Volunteering and the Strategic Value of Ignorance

(b) For рн = 0.5, c_l∕2 — c_l Inр_н = 1 — 2 In 0.5 < 3 = c∕2. With (15),

Hence, if in case (N, I) the pure strategy equilibrium is selected, T = c∕2 = 3. For
the mixed strategy equilibrium, with (17), the value of information for T > c∕2 is

(0.5)² (τ + ¹2⁰ — 2 + 21n0.5^

which is equal to zero for ^rT = 3.56.

(c) For р_н = 0.25, c_l∕2 — c_l 1nр_н = 1 — 21n0.25 > 2 = c∕2. At T = c∕2,

0.25 , _x 1 2          /4     ∖

_ (10 — 2) e2- 2 +0.25 Ç2 + 2J

Thus VZ¹ is negative for all T < c∕2. This implies that T = c∕2 = 2 for the pure
strategy equilibrium in case (N, I). Now consider the mixed strategy equilibrium in
case (N, I) and suppose that T approaches c_h∕2 = 5. Since 1 — 21n0.25 < 5, V_zⁱ is
given by (17) and approaches

1 .      1.5            _____

+ -4_e^-2⁺ 4 ≈ —0.2097.

By monotonicity, V_zⁱ is negative for all T ∈ (1, 5).

References

[1] Amann, E., Leininger, W., 1996. Asymmetric all-pay auctions with incomplete
information: the two-player case. Games and Economic Behavior 14, 1-18.

[2] Bilodeau, M., Slivinski, A., 1996. Toilet cleaning and department chairing: vol-
unteering a public service. Journal of Public Economics 59, 299-308.

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