Volunteering and the Strategic Value of Ignorance

and if t_i = T, i gets

Thus i is indifferent between all ti E (O, —c/2 + T] U {T}, and Fi and Fj are mutually
best responses.

A.3 Proof of Lemma 3

Parts a), b(i) and c) follow directly from the analysis in the main text. It remains
to prove part b(ii). If there is an equilibrium in mixed strategies, the equilibrium
strategies must exhibit similar properties as in the case of complete information. In
particular, for waiting times t_i and tj_L in the support of the mixed strategies, it has
to hold that

^FjL ^(tjL ⁾ = ~ ^{1 — (1 —} Pl⅛l ^)exp f^—-jτ^ ,                           ⁽¹¹⁾

^pL L                         ∖ ^c /_

where the constants b_i and bj_L correspond to the mass points at zero, F_i (O) and
Fj_L (O), and remain to be determined. The factor 1/p_; in Fj_L takes into account the
probability p_; that i faces a rival with cost c_l. It has to hold that O ≤ b_i, bj_L < 1, and
min (b_i, bj_L ) = O : if i (j_;) concedes immediately with strictly positive probability, j_L
(i) strictly prefers a concession in ε > O, ε infinitesimally small, to a concession in O.

Assumption 1 implies that no tj_H < T with F_i (t j_H ) < 1 can be part of j_H’s
equilibrium strategy: j_H won’t choose any tj_H < T in the support of F_i. In turn,
for any t_i < T, we must have Fj_H (t_i) = O, and thus i strictly prefers t_i = T to all
t_i E (—c/2 + T, T). Moreover, F_i must be continuous on (O, T). To see why, suppose
that i concedes in t_i E (O, —c/2 + T] with strictly positive probability. Then, there
are δ > O, ε > O such that j_; strictly prefers tj = t_i + ε to any tj E (t_i — δ, t_i), hence
i is strictly better off by choosing t_i — δ∕2 instead of t_i. Therefore, possible mass

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