Volunteering and the Strategic Value of Ignorance

A.10 Proof of Proposition 4

First, consider the war of attrition fixing the decisions on information. We only
analyze the case where T < — (υ — c). Note that this implies that, in the war of
attrition, uninformed individuals prefer to wait until T. Thus, in case (N, N) where
no individual is informed, both get an expected payoff of —T. Moreover, if individual
i is uninformed and individual j knows his cost of provision (case (N, I)), i and j_H
prefer waiting until T to any concession before T. Since υ — c_l > —T, j_L will concede
immediately. Ex ante expected payoffs in case (N, I) are

E (π_i) = p_bυ + (1 — p_L) (—T)

^{e (}^j⁾ = Pl ⁽ⁱ' ^{— c}l) ^{+ (1 —} Pl^{) (—t)}

Now consider the war of attrition if both individuals know their provision cost.
Due to Assumption 1’ on the time limit, there is no equilibrium where an individual
concedes immediately or waits until T for both types. In the symmetric equilibrium,
q_iH (T) = Qj_H (T) = 1, and low types randomize according to F_iL (t) = ɪΦ (t; c_l, t, 0)
where t = min {T — (-υ + c_l) , —c_l Inp_H} . Similar to Lemma 4, if T is small, low-
cost types put a mass point at T, and if T is large, they concede before T with
probability one.²⁸ This leads to an ex ante expected payoff in case (I, I) equal to

υ — cl,

^{(1 —} Ph⁾ (^{υ — c}l) ^— pH ^(cl lⁿPh ^{+ t) ,}

In the game of information acquisition, suppose that j acquires information and
T < — (υ — c_l) — c_l lnp_H. If i acquires information, his expected payoff is υ — c_l
(compare (20)). If i remains uninformed, by (19), he gets an expected payoff of

²⁸A proof that these strategies constitute an equilibrium is omitted.

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