Volunteering and the Strategic Value of Ignorance

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,
i_H, provides the public good in t_iH < T with strictly positive probability. If i_H
chooses a time of concession t_iH < T with strictly positive probability, then j_H must
concede before t_iH with probability one, contradicting Lemma 1. Therefore, in any
equilibrium, q_iH (T) = Qj_H (T) = 1.

It remains to characterize the individuals’ equilibrium strategies for a low pro-
vision cost. As before, denote by i_L an individual i with low cost. There can’t be
an equilibrium where i_L 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 i_L chooses t = 0 with probability one. j_L’s best response is to concede
in t' = ε, ε infinitesimally small, knowing that i_H will wait until T. But then, i_L 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 (—c_L/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, q_iH (T) = 1 and F_ib (t) = Φ (t; c_L,t, 0) where t =
min { — ≤b + T, — c_L InPh}, i = 1, 2.

If the probability p_H 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 p_H,
i_L and j_L concede before T with probability one. This holds if — ɪ + T ≥ — c_L Inp_H
or

T ≥ CL — Cl In Ph ∙

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