Volunteering and the Strategic Value of Ignorance

small рн.) We get

^vi (T) = ~^cl - pH (t + ~H ~ ^{Cl + Cl ln}Рн) + Рнce^-2+1              (17)

and hence

∂V_i¹ (T )      ₂       _ 1 ₊1      _{2          n}

—= ^-Рн ⁺ Рнe ^{2 +} ð > ^-Рн ⁺ Рн > O.

OT

Continuity of V? follows directly from continuity of the expected payoffs.

A.6 Proof of Proposition 1

From Lemma 5(i), it follows that the best response to σ_j∙ = N is to become informed.
Now suppose that in case (N, I) the pure strategy equilibrium is selected. With
Lemma 5(ii)-(iii), there exists a unique T < c/2 such that the best response to
σj = I is to remain uninformed if and only if T < T. In this case, there are
two asymmetric equilibria where one individual acquires information and the other
individual remains uninformed. In addition, there is a symmetric equilibrium where
the individuals randomize their information choice and learn their provision cost with
probability Vⁿ/ (V(ⁿ — V(^r) ∈ (0,1). If T > ^rT, there is a unique equilibrium where
both individuals find out about their provision cost.

For the mixed strategy equilibrium in case (N, I), this result follows from monotonic-
ity of V_i¹ (Lemma 5ii÷iv). Note that T < c/2 — Clnр_н as, for T → c/2 — Clnр_н,
V? converges to the value of information in the pure strategy equilibrium and hence
is strictly positive.²⁵ Therefore, whenever c/2 — clnр_н < c_h/2, there exists an in-
terior T ∈ (cl,C/2 — clnр_н) such that the best response to σ_j∙ = I is to remain
uninformed if and only if T < T, and information acquisition is strictly dominant if
T > T. If C/2 — Clnр_н > c_h/2, the interval where in equilibrium both individuals
acquire information can be empty which is the case if limʃ .,∙_7z ∕₂ V_i¹ is negative.

²⁵This follows from the convergence of i’s expected payoff in case (N, I) for T → c∣^cl — clnp∏.

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