The War of Attrition and the Revelation of Valuable
Information∗
Maciej H. Kotowski†
Fei Li‡
March 30, 2014
Abstract
We provide a simple example demonstrating that the unconditional revelation information in a war of attrition with private budget constraints can decrease expected
revenue. Our example suggests that information non-revelation can counteract the adverse revenue impact of budget constraints and almost counterbalance their otherwise
negative effects.
Keywords: War of Attrition, Budget Constraints, Linkage Principle, Auctions, Information Disclosure, Contests, Patent Race
JEL: D44
∗
We thank Sergio Parreiras for comments on a preliminary draft of this manuscript.
John F. Kennedy School of Government, Harvard University, 79 JFK Street, Cambridge MA 02138.
E-mail: <[email protected]> (Corresponding author.)
‡
Department of Economics, 107 Gardner Hall, CB 3305, University of North Carolina, Chapel Hill NC
27599. E-mail: <[email protected]>
†
1
Milgrom and Weber (1982) establish that in many circumstances an auctioneer should
unconditionally reveal privately-held information that is relevant to bidders. In expectations
such a policy cannot adversely affect revenue. This result is often noted as a case of the linkage principle. Building on Fang and Parreiras (2002), Fang and Parreiras (2003) construct
an example of a second-price auction where the linkage principle holds if bidders are mildly
budget-constrained, but fails once those constraints are sufficiently severe.1
In this note we provide a new example demonstrating the drawbacks of information
revelation when agents are budget constrained, even if those constraints are arbitrarily mild.
In contrast with other such examples, we focus on the war of attrition.2
In environments where agents are budget-constrained, this mechanism is of interest for
several reasons. First, allocation mechanisms with an all-pay flavor are generally considered
to be revenue-superior to their “winner-only”-pay counterparts, such as first- or second-price
auctions, when budget constraints are present. (Maskin, 2000; Pai and Vohra, 2014). Thus,
our is a tentative step toward incorporating elements of information disclosure into the study
of optimal mechanisms with budget-constrained bidders.3 We argue that a policy of nonrevelation can in some cases almost fully counteract the adverse revenue implications of
budget constraints.
Second, our results inform the design of contests when participants face private limits
on the effort or resources that they can expend. Many situations, such as patent races,
have a natural formulation as a war of attrition where budgets are a key consideration
(Leininger, 1991). If the goal is to maximize expected total effort (or revenue), then our
results suggest that withholding relevant information from participants may be a worthwhile
policy. Conversely, if effort or expenditures are otherwise wasteful, information provision
can limit profligacy.
Finally, our study highlights a comparatively tractable class of examples for the study
of auctions with private budget constraints.4 Noting the scarcity of such instances in the
literature, this observation may be of interest to others examining related questions in auction
and mechanism design.
Our note is organized simply. The next section introduces our model, but without budget
constraints. Section 2 incorporates private budget constraints and examines revenue under
1
The linkage principle can fail in other circumstances as well (Perry and Reny, 1999; Board, 2009).
We model the war of attrition as a static game, as in Krishna and Morgan (1997).
3
Gershkov (2009) shows that in an environment without budget constraints, an optimal selling mechanism
involves full information disclosure.
4
Here we mean the war of attrition where budgets have an exponential distribution.
2
2
both disclosure and non-disclosure regimes.
1
A War of Attrition
Consider a war of attrition with two ex ante symmetric bidders competing for a single prize.5
Each bidder i submits a bid, bi ≥ 0, and the highest bidder is the winner. If bidder i is the
winner, his payoff is Vi − bj where Vi is his valuation and bj is the bid of bidder j 6= i. Bidder
j’s payoff in this scenario is −bj . A fair coin flip resolves ties.
Bidder i’s valuation for the prize is a product of two pieces of information: Vi = Si S0 . Si is
bidder i’s private signal concerning the prize’s value. It is known only to him and is unknown
to others. S0 is information held by an outside party (the “seller”). It is initially unknown to
either bidder, but may become publicly known depending on the disclosure regime. For all
i, Si is distributed independently according to the cumulative distribution function (c.d.f.)
F (s) : [0, s̄] → [0, 1] with corresponding density f (s) > 0. The seller’s information satisfies
Pr[S0 = 1] = p and Pr[S0 = 0] = 1 − p. All prior distributions are common knowledge.
A motivation for this setup is a patent race. Si is firm i’s private benefit from securing the
patent. S0 describes the patent authority’s information about the new good’s patentability.
Alternatively, it may describe the authority’s willingness (or lack thereof) to enforce a patent.
