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 —} Pl^cl ^— Ph (t H—^^) ∙                                            ⁽¹⁸⁾

In case (N, I), if T < c∕2, in equilibrium of the war of attrition, we have q_i (T) =
Q-Jh (T) = 1 and qj_L (0) = 1. The uninformed individual i gets an expected payoff of

^v—рн (^t+^c2^h )

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

^{v — p}L^cL ^{— р}н (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 ≥ c∕2, the same is true if in case (N, I) the equilibrium with q_j∙_l (0) = 1
is selected; then, in the war of attrition, equilibrium payoffs are as above. If in case
(N, I) the equilibrium with q⅛ (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 (π⅛) in some interval (τ — δ, T^, δ > 0.

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