Wars of Attrition with Endogenously Determined
Budget Constraints∗
Joshua Foster†
Economics Department
University of Wisconsin - Oshkosh
November 7, 2016
Abstract
This paper models a war of attrition where participants first choose contest investment
levels that act as a constraint for how long they can compete. To include a measure of the
resource’s transferability to other uses (e.g. other contests), expenditures are a convex combination of investment decisions and the contest’s duration. It is shown in the symmetric
equilibrium that participants use a mixed strategy for their resource investments and plan
to exhaust those resources in the contest. Implications of participants’ abilities to compete
in subsequent contests are explored. These modifications to the standard model allow for
important insights into a variety of pre-calculated and budgeted all-pay contests.
Keywords: War of attrition, All-pay auction, Contest design, Budget constraints
JEL Classification: D44, D82, L13
∗I
would like to thank Cary Deck, Amy Farmer, Jeffrey Carpenter, Salar Jahedi, Shannon Rawski, and seminar
participants at the University of Arkansas, ESA 2013 North America Meetings, and SEA 2013 Annual Meetings for
their helpful comments at various stages of the development of this project.
† You may contact the author at [email protected].
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WARS OF ATTRITION WITH ENDOGENOUSLY DETERMINED BUDGET CONSTRAINTS
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This paper presents a theoretical variation of the common value war of attrition in which
participants must make initial investments to claim an indivisible prize. It is applicable to
pre-calculated all-pay contests for which participants have planned strategically. There are
several examples of large conflicts best described as wars of attrition, and most of them are
non-spontaneous. This includes military tactics where participants must make costly outlays
over time in pursuit of claiming a prize. For defensive purposes, planning for and investing
in these contests is critical when the prize is currently in your possession. Another attribute
of these contests is that it can be difficult to repurpose the unused resources invested in the
contest when it is over. For these reasons it is important to understand how endogenizing
participants’ partially sunk investments impacts contests as it will change the contest duration
and individual expenditures, both of which become increasingly meaningful in the presence of
time-based externalities.
The model proceeds in two stages. In the first stage, participants choose an initial investment that determines the maximum duration for which they may compete, thereby endogenizing their contest budget constraints. In the second stage, participants then decide for how long
they are willing to compete in the contest given their resources. If, when the contest ends, a participant did not exhaust their budget, then the unused portion has a repurpose value at some
exogenously determined fraction of their original cost. The model is solved using backward
induction. It is shown there is a unique symmetric perfect-Bayesian equilibrium where contest
duration increases as it is cheaper to repurpose unused contest resources. Additionally, it is
shown that contest duration decreases as the number of participants increases. Finally, the only
participant who will have unused resources in equilibrium is the contest winner. The unused
portion of their investment increases as it becomes cheaper to repurpose said investment; the
unused portion decreases as the number of participants increases. This model is extended to
explore investment and bidding strategies over a sequence of contests. The symmetric equilibrium is presented for a special case of this extension.
1
Literature Review
The variety of applications for the war of attrition is impressive. While suitable for modeling
fighting among fiddler crabs (Morrell et al. 2005), it is equally suited for modeling macroeconomic stabilization (Alesina and Drazen 1991). Economists have gleamed theoretical insights from this model in funding public goods (Andreoni 2006), markets facing decreasing net
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FOSTER, JOSHUA R
present value (Ghemawat and Nalebuff 1985, 1990; Fudenberg and Tirole 1986), labor strikes
(Kennan and Wilson 1989), patent races (Leininger 1991), political lobbying (Dekel et al. 2008,
2009) and economic diplomacy (Ponsatí 2004).
In general, the war of attrition models n + k contestants vying for k indivisible prizes. It is
assumed that contestants must make unrecoverable investments through time until n contestants choose to exit, and the prizes are awarded to the k contestants who are willing to stay in
the contest the longest. Originally introduced by biologists to describe animal conflicts (Smith
1974), an apparent asymmetry between the the animals’ abilities to compete allowed for costless determination of the victor (the relatively strong contestant). This was first defined as the
handicap principle in Zahavi (1975, 1977), which states that costly signals can help minimize animal conflicts. On the other hand, when there is no salient asymmetry between the contestants
the conflict can result in costly investments to determine the winner.