If the good is not patentable, the invention is worthless.
The seller can adopt one of two disclosure policies. Under a policy of no disclosure (N)
the seller commits to not informing the bidders of the realized value of S0 prior to bidding.
Under a policy of disclosure (D), the seller always informs the bidders of the realized value
of S0 prior to bidding.6 We study each disclosure regime in turn. To compute expected
revenue under each policy we appeal to Krishna and Morgan’s (1997) analysis of the war of
attrition. The equilibrium bidding functions defined below follow from their analysis.
No Disclosure The expected value of the prize to bidder i in a non-disclosure regime is
vN (s) = E[Si S0 |Si = s] = ps. Thus, the equilibrium bidding strategy of each bidder is
bN (s) =
Z
0
s
f (y)
dy =
vN (y)
1 − F (y)
5
Z
0
s
py
f (y)
dy.
1 − F (y)
For expositional ease we adopt a phrasing from auction theory. Agents as bidders, actions are bids, etc.
The subscript “N ” denotes values associated with a non-disclosure regime. The subscript “D” is used in
the case of disclosure.
6
3
Given two independent random variables each with c.d.f. F (s), H(s) = F (s)2 + 2(1 −
F (s))F (s) is the c.d.f. of the second-highest order statistic. Let h(s) be the associated
density. Therefore, the expected revenue is
RN = 2 ×
Z
s̄
bN (s)h(s)ds = 2p
0
Z s̄ Z
0
s
yf (y)
dy h(s) ds.
1 − F (y)
0
Disclosure Suppose instead that the seller adopts a policy of unconditional disclosure. If
S0 = 0, the prize’s value is zero and all bidders bid zero. If instead S0 = 1, the prize’s value
is vD (s) = E[Si S0 |Si = s, S0 = 1] = s. Thus, a bidder with private signal s bids
bD (s) =
Z
s
0
f (y)
vD (y)
dy =
1 − F (y)
Z
s
y
0
f (y)
dy.
1 − F (y)
Therefore, the expected revenue is
RD = 2 × p ×
Z
1
bD (s)h(s)ds + (1 − p) × 0 = 2p
0
Z s̄ Z
0
s
0
yf (y)
dy h(s) ds.
1 − F (y)
Both regimes yield the same revenue and the seller is indifferent between disclosure or concealment of private information.
Example 1. Suppose s̄ = 1 and F (s) = s. Thus, bidders’ types are uniformly distributed.
In this case, RN = RD = p/3.
2
A War of Attrition with Budget Constraints
Expanding our model, suppose additionally that bidders face a private budget constraint, as
in Che and Gale (1998). A bidder’s private information consists of his signal Si , as above,
and a private budget, Wi . A bidder with a budget of Wi = w must bid less than w. Private
budget constraints feature in many applications. For example, in a patent race budget
constrains are reflected by firms’ overall financial capacity or by budgetary allocations to
research groups (Leininger, 1991). Si and Wi are independent and each Wi is independently
distributed according to the c.d.f. G(w) with density g(w).
Kotowski and Li (2014) study the war of attrition with private budget constraints. They
show that (under appropriate regularity conditions) there exists a symmetric equilibrium in
nondecreasing, continuous strategies of the form β(s, w) = min{b(s), w}. If v(s) denotes the
4
prize’s expected value given the available information, then b(s) is a solution of
b′ (s) =
(1 − G(b(s)))v(s)f (s)
.
(1 − G(b(s)))(1 − F (s)) − g(b(s))(1 − F (s))v(s)
(1)
subject to an appropriate boundary condition.7
Except in special cases, as in the continuation of Example 1 below, (1) rarely admits an
elegant solution. Moreover, each bidder’s strategy β is a function of his multi-dimensional
type. To compute revenues, therefore, we first define a distribution of “pseudo-types” who
(given the distribution of budgets and the equilibrium bidding strategy) bid less than b(s).
We use this distribution to compute revenues.8
No Disclosure Let bN (s) be the solution to (1) when v(s) = vN (s) = ps. Given the strategy βN (s, w) = min{bN (s), w}, the probability that a bidder with signal Si = s bids less than
bN (s) is F̂N (s) = F (s) + (1 − F (s))G(bN (s)). If ĤN (s) = F̂N (s)2 + 2(1 − F̂N (s))F̂N (s), then
R s̄
the expected revenue generated by the non-disclosure regime is R̂N = 2 × 0 bN (s)ĥN (s)ds.
