Volunteering and the Strategic Value of Ignorance

PlV +(1 — Pl) (—T) which is strictly larger than

p_Lv + (1 — Pl) ((v — Cl) + ⅛ Inp_H)

= v — Cl + Cl (1 — Ph + Ph In Ph )

> v — Cl

where the inequality follows from the fact that 1—p_H +p_H Inp_H > 0 for all p_H ∈ (0,1).

Thus, if T is sufficiently small, only one individual acquires information. For
T ≥ — (v — c_l) — c_l Inp_H, by continuity of the expected payoff in (20), the same
result holds if T is sufficiently close to — (v — c_l) — c_l Inp_H. For larger T, equilibrium
information acquisition depends on the parameter values.²⁹

B Supplementary appendix

B.l Details on Example 1

Note first that, for c_l = 2 and c_h = 10, c/2 — clnp_H > c_h/2 for all p_H ∈ (0,1).
Hence, the mixed strategy equilibrium exists for all T ∈ (c/2, c_h/2).

(a) For p_H = 0.75, we have c_l∕2 — c_l lnp_H = 1 — 2ln0.75 < 4 = c/2. Hence, using
(15), the value of information if T → c/2 converges to

^—cl ^{— p}H (j^{T +}—2   ^cl ^{+ c}l ^lnрн) ^{+ p}H 2^2 ^{+ T})

= —2 — -⁹- (4 + 5 — 2 + 2ln0.75) + 0.75 (4 + 4) ≈ 0.386 > 0
16

Moreover, V_i¹ is negative for all T < c_l/2 — c_l lnp_H = 1 — 2 ln 0.75. By monotonicity,
there is a unique T ∈ (1 — 2ln0.75, 4) such that h)ʃ = 0. Setting (15) equal to zero
yields ^rT = 1.94.

²⁹Note that, while both (19) and (20) are decreasing in T, the slope of (19) is steeper. As in
Corollary 1, whenever p∏ is small, it is more likely that only one individual will acquire information
in equilibrium.

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