Volunteering and the Strategic Value of Ignorance



A.9 Proof of Proposition 3

First of all, we solve for the payoffs in the war of attrition. In case (N, N), expected
payoffs are as in (3). In the symmetric equilibrium in case (
I, I), if it is common
knowledge that both have a low cost, both individuals randomize and get an expected
payoff of
V Cl. If both have a high cost, they wait until T. This leads to an ex ante
expected payoff of

v Plcl Ph (t H—^^) ∙                                            (18)

In case (N, I), if T < c2, in equilibrium of the war of attrition, we have qi (T) =
Q-Jh (T) = 1 and qjL (0) = 1. The uninformed individual i gets an expected payoff of

v—рн (t+c2h )

which is strictly larger than (18). Thus, the best response to information acquisition
of
j is to remain uninformed. The informed individual gets

vpLcL — рн (τ +

which is strictly larger than the payoff in case (N, N). Therefore, in the equilibrium
of the game of information acquisition, only one individual acquires information.

For T c2, the same is true if in case (N, I) the equilibrium with qjl (0) = 1
is selected; then, in the war of attrition, equilibrium payoffs are as above. If in case
(
N, I) the equilibrium with q(0) = 1 is selected, the uninformed individual gets an
expected payoff of
V c; in the game of information acquisition, both individuals
acquire information.

E (πj) E (π) in some interval δ, T^, δ0.

35



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