Volunteering and the Strategic Value of Ignorance



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(nV(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 Tc/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 lim
ʃ .,∙7z2 Vi1 is negative.

25This follows from the convergence of i’s expected payoff in case (N, I) for T ccl clnp.

33



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