Several papers consider situations where there is asymmetry in the value of the prize to the
contestants, which is public knowledge, such as Smith and Parker (1976) and Hammerstein and
Parker (1982). These papers demonstrate that the stronger contestant tends to win uncontested,
as well as validate that if there is no heterogeneity among the contestants then there are no
asymmetric equilibria. If, instead, the identities of valuations are unknown then the continuum
of asymmetric equilibria is maintained and a “pecking order” emerges among the contestants
(Nalebuff and Riley 1985). Here, contests often resolve with one very aggressive contestant and
one very passive contestant. However, it is possible (though still improbable) for those lower in
the pecking order to make serious challenges. More recently, Myatt (2005) looks at asymmetric
wars of attrition where contestants have different distributions from which they draw their
private values. Contestants’ ‘stochastic strengths’ are compared based on these distributions,
where a contestant is stochastically strong if she has a larger ex ante valuation - even if her
realization ex post is smaller. The standard design is modified by, for example, setting a time
limit (then a winner is picked randomly). Here, the author shows that the stochastically weaker
contestant exits instantaneously.
Economists began employing the war of attrition by applying it to firm exits in declining
future value markets. Beginning with Ghermawat and Nalebuff (1985), it was assumed that
a potentially heterogeneous duopoly resulted in negative firm profits, but a monopoly was
sustainable. In their paper, it was further assumed that production was rigid, leading to firms
to exit in an ‘all-or-nothing’ manner. This led to the prediction that the biggest firms in the
market will, in expectation, exit first. The somewhat unrealistic assumption placed on firms’
WARS OF ATTRITION WITH ENDOGENOUSLY DETERMINED BUDGET CONSTRAINTS
4
all-or-nothing strategy for production was relaxed in Whinston (1988), where firms instead had
multiple plants allowing for shifts down in production. That paper corroborated the original
prediction that larger multi-plant operations exit the declining market first. In a third paper,
Ghemawat and Nalebuff (1990), the production assumption was relaxed further, allowing for
firms to continually adjust production levels. The results of their theoretical model predict that
relatively large firms will shrink to the sizes of their (formerly) smaller rivals.
In a slightly different context, Fudenberg and Tirole (1986) offers a theory of exiting in an
unsustainable duopoly. The model is largely based on differing fixed or opportunity costs,
which are private knowledge to the firms. In their model, the only strategic variable for firms is
the time of exit, the last of which receives monopoly profits. To generate a unique equilibrium it
is also assumed that firms could potentially compete forever - which is justified by suggesting
that the market may at some point be able to sustain the current duopoly. This model of firm
exit was directly tested by Oprea et al. (2013), where it is reported that firm exits are efficient
(76% of the time higher cost firms exit first) and very closely resemble the point predictions of
the model. It is further reported in Oprea et al. that there is little difference in strategies when
payoffs are framed as losses or as gains.
Almost simultaneously to Fudenberg and Tirole (1986), applications for the war of attrition
were being fashioned within firms’ research and development (R&D) strategies. Among the
first in this vein, Harris and Vickers (1985) model two contestants seeking to claim an indivisible
prize. The prize is awarded to the contestant to reach the ‘finish line’ first (i.e. to complete some
task first), which involves several stages of making unrecoverable investments. In their model,
it is shown that the winner behaves exactly as if she were the only contestant. Moreover, it
is shown that another contestant has the potential to impact the winner’s behavior in the first
stage of the game only, and not thereafter. Building from this base, Leininger (1991) models
a patent race using a dynamic, turn-based all-pay auction. Here, only by outbidding one’s
opponent can one remain in the race. Because that paper abstracts from discounting payouts
through time, the characterization of equilibrium becomes quite complicated.
