Volunteering and the Strategic Value of Ignorance

A.7 Proof of Corollary 1

Part (i) follows from Lemma 5(iii). For part (ii), the threshold T is larger than c/2
if and only if the value of information is negative as T approaches c/2. Moreover,
V_i¹ may even be negative for all T. Note that c/2 — Clnp_H > o_l/2 — o_l Inp_H, and
for small p_H, we have o_l/2 — o_l lnp_H > o_h/2. With monotonicity of (16), it follows
that, for all T ∈ (o_l/2,o_h/2), V)^j is smaller than

_                    Pn              1_ ^cH            1 с_и

lim V/ (T ) = — c_L — —(oh — c_l) e^{2 2c}r + рн ce 2 + ^2c
T →c_h/2                   2

which is negative for small p_H since the second and the third term approach zero if
p_H → 0. For intermediate values of p_H, we have T ∈ (c/2, o_h/2).

A.8 Proof of Proposition 2

Suppose that i remains uninformed and j acquires information. If T < c/2, i’s

expected payoff is strictly higher than j^,s expected payoff (compare the first row in

Thus, if T < min {c/2,t},

game of sequential information acquisition, the first mover remains uninformed.

If T > c/2 and in case (N, I) the mixed strategy equilibrium is selected, by
continuity of the expected payoffs, E (π⅛) > E (πj) also holds for T = c/2 + δ, δ > 0
sufficiently small (compare the second rows in (6) and (7)). As T → c/2 — clnp_H,
the payoff of the uninformed player i approaches υ — C, while the informed player j
gets strictly more than v — C.²⁶ The difference in payoffs E (π_j∙) — E (π⅛) is strictly
increasing in T, and there is a critical value T_s where E (π_j∙) = E (π_i). Depending
on the parameters of the model, T_s can be smaller or larger than the threshold T
for information acquisition. If T_s > T, in all equilibria where one player chooses
to remain uninformed, this will be the first mover. If T_s < T, there can also be
equilibria of the game with sequential information acquisition where the first mover
chooses to acquire information and the second mover remains uninformed.²⁷

²⁶By choosing qj_L (0) = 1 and q_jH (T) = 1, j can ensure a payoff of at least υ — Plc^ —
Ph (ch/2 + T), which is strictly larger than υ — c; thus, his equilibrium payoff cannot be smaller.

²⁷This is the case, for instance, if ¾ = 2, ch = 10, and рн = 0.3. There, in case (N, I),

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