small рн.) We get
vi (T) = ~cl - pH (t + ~H ~ Cl + Cl lnРн) + Рнce-2+1 (17)
and hence
∂Vi1 (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 σj∙ = 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 Vi1 (Lemma 5ii÷iv). Note that T < c/2 — Clnрн as, for T → c/2 — Clnрн,
V? converges to the value of information in the pure strategy equilibrium and hence
is strictly positive.25 Therefore, whenever c/2 — clnрн < ch/2, there exists an in-
terior T ∈ (cl,C/2 — clnрн) such that the best response to σj∙ = I is to remain
uninformed if and only if T < T, and information acquisition is strictly dominant if
T > T. If C/2 — Clnрн > ch/2, the interval where in equilibrium both individuals
acquire information can be empty which is the case if limʃ .,∙7z ∕2 Vi1 is negative.
25This follows from the convergence of i’s expected payoff in case (N, I) for T → c∣cl — clnp∏.
33
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