It is easy to envision that the repeated play of these games requires some consideration for
the potential of reputational effects. One of the first papers to consider the role of reputations
in finitely repeated games was Kreps and Wilson (1982) in the context of “the chain-store paradox” (Selten 1978). Their paper considers how imperfect information acts as a mechanism for
how strategies in one period can influence payoffs in the next. Allowing for one-sided and
two-sided uncertainty, they demonstrate that the development of a reputation can, in fact, be
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FOSTER, JOSHUA R
advantageous in the way one intuitively expects.
The lobbying of legislatures is a more recent application of the war of attrition in political
science. In a pair of papers, Dekel et al. (2008, 2009) consider, first, how candidates for public
office would solicit votes in an election when the notion of buying votes is publicly acceptable
and legal. While the model relies on an all-pay auction to describe investments from candidates,
this notion of “buying” can be framed as designing one’s platform for the campaign. Here, the
authors focus primarily on a sequential and complete information design. Second, the authors
consider the payments lobbyists would make to legislatures in a similar design, again, if doing
so were publicly acceptable and legal.
There are several theoretical considerations that are appropos to the war of attrition include
the dollar auction, the tug of war and the ability to jump bid in a sequential all-pay auction. The
dollar auction is an all-pay auction where the two highest bidders must pay their bids, but only
the higher of the two gets the dollar. This auction was designed to show that individuals who
have complete information in a sequential move game will indeed make economically irrational
decisions. In O’Neill (1986) and Leininger (1989), the authors modify the original dollar auction
presented in Shubik (1971). These modifications both assume that there are known budget
constraints that allow for backward induction to take place, which does not allow for escalation
to take place. Here, only one bid is offered and the level of this bid is sensitive to the budget
constraints.
The ‘tug of war’ is a modification of the war of attrition. Here, a player must win a series of
contests to claim a prize, whereas in the war of attrition a contestant may quit at any time. It is
called a tug of war because, reminiscent of the gym class activity, each contest won brings the
participant closer to a victory. Konrad and Kovenock (2005) reports a unique Markov perfect
eqiulibrium where contestants focus their effort on “tipping states” which are determined, in
part, by their relative strength. They also show that each contest outcome in the tug of war is
stochastic, but the eminence of one’s victory increases the probability that they win. Moreover,
Agastya and McAfee (2006) corroborate the prediction that one’s likelihood of losing the tug of
war decreases the likelihood they win succeeding battles.
As in this paper, others have also given consideration to relaxing some of the common assumptions associated with the war of attrition. For instance, with respect to jump bidding,
Hörner and Sahuguet (2007) suggest that a single and continuous cost paid per time unit is an
unreasonable assumption to make for many situations currently modeled as a war of attrition.
In an alternative proposition, the authors evaluate how strategies would change when contes-
WARS OF ATTRITION WITH ENDOGENOUSLY DETERMINED BUDGET CONSTRAINTS
6
tants are allowed to vary the amount they choose to spend in a given time period. In their
setup, these bids reveal information about valuations, and contestants must match their opponents’ total expenditures or else exit. Games of complete information are given the primary
focus, along with the mixed strategies thereof - though there are asymmetric equilibria where a
contestant will give up with a high probability at the beginning. The authors find that expected
delay is shorter and rent dissipation is smaller with these assumptions. This is partly due to
the ability for contestants to jump bid. After several decades of considering special cases of
the war of attrition, Bulow and Klemperer (1999) offers a generalized model that provides the
unique perfect Bayesian Nash equilibrium for any n contestant and k prize contest with private
information. Their paper, built from the contributions of Hendricks et al. (1988), shows that for
any war of attrition with no continuation cost (i.e. contestants who exit don’t incur any costs
while the contest is ongoing) all but k + 1 contestants will drop out instantaneously. It is further
shown that this result is efficient.
