j = 1; 2; i 6= j;
vfi (~
'i (~b))
This is,
~
~
b 1
'0i ( )
=
This reduces to
~b
'0i ( ) =
'j ( b )
~
vfi ('i ( b ))
; i; j = 1; 2; i 6= j;
~
'j ( b )
~
vfi ('i ( b ))
' (b)
; i; j = 1; 2; i 6= j;
j
; i; j = 1; 2; i 6= j:
which is the same as (18), i.e., '0i (b) = vfi ('
i (b))
We thus have bi (ci ; v) = bi (ci ; v) for any v. Setting v = 1; we have bi (ci ; ) =
bi (ci ; 1) = i (ci ) . This shows that bi (ci ; ) is linear in .
30
Case 3: Incomplete-Information All-Pay Auction with One-Sided Asymmetric
Information
Proof. In this case, we assume without loss of generality that player 1’s bidding cost,
c1 > 0, is common knowledge, while player 2’s cost, c2 , is privately known to himself. Let c2
be continuously distributed over an interval [c; c]. The distribution of c2 , F (c2 ), is commonly
known. This all-pay auction is in principle similar to the one-sided asymmetric contest
proposed and studied by Morath and Münster (2010). The following proof adopts a similar
approach to that of Morath and Münster (2010).
We de…ne Bi (xi ) as the ex ante bid distribution of player i. The function must demonstrate the following properties:
P1: Bi ( ) is atomless above zero.
P2: The distribution of bids must have the same supports, and the lower support is zero.
P3: min(B1 (0); B2 (0)) = 0.
P4: Bi ( ) is strictly increasing, 8i 2 f1; 2g.
These properties are all standard in auction and contest games. We verify them brie‡y.
We …rst verify the claim of P1. Suppose for contrary that Bi (xi ) has a positive probability
mass at x^ > 0. It implies that player j’s expected payo¤ function has an upward jump at
x^. Player i would strictly gain by choosing e¤ort in some -neighborhood above x^ to e¤ort
in an "-neighborhood below x^. That is, player i would not bid xi 2 (^
x "; x^]. Given that,
player j would not place positive probability mass at x^.
We then verify P2. We …rst argue that they have the same upper support b. Given that
one player’s bid is bounded by b, the other player has no incentive to bid more that, because
an e¤ort b allows him to win with probability one. We then argue that they have the same
lower support b. Suppose for contrary that one player i’s minimum bid bi > b. Then player
j has no incentive to exert an e¤ort xj 2 (b; bi ]. We then argue b = 0. Suppose for contrary
b > 0. Then a player i would prefer simply bidding zero to bidding xi in a "-neighborhood
above b. Contradiction results.
We continue to verify P3. Suppose for contrary that both B1 (0) and B2 (0) are positive.
Given Bj (0) > 0, player i strictly prefers bidding xi within a "-neighborhood above zero to
zero. Contradiction results.
We again de…ne the player i with Bi (0) > 0 to be the ”weaker player”, and the other to
be the “stronger bidder”. We assume that in the zero probability event that x1 = x2 = 0,
the stronger bidder is favored under the tie-breaking rule.
We …nally verify P4. Suppose for contrary that there exists an interval (x; x) over which
Bi ( ) does not strictly increase with Bi (x) = Bi (x) and Bi (x) > Bi (x) for x > x. It implies
that player i does have a bid over this interval. Hence, for player j, an e¤ort x must be
strictly preferable to any bid within this interval, also that within an "-neighborhood above
31
x. That is, player j does not bid xj 2 (x; x + "). Given that, player i would not bid an
e¤ort with the "-neighborhood above x. It thus contradicts with the fact of Bi (x) > Bi (x)
for x > x.
We then claim that players have the following strategies. Player 1 has a mixed-bidding
strategy, which is depicted by a distribution function B1 (x1 ). Player 2 plays a pure bidding
strategy b(c2 ), which is mapping b : [c; c] ! R+ . Player 2 may stop placing positive bid if c2
exceeds a threshold c^. The bidding function b(c2 ) is continuous and di¤erentiable over for
c2 2 [c; c^]. It is also strictly decreasing over this interval.
We now verify that player 2 does not randomize his bid while player 2 has a pure strategy.
