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,
Vi1 may even be negative for all T. Note that c/2 ClnpH ol/2ol InpH, and
for small
pH, we have ol/2ol lnpH oh/2. With monotonicity of (16), it follows
that, for all
T (ol/2,oh/2), V)j is smaller than

_                    Pn              1_ cH            1 си

lim V/ (T ) = cL —(ohcl) e2 2cr + рн ce 2 + 2c
T ch/2                   2

which is negative for small pH since the second and the third term approach zero if
pH0. For intermediate values of pH, we have T (c/2, oh/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

(6) to the first row in (7)).


Thus, if T < min {c/2,t},

in the equilibrium of the


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 (π) 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 clnpH,
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 (πj) E (π) is strictly
increasing in
T, and there is a critical value Ts where E (πj) = E (πi). Depending
on the parameters of the model,
Ts can be smaller or larger than the threshold T
for information acquisition. If Ts T, in all equilibria where one player chooses
to remain uninformed, this will be the first mover. If
Ts < 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 qjL (0) = 1 and qjH (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),

34



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