Team Contests with Multiple Pairwise Battles∗
Qiang Fu†
Jingfeng Lu‡
Yue Pan§
November 2013
Abstract
Two teams with an equal number of players compete in a contest. Players from rival teams
form pairwise matches to fight in a series of component battles. A team wins if and only if its
players secure a sufficient number of victories. Each player benefits from his team’s win, while
he can also receive a private reward for winning his own battle. We find that the outcomes of
past battles do not distort the outcomes of future battles. Neither the total expected effort nor
the overall outcome of the contest depends on the contest’s temporal structure or its feedback
policy.
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 co-editor Andy Skrzypacz and three anonymous reviewers for very thoughtful and
detailed advice on revising our paper. We are grateful to Kai Konrad for an inspiring and encouraging
discussion. We thank Michael Baye, Yan Chen, John Conley, Luis Corchón, Andrew Daughety, Jeffrey Ely,
Tilman Klumpp, Dan Kovenock, Roger Lagunoff, David Malueg, Jennifer Reinganum, Brian Roberson, Ron
Siegel, Quan Wen, Myrna Wooders, and audiences at the 2012 Spring Midwest Economic Theory Conference
(Indiana University); the 2012 Society for the Advancement of Economic Theory Conference (Queensland
University); and the economics departmental seminar at Vanderbilt University for helpful comments and
suggestions. The usual disclaimer applies. Qiang Fu thanks the Max Planck Institute of Public Finance
and Tax Law for its support and hospitality; part of the project was executed when Fu visited the institute.
Jingfeng Lu is grateful to Vanderbilt University, Ohio State University and Northwestern University for kindly
hosting his sabbatical leave visits.
†
Qiang Fu: Department of Business Policy, National University of Singapore, 1 Business Link, Singapore,
117592; Tel: (65)65163775, Fax: (65)67795059, Email: [email protected].
‡
Jingfeng Lu: Department of Economics, National University of Singapore, Singapore 117570. Tel :
(65)65166026. Email: [email protected].
§
Yue Pan: Department of Economics and Department of Chemical Engineering, National University of
Singapore. Email: [email protected].
1
Introduction
In many competitive situations, contenders meet each other on multiple fronts and the ultimate outcome is determined by aggregating the parties’ performance in a series of confrontations (see Konrad, 2009). These competitions often take place between teams or groups,
as coalitions of affiliated but independent players: Players from rival teams form pairwise
matches and compete head-to-head; they take decentralized actions to vie for victories in
their own battles, which contribute to their teams’ eventual win.
This phenomenon of collective competitions is widespread, and we label them “team contests with multiple pairwise battles.” Many contentions, conflicts, and disputes take place
between groups or camps with diverging social, political, or economic interests. Consider,
for instance, general elections in most democracies. Candidates representing rival parties
compete head-to-head for legislative seats in each constituency; a party is able to form a
government only if it achieves majority status.1 Various collective decision-making processes
also resemble a team contest. As another example, consider the often observed policy debates on regional or international issues, e.g., pollution standards, free-trade agreements,
cooperation on defense or anti-terrorism security, nuclear disarmament, cross-border banking
regulatory frameworks, forming/maintaining a monetary union, etc. Within each country
involved, the final consensus on a given issue will affect various domestic interest groups
differently, and competing groups lobby to advocate for their preferred policy alternatives in
individual countries. Final agreement is reached by aggregating member countries’ decisions
across the region or international community. Analogously, a large-scale military operation
usually includes a series of separate battles between matched individual units, e.g., as in
World War II.
Such collective competitions are also prevalent in industry–for instance, the rivalry between multinational corporations, such as Sony vs. Samsung, Toyota vs. Hyundai, etc.
Globalized competition is typically executed by firms’ regional divisions in separate market
segments, which are usually independently operated profit centers. 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. A project–e.g.,
the development of a new product–is typically split and sourced to member entities within
each network. Moreover, team contests are exemplified by many sports events with team
titles, in which the competitions consist of a series of component individual matches, such
as the Ryder Cup in men’s and women’s golf, the Davis Cup and the Fed Cup in men’s and
1
Cox and Magar (1999) and Hartog and Monroe (2008) demonstrate empirically that a political party’s
majority status in a national legislative body is a valuable asset worth considerable collective effort to attain.
The competition for majority status between political parties thus resembles a team contest in which a team’s
victory benefits all individual members.
2
women’s tennis, the Thomas Cup and the Uber Cup in men’s and women’s badminton, and
the Swaythling Cup and the Corbillon Cup in men’s and women’s table tennis.
Two critical elements can commonly be identified in these contexts. First, externalities typically breed among individual players within a team: A team’s win benefits all its
members, while each bears the cost of his own contributions in the contest. Central coordination mechanisms, which facilitate cost sharing and internalize players’ interests, are typically
absent or infeasible. Alliances or interest groups often develop informally; players can be
categorized into different camps based solely on their interests, opinions, or ideological positions. Even in formally organized activities, central coordination usually plays only a limited
role–for instance, U.S. congressional races. As pointed out by Snyder (1989), and confirmed
by recent statistics, political parties supply only a small fraction of the candidate’s total
funding, due to institutional constraints such as contribution limits.2 A candidate’s success
largely depends on his/her own fund-raising and campaign input, instead of centrally coordinated resources.3 Moreover, players’ nonpecuniary efforts often contribute largely to the final
outcome. Central coordination becomes less feasible, because (1) players’ specific resources–
e.g., individual players’ expertise, time or physical stamina–may not be fully transferrable
across disjoint battlefields or distinct tasks, and (2) efforts cannot be perfectly verified, which
leads to the usual moral-hazard problems.
Second, in these contexts, team members’ efforts interact in a subtle manner; they do not
simply sum up to produce a joint output, as in the conventional team production process.
Each individual player carries out a distinct task on behalf of his team, i.e., rivaling his
matched opponent on one particular disjoint front. The collection of multiple battles, which
are fought by different pairs of contenders, forms a series. One’s victory discretely increases
the team’s lead, while his input cannot directly substitute for that of his teammates, and vice
versa. Consider, for instance, the aforementioned R&D races between alliances or networks:
A project is split and sourced to individual units within each network, and units with similar
expertise from rival networks compete on each specific component to advance.4
2
Party contributions (including coordinated expenditures) comprised 6% of total campaign expenditures
for Senate races in 2010, while donations and contributions from individuals and political action committees
(PACs) accounted for the lion’s share. Even candidates’ own funds accounted for only 13%. In addition,
we believe that candidates’ nonpecuniary efforts also play an important role, e.g., time and energy spent
on campaigning and fund-raising. (Source: Campaign Funding Sources for House and Senate Candidates,
1984-2012, The Campaign Finance Institute.)
3
One recent example is Senator Patty Murray, a three-term Democratic incumbent who has been accused
of spending too much time on campaigning: During her last race, she raised 10 times more than her opponent.
4
The imperfect substitutability between players’ efforts within a team/group/alliance can be exemplified
by the Sino-U.S. alliance in World War II. President Roosevelt famously commended the indispensable role
played by China on the Western Front of the Pacific Theatre in World War II, and emphasized forcefully to
the U.S. public the significance of keeping China as its ally (See Plating, 2011).
3
These widespread observations of team contests compel us to contemplate the implications of the distinct competition structure–i.e., pairwise-matched confrontations–for players’ strategic mindset and the outcome of the contest. Existing studies on multi-battle
contests have conventionally focused on the rivalry between individual contenders, which
abstracts away the possibility of collective contentions and the ensuing externalities among
team players. The literature on contests between allied players, however, typically assumes
that contestants in each coalition simply join forces to carry out a single task, which ignores
the possibility of competitions on multiple fronts. It remains unclear how players’ behavior in this unique team-production setting would deviate from existing predictions when (1)
team members are only linked indirectly through the prospect of winning a common trophy
and not direct coordination, and (2) their efforts do not additively produce a joint output,
and the aggregation of their discrete advances on multiple fronts determines the ultimate
outcome. To close this gap, we conduct a benchmark theoretical analysis of a multi-battle
team contest, which allows us to identify the role played by this competition structure while
isolating other confounding factors.
A snapshot of the game follows. Two teams consisting of an equal number of players
compete for a common object. Players from rival teams form pairwise matches, and each
pair competes head-to-head in one component battle.5 In our baseline setting, component
battles are carried out successively. Players observe the outcomes of previous battles and
strategically place their bids to maximize their own expected payoffs. A team wins if its
players prevail in the majority of component battles. A player may also secure a private
trophy for winning his own battle. Three main neutrality results can be highlighted:
History Independence. The outcomes of early battles do not distort the balance of future
confrontations. Throughout the contest, a player’s probability of 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 or how many additional wins each team will need to reach the finish line. In
each battle, a player’s winning odds are purely determined by participating players’ innate
abilities.
Sequence Independence. Fix the matching of players and let each battle be contested
between a fixed pair of players. We consider a thought experiment in which the sequence of
these heterogenous battles is allowed to be reshuffled. The expected outcome and performance
of the contest are immune to the reshuffling of the sequence. That is, each battle yields a
fixed amount of ex ante expected overall effort; the overall expected effort of the contest, as
well as the probability of each team’s winning the contest, is not affected. Each individual
5
It should be noted that the organizing rules of the Davis Cup and the Fed Cup are not entirely different
from our model. In our model each player participates in one battle, while a player in the Thomas Cup, for
instance, may play in two “rubbers” (component individual matches) during a “tie” (the whole best-of-five
team match), with one in the first two and the other in the last three.
4
player’s ex ante expected payoff is also independent of the sequencing arrangement.6
Temporal-Structure Independence (or Disclosure Independence). The ex ante expected
effort of a battle between a fixed pair of players would not vary, even when the battles of
the contest–or a portion of these battles–are carried out simultaneously. Neither would
its stochastic outcome. It thus follows that the stochastic outcome and expected effort of
the whole contest are also independent of the prevailing information-disclosure or feedback
policy, i.e., whether players observe the history of past battles or how much they observe.