This paper is most similar to Amann and Leininger (1996), which makes a participant’s
total expenditure a convex combination of the contest’s duration and their own investment
strategy. However, a few fundamental differences exist between the previous literature and this
paper. First, resource investment decisions are endogenously determined by each participant
at the beginning of the contest. Second, the payments made by all participants are a convex
combination of the loser’s bid and their own resource investment decision according to an
exogenously determined repurpose value. It will be shown that by setting the repurpose value
of unused investments equal to zero (i.e., investments are fully sunk into the contest) that direct
comparisons can be made to the more familiar first-price all-pay auction. In contrast, setting the
repurpose value of unused investments equal to their original cost (i.e., resources can be fully
transferred to other contests) one recreates the war of attrition. Regardless of the repurpose
value of unused resource investments, in the symmetric equilibrium participants will use a
mixed strategy when making their resource investments and will intend to compete in the
contest until that investment is exhausted.
Setup
In this contest there are k + 1 risk-neutral participants looking to claim a single prize with
a pure common value, v. Figure 1 outlines the timing of the decision stages of this contest.
Let t = 0 be the investment stage where participants privately choose a resource investment
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FOSTER, JOSHUA R
x ∈ [0, ∞), which they will draw from during the contest stage. It is assumed t = 0 is the
only time participants can contribute to their investment resource for the contest. For the same
reasons there are symmetric mixed strategies in all-pay auctions under complete information
(see Smith 1974, Baye et al. 1996), so too are strategies in the investment stage.1 As such, let x be
drawn from a continuous distribution F with density f .
Figure 1: Timing of Decision Stages
Investment Stage:
Contest Stage:
Participants privately choose
their investments, x.
Participants choose their exit time, b(x).
Time t = 0
Time t > 0
Due to the private information from the investment stage, analysis of the contest stage will be
restricted to symmetric perfect-Bayesian equilibria by modifying Bulow and Klemperer (1999).2
Let b(x) be the ‘bidding’ function used to define exit times in the contest stage. Participants incur
a continuous cost to compete for the prize at a normalized rate of 1 per unit time, t. If, after
the contest has been resolved, a participant has any investment remaining (i.e. not utilized
during the contest) it may be repurposed for another use at an exogenously determined rate
θ ∈ [0, 1] per unit of x. Practically, this may occur in a contest for several reasons: one is
participants may choose to exit without using their entire investment; another is a participant
may win the contest before their investment is fully exhausted. Both cases represent important
considerations for pre-planned wars of attrition when unused resources may later be applied
to another contest. In the following analysis the symmetric equilibrium will be established via
backward induction, beginning with the contest stage.
The Contest Stage
According to the setup proposed by this paper, participants will have privately chosen their
resource investments according to a mixed strategy F (x) before the contest stage begins. Since
unused investments can be repurposed at the rate θ after the contest, this means the cost of
1 There
is no equilibrium involving pure symmetric investment strategies, since for any investment x̃ > 0 each
participant could deviate to either 0 or x̃ + and be strictly better off.
2 For discussions of asymmetric equilibria, see Nalebuff and Riley (1985) regarding wars of attrition, see Baye et al.
(1996) regarding first price all-pay auctions.
WARS OF ATTRITION WITH ENDOGENOUSLY DETERMINED BUDGET CONSTRAINTS
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competing dt longer is reduced to θdt.3 The payout to a participant with an investment of x
who bids as though they have an investment of r is
Z
u(r | θ, k, v) =
r
[v − θb(z) − (1 − θ)x]kF (z)k−1 f (z)dz − (θb(r) + (1 − θ)x) [1 − F (r)k ]
(1)
0
s.t. x − b(r) ≥ 0 and r ≥ 0.
