Volunteering and the Strategic Value of Ignorance

As the time limit approaches, it becomes less costly to wait until the end, and
thus there is a point in time after which the individuals are inactive (for all t ∈
(—c/2 + T, T)) because they prefer the random selection at T. Finally, they put the
remaining probability mass on a concession at T.

From Lemma 2, we can compute the individuals’ expected payoff in the symmetric

equilibrium, which is equal to

One individual knows his cost of provision. Suppose that only individual j
has become informed about his provision cost, while i = j remained uninformed. j’s
strategy is now contingent on his type (denoted by j_L or j_H), and i’s optimal strategy
is to choose his concession time as if his cost was c. Recall that we still assume that
Assumption 1 holds.

Lemma 3 (One individual is informed.)

a) If T ≤ c/2, in equilibrium, q_i (T) = qj_H (T) = 1, and qj_L (0) = 1. (If T = c/2,
there is an additional equilibrium where q_i (0) = 1 and qj_L (T) = 1.)

b) If c/2 <T < c/2 — cln p_H,

(i) there is a pure strategy equilibrium where q_i (0) = 1;

(ii) there is a mixed strategy equilibrium where Fi (t) = Φ (t; c_L, — ⅜ + T, θ},
^Fj l ^(t) = J_;$ (^{t; c, —} 2 ^{+ t, 1 — (1 — Pl) e 2 + 2})^{, and q}jH ^(T) = ¹∙

c) If T > c/2 — clnp_H, in equilibrium, q_i (0) = 1.

If T < c/2, both i and j_H prefer a random selection at T to any concession before
T, and this makes it optimal for j_L to concede immediately. Since there is positive
probability that the time limit T is reached, the equilibrium strategies of i and j_H
are uniquely pinned down.

If T > c/2, the structure of the equilibrium reverses, and there is a ‘pure strategy
equilibrium’ where i concedes immediately and both j_L and j_H wait until T. To

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