Our results are in stark contrast to those for multi-battle contests between individuals.
These dynamic individual contests typically demonstrate a feature of “history-dependence.”7
A player’s (perhaps purely accidental) victories in early battles generate strategic momentum
for himself while discouraging the other. The more motivated leader thus attains easy wins in
subsequent confrontations with the laggard; the outcomes of earlier battles distort subsequent
competitions and determine the ultimate winner.
Our model’s contrasting prediction can be attributed intuitively to the pairwise-matched
nature of this team contest. In a dynamic multi-battle contest between two individual players,
each player participates in a series and takes into account the outlays in future battles when
placing his bid. For the leader, slackening in the current battle implies elevated outlays in
the future, which compels him to step up his current effort to save on his future costs. For
the laggard, winning the current battle implies a prolonged series and continued input, which
attenuates his incentives to remain in the contest. The former mechanism gives rise to a
“strategic momentum,” while the latter creates “discouragement.” The concern about future
cost savings, however, loses its bite in our context. In the unique pairwise-matched team
contest, each player participates in one distinct battle, which allows him to not internalize
outlays for future battles, as they are to be borne by his teammates instead of himself. A
team’s lead does not provide its players with additional momentum on the one hand, while
its lag would not discourage them on the other. Neutrality results thus emerge.
Our analysis reveals the distinct incentives created by the competition structure of such
team contests. It thus provides a novel perspective for understanding the strategic implications of team productions or collective actions in contests with multiple pairwise battles. In
addition, our results yield several useful implications for contest design and practical policy
6
It should be noted that the sequence-independent result does not claim that the equilibrium outcome of
a battle between a fixed pair of players is independent of the order of battles. That is, a given pair of players’
effort distributions could differ when their battle takes place earlier in the series than when it does later.
The result, however, confirms that the order of battles does not affect the game’s equilibrium outcomes on
expected dimensions. For instance, every player’s ex ante expected efforts would be the same regardless of
the sequence in the contest. More details will be provided in Section 5.2.
7
Readers are referred to Harris and Vickers (1987), Klumpp and Polborn (2006), Konrad and Kovenock
(2009a), and Malueg and Yates (2010).
5
experimentation, which will be discussed after we carry out the formal analysis. It should
be noted, however, that our study, as a benchmark theoretical analysis, is conducted in a
relatively stylized setting. This framework allows us to identify a number of additional strategic or behavioral elements, which, as discussed later in the paper, may either strengthen or
nullify the predictions obtained from the benchmark setting and inspire future work.
The rest of the paper is organized as follows. In Section 2, we briefly review the relevant
literature. In Section 3, we present a simple and stylized example to illustrate the fundamental
logic. It highlights the current setting’s unique strategic trade-offs, which distinguish a team
contest from its counterpart between individuals. Section 4 sets up a general model of a
multi-battle contest between teams. Section 5 conducts the main analysis and presents a
complete set of results. In Section 6, we discuss the main implications of our results and
several possible extensions and illustrate several caveats. Section 7 concludes the paper.
2
Link to Literature
This paper primarily belongs to the literature on multi-battle contests. Harris and Vickers
(1987) identify the discouragement effect in a dynamic R&D race, in which two firms compete against each other on a series of component technologies and one secures the patent by
accumulating a sufficient number of advances ahead of the other. Malueg and Yates (2010)
demonstrate theoretically the distortion caused by early outcome in a best-of-three contest
between two symmetric contestants and test it empirically. Konrad and Kovenock (2009a)
provide a complete characterization of the unique subgame perfect equilibrium of a generalized sequential multi-battle contest between two players, with each component battle being
modeled as an all-pay auction. Konrad and Kovenock allow 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. Their analysis shows that equilibrium individual and aggregate efforts, as
well as players’ winning probabilities, can be non-monotonic in the closeness of the race when
bidders are asymmetric. When a component battle awards a positive intermediate prize, even
a large lead by one player does not fully discourage the other. In a further study, Konrad
and Kovenock (2010) demonstrate that the aforementioned discouragement effect can also
be attenuated by the uncertainties in the players’ bidding costs.
Klumpp and Polborn (2006) focus on U.S. presidential primaries and compare the efficiency of sequential campaigning to that of its simultaneous counterpart, which provides a
rationale for the New Hampshire effect observed in presidential primaries. They demonstrate
that a sequential arrangement outperforms a simultaneous one, as the former helps reduce
campaign expenditures. In a war-of-attrition model, Strumpf (2002) highlights the distorting
effect of a particular sequence of heterogeneous battles on the outcome of a dynamic contest
6
between two individual candidates when continued participation is costly: The winner who
stands out in a sequential election, e.g., U.S. presidential primaries, could have been the loser
if state primaries had been ordered differently.
These studies typically assume that players compete with each other on disjoint battlefields, and the grand winner is picked by simply aggregating the outcomes of component
battles. Kovenock, Sarangi, and Wiser (2011) introduce interdependence between component
battles: A contestant succeeds if he wins a certain combination of component battles. Snyder
(1989) studies a simultaneous multi-battle contest, in which two political parties compete in
the parallel elections of multiple constituencies. In contrast to the aforementioned study,
a party derives rent from winning each individual battle (election in a constituency), and
maximizes its expected aggregated payoffs. Kovenock and Roberson (2009) extend Snyder
by letting the competition repeat in another period and introduce another type of interdependence: One player’s win on one battlefield in the first period grants him a head start
when the competitors meet again on the same front.
These studies assume that two players meet and compete against each other in all component battles. Our paper, in contrast, explores team contests in which each component battle
is fought between a particular pair of matched players from rival teams. Our results are
unique to the setting of competitions among players in coalitions, and thus have a different
scope of applications. For instance, while Klumpp and Polborn (2006) and Strumpf (2002)
focus on the impact of electoral institutions on candidates’ behavior, our study yields useful
implications for campaigning races between political parties and, in this regard, complements
the literature.
Our paper studies a distinct and important form of team production. Hence, it naturally
contributes to the extensive literature on group contests or contests between allied players,
which includes Skaperdas (1998), Nitzan (1991), Esteban and Ray (2001, 2008), Nitzan
and Ueta (2009), Münster (2007), Konrad and Leininger (2007), and Konrad and Kovenock
(2009b), among many others. As mentioned previously, these studies assume that group
members contribute perfectly substitutable efforts to collectively produce a single output.8
Our analysis concludes that the expected effort of the team contest does not depend on
whether players learn the outcomes of previous battles, or on how much they have learned. It
can thus be linked to the burgeoning literature on communication and interim performance
feedback in dynamic contests, e.g., Gershkov and Perry (2009), Aoyagi (2010), Ederer (2010),
Gürtler and Harbring (2010), and Goltsman and Mukherjee (2011). In contrast to the present
paper, players in those studies compete on single tasks and supply continuing efforts that
8
Kovenock and Roberson (2012) and Rietzke and Roberson (2013), as two notable exceptions, consider
rivalry between two allied players and one independent player. In their settings, a player may participate in
multiple battles and maximize the sum of the private payoffs he receives from all his battles; players form
alliances only to exchange resources, instead of vying collectively for a shared trophy.
7
add to their bids. The winner is determined by players’ accumulated output in carrying out
a single task, instead of by the accumulated number of wins in disjoint battles.
3
Example: Symmetric “Best-of-Three” Contest
We use a simple complete-information best-of-three contest to illustrate the distinct features
of pairwise-matched team contests.9 We first consider a benchmark contest between two
symmetric individual players. They meet in three successive battles, and one wins the contest
by winning two of them. We then consider a contest between two teams. Each team has
three players, with each confronting his matched opponent from the rival team in one of three
successive battles. A team wins if and only if its players secure two victories.
In both settings, we assume (1) that one’s effort incurs a unity marginal cost, which is
commonly known, and (2) that players simultaneously exert their efforts in each component
battle. For the moment, we do not fix a particular form of contest technology. Instead,
we only impose mild restrictions on bidding behavior. First, in a battle, the player with a
higher valuation of the win tends to place a higher bid, and therefore is more likely to prevail.
Second, a symmetric bidding equilibrium exists when two players in a battle equally value
the win, in which case players adopt the same bidding strategy and win the battle with equal
probability.10
We next compare the observations obtained from the two settings (individual contest vs.
team contests).
3.1
Benchmark: Individual Contest
Two individual players, indexed by i = A, B, compete for a prize of value 1. They confront
each other in three successive battles, which are indexed by t = 1, 2, 3.
The game can be analyzed by backward induction. Suppose that one player wins the first
two battles. Battle 3 becomes irrelevant. Next, suppose that each has won one battle. The
outcome of battle 3 determines the ultimate winner. In a symmetric equilibrium, each wins
with a probability of 1/2. Let E(x3 )(> 0) be one’s expected effort in the equilibrium. Each
player thus expects a payoff of v3 = 1/2 − E(x3 ) for participating in this battle.
We then consider players’ incentives when battle 2 is to be carried out. We assume,
without loss of generality, that player 1 has won the first battle. For player 1, winning battle
2 ends the contest and allows him to collect a prize of value 1. Losing the battle would force
9
Our general setup, which is presented in Section 4, allows each team to consist of 2n + 1 players. Players’
effort costs are allowed to be heterogeneous and privately known.
10
A standard Tullock contest or an all-pay auction clearly satisfies these requirements, and thus serves our
purpose.
8
him to participate in the deciding battle, from which he expects a gain of v3 . Hence, player
1 responds to an “effective prize spread” of 1 − v3 = 1/2 + E(x3 ) when placing his bid. In
contrast, player 2 has an effective prize spread of exactly v3 − 0 = 1/2 − E(x3 ): The contest
proceeds to a deciding battle if he wins and ends if he fails to even the score.
