Volunteering and the Strategic Value of Ignorance

Otherwise, i’s expected payoff from a concession in t_i is

+ ^{(1 - φ}j∙ ^{(T)) (v - t}i ^- <⅛) + (Φj ^{(T) -} φj∙ ^(ti^{)) (}v ^{- t}_i ^- c_i)

= Λ (v - t) dΦj (t) + (1
Jo

Φj (t_i) < 1 implies that Φ_j∙ (T) - Φ_j∙ (t_i) > 0 or∕and 1 - Φ_j∙ (T) > 0. Therefore, for
all t_i ∈ (-c_i∕2 + T, T), (9) is strictly smaller than

which is i’s expected payoff for t_i = T.

A.2 Proof of Lemma 2

(i) As argued in the main text, the best response to tj = T is t_i = T, and q_i (T) =
Qj (T) = 1 is an equilibrium. Moreover, since -c_i∕2 + T < 0, Lemma 1 rules out
any further equilibrium because any individual who contributes with strictly positive
probability in t' ∈ [0,T) would strictly prefer a concession in T to a concession in t'.
(ii) The structure of the equilibrium strategies follows from Hendricks et al. (1988)
and the analysis in the main text. We only show that the strategies constitute an
equilibrium. Suppose that j randomizes according to Fj (t) = Φ (t; c, - ^ + T, 0j,
where Φ is defined in (2). Then, by Lemma 1, i strictly prefers t_i = T to any
t_i ∈ (-c/2 + T, T). For t_i ∈ [0, -c/2 + T], i’s payoff is

ʃ (v - x) - exp (-—) dx + exp ^-_τθ

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