Both individuals know their cost of provision. Suppose that both individuals
have decided to acquire information about their provision cost. By Lemma 1 together
with Assumption 1, there can’t be an equilibrium where a type of i with high cost,
iH, provides the public good in tiH < T with strictly positive probability. If iH
chooses a time of concession tiH < T with strictly positive probability, then jH must
concede before tiH with probability one, contradicting Lemma 1. Therefore, in any
equilibrium, qiH (T) = QjH (T) = 1.
It remains to characterize the individuals’ equilibrium strategies for a low pro-
vision cost. As before, denote by iL an individual i with low cost. There can’t be
an equilibrium where iL chooses a pure strategy. In particular, there can’t be an
equilibrium where an individual with low cost volunteers immediately. To see why,
suppose that iL chooses t = 0 with probability one. jL’s best response is to concede
in t' = ε, ε infinitesimally small, knowing that iH will wait until T. But then, iL is
strictly better off by choosing t'' = 2ε.
Hence, individuals randomize their waiting time if they have a low provision
cost. By Lemma 1, there must be zero probability that an individual volunteers in
the interval (—cL/2 + T, T), and at most one individual can have a mass point at
zero. As it is a typical feature of the war of attrition, there may be a continuum of
equilibria which differ in the size of the mass point at zero. Since the individuals are
symmetric ex ante, we focus on the symmetric equilibrium.
Lemma 4 (Both individuals are informed.)
In the symmetric equilibrium, qiH (T) = 1 and Fib (t) = Φ (t; cL,t, 0) where t =
min { — ≤b + T, — cL InPh}, i = 1, 2.
If the probability pH that the other individual has a high cost is large, it is more
attractive for an individual with low cost to volunteer early. For sufficiently high pH,
iL and jL concede before T with probability one. This holds if — ɪ + T ≥ — cL InpH
or
T ≥ CL — Cl In Ph ∙
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