Player 1 is more likely to win the second battle simply because of his earlier victory.
A usual strategic-moment effect or discouragement effect arises: The outcome of battle 1
amplifies the incentive of the leader to win the subsequent battle, while it attenuates that of
the laggard. The ex ante symmetric contest is thus diverted onto an asymmetric path after
one obtains a lead.
3.2
Team Contest
We now demonstrate that the ex post asymmetry illustrated above for individual contests
does not appear in a pairwise-matched team contest. Two teams, each consisting of three
players, compete for a trophy. The trophy is a public good and has a value of 1 to all players.11
A player is indexed by i(t) if he is affiliated with a team i, i ∈ {A, B}, and is assigned to the
t−th battle.
If one team has won the first two battles, battle 3 becomes irrelevant. If the two teams
score evenly in the first two, battle 3 is a symmetric match, and each player wins with a
probability of 1/2. We then consider the immediately preceding battle. Assume without loss
of generality that player A(1) has won battle 1 on behalf of team A.
If player A(2) wins the battle, the contest ends and he receives an immediate reward of
1. If he loses, the contest proceeds to battle 3. Player A(2) expects a payoff of 1/2 from
the latter event, because his teammate–i.e., player A(3)–may win the battle and secure
the trophy for his team with probability 1/2. Hence, player A(2)’s effective prize spread is
1 − 1/2 = 1/2.
If player B(2) wins the battle, the contest proceeds to the deciding battle. Player B(2)
would stand a chance of winning the contest with probability 1/2 (if player B(3) wins battle
3). He thus expects a payoff of 1/2 from winning his battle. If he loses, the contest ends.
Hence, player B(2)’s effective prize spread is 1/2 − 0 = 1/2.
Player B(2) values his win as much as his rival A(2) does. Battle 2 remains symmetric in spite of team A’s lead. One’s initial lead does not distort subsequent competitions.
The strategic-momentum or discouragement effect, which is identified in the multi-battle
individual contest, does not arise in the current context.
11
For the moment, players are assumed to not receive a private reward for winning a component battle.
9
3.3
Intuition
We first elaborate on the nature of the strategic-momentum or discouragement effect, as
illustrated in the benchmark example. For a laggard in a best-of-three individual contest,
the benefit of winning the second battle is to even the score; no actual reward would accrue
unless he wins the third. He has to continue to sink costly effort into the third battle if he wins
the current one, which dissipates his future rent, thereby attenuating his incentive to remain
in contest. As shown in Section 3.1, his incentive to win battle 2 is given by a prize spread
1/2 − E(x3 ). For his leading opponent, the contest would be prolonged if he loses the second
battle, which forces him to endure another uncertain and costly battle in the future while
continuing to supply effort. The additional outlay burns the rent, and therefore aggravates
the pain from the current loss. As shown by our example, he receives 1 immediately if he
wins, but only 1/2 − E(x3 ) if he loses. In summary, the laggard has less to win, while the
leader has more to lose. Ex post asymmetry thus arises.
The same, however, cannot be said of team contests. Consider player A(2) on the leading
team (i.e., team A). The team’s lead, paradoxically, allows him to slack off: His team would
not lose the contest immediately because of his loss, and he would still receive the prize as
long as his teammate prevails. He suffers less from a prolonged struggle than his counterpart
in individual contests, because a subsequent battle does not require his own contribution.
The third battle becomes an effortless fair draw. He is punished less if he loses because of
the ability to (partially) free-ride, which, in turn, suppresses the value of his win.
We then consider player B(2), who is on the lagging team. Analogous to his counterpart
in individual contests, he cannot secure the prize immediately, even if he wins. 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 exactly 1/2
from his own win, as it leads to an effortless fair draw. This effect thus amplifies the effective
prize spread in his battle.
These forces neutralize the impact of the discouragement effect on the lagging team,
thereby maintaining an even battlefield, regardless of the outcome of past plays. In subsequent sections, we provide a complete analysis of pairwise-matched team contests in a general
setup. We demonstrate that the fundamentals we observe in this stylized example remain
valid.
4
General Setup
Two teams, indexed by i = A, B, compete in a contest for an indivisible object. Each team
consists of 2n + 1 risk-neutral players.
10
Each player on one team is matched to a player on the rival team.12 Each pair of matched
players competes head-to-head in a component battle on one disjoint battlefield. In a component battle, the players involved simultaneously place their bids. A team is awarded the
object if and only if it accumulates at least n+1 victories from the 2n+1 pairwise component
battles.
4.1
Temporal Structure
We begin with a setting in which component battles are carried out successively.13 Battles
are indexed by t = 1, 2, ..., 2n + 1, i.e., their orders in the given unfolding sequence. A player
on team i, where i = A, B, is indexed by i(t) if he is assigned to the t-th battle.
The state of the contest before battle t is carried out is summarized by a tuple (kA , kB )
with t = kA + kB + 1, where ki is the number of wins secured by team i. The state (kA , kB )
is observed by players A(t) and B(t) before they sink their efforts.
The contest reaches a terminal state (kA , kB ) once a team has accumulated exactly n + 1
victories, i.e., max{kA , kB } = n+1. Once the winning team has been determined, subsequent
battles are irrelevant to the outcome of the team contest. A component battle t is called trivial
if the contest is carried out in a state (kA , kB ) with max{kA , kB } ≥ n + 1. However, players
may still put forth positive efforts if they receive a private reward for winning their own
battles. More details are provided when we depict the payoff structure.14
4.2
Payoffs
Rewards for each player arise from two sources: team prize and battle prize. First, one
benefits from his team’s win. The winning team’s trophy is a public good, and all players
on the team equally value it. We normalize the common valuation to 1. Second, a player
may reap a private reward from winning his own battle, irrespective of his team’s success or
failure. Specifically, a player i(t) receives a private reward π t ≥ 0 if he wins his own battle.
The value of the private reward π t is common to the fixed pair of matched players A(t)
and B(t). It is determined by the characteristics of the particular battlefield and allowed to
12
Throughout the paper, we assume that the two players involved in a particular battle remain unchanged
once the matching is complete. The model does not allow teams to strategically set the order in which their
players turn up in a sequential contest. However, the sequence of battles is allowed to vary in Section 5.2.
13
In Section 5.3, we will present formal analysis of an alternative setting in which component battles are
contested (partially or completely) simultaneously.
14
In our setting, the contest would not be terminated unless all 2n + 1 component battles have been carried
out. Konrad and Kovenock (2009a) assume that the contest ends immediately once it reaches a terminal
state. Our main results do not depend on the prevailing termination rule.
11
vary across battlefields.15 For instance, U.S. Senate membership could have higher value for
Californian politicians than for their counterparts in North Dakota.
A trivial battle t is no different from a standard static contest in which two players
compete for a single prize πt . If π t = 0, a trivial battle elicits zero effort.
4.3
Component Battles
In each component battle t, two matched players simultaneously exert their efforts xi(t) .
Following Konrad and Kovenock (2009a, 2010), we model each component battle as an allpay auction: One wins the battle with certainty if he contributes a higher effort than his
opponent. The winner is picked randomly when there is a tie.
Players can be heterogeneous in terms of their innate abilities. The ability differential is
reflected by the possibly different marginal effort costs. Each player i(t)’s effort entry xi(t)
incurs a constant marginal cost ci(t) > 0. The marginal costs ci(t) , ∀i, t, are independently distributed over intervals [ci(t) , c̄i(t) ], respectively, with cumulative distribution functions Fi(t) (·)
and continuously differentiable density functions fi(t) (·) > 0. The distribution of each ci(t) is
commonly known, while its realization is privately observed by player i(t) only. Hence, each
pairwise battle is an incomplete-information all-pay auction.
5
Analysis
Let vi (kA , kB ) denote the continuation value to team i when the contest is in a state (kA , kB ).
It denotes the gross payoff players of team i expect from winning the team trophy when it
is rationally assessed in a given state. As the value of the trophy is normalized to 1, the
continuation value coincides with the team’s conditional winning probability. Apparently,
we must have vA (kA , kB ) = 1 if kA ≥ n + 1, and vA (kA , kB ) = 0 if kB ≥ n + 1. In the former
case, team A has won the contest, and all its players receive a gross benefit of value 1 from
the team trophy. In the latter case, team A has lost, and the trophy is awarded to the rival
team. For the same reason, vB (kA , kB ) = 1 if kB ≥ n + 1, and vB (kA , kB ) = 0 if kA ≥ n + 1.
Consider an arbitrary battle t in a state (kA , kB ), with t = kA + kB + 1. For player A(t),
the contest reaches a state (kA + 1, kB ) if he wins, and the continuation value for team A’s
players becomes vA (kA + 1, kB ). In contrast, the contest reaches a state (kA , kB + 1) if he
loses, and the continuation value becomes vA (kA , kB + 1) in the corresponding state. Hence,
player A(t) has an effective prize spread π t + [vA (kA + 1, kB ) − vA (kA , kB + 1)]. Similarly, the
prize spread for player B(t) is π t + [vB (kA , kB + 1) − vB (kA + 1, kB )].
15
For the sake of expositional efficiency, we collapse the asymmetry among players into a setting of heterogeneously distributed effort costs.
12
We now present two key observations about the game, which pave the way for all main
analyses.
Observation 1 In a battle t at state (kA , kB ), players A(t) and B(t) have a common effective
prize spread of Vt (kA , kB ) = π t +∆v(kA , kB ), where ∆v(kA , kB ) = vA (kA +1, kB )−vA (kA , kB +
1) = vB (kA , kB + 1) − vB (kA + 1, kB ).