The first term in (1) is the expected payout from winning the contest when the k other participant choose to bid an amount less than b(r), while the second term is the expected payout from
losing by being outbid by at least one other participant. The costs incurred are a convex combination of the participant’s bid and the contest’s duration. A constraint is placed on participants
in this setting, namely bidding strategies must reflect that no one may bid more than their investment. The necessary conditions for the optimal bidding function, based off the Lagrangian
function associated with (1), are
Lr = vkf (r)F (r)k−1 − θb0 (r)[1 − F (r)k ] − λb0 (r) ≤ 0
b(r) − x ≤ 0
r≥0
λ≥0
r vkf (r)F (r)k−1 − θb0 (r)[1 − F (r)k ] − λb0 (r) = 0
λ [x − b(r)] = 0
where λ is the Kuhn-Tucker multiplier. Solving this set of inequalities begins by assuming the bidding constraint is binding, allowing λ ≥ 0 and b(r) = x. This implies −θ +
(r)
≥ 0. Solving this inequality in terms of F (r) and imposing r = x according
F (r)k θ + kvf
F (r)
to the revelation principle (Myerson 1981) leads to Finding 1.
Finding 1: A participant intends to expend their entire investment of x in the contest when
the cumulative density over investments F (x) is sufficiently large. Specifically,
b(x | θ, k, v) =


x
1
if F (x) ≥ 1 − exp(− vθ x) k
f (z)F (z)k−1
dz
0
1−F (z)k

 vk R x
θ
(2)
otherwise.
3 Another way to understand this is to say for an investment of x a participant automatically forfeits (1 − θ)x of it
by the exogenous nature of the contest environment.
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FOSTER, JOSHUA R
Finding 1 demonstrates that bidding constraints are binding when the probability other
participants are choosing smaller investments is sufficiently large. If the condition in (2) is
not met, then participants will bid less than their investment according to the hazard function
vk
θ .
presented in (2), adjusted by
In this case we find, ceteris paribus, that increasing the prize
value (v) or decreasing the redemption value of unused investments (θ) leads to larger bids.
In the next section we backward induct optimal investment decisions given this bidding
function. We begin by first assuming that the bidding constraint is binding such that b(x) = x.
The Investment Stage
Given the optimal bidding behavior from the contest stage, it is possible to backward induct
optimal investment behavior. Substituting the bid function b(x) = x into (1) yields
Z
u(x | θ, k, v) =
x
[v − θz − (1 − θ)x]kF (z)k−1 f (z)dz − x[1 − F (x)k ].
0
Finding the first order condition yields the differential equation
0 = vkF (x)k−1 f (x) + θF (x)k − 1
(3)
Solving the differential equation in (3) and imposing F (0) = 0 gives us Finding 2.4
Finding 2: There is a unique symmetric perfect-Bayesian equilibrium in which
k1
1
θ
, and
F (x | θ, k, v) =
1 − exp − x
θ
v
(4)
b(x) = x.
Comparing this equilibrium distribution over investments to that in (2) we find the bidding
constraint is, in fact, binding for all θ ≤ 1.5 Moreover, it can be shown there is no symmetric mixed strategy equilibrium over investments when the bidding constraint is not binding.
Specifically, there is a constant negative return to investments of −(1 − θ).
The equations in (4) show the expected investment from the investment stage is increasing
4 To demonstrate that the lower support of the mixing distribution is atomless (F (0) = 0), consider a symmetric
equilibrium where both players put positive mass on an investment of zero. It is straight-forward to see there is an
incentive to deviate one’s investment decision of zero to and win those contests that previously would have been ties.
5 For θ = 1, this strategy is weakly dominant since unused investments are costless. Because this setting is equivalent
to the traditional war of attrition participants would be indifferent between the strategy in Finding 1 and the traditional
equilibrium strategy (i.e. choosing an infinite investment level while exiting according their mixed strategy).
WARS OF ATTRITION WITH ENDOGENOUSLY DETERMINED BUDGET CONSTRAINTS
10
in θ. In the limit as θ approaches zero, and all resource investments are fully sunk, the mixing
distribution becomes the familiar first-price all-pay equilibrium over bids for k + 1 participants,
1
F (x) = xv k , as shown in Baye et al. (1996). When θ = 1 and unused investments may be
fully repurposed, F (x) becomes the standard war of attrition bidding equilibrium for k + 1
1
participants, F (x) = 1 − exp(− xv ) k .
