Volunteering and the Strategic Value of Ignorance

t = -Cl In (1 — p_L) and Fi_L (t) = 1, that is, i_L concedes with probability one before
t < T. Indeed, waiting until T would lead to a payoff lower than V — c_l. Since
any symmetric equilibrium must be in mixed strategies, this is the only symmetric
equilibrium. Expected payoff of i_L is V — c_l, and expected payoff of i_H is

Hence, the ex ante expected payoff is

Inserting t leads to (8).

A.5 Proof of Lemma 5

(i) Suppose that T < c∕2. Together with (3) and (5),

^v /ʌ = ^-pL^cL ^{— p}H f— ^{+ t} ) ⁺ 5^{+ t} = ^-pL~χ- ^{+ (1 — p}H ) ^t > 0.
∖ 2      2    2                2

If T > c∕2, expected payoff is V — c in case (^ Λ^r) which is the payoff an informed
individual i can ensure by conceding immediately for both possible contribution costs.
Since for a high contribution cost, i strictly prefers waiting until T, his payoff must
be strictly higher. Thus, V°^j ^ > 0 for all T ∈ (c_l∕2,c_h∕2).

(ii) If T < c_l∕2 — c_l Inp_H, subtracting the first row in (4) from the first row in (8)
leads to

^v/ ^(T) = ^—cl ^— ɪ ^(ch ^{— c}l) ^{e2 cl + p}H (ɪ ^{+ t}) .
2                               ∖ 2      /

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