Suppose for the contrary. Let x(c) and x(c) be the upper and lower supports of the e¤ort
bidding strategy of a player 2 with e¤ort cost c, respectively, with x(c) > x(c). By bidding
x(c), his payo¤ is B1 (x(c))v cx(c). If his bidding strategy involves a bid x > x(c), the payo¤
requires B1 (x)v cx = B1 (x(c))v cx(c) , [B1 (x) B1 (x(c))]v = [x x(c)]c. Alternatively,
if the bidding strategy involves a bid x0 < x(c), the payo¤ requires [B1 (x(c)) B1 (x0 )]v =
[x(c) x0 ]c.
We then consider a player 2 of type c0 > c. A player of this type will strictly prefer
bidding x(c) to bidding x > x(c): In this case, B1 (x)v c0 x < B1 (x(c))v c0 x(c) must
hold. Hence, we can conclude that x(c0 ) x(c). We then consider a player of type c00 < c.
A player of this type will strictly prefer bidding x(c) to bidding x0 > x(c): In this case,
B1 (x0 )v c00 x0 < B1 (x(c))v c00 x(c) must hold. Hence, we can conclude that x(c00 ) x(c).
As a result, we see that only type c’s strategy may involve a bid in the interval (x(c); x(c)).
Because players’types are continuously distributed, B2 ( ) cannot be strictly increasing over
the interval (x(c); x(c)). Contradiction results.
We then verify that player 1 does not play a pure strategy. Suppose …rst the contrary
that player 1 bids zero with probability one. Then player 2 would win with certainty by an
e¤ort ". Then player 1 would place positive e¤ort instead. Contradiction. We then suppose
that player 1 bids a positive e¤ort x with probability one. First assume x = vc . Then player
2 of all types would give up by bidding zero. In response, player 1 would not place that
e¤ort with probability one. We then assume x < vc . De…ne c~ = xv . Then player 2 of all types
c < c~ would place an e¤ort x + ". Apparently, it cannot be a part of the equilibrium.
We are now ready to characterize the equilibrium.
We …rst consider player 1. His expected payo¤ when bidding x1 is given by
E( 1 (x1 )) = [1
F (b2 1 (x1 )]v
c1 x1 :
(19)
As player 1 randomizes over his bids, the equilibrium must satis…es the …rst order condition
F 0 (b2 1 (x1 ))v = c1 b02 (b2 1 (x1 ));
32
(20)
for all x1 2 [0; b]. A solution to the di¤erential equation must have
c1 b2 (c2 ) = d
(21)
F (c2 )v:
Let us de…ne 2 v d. The threshold for player 2’s exit would exist if F
We now turn to player 2. By placing a bid x2 , his payo¤ is given by
E( 2 (x2 )) = B1 (x2 )v
c2 x2 .
1
(
2)
< c.
(22)
The equilibrium must satisfy
B10 (x2 )v = c2 :
(23)
By results obtained above, it is further rewritten as
B10 (x2 )
=
1 d c 1 x2
( v )
F
v
(24)
;
which leads to
R x2 F
B1 (x2 ) =
0
R min(F
=
v
1(
1
b2 (x2 )
What remains is to determine 1 and
support. It requires B1 (b2 (c)) = 1. That is
R min(F
1 d c1 z
( v )
1(
2 );c)
c
2.
dz + 1
2 );c) c
dF (c) +
c1
(25)
1:
Recall that B1 ( ) and B2 ( ) have the same
c
dF (c) +
c1
1
(26)
= 1:
Further recall that it is impossible to have both B1 ( ) and B2 ( ) have positive probability
mass at zero bid. B2 ( ) has a positive probability mass if and only if F 1 ( 2 ) < c. In that
case, 1 = 0, and 2 is the solution to the equation
RF
c
If F 1 (
equation
2)
1(
2)
c
dF (c) = 1.
c1
c, B2 ( ) has no probability mass at zero. Then
(27)
1
is the solution to the
Rc c
dF (c) + 1 = 1:
(28)
c
c1
Apparently, neither 1 nor 2 depends on v.
Rc
To further understand that, rewrite (10) as [ c cdF (c)]=c1 + 1 = 1 , E(c2 )=c1 + 1 = 1.
If E(c2 ) < c1 , B1 (x1 ) must have a positive probability mass at zero. However, if E(c2 ) > c1 ,
R F 1( 2) c
dF (c) = 1 must yield a solution to 2 with F 1 ( 2 ) < c.