Finally, we show that the expected total expenditure by all participants, E, is equal to v, a
common feature of contests for prizes with pure common values. This is defined by
Z
x̄
x f (x) − θ f(k+1) (x) − f(k) (x) dx
E = (k + 1)
0
Z
= (k + 1)
x̄
x 1 + θF (x)k−1 (k − (k + 1)F (x)) f (x)dx
(5)
0
= v.
Here, f(k+1) (x) and f(k) (x) define the probability density function of the (k + 1)th and k th smallest of k + 1 independent draws from F , and where x̄ gives us F (x̄) = 1. The value of x̄ varies
with θ. To explicitly define the value of x̄ for all θ we solve for x̄ when F (x̄) = 1 which gives us
x̄ = − vθ ln[1 − θ]. We find that limθ→0+ x̄ = v and when limθ→1− x̄ = ∞.
One Investment, Sequential Contests
A natural extension of this model is to consider equilibrium resource investment strategies
when a series of contests is expected. To explore this, we assume there is a sequence of C
contests. In a given contest c ∈ {1, 2, ..., C} there is a prize of vc awarded to one of kc + 1
participants. In order for a participant to compete in the cth contest they must win the (c − 1)th
contest. Therefore, participants in subsequent contests are utilizing residual resources from the
previous contests based on their original investment of x.
If participants begin the first contest with an investment drawn from F , then for each subsequent contest a new distribution over residual investment resources of winners must be established. To do this, we calculate the cumulative density function for the winning participant
Pc−1
in contest c − 1 retaining xc investment resources or less, where xc = θc−1 x − i=1 θi b(xki ,i )
and b(xki ,i ) is the bid from participant with the kith highest investment in the ith contest. Thus,
there is a distribution over the expected residual investments in contest c > 1 defined by Fc (xc ).
Next we describe the determination of Fc (xc ).
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FOSTER, JOSHUA R
Determination of the cth Contest’s Resource Distribution
Using the joint distribution of the kc−1 and kc−1 + 1 order statistics we can calculate the distribution of differences, d, between the largest investment and the bid from the second largest
investment in the (c − 1)th contest for any underlying distribution of investments, Fc−1 (x). For
any two largest investments x < y from kc−1 + 1 draws of Fc−1 , this difference is d = y − b(x).
The joint density over these investments is given by
fc (x, y) = kc−1 (kc−1 + 1)fc−1 (x)fc−1 (y)Fc−1 (x)kc−1 −1 .
The figure below symbolically describes how this distribution is determined by exploring level
curves of d = y − b(x).
Figure 2: Level Curves of Differences in the Joint Distribution of Order Statistics
d = y − b(x)
ȳ = d¯
y=d
x̄
ȳ − b(x) = d
x
ȳ
y
Because b(x) < y, analysis in the figure in restricted to the area left of the vertical partition.
The range of possible values of d starts at zero, where investments are the same, and increases to
¯ where the largest investment is at its maximum value (ȳ) and the second largest investment
d,
is at its minimum value (x = 0, thus b(0) = 0). For some difference in the largest investment
and the bid from the second largest investment d = y − b(x), the joint probability distribution
WARS OF ATTRITION WITH ENDOGENOUSLY DETERMINED BUDGET CONSTRAINTS
12
can be re-expressed in terms of y as
f (d, y) = kc−1 (kc−1 + 1)fc−1 (b−1 (y − d))fc−1 (y)Fc−1 (b−1 (y − d))kc−1 −1
where b−1 is the inverse of the bid function. Finally, to determine the probability with which
some difference d occurs we integrate over all combinations of b(x) and y where d = y − b(x).
This is equivalent to integrating over the solid line in the figure from d to ȳ. Thus, the new
distribution is
ȳ
Z Z
fc−1 (b−1 (y − d))fc−1 (y)Fc−1 (b−1 (y − d))kc−1 −1 dydd
Fc (d) = kc−1 (kc−1 + 1)
(6)
d
which includes a constant that ensures F (0) = 0 since ties are impossible.