Observation 1 says that each battle t ∈ {1, 2, . . . , 2n + 1} is a common-valued all-pay
auction in which the two matched players, A(t) and B(t), equally value the win, regardless
of the state of the contest. Observation 1 stems from the fact that the sum of teams’
continuation values always amounts to one, regardless of the prevailing state of the contest,
i.e., vA (kA , kB ) + vB (kA , kB ) ≡ 1, ∀kA , kB ∈ {0, 1, 2, . . . , 2n + 1}, with kA + kB ≤ 2n + 1.
The outcome of a battle shifts the state of the contest and changes teams’ prospects for their
eventual success. However, because the team prize has to be won by one and only one team,
the shift of state only affects the probabilistic division of the team trophy between the two
teams and not the sum of the rent. It thus follows that ∆v(kA , kB ) = vA (kA + 1, kB ) −
vA (kA , kB + 1) = vB (kA , kB + 1) − vB (kA + 1, kB ).
Observation 1 generalizes the insights illustrated by the best-of-three example. To a fixed
pair of matched players, the outcomes of preceding battles vary their valuations of winning
the battle. The impact, however, is constantly symmetric. As a result, matched players are
equally motivated over all possible paths as the contest proceeds.
By Observation 1, each battle in the contest boils down to an incomplete-information
common-value all-pay auction between asymmetric players. Amann and Leininger (1996)
analyze a two-player all-pay auction in which players’ valuations are drawn from different
distributions, but they bear the same constant marginal effort cost. A battle t in our team
contest can be viewed as equivalent to the game studied by Amann and Leininger. Their
result and approach thus immediately extend to our context. They establish the existence
of a unique pure-strategy Bayesian Nash equilibrium. Moreover, one can further verify that
in our setting, both players’ equilibrium bidding strategies are linear in the common prize
spread in each battle t.16
Observation 2 In the unique pure-strategy Bayesian Nash equilibrium of battle t with a
symmetric prize spread of Vt > 0, each player i(t), i = A, B, exerts an effort bi(t) (·; Vt ) =
Vt ξ i(t) (·). The function ξ i(t) (·) depends solely on the two players’ effort cost distributions.
With these preliminaries, we are ready to present our main results.
16
The proof is omitted to save space, but is available from the authors upon request.
13
5.1
History Independence
Denote by µi(t) the expected probability of a player i(t)’s winning his battle t, provided that
the common prize spread Vt is strictly positive. We call the equilibrium winning likelihoods
(µA(t) , µB(t) ) the stochastic winning outcome of battle t. For expositional efficiency, we further
define θi ( h| m)|tt+m−1 , with t ∈ {1, 2, . . . , 2n + 1}, m ∈ {0, 1, 2, ..., (2n + 1) − (t − 1)} and
h ∈ {0, 1, . . . , m}, to be the probability that players i(t) to i(t + m − 1) will win exactly h
of m battles (battles t to t + m − 1), provided that all these m battles have strictly positive
prize spreads.17
By the two fundamental observations presented above, the following result readily obtains.
Theorem 1 Consider an arbitrary battle t in a state (kA , kB ) with t = kA + kB + 1.
(a) (History Independence of Winning Outcome) Whenever Vt (kA , kB ) > 0, the stochastic
outcome (µA(t) , µB(t) ) of the battle is determined solely by the matched players’ effort cost
distributions (FA(t) (·), FB(t) (·)), and thus is independent of (kA , kB ).
(b) (Contingent Prize Spread) The common prize spread can be written as follows:
Vt (kA , kB ) = π t + θi ( n − ki | 2n − kA − kB )|2n+1
t+1 , i = A, B.
Proof. See Appendix.
Theorem 1(a) directly stems from Observations 1 and 2. For a given battle t, the (common) prize spread Vt (kA , kB ) depends on the state of the contest. Its particular size, however,
does not affect players’ equilibrium winning likelihoods so long as the battle awards a positive prize spread Vt , because the equilibrium bidding strategies are linear in Vt as shown by
Observation 2. The contest can then be viewed as a series of independent random draws:
Each battle t with Vt > 0 is resolved by fixed winning likelihoods (µA(t) , µB(t) ), which are
predetermined by the profiles of contenders’ bidding competence. The outcome of earlier
battles does not distort the outcomes of subsequent competitions.
However, players’ equilibrium outlays must be contingent on the exact state (kA , kB ) of
the contest, as it determines the prize spread Vt (kA , kB ). Theorem 1(b) provides a universal formula for a battle t’s prize spread Vt (kA , kB ) in all possible states. The expression
θi ( n − ki | 2n − kA − kB )|2n+1
is the probability that player i(t)’s teammates will win ext+1
actly n − ki out of the remaining 2n − kA − kB battles after battle t, provided that all
these battles have strictly positive prize spreads. Given the history-independence result, the
probability θi ( h| m)|t+m−1
can be computed explicitly based on the set of (fixed) probabilt
ities µ = {(µA(t) , µB(t) ), t = 1, 2, ..., 2n + 1}.18 Obviously, θA ( n − kA | 2n − kA − kB )|2n+1
t+1 =
We define θi ( 0| 0)|t−1
= 1, θi ( h| m)|t+m−1
= 0 for h < 0 and m ≥ 0, and θi ( h| m)|t+m−1
= 0 for h >
t
t
t
m ≥ 0.
18
Consider, for example, the special case of symmetric team contests, in which the marginal effort costs of
17
14
θB ( n − kB | 2n − kA − kB )|2n+1
t+1 ) must hold, because a team i’s winning n−ki battles is equivalent to its rival team j’s winning n − kj of them.19
In the above event, battle t turns out to be pivotal to the entire contest: Its outcome ex
post determines the winning team. On one hand, the formula can be interpreted intuitively.
Winning a battle allows a player to derive additional benefit from the team trophy ex post if
and only if this battle turns out to be pivotal. The incentive provided by the team trophy must
be discounted ex ante accordingly. On the other hand, the result of Theorem 1(b) remains
subtle in our dynamic setting. Calculating θi ( n − ki | 2n − kA − kB )|2n+1
t+1 uses the winning
likelihoods (µA(t′ ) , µB(t′ ) ) for subsequent battles, from t + 1 to 2n + 1, which is equivalent to
assuming that all these battles have strictly positive prize spreads. Note, however, that the
current battle t’s outcome affects the likelihoods of subsequent battles’ being nontrivial. If a
battle t′ > t becomes trivial and awards no private prize, i.e., π t′ = 0, its prize spread drops
to zero: Players would exert zero effort, and the winning probabilities (µA(t′ ) , µB(t′ ) ) could
lose their bite. Nevertheless, the formula of Theorem 1(b) implies that the dynamic path
beginning from t can be analyzed as if all the subsequent battles took place concurrently with
battle t, such that the equilibrium stochastic outcomes (µA(t′ ) , µB(t′ ) ) remain valid.
It is innocent, though, to assume strictly positive prize spreads for all these battles. The
intuition is as follows. Consider, without loss of generality, team A. In a given state (kA , kB )
with t = kA + kB + 1, its conditional winning odds, or its continuation value, can be written
as
(m−1)
vA (kA , kB ) = 2n+1
].
(1)
m=kB +n+1 [µA(m) · θ A ( n − kA | m − t) t
To win the contest, it could secure its (n + 1)th win in any battle m, with m ∈ {kB +
n + 1, . . . , 2n + 1}. The expression (1) adds up all these possibilities: Within the sum,
(m−1)
each item µA(m) · θA ( n − kA | m − t)t
is the probability of securing its (n + 1)th win in a
particular battle m. In each of the possible events, all the m battles, which are involved in
(m−1)
calculating µA(m) · θA ( n − kA | m − t)t
, are nontrivial and thus must have strictly positive
prize spreads. Winning probabilities in subsequent trivial battles, i.e., battles t′ > m, in fact
become irrelevant. Hence, the current battle’s impact on the subsequent dynamic path can
be innocently assumed away when calculating the common prize spread of a battle t. The
probability of battle t’s being ex post pivotal can thus be established as if all subsequent
battles were concurrent with it, which leads to the formula stated in Theorem 1(b).
Recall that in a state (kA , kB ), a battle t has an effective prize spread Vt (kA , kB ) =
π t + θi ( n − ki | 2n − kA − kB )|2n+1
t+1 , i = A, B. Observation 2 and Theorem 1(b) further imply
the following result.
matched players are identically distributed. The contest renders a series of fair draws, with µA(t) = µB(t) = 12 ,
h 1 m
for all t ∈ {1, . . . , 2n + 1}. We then have θ i ( h| m)|t+m−1
= Cm
(2) .
t
19
The expression boils down to one if kA = kB = n, as the current battle immediately closes the team
contest; it reduces to zero if max{kA , kB } ≥ n + 1, as the battle no longer matters.
15
Corollary 1 (Contingent Expected Effort) Consider an arbitrary battle t in a state (kA , kB ).
(a) (Individual Effort) Player i(t) exerts an expected effort
E(xi(t) (kA , kB )) = ρi(t) Vt (kA , kB ), i = A, B,
where the constant ρi(t) = Eci(t) [ξ i(t) (ci(t) )] depends solely on the two players’ effort cost distributions.
(b) (Total Effort) The battle elicits a total expected effort
E( xt | (kA , kB )) = ρt Vt (kA , kB ), i = A, B,
where xt = xA(t) + xB(t) , ρt = ρA(t) + ρB(t) .
5.2
Sequence Independence
Fix the pairwise matching between players on the two teams, and let each pair be assigned to
a fixed battlefield. We now consider reshuffling the sequence of these (heterogeneous) battles.
The t-th battle in an existing sequence, which is fought between a given pair of players on a
fixed battlefield, can be carried out in a different order t̃ under a reshuffled sequence. Each
given pair of players, however, continues to compete for a given private trophy.
For a fixed pair of players, the stochastic outcome of their battle is independent of the
particular contest sequence–provided that the battle has a positive prize spread, in which
case players’ winning odds are simply determined by their marginal effort cost distributions.