1 = 0, and c
c1
Player 2’s bidding strategy is written as b2 (c2 ) = 1 cF1 (c2 ) v: His bid linearly increases
with v. Consider an alternative contest with a prize purse v 0 = v, with > 0. Then player
33
2’s bidding strategy would become ~b2 (c2 ) = 1 cF1 (c2 ) v. As a result, player 1’s winning
probability by bidding x1 would be the same as that in the original contest where he bids
x1 .
~1 ( x1 ) in the alternative contest
What remains is to show player 1’s bidding strategy B
~1 ( x1 ) = B1 (x1 ). The probability of player 1’s bid falling below an arbitrary level
satis…es B
x1 is given by
R
1(
2 );c) c
~1 ( x1 ) = ~min(F
dF (c) + 1 :
(29)
B
1
b2 ( x1 )
c1
R
1(
2 );c) c
~1 ( x1 ) = min(F
Because ~b2 1 ( ) is linear in v, ~b2 1 ( x1 ) = b2 1 (x1 ). As a result, B
dF (c)+
c1
b2 1 (x1 )
1 = B1 (x1 ), which demonstrates homogeneity of degree zero. That is, he places each bid
x1 with the same probability as he does x1 in the original contest. In conclusion, the ex
ante winning likelihood of either player does not depend on v.
Proof of Theorem 3
Proof. Suppose that the contest is in a state (k; l), and a total of t 1 battles have been
carried out, with t 1 k + l. Further assume that the upcoming battle t is nontrivial, i.e.,
k; l < n + 1. Because we normalize the team prize to one, the continuation value v1 (k; l) is in
fact equivalent to the probability of team 1 winning the contest assessed at a state (k; l), i.e.
v1 (k; l) = p1 (k; l). Team 1 may win the contest in any of n l + 1 terminal states: f(n + 1; l),
(n + 1; l + 1); : : : ; (n + 1; n)g. From a state (k; l), players of team 1 needs to win another
n + 1 k battles, while allowing those of team 2 to win no more than n l battles.
Let Pr( (n + 1; l0 )j (k; l)) denote the conditional probability of team 1’s winning the contest in a terminal state (2n+1; l0 ), with l0 2 fl; : : : ; ng, when the contest is currently in a state
Pn
0
(k; l). The continuation value v1 (k; l) can be simply written as v1 (k; l)
l0 =l Pr((n + 1; l )j (k; l)).
The probability Pr( (n + 1; l)j (k; l)), i.e., that of team 1’s winning the next n + 1 k battles
k
consecutively, is simply given by 1 (n + 1 kj n + 1 k)jt+n
. The probability of its clost
k
ing the team contest in a terminal state (n+1; l+1) is given by 1 (n + 1 kj n + 2 k)jt+n+1
t
k
(1 t+n+1 k )
kj n + 1 k)jt+n
. It should be noted that 1 (n + 1 kj n + 2 k)jt+n+1
1 (n + 1
t
t
is the probability of the event that team 1 wins any n+1 k battles in battles t to t+n+1 k.
To compute the probability of the team contest ending in state (n + 1; l + 1), we have to exclude from it the probability of the event that team 1 wins all battles in battles t to t + n k,
but loses battle t + n + 1 k.
By the same token, the probability of the team contest ending in a terminal state (n+1; l+
k
k
2) is given by 1 (n + 1 kj n + 3 k)jt+n+2
(1 t+n+2 k )
kj n + 2 k)jt+n+1
.
1 (n + 1
t
t
We sum them up, and obtain the formula stated in Theorem 3.
34
k
Proof of Theorem 4
Proof. The following fact must hold:
v1 (k; l) =
t v1 (k
+ 1; l) + (1
t )v1 (k; l
(30)
+ 1);
which leads to
+ 1; l) v1 (k; l + 1)] ,
v1 (k; l) v1 (k; l + 1)
:
v1 (k; l + 1) =
v1 (k; l)
v1 (k; l + 1) =
v1 (k + 1; l)
t [v1 (k
(31)
t
By Theorem 3, we must have
P2n
(k+l+1)
m=n+1 k
v1 (k; l + 1) =
+
1 (n
[
1 (n
t+m+1
+1
kj 2n
+1
kj m)
l)j2n+1
t+1 ]:
k
(32)
The following facts must hold:
kj m 1)jt+m
Fact 1 1 (n + 1 kj m)jt+m
1 (n
t+1
t+1 = t+m
t+m 1
)
(n
+
1
kj
m
1)j
+(1
1
t+m
t+1
1
1
kj m 1)jt+m
Fact 2 1 (n + 1 kj m)jt+m
=
1 (n
t
t+1
t
1
kj m 1)jt+m
.