Establishing the distribution of residual investments allows for the consideration of bidding
behavior in sequential contests according to (1). In the next subsection, we apply this analysis
to a special case to show how it affects investment decisions in equilibrium.
Special Case: C = 2, kc = 1, vc = v, θ = 1
In this special case there are C = 2 contests, both with kc + 1 = 2 participants and a pure
common value prize of v. At the end of each contest, unused investments have a full redemption value of θ = 1.
Relying on backward induction, analysis begins with the last subgame, the second contest.
In this subgame, (1) once again derives Finding 1 to determine the equilibrium bidding behavior
in the second contest. Given this bidding strategy, we now consider bidding strategies in the
first contest. To do so, we modify (1) to include a continuation payoff of u2 (r − z | 1, 1, v) - the
expected return of the second contest according to F2 (x). This gives us
Z
u1 (r | 1, 1, v) =
r
[v − b(z) + u2 (r − b(z) | 1, 1, v)]f (z)dz − b(r)[1 − F (r)]
(7)
0
s.t. x − b(r) ≥ 0 and r ≥ 0.
We begin by assuming F (x), F2 (x) ≥ 1 − exp(− v1 x), thus making the bidding constraints
binding and giving b(x) = x in both contests. Using (6) to define F2 (x) where b−1 (x) = y −
d under our temporary assumption, it can be shown there is a symmetric perfect-Bayesian
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FOSTER, JOSHUA R
equilibrium in which
1
F (x | 1, 1) = 1 − exp − x , and
v
1
F2 (x | 1, 1) = 1 − exp − x , and
v
b(x) = x for both contests.
Interestingly, the results of this special case indicate that the initial investment distribution
F (x) leads to an identical residual investment distribution for the second contest. Moreover,
this distribution is identical to the distribution we would have anticipated for the case where
C = 1 - just a single contest - which we know from previous analysis incentivizes binding
bidding constraints. Finally, since the original investment distribution is self-replicating in the
residual investment distributions, it is clear from this special case that this result will hold for
any number of contests C. As a result, we would expect empirical tests of this model to show
no difference in the investments between participants in one contest or a sequence of contests
of any size.
Conclusion
In pre-calculated wars of attrition an important strategic decision for participants is to decide how much they are willing to invest in the contest. This investment decision can also
constrain a participant’s ability to compete. Here, symmetric equilibrium strategies are provided for when resource investments are endogenized into the contest. It does so using a two
stage design where, in the first stage, participants privately choose their resource investments
for the contest, and in the second stage choose for how long to compete given their investment
levels. Additionally, it is assumed that any resources invested in the contest but never utilized
have some repurpose value afterward. This basic model is extended to a tournament where
winning participants compete in a series of sequential contests.
The model presented in this paper is important in understanding behavior in pre-calculated
and purposefully budgeted all-pay contests. It demonstrates for this modified war of attrition
there is a unique symmetric perfect-Bayesian equilibrium where participants rely on a mixed
strategy for their resource investments and use the pure strategy of bidding all of that investment in the contest. This theoretical result suggests that the preceding stages to a pre-calculated
WARS OF ATTRITION WITH ENDOGENOUSLY DETERMINED BUDGET CONSTRAINTS
14
war of attrition may be the most important ones, as in equilibrium a majority of the expenditure often comes directly from investments, and the contest itself is a formality of the strategies
employed in its preparation.
References
[1] Agastya, Murali and R. Preston McAfee (2006): “Continuing Wars of Attrition,” Working
Paper, Available at SSRN.
[2] Alesina, Alberto and Allan Drazen (1991): “Why are Stabilizations Delayed?” American
Economic Review, 81 (5), 1170-1188.
[3] Amann, Erwin and Wolfgang Leininger (1996): “Asymmetric All-Pay Auctions with Incomplete Information: The Two-Player Case,” Games and Economic Behavior, 14, 1-18.
[4] Andreoni, James (2006): “Leadership Giving in Charitable Fund-raising” Journal of Public
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