Volunteering and the Strategic Value of Ignorance

an incentive for the rival to remain uninformed of his own cost. For intermediate
values of T, a mixed strategy equilibrium exists that exhibits similar properties to
the one characterized in Lemma 3b(ii). The value of information is then determined
by the shape of the probability distribution of the provision cost. The resulting
effects are qualitatively the same, but can be most clearly demonstrated by using a
two-point distribution and varying the probabilities that the cost of provision is high
and low, respectively. The key assumption remains that with positive probability
individuals face a rival who prefers to wait until the time limit is reached. In this
sense, our approach is similar to Fudenberg and Tirole (1986), who assume that there
is a positive probability that the rival never concedes. For the incentives to find out
about the cost of provision, the time limit is of additional strategic importance.

A Appendix

A.l Proof of Lemma 1

Denote by Φ_j∙ the distribution of j’s waiting times, from the point of view of i, that
is, Φ_j∙ (t) = a means that, from the point of view of i, j concedes before t with
probability a.²³ Consider a concession of i in t_i ∈ (—c_i∕2 + T₁T) and suppose that
there is a strictly positive probability that i provides the good in t_i, i.e. Φ_j∙ (t_i) < 1.
If Φ_j∙ exhibits a discontinuity at C, then there is an ε > O such that i is strictly better
off by conceding in t + ε instead of in t⅛, because this would strictly decrease the
expected contribution cost at only an infinitesimally higher expected waiting cost.

²³Note that Ψ_j∙ captures both uncertainty of i over j’s contribution cost and possible randomiza-
tion of j.

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