+(1
1 (n + 1
t)
t+1
Fact 1 leads to
1 (n
t+m
=
1 (n
By Fact 2, for m
[
=
t+m
t+m
n+2
"
+(1
t+m
t
+1
1 (n
1
1
(1
t+m )
1 (n
+1
1)jt+m
t+1
kj m
1
:
k,
kj m)
t+m 1
t
1 (n
t+m
1
kj m 1)jt+m
t+1
kj m 1)jt+m
1 (n + 1
t+1
1 (n
t
t+m
=
kj m)jt+m
t+1
+1
1 (n
1)jt+m
t+1
kj m
t+m
]
t+1
t)
kj m
1)t+m
t+1
1 (n
kj m
1)jt+m
t+1
1
1 (n
kj m
1)jt+m
t+1
1
kj m)jt+m
t+1
(1
+1
1
+1
#
1)t+m
t+1
kj m
1
1
t+m
1 (n
+1
kj m
1)jt+m
t+1
t+m
1 (n
+1
kj m
1)jt+m
t+1
1
:
Further, by Fact 1,
t
=
=
t
t
"
t+m
1 (n
+1
t+m
1 (n
+1
kj m)jt+m
t+1
t+m )
+1
kj m
+ 1 kj m
1
1)jt+m
t+1
+1
kj m
1)jt+m
t+1
1 (n
1 (n
35
1 (n
1
:
1)jt+m
t+1
1
1
#
(33)
Hence,
v1 (k; l)
( P
v1 (k; l + 1)
t+m 1
2n (k+l)
m=n+1 k
=
+
( P
2n
[ t+m
kj m) t
1 (n + 1
kj 2n + 1 k l)j2n+1
]
1 (n + 1
t
(k+l+1)
m=n+1 k
[ t+m+1
kj m)
1 (n + 1
2n+1
kj 2n k l)jt+1 ]
1 (n + 1
+
]
)
)
t+m
]
t+1
(34)
n+1+l
]
k+l+1
+[ 1 (n + 1 kj 2n + 1 k l)j2n+1
]
kj 2n
1 (n + 1
t
)
( P
t+m
1
2n (k+l)
]
(n
+
1
kj
m)
[
1
t+m
k
t
:
+
Pm=n+2
t+m
2n (k+l+1)
kj m) t+1 ]
1 (n + 1
m=n+1 k [ t+m+1
= [
1 (n
n+2+l
+1
kj n + 1
k)
k
l)j2n+1
t+1 ]
(35)
t+m 1
2n (k+l)
We match the ith item of the series f[ t+m
kj m) t
]gm=n+2
1 (n + 1
t+m 2n (k+l+1)
ith item f[ t+m+1
kj m) t+1 ]gm=n+1 k . By (35), we then obtain
1 (n + 1
( P
)
t+m 1
2n (k+l)
(n
+
1
kj
m)
[
]
1
t+m
k
t
Pm=n+2
t+m
2n (k+l+1)
(n
+
1
kj
m) t+1 ]
[
1
t+m+1
m=n+1 k
)
(
P2n (k+l)
t+m 1
(n
+
1
kj
m)
]
[
1
t+m
k
t
=
P2nm=n+2
t+m 1
(k+l)
(n
+
1
kj
m
1) t+1 ]
[
1
t+m
m=n+2 k
P2n (k+l)
1
kj m 1)jt+m
=
kj m)jt+m
1 (n + 1
1 (n + 1
t
m=n+2 k
t+1
t+1
Further, note [ n+2+l
Hence, we must have
v1 (k; l)
=
t
+
P2n
1 (n
kj n + 1
k)
n+1+l
k+l+1
=
1 (n
t
+1
to the
.
k)n+2+l
t+1 .