However, if a battle does not award a private trophy, its stochastic outcome is not immune
to reshuffling. When the battle is trivial, the winner is picked by the prevailing tie-breaking
rule, as neither would place a positive bid. The prevailing sequencing arrangement does affect
the probability of a battle’s being trivial. For instance, the t-th battle under a sequence, with
t ≤ n + 1, would never be trivial. However, when it is ordered at t̃-th place under an
alternative sequence, with t̃ > n + 1, it ends up as trivial with positive probability. Despite
this nuance, we will show that the sequencing arrangement does not affect the outcome of
the entire contest, i.e., the probability of each team’s winning the trophy. Further, neither
players’ ex ante expected equilibrium efforts nor their ex ante expected payoffs depend on it.
We begin by deriving the ex ante expected effort of a battle t(g) assigned to pair g under a given sequence. By Corollary 1, a player i(g) exerts in equilibrium an expected effort
E( xi(g) (kA , kB )) = ρi(g) Vt(g) (kA , kB ), where the constant ρi(g) is a function of (FA(g) (·), FB(g) (·)),
his and his rival’s marginal effort cost distributions. Let ρg = ρA(g) + ρB(g) . Further define
xg ≡ xA(g) + xB(g) , which is the overall effort of battle t(g).
We begin by deriving the ex ante expected effort of a battle t(g) assigned to pair g under a
given sequence. By Corollary 1, in a state (kA , kB ), the battle yields an expected overall effort
16
E( xg | (kA , kB )) = ρg Vt(g) (kA , kB ), where the constant ρg is a function of (FA(g) (·), FB(g) (·)),
his and his rival’s marginal effort cost distributions.
Let Pr( (kA , kB )| t(g)) denote the ex ante probability that battle t(g) will take place in a
state (kA , kB ). Summing up E( xg | (kA , kB )) over all possible states in which the battle could
take place, we obtain the ex ante expected overall effort of this battle:
E( xg | t(g)) =
kA +kB =t(g)−1
Pr( (kA , kB )| t(g))E(xg | (kA , kB )) = ρg E(Vt(g) ),
where E(Vt(g) ) = kA +kB =t(g)−1 Pr( (kA , kB )| t(g)) · Vt(g) (kA , kB ) is the ex ante expected prize
spread for battle t(g) under this given sequence. The expected total effort of the entire contest
amounts to
E(x) = 2n+1
g=1 ρg E(Vt(g) ).
Define θ( n| 2n)|−g ≡ θi (n| 2n)|−t(g) , ∀i ∈ {A, B}, which is the probability that each team
will win exactly n out of the 2n battles other than the battle of pair g, assuming that the
equilibrium winning probabilities (µA(g′ ) , µB(g′ ) ) established by Theorem 1(a) apply to all
these 2n battles with g ′ = g. The value of θ( n| 2n)|−g can then be computed based on this
set of (fixed) probabilities. Note that we must have θA ( n| 2n)|−t(g) = θB ( n| 2n)|−t(g) : When
one team wins n out of the 2n battles, the rival team must prevail in exactly n other battles as
well. Moreover, the expression θ( n| 2n)|−g is obviously independent of the contest sequence
due to Observation 2. The following obtains.
Theorem 2 (Sequence Independence) (a) The ex ante expected prize spread in the battle of
pair g is
E(Vt(g) ) = π g + θ( n| 2n)|−g ,
which is independent of the sequence of battles. Thus, neither the ex ante expected effort of
the battle between pair g nor that of the whole contest depends on the sequence of battles.
(b) The ex ante expected likelihood of each team’s winning the contest is independent of the
sequence of battles.
(c) For any given pair of players, their ex ante expected payoffs are independent of the sequence
of battles.
Proof. See Appendix.
The prize spread Vt(g) (kA , kB ) in battle t(g) is contingent on the exact state of the contest.
Theorem 2(a), however, shows that for a fixed pair of players, its ex ante expectation, when
aggregating over all possible contingencies, is simply the odds of each team’s winning n of the
other 2n battles, i.e., the probability that the battle of group g will be ex post pivotal. By
Theorem 1, the probability θ( n| 2n)|−g is determined solely by other players’ marginal cost
distributions and not the prevailing sequencing arrangement. Ex ante expected equilibrium
17
effort would remain invariable if battles were ordered differently, either for individual battles
or for the overall contest.
As mentioned above, if a battle between a pair g does not award a private trophy, i.e.,
π g = 0, its stochastic outcome would be determined by the prevailing tie-breaking rule,
instead of the winning probabilities (µA(g) , µB(g) ). Hence, the sequencing arrangement does
affect the conditional winning outcome distribution of a battle between pair g. This nuance,
however, does not distort each team’s ex ante expected winning odds. Under a given sequence,
a team’s ex ante expected winning odds are given by
vi (0, 0) =
2n+1
m=n+1
[µi(m) · θi ( n| m − 1)
(m−1)
1
], i = A, B.
(2)
A team could secure its (n + 1)th win in any battle m, with m ∈ {n + 1, . . . , 2n + 1}, and
the expression aggregates all these possibilities. The same argument as that in Section 5.1
applies here: The value vi (0, 0) does not depend on how a trivial battle is resolved. In other
words, vi (0, 0) remains the same even if the winning likelihoods (µA(g) , µB(g) ) established by
Theorem 1(a) were assumed to apply for trivial battles. The contest can conveniently be
viewed as a series of independent lotteries, with each battle to be resolved stochastically by
fixed probabilities (µA(g) , µB(g) ). Reshuffling does not vary the probability of each team’s
being drawn in at least n + 1 independent lotteries.
Theorem 2(c) is then straightforward. A player i(g)’s ex ante expected payoff, under a
given sequence, includes three components: (1) the discounted payoff from the team trophy,
i.e., vi (0, 0); (2) the expected payoff from winning the private reward, i.e., µi(g) π g ; and (3)
his ex ante expected effort ρi(g) [π g + θ( n| 2n)|−g ]. As shown above, none of them would vary
if battles were ordered alternatively.
5.3
Temporal-Structure Independence
We now relax our assumption of sequential contests and explore the ramifications of alternative temporal structures. We allow component battles to be carried out (partially)
simultaneously.
More specifically, the 2n + 1 component battles are partitioned into Z ≤ 2n + 1 clusters.
Battles included in the same cluster are carried out simultaneously. Players in a battle do
not observe the outcomes of parallel battles within the same cluster, while they learn those of
battles contained in previous clusters. This setup accommodates various possible temporal
structures. With Z = 2n + 1, the sequential setting considered in the baseline setting is
restored. With Z = 1, a completely simultaneous setting obtains, in which all battles are
carried out at the same time.
We then explore how alternative temporal structures affect the outcome and efficiency of
the contest. The following obtains.
18
Theorem 3 (Temporal-Structure Independence) (a) Provided that a battle t is nontrivial or
the private award π t > 0, the winning likelihoods of this battle are given by (µA(t) , µB(t) ),
which are independent of the history or the temporal structure of the contest.
(b) The ex ante expected prize spread in battle t is E(Vt ) = π t + θ( n| 2n)|−t , which is independent of the temporal structure. Thus, neither the ex ante expected effort of battle t nor
that of the whole contest depends on the temporal structure.
(c) Each team’s ex ante probability of winning is independent of the temporal structure.
(d) The ex ante expected payoffs for the players in each battle t are independent of the temporal
structure.
Proof. See Appendix.
Theorem 3 extends and generalizes Theorems 1 and 2. The equilibrium outcome (µA(t) , µB(t) )
does not depend on the prevailing temporal structure of the contest. Regardless of the temporal structure, the winner of each nontrivial battle is an independent draw, with one’s winning
odds to be predetermined by players’ cost distributions. By the same arguments laid out in
the previous subsection, the prevailing temporal structure does not affect each team’s ex ante
expected winning odds or players’ ex ante expected payoffs.
By Corollary 1 and Theorem 3(b), the team prize lures a fixed pair of players in a battle
t to contribute an ex ante expected effort ρt θ( n| 2n)|−t , where ρt is determined purely by
the characteristics of the players. The team trophy is discounted by θ( n| 2n)|−t , which, as
previously stated, is the probability that each team will win exactly n of 2n nontrivial battles
other than t. In other words, this is the probability that battle t will be ex post pivotal.
In summary, we conclude that the ex ante expected outcome, the ex ante expected effort
of the contest, and players’ ex ante expected payoffs are entirely independent of contest
sequence, the temporal structure of the contest, or the information available to the players
about the outcomes of preceding battles.
5.4
Symmetric Players in Each Battle
Our general results allow us to obtain immediate predictions of the equilibrium behavior and
outcome when matched players in each battle are symmetric, i.e., their marginal effort costs
are identically distributed. However, players can still be heterogeneous across battles.
Recall Theorems 1(b) and 2(a), which say that for a fixed battle t, the contingent prize
spread in state (kA , kB ) is Vt (kA , kB ) = π t + θi ( n − ki | 2n − kA − kB )|2n+1
t+1 , i = A, B. The ex
ante expected prize spread is E(Vt ) = π t + θ( n| 2n)|−t . Further note that every nontrivial
battle renders a fair draw on winning outcomes when the players involved are symmetric,
i.e., µA(t) = µB(t) = 12 . The following thus obtains.
Corollary 2 Consider a symmetric team contest, in which matched players in each battle
19
are homogeneous.
(a) (Contingent Prize Spread) The common prize spread in a battle t in state (kA , kB ) is
1
(2n − kA − kB )!
Vt (kA , kB ) = π t + ( )2n−kA −kB ·
.
2
(n − kA )!(n − kB )!
(b) (Expected Prize Spread) The ex ante expected prize spread in a battle t is
E(Vt ) = πt +
(2n)! 1 2n
( ) .
n!n! 2
Recall ρt = ρA(t) + ρB(t) , which depends solely on the two players’ effort cost distributions.
