Volunteering and the Strategic Value of Ignorance

If T > c/2 and, in case (N, I), the pure strategy equilibrium is selected, learning
the own provision cost is strictly dominant, and thus the threshold T is (weakly)
smaller than c/2.¹⁵ However, if we focus on the mixed strategy equilibrium, T > c/2
for a small p_H, and the value of information V_i¹ can even be negative for all T ∈
(⅛/2,c_h/2). Thus, the strategic value of remaining uninformed is not only present
in the case where an uninformed individual i has a dominant strategy not to concede
before T (as in Lemma 3a), but also when the individuals randomize their concession
time (as in Lemma 3b(ii)). The sufficiently high probability that the rival has a low
cost and volunteers immediately with positive probability makes it optimal for i to
disregard information that is available without cost. This strategic value disappears
only if the probability of having a high contribution cost, p_H, is large, because, from
the point of view of the rival, an early concession of the individual who knows his
provision cost is less likely.

Example Consider the following example where ⅛ = 2, and c_h = IO.¹⁶ As-
sumption 1 requires that 1 < T < 5.

(a) Suppose that p_H = 0.75. If T → c/2 = 4 from below, the value of information
V_i¹ is positive. Hence, the critical threshold T < c/2. Setting V_i¹ (T) = 0 yields
T = 1.94. Thus, for all T < 1.94, only one individual learns his cost of provision,
and for all T > 1.94, both individuals learn their cost of provision.

(b) Now suppose that p_H = 0.5. V_i¹ is negative if T approaches c/2 = 3 (from below).
Hence, if in case (N, I) the pure strategy equilibrium is selected, T = c/2 = 3, and
if the mixed strategy equilibrium is selected, T > c/2. In the latter case, T = 3.56.

(c) If p_H = 0.25, again V_i¹ is negative if T approaches c/2 = 2, and T = c/2
if in case (N, I) the pure strategy equilibrium is selected. If the mixed strategy
equilibrium is selected, V_i^l is negative for all T satisfying Assumption 1, and thus
there is no equilibrium where both individuals find out about their cost of provision
with probability one.

¹⁵Concretely, if V°^³~¹ is negative for all T < c/2, T? = c/2. Otherwise, T is defined as the
solution to V_i^σ^³~¹ (t^ = 0.

¹⁶Details on this example are in Appendix B.

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