kj n + 1
v1 (k; l + 1)
1 (n
+1
1 (n
+1
k)n+2+l
t+1
kj n + 1
(k+l)
m=n+2 k
+[
+1
k
t
1 (n
kj 2n + 1
kj m)jt+m
t+1
+1
l)j2n+1
t
k
1 (n
1 (n
+1
+1
kj 2n
kj m
1)jt+m
t+1
k
l)j2n+1
t+1 ]:
1
First consider the sum
t
+
P2n
1 (n
+1
(k+l)
m=n+2 k
kj n + 1
t
1 (n
k)n+2+l
t+1
+1
All elements are canceled out, except
boils down to
v1 (k; l)
=
1 (n
1 (n
kj 2n
t
+1
+1
kj m
1)jt+m
t+1
1
:
l)j2n+1
t+1 . The entire sum
k
v1 (k; l + 1)
1 (n
t
kj m)jt+m
t+1
+1
kj 2n
k
l)j2n+1
t+1
[
1 (n
+1
kj 2n + 1
k
l)j2n+1
t
= [
1 (n
+1
kj 2n + 1
k
l)j2n+1
t
1 (n
(1
36
+1
t)
kj 2n
1 (n
k
+1
l)j2n+1
k+l+2 ]
kj 2n
k
l)j2n+1
t+1 :
Again, by Fact 2,
v1 (k; l)
=
v1 (k; l + 1)
1 (n
t
kj 2n
l)j2n+1
t+1 :
k
We then conclude
v1 (k + 1; l)
v1 (k; l)
v1 (k; l + 1) =
v1 (k; l + 1)
t
=
1 (n
kj 2n
k
l)j2n+1
t+1 ;
(36)
which completes the proof.
Proof of Theorem 5
Proof. From (11), we have
Pt
1
k=0
Pt 1
k=0
E(xt ) =
=
=
t
(
It should be noted that
E(xt ) =
Pr( (k; l)j t)E(xi(t) (k; l))
fPr( (k; l)j t) t [ t + v1 (k + 1; l) v1 (k; l + 1)]g
)
Pt 1
k=0 Pr((k; l)j t) t
:
P
+ tk=01 Pr((k; l)j t)[v1 (k + 1; l) v1 (k; l + 1)]
Pt
1
k=0
t t
+
t
Pr((k; l)j t) boils down to one. Hence, (12) reduces to
Pt
1
k=0
Pr((k; l)j t)[v1 (k + 1; l)
By Theorem 4, we must have, for k; l
1 (kj k
+ l)jt1
v1 (k; l + 1)]:
(38)
n, min(k; l) < n.
Pr( (k; l)j t)[v1 (k + 1; l)
=
(37)
1
v1 (k; l + 1)]
1 (n
kj 2n
k
l)j2n+1
t+1 ;
If max(k; l) n + 1; Pr( (k; l)j t)[v1 (k + 1; l) v1 (k; l + 1)] boils down to zero, in which
case team 1 has won. If t > n + 1, and t 1 k > n + 1, Pr((k; l)j t)[v1 (k + 1; l) v1 (k; l + 1)]
also boils down to zero, in which case team 2 has won.
In the knife-edge case of t = 2n + 1 and k = l = n, Pr((k; l)j t)[v1 (k + 1; l) v1 (k; l + 1)]
is simply given by 1 (nj 2n)j2n
1 , with t = k + l + 1.
Hence, we must have
E(xt ) =
t t
+
t
=
t t
+
t
=
t t
+
t
Pn
k=0
Pn
k=0
Pr((k; l)j t)[v1 (k + 1; l)
1 (kj k
1 (nj 2n)j t
+
l)jt1 1
1 (n
v1 (k; l + 1)]
kj 2n
k
l)j2n+1
t+1
(39)
:
37
where 1 (nj 2n)j t is the overall probability of team 1’s winning exactly n out of the 2n
battles, excluding battle t. Because the history-independence result of Theorem 2, the
outcome of each of the 2n battles is an independent draw. Reordering these battles does not
change 1 (nj 2n)jt0 6=t . Apparently, neither t nor t depends on the order, which completes
the proof.
Proof of Theorem 6
Proof. The case of max(k; l) n + 1 is trivial. So we focus on the case of k; l < n + 1. Let
#z be the number of battles included in a cluster z.
We consider …rst the last cluster z that contains more than one battle. Without loss of
generality, let t be the battle with smallest index in the cluster. Aparently, the stochastic
outcomes of battles subsequent to those in this cluster, which are not clustered, will not be
a¤ected by previous outcomes either (by Theorem 2). The win probability of other battles
in the same cluster does not depend on battle t’s outcome either. Let i(t) (x1(t) ; x2(t) ) be the
probability of player i(t)’s winning the battle for a given e¤ort pro…le (x1(t) ; x2(t) ).