Disclosure As above, both bidders bid zero when S0 = 0. Thus, let bD (s) be the solution
to (1) when v(s) = vD (s) = s, i.e. it is known that S0 = 1. Defining terms analogously to
R s̄
the previous paragraph, the expected revenue is R̂D = 2 × [p 0 bD (s)ĥD (s)ds].
Example 1 (Continued). Continuing to assume that F (s) = s, suppose additionally that
G(w) = 1 − e−w/α , α ≥ 1, is the distribution of budgets. Solving (1) with v(s) = vN (s) = ps
and the boundary condition bN (0) = 0 gives
bN (s) =
α2 log 1 −
ps
α
− αp log(1 − s)
.
α−p
The expected revenue without disclosure is
R̂N =
Z
1
0
2
α log 1 −
ps
α
− αp log(1 − s)
α−p
7
!

2
α+p
α−p
1−
4α (1 − s)
(α − ps)
ps
α

2α
p−α
 ds.
Kotowski and Li (2014) outline how to identify the boundary condition.
Our approach to compute revenues depends on a single-dimensional re-parameterization of the model.
See Araujo et al. (2008) for a general discussion of re-parameterizations of auction models where agents
have multi-dimensional types. Che and Gale (1998) and Che and Gale (2006) employ variations of similar
procedures to compute revenues.
8
5
Similarly, solving (1) with v(s) = vD (s) = s and the boundary condition bD (0) = 0 gives
α2 log 1 −
bD (s) =
s
α
− α log(1 − s)
.
α−1
Expected revenue is
R̂D =
Z
2
1
α log 1 −
0
s
α
− α log(1 − s)
α−1
!
4αp(1 − s)

α+1
α−1
1−
α−s
2α
s − α−1
α

 ds.
Expected revenues under both regimes—R̂N and R̂D —are plotted as a function of p when
α = 1 in Figure 1.9 Also illustrated are the revenues from the no-budget constraints case.
The non-disclosure regime generates greater expected revenues for all p ∈ (0, 1).
We conclude with two remarks drawing on Example 1.
Remark 1. An appropriate choice of disclosure regime can almost entirely mitigate the relative revenue loss due to budget constraints. Notably,
R̂N
=
lim
p→0 RN
but R̂D /RD = 6
1
2
− e2
R∞
2
Z
1
12(s − 1)(s + log(1 − s))ds = 1
0
e−t
dt
t
≈ 0.832. Therefore, when p is small the relative revenue
loss due to budget constraints is negligible under a non-disclosure regime.10 The loss is
always non-negligible under a disclosure regime.
Under a non-disclosure regime, each agent’s strategy is a function of p. When p is small,
the distortion in the bidding strategy due to budget constraints becomes small allowing expected revenues to approximate the outcome in the absence of budget constraints irrespective
of the true state. Under disclosure, however, bidding does not depend on p, but expected
revenue does. Its a weighted average of revenue from different contingencies, thereby preserving the relative gap in revenues. The right disclosure regime allows the auctioneer to
exploit these subtle differences to secure a large fraction of the expected revenue form the
benchmark, no budget constraints scenario even when bidders are budget-constrained.
Remark 2. In the second-price auction, Fang and Parreiras (2003, Proposition 2) show that
the linkage principle holds when budget constraints are sufficiently lax; hence, a disclosure
9
10
When α = 1, then bD (s) = log(1 − s) + s/(1 − s) and R̂D = 2p
Actually, for all p < 0.1,
is relatively small.
R̂N
RN
1
2
− e2
R∞
2
e−t
t dt
.
≥ 0.98. Therefore, the relative revenue loss for a wide range of parameters
6
Expected
Revenue
0.333
0.277
0.25
0.125
RD = RN
R̂N
R̂D
p
0
0.5
1
Figure 1: Expected revenue as a function of p in Example 1 (α = 1).
regime is optimal. Our example qualifies this intuition. Even if budget constraints are
arbitrarily lax, non-disclosure continues to be the revenue-superior policy in our model. As
an illustration, consider Figure 2 which plots revenues as a function of 1/α assuming p = 1/2.
As 1/α → 0 the severity of budget constraints vanishes. However, R̂N > R̂D for all 1/α > 0.
Hence, the linkage principle is far more sensitive to the presence of budget constraints than
previously identified.
7
Expected
Revenue
0.167
0.153
RD = RN
R̂N
R̂D
0.139
1
α
0
1
8
1
4
1
2
1
Figure 2: Expected revenue as a function of 1/α in Example 1 (p = 1/2).
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