A fixed pair of players in their battle thus exerts a contingent expected effort of
1
(2n − kA − kB )!
xt (kA , kB ) = ρt [π t + ( )2n−kA −kB ·
],
2
(n − kA )!(n − kB )!
and an ex ante expected effort of
E(xt ) = ρt [π t +
6
6.1
(2n)! 1 2n
( ) ].
n!n! 2
Implications, Extensions, and Caveats
Implications
Our analysis provides a thorough account of the equilibrium behavior and outcomes of contests between teams. The three neutrality results add to our understanding of both multibattle contests and team contests. They sharply contrast with conventional wisdom and reveal the subtle and important roles played by the distinct competition structure of pairwisematched team contests. Two remarks are in order to highlight the theoretical subtleties.
First, the unique form of team production considered in our analysis largely contributes to
the results. Consider, for instance, a hypothetical situation in which two soccer teams meet
each other repetitively. In that context, players on a team simply join forces in each joint
battle, and every player has to participate in a series of battles. The usual discouragement
would still loom large, as one is required to supply his effort repetitively on the team. In
contrast, in our setup, each player stands alone on behalf of his team in only one battle,
which severs the temporal linkage in cost trade-offs. Second, the same fact renders our study
complementary to that of Konrad and Kovenock (2010). They consider a situation in which
two players meet repetitively, but their effort costs are redrawn in every battle. They find
that the uncertainty regarding costs mitigates dynamic discouragement, but does not entirely
eliminate it: A player is still concerned about his future cost, although his marginal effort
cost has to be redrawn. The contrasting predictions of our model reveal that it is the turnover
20
of players across different battles, rather than the variations in their abilities, that nullifies
the discouragement effect.
These results shed light on contest design in various contexts. For instance, we address a
classical issue on intermediate feedback policy: Does it pay to provide intermediate feedback
to contestants in a dynamic contest? A (partially) simultaneous temporal structure in our
setting can be equivalently interpreted as a contest with limited intermediate feedback, which
prevents players from perfectly observing past plays. Our analysis in Section 5.3 thus implies
that in our context, the feedback or disclosure policy does not affect the performance or the
outcome of the contest.
Our results provide useful insights into administering practical competitive events. For
instance, they provide a useful perspective for evaluating the merits of various sports competitions’ organizing rules. History-independence results can be viewed as an alternative
rationale for the ascendancy and popularity of various team tournaments, e.g., the Sudirman
Cup in mixed badminton: As implied by our history-independence result, suspense in each
individual match can still be preserved regardless of the history of the accomplished matches,
which can trigger even more excitement, thereby appealing more to the audience. In these
team tournaments, e.g., the Thomas Cup (men’s badminton), athletes are typically sorted
and matched by their professional rankings. The fixed matching of players between two
teams leaves open the question of how to order these matches. For instance, should the
match between the top-ranked players be scheduled for an earlier or a later round? Our
sequence-independence result sheds light on these issues.
Our analysis also provides a novel perspective on the design of electoral systems. For
instance, how does the temporal format of an electoral competition affect its outcome and
the resulting campaign expenditures? Klumpp and Polborn (2006), focusing on electoral
competitions between two individual candidates (e.g., U.S. presidential primaries), show that
a sequential multi-battle contest (e.g., the format for U.S. primaries) leads to lower campaign
expenditures than the institution of a national election day (a simultaneous contest), due to
the discouragement effect. These issues, however, are less clear in the context of electoral
competitions between political parties. National general elections, for instance, resemble
a simultaneous multi-battle contest: Political parties, as coalitions of affiliated politicians,
compete for majority status, and votes are cast on election day across all constituencies. In
the U.S., the election of House members has a similar structure. In contrast, Senate elections
are partially sequential: One third of the seats are up for election every other year, and all
elections for those seats are carried out simultaneously. Our model and the neutrality results
provide a theoretical benchmark to identify the distinctive strategic elements involved in such
contexts.
It must be noted that our analysis is conducted in a relatively stylized setting, and our
21
results should be interpreted with sufficient caution when being applied to practical competitive situations. Our results’ robustness and caveats should also be examined carefully to
identify the limits of our analysis and the scope of its applications, which are discussed in
the remainder of this section.
6.2
Extensions and Caveats
Several extensions will expand our understanding of the robustness of our main results and
allow us to identify their limits and the factors that would nullify the current predictions.
6.2.1
Unequal “Finish Line”
One straightforward extension is to allow teams to race toward different finish lines. Consider
a T -battle first-past-the-post contest, in which team A wins by securing k̄A victories before
team B wins k̄B , with T ≡ k̄A + k̄B −1 and k̄A = k̄B .20 This race can be viewed as a subcontest
of a hypothetical “best-of-(2n + 1)” contest in which team i has secured (n + 1) − k̄i wins,
with k̄i ≤ n + 1, i = A, B. Specifically, an arbitrary battle t in the original race is equivalent
to the (2n + 1 − T + t)-th battle in the hypothetical best-of-(2n + 1) contest, because the
history-independence result always holds. All of the primary predictions obtained in our
baseline setting naturally extend here.
6.2.2
Uneven Winning Scores and Unequally Weighted Battles
Another direct extension is to allow component battles to carry different weights for teams’
wins. Consider a contest with T ≥ 2 successive component battles. A team t is awarded
a score sit if its player i(t) wins the battle. Each team can maximally score 1 if its players
prevail in all battles, i.e., Tt=1 sit = 1, ∀i. A team wins the entire contest if and only if it
obtains a higher accumulated score s̃i than the rival. The score awarded for winning a battle
may vary across battles for a given team, i.e., sit = sit′ for t = t′ . It may also vary across
B
teams for a given battle, i.e., sA
t = st . Our baseline model becomes a special case of this
1
B
′
extended setting, with sA
t = st′ = 2n+1 , ∀t, t ∈ {1, . . . , 2n + 1}.
The fundamentals of our analysis remain valid. Let the state of the contest be summarized
by the tuple (s̃A , s̃B ), which indicates each team’s accumulated score. Consider an arbitrary
battle t in a state (s̃A , s̃B ). Again, let vi (s̃A , s̃B ) be the continuation value of team i assessed
B
A B
B
in this state. In battle t, A(t)’s prize spread is VA(t) = π t + [vA (s̃A + sA
t , s̃ ) − vA (s̃ , s̃ + st )],
A
A B
while B(t)’s prize spread is VB(t) = π t + [vB (s̃A , s̃B + sB
t ) − vB (s̃ + st , s̃ )].
20
For instance, in an R&D race toward a final product, one firm may have secured patents on a few critical
component technologies, to which the other has no access.
22
Note that the simple fact still holds that the expected winning chances of the two teams
must sum up to 1 in any state. Observation 1 thus remains intact, i.e., VA(t) = VB(t) . Analogous to the baseline setting, each battle would be a common-valued static contest, and
therefore Observation 2 must also hold. These two key observations lead to the same qualitative predictions as those in our baseline setting, while the exact formula for effective prize
spread and expected effort must be adapted quantitatively to accommodate the alternative
scoring rule.
6.2.3
Alternative Contest Mechanisms
The baseline analysis models each component battle as an all-pay auction. As shown by
Observation 2, such a contest technology leads equilibrium bidding strategies to be linear
in players’ common values for winning their battles. This property critically contributes
to the history-independence result, and eventually to the sequence-independence and the
temporal-structure-independence results as well.
Our main results would continue to hold as long as an alternative contest mechanism
renders linear bidding strategies. This requires that the mechanism be homogeneous of degree
zero, if we focus on the well-adopted ratio-form contest success functions, while assuming that
players’ effort marginal costs are commonly known. For such a contest technology, a player
i(t)’s winning probability is
pi(t) (xA(t) , xB(t) ) =
ft (xi(t) )
, i = A, B,
ft (xA(t) ) + ft (xB(t) )
where impact function ft (·) is strictly increasing with ft (0) = 0.21
As established by Skaperdas (1996), a ratio-form contest success function is homogenous
of degree zero, i.e., pi(t) (xA(t) , xB(t) ) = pi(t) (θxA(t) , θxB(t) ), ∀xA(t) , xB(t) , ∀θ > 0, if and only
t
if ft (xi(t) ) = at xri(t)
, with at , rt > 0. In particular, when rt → +∞, the contest converges
to an all-pay auction. The recent work of Alcalde and Dahm (2010) and Wang (2010) has
established that the equilibrium bidding strategies of both (heterogenous) players are linear
in their common value for the whole range of rt > 0.22
Homogeneity of degree zero ensures that a player’s effort is a linear function of the common
prize spread. Absent this property, the stochastic outcome would depend on the particular
size of the prize spread, which is ultimately determined by the specific state of the contest,
although matched players still equally value winning the battle. As a result, the stochastic
outcome would no longer be immune to changes in the state of the contest in general. History,
21
This contest success function is axiomatized by Skaperdas (1996) and Clark and Riis (1998). Fu and Lu
(2012) provide a micro foundation based on a noisy-ranking approach.
22
When the equilibrium is in mixed-strategy, each bidder’s equilibrium bid can be written as the product
of a fixed random variable multiplied by the common value.
23
sequence, or temporal structure must distort effort supply and outcome of the contest. By this
logic, one can expect similar implications (1) when players’ effort cost function is nonlinear,
or (2) when players are not risk neutral.
6.2.4
Asymmetry in Valuations
Our analysis has assumed a symmetric prize structure, i.e., each pair of matched players
equally values winning both the overall contest and their individual battle. Our prediction
would not entirely hold if matched players were allowed to value the prizes unequally. In such
a case, assuming the same contest technology (i.e., all-pay auctions), our model’s predictions
would persist if and only if players receive no private reward from winning their own battles.
