Volunteering and the Strategic Value of Ignorance

For T → c_l/2, (14) converges to

^—Cl —    ^(ch — c_l) + ʒ- (c + c_l) — ~PlCl —    ^(ch — c) < 0.

2                2                         2

Moreover, deriving (14) with respect to T yields

1 Ph /        x i-ɪ          „

-—(ch — Cl) e^{2 c}l + рн > 0.
^{2 c}l

If c_l/2 — c_l In p_H <T < c/2, using again (4) and (8) we have

Vi ^{(T) — —c}l ^{— p}H (^{T +} “ù--^cl ^{+ c}l ^lnpH) ^{+ p}H (^ ^{+ T})

and

(iii) With (4), i gets υ — c if he does not become informed. By the same argument as
in (i) for T > c/2, z’s ex ante payoff in case (I, I) must be strictly larger than υ — c,
and thus V_i¹ > 0 for all T ∈ (c/2, c_h/2).

(iv) In (ii), monotonicity has been shown for T < c/2. Now suppose that T >
c/2. Consider first the case where T is smaller than c_l/2 — c_l lnp_H, i.e. T ∈
(c/2,c_l/2 — c_llnp_H). (For a sufficiently large p_H, this interval is empty.) Then,
with (6) and (8),

_p                    Prr                i_τ              i τ

V_i¹ (T) — —Cl — -^~(ch — Cl) C^{2 c}L + рнce ^{2 +} ð                           (16)

2

and

Ph ,         ⁱ i--T-        _i₊τ „

— — (ch — Cl) e^{2 cl} + Phe ^{2 +} ð > 0.

2cl

Now suppose T is larger than c_l/2 — c_llnp_H, but smaller than c/2 — clnp_H, i.e.

T ∈ (c_l/2 — c_l lnp_H, min {c/2 — clnp_H,c_h/2}). (This interval may be empty for a

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