In this case, a player 1(t) receives t if he wins. If he wins, the contest may enter any
state (k + 1 + g; l + #z 1 g), with g 2 f0; : : : ; #z 1g, after all the battles in z are fought.
If he loses, then the contest may enter any sate (k + g; l + #z g), with g 2 f0; : : : ; #z 1g,
after all the battles in z are fought. As a result, his e¤ective spread amounts to
9
8
>
> P#z 1 [v (k + 1 + g; l + #z 1 g) ~ (gj #z 1) t+#z 1 ] =
<
1
1
g=0
t+1
;
(40)
t+
t+#z 1
P#z 1
>
>
~
;
:
[v
(k
+
g;
l
+
#z
g)
(gj
#z
1)
]
1
1
g=0
t+1
t+#z 1
where ~ 1 (gj #z
#z
is the probability of the event that team 1 wins g out of the
1)
t+1
1 simultaneous battles in cluster z, excluding battle t. We then have
#z 1
(42) =
t+
X
[v1 (k + 1 + g; l + #z
1
g)
v1 (k + g; l + #z
g)]
~ 1 (gj #z
t+#z 1
1)
:
t+1
g=0
(41)
Similarly, the e¤ective prize spread of player 2(t) is
#z 1
t+
X
[v2 (k + g; l + #z
g)
v2 (k + 1 + g; l + #z
g=0
1
g)]
~ 1 (gj #z
t+#z 1
1)
:
t+1
(42)
We apply Lemma 1, and obtain that the two players must have (43) = (44). It implies
that all battles in z are common-valued, and each battle t0 in this cluster has a stochastic
outcome ( t0 ; 1 t0 ), which is purely determined by matched players’cost conditions. Hence,
t+#z 1
~ 1 (gj #z 1)
= 1 (gj #z 1)jt+#z 1 .
t+1
t+1
38
We further consider the sum in (43). Recall
v1 (k + 1 + g; l + #z
=
if k + g; l + #z
rewritten as
1 ( (n
g=max(0;l+#z 1 n)
=
1 (n
kj 2n
g)
v1 (k + g; l + #z
(k + 1 + g)j 2n + 1
(t + #z
g)
1))j2n+1
t+#z
g < n + 1, and it boils down to zero otherwise. Hence, the sum is
1
min((#z
P1;n k)
+ 1)
1
n
k
1 (n
(k + g)j 2n + 1
(t + #z
1))j2n+1
t+#z
1 (gj #z
1
1)jt+#z
t+1
l)j2n+1
t+1 :
Apparently, all battles in z are no di¤erent from each other as they are fought simultaneously. The e¤ective prize spread for an arbitrary battle t0 can be written as t +
kj 2n k l)j t0 ;>k+l , where 1 (n kj 2n k l)j t0 ;>k+l is the probability of team
1 (n
1 winning exactly n k out of the 2n k l battles after the contest enters the state (k; l),
excluding battle t0 .
We can then move back to the immediately preceding cluster. Suppose that the cluster
contains more than one battle. Consider an arbitrary battle t in the cluster. The stochastic
outcomes of battles subsequent to those in the cluster do not depend on the outcomes of
the battles in this cluster. Hence, we can repeat the above exercise. For a given state
(k; l), we conclude that the e¤ective prize spread for players i(t), i = 1; 2, is given by
kj 2n k l)j t;>k+l . Again, the stochastic outcome of a battle t in the cluster
t+
1 (n
is predetermined by players’cost conditions, which is given by ( t ; 1
t ).
If the cluster contains only one battle, the analysis would be trivial.
The same exercises can be repeated to analyze all preceding clusters. It can be concluded
that all battle’s winner is picked in an independent draw, with a stochastic outcome of ( t ; 1
t ). At an arbitrary state (k; l) with k; l < n+1, an arbitrary battle t elicits an expected e¤ort
kj 2n k l)j t;>k+l ]. By repeating the exercise in the proof of Theorem 5,
1 (n
t[ t +
we can conclude that each battle elicits an expected total e¤ort t t + t 1 (nj 2n)j t , which
does not depend on how these battles are clustered. We then complete the proof.
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