Let each player i(t) have a valuation of τ i(t) > 0 for the team trophy. Recall that vi (kA , kB )
is the probability of a team i’s winning the contest assessed in a state (kA , kB ). For a
battle t, the two matched players’ effective prize spreads in the current state are given,
respectively, by VA(t) (kA , kB ) = π t +[vA (kA + 1, kB ) − vA (kA , kB +1)]τ A(t) , and VB(t) (kA , kB ) =
π t + [vB (kA , kB + 1) − vB (kA + 1, kB )]τ B(t) .
Recall that ∆v(kA , kB ) = vA (kA +1, kB ) −vA (kA , kB +1) = vB (kA , kB +1) −vB (kA +1, kB )
by Observation 1. When a battle t involves no private reward, i.e., π t = 0, we must have
Vi(t) (kA , kB ) = ∆v(kA , kB )τ i(t) , i = A, B. In other words, the ratio between their prize spreads
must remain constant regardless of the prevailing state (kA , kB ) of the contest. The following
thus obtains.
Observation 3 Suppose that a nontrivial battle t is in state (kA , kB ) and π t = 0. At equilibrium, player i(t) exerts an effort bi(t) (·; (kA , kB )) = ∆v(kA , kB ) · ς i(t) (·), where ς i(t) (·) is i(t)’s
equilibrium bidding strategy when effective prize spreads are Vi(t) = τ i(t) , i = A, B. Hence,
the probability of each player’s winning this battle is independent of the current state of the
contest (kA , kB ).
The state of the contest only changes the common factor of ∆v(kA , kB )(≤ 1) in players’
effective prize spreads ∆v(kA , kB )τ i(t) . In a two-player all-pay auction with private marginal
effort costs, one can verify that when players’ prize valuations Vi(t) are scaled up or down
by a common factor γ(> 0), their equilibrium bids would be scaled up or down by the same
factor γ.23 Without private rewards, each pair of matched players’ winning odds are solely
determined by their initial valuations of the team trophy and the distributions of their effort
cost distributions; they remain independent of the current state of the contest.
The same, however, does not hold if positive private rewards exist. Note that one’s private
reward is unaffected by the state of the contest. As a result, matched players’ overall prize
23
This result can be obtained by verifying that the differential equation system that determines equilibrium
bidding strategies admits such a solution. A formal proof is available from the authors upon request.
24
spreads would vary by different proportions when the state of the contest changes; a net
discouragement effect thus emerges. This effect is partially muted, however, because players’
expected payoff from the team trophy is still discounted “symmetrically.”
6.2.5
Alternative Information Structures
Our results would continue to hold when either of two alternative information structures is
assumed in each component battle: (1) complete information, i.e., when matched players’
effort marginal costs are commonly observed, and (2) one-sided asymmetric information, i.e.,
one player learns the costs of both himself and his opponent, while the other knows only his
own.24 The two key observations of Section 5 continue to hold, and thus all our main results
extend to these alternative settings.
However, additional complications arise when information asymmetry exists between
teams. We have implicitly assumed that the information about a player’s cost characteristics is symmetrically known to all other players, both his own teammates and the players
on the rival team. If, instead, each player’s type is (partially) known to his teammates–but
remains completely unknown to his opponents–players’ strategic trade-offs and their behavior would change dramatically. For instance, athletes on a team often attend training camp
together before major tournaments, which affects all participants’ competence and allow them
to learn about others’ skill levels. In this situation, a player may give up prematurely if he
knows that his (weak) teammates are unlikely to win future battles. In a dynamic setting,
players’ interactions would involve a complex process of information updating.
7
Concluding Remarks
In this paper, we study a pairwise-matched multi-battle contest between two teams. The
contest consists of a series of pairwise battles fought between matched players from rival
teams. We demonstrate in an incomplete information all-pay-auction setting that the wellstudied strategic-momentum or discouragement effect in individual multi-battle contests no
longer prevails. The history of past plays in the sequential contest does not cause distortion
to the stochastic outcomes of future battles. In addition, the expected effort 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.
Our paper contributes to the burgeoning literature on multi-battle contests by introducing
a setting that involves collective action. Team competitions, as stated in the Introduction,
are pervasive in the real world, yet they have rarely been studied using formal modeling.
24
Morath and Münster (2010) study an all-pay auction in which one player possesses superior information.
25
Our analysis illustrates the fundamental difference in players’ strategic mindsets between
individual contests and pairwise-matched team contests.
As we have demonstrated, the main logic of our baseline analysis remains robust in many
variations of the basic model. On the other hand, the limits of our analysis and the scope of
its applications should be examined carefully. For this purpose, we have pointed out some
caveats that may nullify the current predictions. Our results, therefore, should be interpreted
with caution when being applied to specific contexts.
Appendix
Proof of Theorem 1
Proof. Theorem 1(a) is immediately implied by Observations 1 and 2. To establish Theorem
1(b), we need to show that
∆v(kA , kB ) = vA (kA + 1, kB ) − vA (kA , kB + 1) = θA ( n − kA | 2n − kA − kB )|2n+1
t+1 ,
where t = kA + kB + 1. When kA = kB = n, clearly we have ∆v(kA , kB ) = 1 that equals
θA ( n − kA | 2n − kA − kB )|2n+1
t+1 = 1; when max{kA , kB } ≥ n+1, clearly we have ∆v(kA , kB ) =
0 that equals θA ( n − kA | 2n − kA − kB )|2n+1
t+1 = 0. We now focus on the case where kA , kB ≤ n
and min{kA , kB } < n. Note that in this case, battle kA + kB + 1 is nontrivial. Thus we have
vA (kA , kB ) = µA(kA +kB +1) vA (kA + 1, kB ) + (1 − µA(kA +kB +1) )vA (kA , kB + 1) by definition. One
can then verify that ∆v(kA , kB ) = µA(kA +kB +1) ∆v(kA +1, kB )+(1−µA(kA +kB +1) )∆v(kA , kB +1).
Applying this result, it is clear that ∆v(kA , kB ) = θA ( n − kA | 2n − kA − kB )|2n+1
for all
t+1
kA , kB ≤ n and min{kA , kB } < n can be verified iteratively by mathematical induction.
Proof of Theorem 2
Proof. In this proof, we establish E(Vt(g) ) = π g + θ( n| 2n)|−g , which is the core of Theorem
2(a). The proofs of the other parts of Theorem 2 are covered by the discussion following the
Theorem in the main text. Recall
E(Vt(g) ) =
kA +kB =t(g)−1
Pr( (kA , kB )| t(g))Vt(g) (kA , kB ).
Note that kA +kB =t(g)−1 Pr( (kA , kB )| t(g)) ≡ 1 regardless of t(g).
We first look at the case where t(g) ≤ n + 1. In this case, we must have max{kA , kB } ≤ n
and min{kA , kB } < n. By Theorem 1(b), we have
E(Vt(g) ) =
kA +kB =t(g)−1
= πg +
Pr( (kA , kB )| t(g))Vt(g) (kA , kB )
kA +kB =t(g)−1
θ A ( kA | kA + kB )|t(g)−1
× θA ( n − kA | 2n − kA − kB )|2n+1
1
t(g)+1
= π g + θA ( n| 2n)|−t(g) = π g + θB ( n| 2n)|−t(g) = π g + θ(n| 2n)|−g ,
26
where θi ( n| 2n)|−t(g) is the overall probability of team i’s winning exactly n out of the 2n
battles (nontrivial) other than battle t(g).
We now turn to the case where t(g) ≥ n + 2. This case can be divided into the following
subcases. (i) When max{kA , kB } ≥ n+1, Vt(g) (kA , kB ) = π g ; (ii) When max{kA , kB } ≤ n−1,
Vt(g) (kA , kB ) = π g + θA ( n − kA | 2n − kA − kB )|2n+1
t(g)+1 ; (iii) When kA = kB = n, Vt(g) (kA , kB ) =
2n+1
π g + θA ( n − kA | 2n − kA − kB )|t(g)+1 = π g +1; (iv) When kA = n, kB < n, or kA < n, kB = n,
Vt(g) (kA , kB ) = π g + θ A (n − kA | 2n − kA − kB )|2n+1
t(g)+1 .
Hence, when t(g) ≥ n + 2, we must have
EVt(g) = π g +
kA +kB =t(g)−1
kA ≤n, kB ≤n
θ A (kA | kA + kB )|t(g)−1
· θA ( n − kA | 2n − kA − kB )|2n+1
1
t(g)+1
= π g + θA ( n| 2n)|−t(g) = π g + θB ( n| 2n)|−t(g) = π g + θ(n| 2n)|−g .
Proof of Theorem 3
Proof. Part (a), (c) and (d) of Theorem 3 can be shown following the same arguments for
their counterparts in Theorems 1 and 2. We now focus on part (b).
In this extended setting, we continue to denote by (kA , kB ) the state of the contest. The
tuple indicates the number of victories each team has secured before battles in a cluster z
are carried out.
We first illustrate that for any battle t within any arbitrary cluster, the prize spreads
of the two teams must be symmetric regardless of the prevailing state (kA , kB ). The case
of max{kA , kB } ≥ n + 1 is trivial. In this case, one team has won. The prize spread is
simply πt , which is symmetric. Without loss of generality, hereafter we focus on the case of
kA , kB < n + 1.
Let n(z) be the number of battles included in a cluster z. Let Tz denote the set of battles
in a cluster z. We use t to index a battle in a cluster z with state (kA , kB ). A player A(t)
receives π t if he wins, and the contest may enter any state (kA +1+l, kB +n(z)−1−l), with l ∈
{0, 1, . . . , n(z)−1}, after all the n(z) battles in z are fought. Note that (kA +l, kB +n(z)−1−l)
can be used to denote a stochastic contest state facing battle t. If he loses, then the contest
may enter any state (kA + l, kB + n(z) − l), with l ∈ {0, . . . , n(z) − 1}, after all the battles in
z are fought.
Suppose that after z the contest is in state (k̃A , k̃B ). Let ṽi (k̃A , k̃B ) denote team i’s
conditional winning probability. Clearly, ṽi (k̃A , k̃B ) = 1 and ṽj (k̃A , k̃B ) = 0 when k̃i ≥ n + 1.
27
As a result, player A(t)’s effective spread amounts to
Here, ∆vtA (kA , kB ) =
VtA (kA , kB ) = π t + ∆vtA (kA , kB ), ∀t ∈ Tz .

n(z)−1 
[ṽA (kA + (1 + l), kB + n(z) − (1 + l))
l=0
where θ̃A ( l| n(z) − 1)
Tz \{t}
 −ṽA (kA + l, kB + n(z) − l)] · θ̃ A (l| n(z) − 1)


Tz \{t}

,
is the probability that team A wins l out of the n(z) − 1 simul-
taneous (nontrivial) battles in cluster z, excluding battle t.
Similarly, the effective prize spread for player B(t) is VtB (kA , kB ) = π t +∆vtB (kA , kB ), ∀t ∈
Tz , where




[ṽ
(k
+
l,
k
+
n(z)
−
l)
B A
B

n(z)−1 


B
−ṽB (kA + (1 + l), kB + n(z) − (1 + l))] .
∆vt (kA , kB ) =



l=0 


· θ̃ A (l| n(z) − 1)
)
Tz \{t}
Note that i∈{A,B} ṽi (kA + l, kB + n(z) − l) = 1 and i∈{A,B} ṽi (kA + (1 + l), kB + n(z) −
(1 + l)) = 1 hold for every l ∈ {0, ..., n(z) − 1}. We thus obtain VtA (kA , kB ) = VtB (kA , kB ).
Hence, each battle in z is symmetrically valued, and each nontrivial battle t in any cluster z
has a stochastic outcome (µA(t) , µB(t) ), which is solely determined by matched players’ cost
distributions by Theorem 1(a).
We then consider VtA (kA , kB ) to further pin down the symmetric prize spread Vt (kA , kB )
for a battle t in any cluster. The case of max{kA , kB } ≥ n + 1 is trivial. One team has
won, thus the prize spread is simply π t . We now focus on the case of kA , kB ≤ n. For the
unclustered battles after the last clustered battles, the results of Theorem 1(b) apparently
hold. We now consider the last cluster that contains more than one battle, which is denoted
by z1 . We assume without loss of generality that there are unclustered battles following this
cluster.25 Theorem 1(b) applies to all (unclustered) battles that follow cluster z1 . Recall that
∆v(kA , kB ) is defined in Observation 1 as vA (kA + 1, kB ) − vA (kA , kB + 1) or equivalently
vB (kA , kB + 1) − vB (kA + 1, kB ). Note that from Theorem 1(b), we have ∆v(kA , kB ) =
θi ( n − ki | 2n − kA − kB )|2n+1
t+1 .
For a stochastic state (kA + l, kB + n(z1 ) − 1 − l), let
ξA
t (kA + l, kB + n(z1 ) − 1 − l)
= ṽA (kA + (1 + l), kB + n(z1 ) − (1 + l)) − ṽA (kA + l, kB + n(z1 ) − l).
25
The case in which no more battles follow the cluster z1 is simpler and yields the same result.
28
We have
ξA
t (kA + l, kB + n(z1 ) − 1 − l)
= vA (kA + (1 + l), kB + n(z1 ) − (1 + l)) − vA (kA + l, kB + n(z1 ) − l)
= ∆v(kA + l, kB + n(z1 ) − 1 − l)
= θA ( (n + 1) − (kA + 1 + l)| 2n + 1 − (kA + kB + n(z1 )))|2n+1
kA +kB +n(z1 )+1 ,
if kA + l < n + 1, and kB + n(z1 ) − 1 − l < n + 1; and it boils down to zero otherwise. Hence,
∆vtA (kA , kB ) =
n(z1 )−1
[ξ A
t (kA + l, kB + n(z1 ) − 1 − l) · θ̃ A ( l| n(z1 ) − 1)
l=0
min{n(z1 )−1,n−kA }
Tz1 \{t}
]
[ξ A
t (kA +l,kB +n(z 1 ) − 1 − l)· θ̃ A ( l| n(z1 ) − 1)
=
l=max{0,kB +n(z1 )−1−n}
min{n(z1 )−1,n−kA }
θA ( (n + 1) − (kA + 1 + l)| 2n + 1
=
[
−(kA + kB + n(z1 )))
l=max{0,kB +n(z1 )−1−n}
· θ̃ A (l| n(z1 ) − 1)
Tz1 \{t}
Tz1 \{t}
]
2n+1
kA +kB +n(z1 )+1
]
= θA ( n − kA | 2n − (kA + kB ))|{t̃|t̃≥kA +kB +1,t̃=t} , ∀t ∈ Tz1 ,
where θA ( n − kA | 2n − (kA + kB ))|{t̃|t̃≥kA +kB +1,t̃=t} is the probability of team A’s winning exactly n − kA out of the 2n − kA − kB (nontrivial) battles (excluding battle t) when the contest
enters the state (kA , kB ) after the cluster that precedes z1 is contested.
It follows that Vt (kA , kB ) = VtA (kA , kB ), t ∈ Tz1 can be rewritten as
π t + ∆vtA (kA , kB ) = π t + θ A ( n − kA | 2n − (kA + kB ))|{t̃|t̃≥kA +kB +1,t̃=t} .
The above formula for Vt (kA , kB ) still applies for the case max{kA , kB } ≥ n + 1. In this
case, clearly θ A (n − kA | 2n − (kA + kB ))|{t̃|t̃≥kA +kB +1,t̃=t} = 0. Because kA + l < n + 1, and
kB + n(z1 ) − 1 − l < n + 1, the range of l (i.e. from max{0, kB + n(z1 ) − 1 − n} to min{n(z1 ) −
1, n − kA }) excludes all (and only) the events that the result of battle t does not make a
difference in determining the winning probability of the whole contest.
We then consider the cluster that immediately precedes z1 , which is denoted by z2 . Suppose that it faces an arbitrary state (kA , kB ). Note that we allow it to include only one battle.
Recall that we can focus on the case of kA , kB < n + 1. We have
=
∆vtA (kA , kB )

n(z2 )−1 
l2 =0


[ṽA (kA + (1 + l2 ), kB + n(z2 ) − (1 + l2 ))
 −ṽA (kA + l2 , kB + n(z2 ) − l2 )] · θ̃A ( l2 | n(z2 ) − 1)
29
Tz2 \{t}

.
Suppose that after z2 the contest is in a state (k̃A , k̃B ). Recall that ṽi (k̃A , k̃B ) = 1 and
ṽj (k̃A , k̃B ) = 0 when k̃i ≥ n + 1. Note that we have shown that in cluster z1 , every nontrivial
battle t has winning probabilities (µA(t) , µB(t) ). We now calculate ṽi (k̃A , k̃B ) for k̃A , k̃B ≤ n.
Note in this case, every battle in cluster z1 is nontrivial. Thus
n(z1 )
ṽA (k̃A ,k̃B ) =
[vA (k̃A +l1 ,k̃B +n(z 1 ) − l1 )· θ̃A ( l1 | n(z1 ))
l1 =0
Tz1
], with k̃A +k̃B =kA +kB +n(z 2 ).
Hence, ∀l2 ∈ {0, 1, ..., n(z2 ) − 1},
ṽA (kA + (1 + l2 ), kB + n(z2 ) − (1 + l2 )) − ṽA (kA + l2 , kB + n(z2 ) − l2 )
n(z1 )
=
[vA (kA + (1 + l2 ) + l1 , kB + n(z2 ) − (1 + l2 ) + n(z1 ) − l1 )
l1 =0
−vA (kA + l2 + l1 , kB + n(z2 ) − l2 + n(z1 ) − l1 )] · θ̃A ( l1 | n(z1 ))
Tz1
n(z1 )
=
∆v(kA + l2 + l1 , kB + n(z2 ) − (1 + l2 ) + n(z1 ) − l1 ) · θ̃A ( l1 | n(z1 ))
l1 =0
Tz1
n(z1 )
{ θA ( n − (kA + l2 + l1 )| 2n + 1 − (kA + kB + n(z2 ) + n(z1 ))|2n+1
kA +kB +n(z2 )+n(z1 )+1
=
l1 =0
· θ̃A ( l1 | n(z1 ))
Tz1
}
= θA ( n − (kA + l2 )| 2n + 1 − (kA + kB + n(z2 ))|2n+1
kA +kB +n(z2 )+1 .
(3)
Therefore, ∀t ∈ Tz2 , we have
∆vtA (kA , kB )
n(z2 )−1
{ θ A ( n − (kA + l2 )| 2n + 1 − (kA + kB + n(z2 ))|2n+1
kA +kB +n(z2 )+1
=
l2 =0
· θ̃ A (l2 | n(z2 ) − 1)
Tz2 \{t}
}
= θA ( n − kA | 2n − (kA + kB ))|{t̃|t̃≥kA +kB +1,t̃=t} .
(4)
In view of (3), we can also obtain ∆vtA (kA , kB ), ∀t ∈ Tz3 by considering clusters z3 and z2
while applying the procedure for deriving (4) by considering clusters z2 and z1 . The following
general formula can then be obtained: for any battle t in any cluster zk ,
∆vtA (kA , kB ) = θ A (n − kA | 2n − (kA + kB ))|{t̃|t̃≥kA +kB +1,t̃=t} , ∀t ∈ Tzk ,
(5)
where (kA , kB ) is the contest state before cluster zk is carried out.
By repeating the exercise in the proof of Theorem 2, we can conclude that each battle t
has an ex ante expected prize spread of θA ( n| 2n)|−t , which does not depend on how these
battles are clustered or sequenced. We then complete the proof.
30
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