Volunteering and the Strategic Value of Ignorance

For the analysis of the optimal decision on information acquisition, we have to dis-
tinguish whether or not T > c∕ 2. This distinction does not influence the equilibrium
of the war of attrition in case both individuals know their provision cost, but it is
crucial for the nature of the equilibrium if at least one individual does not know his
cost of provision.¹³

Lemma 5 Suppose that Assumption 1 holds.

(i) Vf^{j N} is strictly positive for all T.

(ii) Vffi' is strictly negative if T is sufficiently small and strictly increasing in T
for T ∈ (c_b∕2,c∕2).

(iii) Suppose in case (N, I) the pure strategy equilibrium is selected. Then Vffi' is
strictly positive for all T > c∕2.

(iv) Suppose in case (N, I) the mixed strategy equilibrium is selected. Then Vffi¹ is
continuous and strictly increasing in T for T ∈ (cffi2, c∕2 — clnp_H).

Provided that the rival does not learn his cost of provision (σ₇∙ = N), learning
one’s own cost always increases one’s expected payoff as the value of information
is positive (Lemma 5 part (i)). If instead the rival decides to learn his cost and
T is small, this result is reversed. However, as long as T < c∕ 2, an increasing
time limit makes waiting more costly in case the rival has a high provision cost,
which increases one’s own value of information (part (ii)). If T > c∕2, the value of
information depends on which equilibrium is selected in case (N, I). For the pure
strategy equilibrium, i’s value of information given that j learns his cost of provision,
V_i¹, exhibits a discontinuity at T = c∕2 and is strictly positive for all T > c∕2 (part
(iii)). For the mixed strategy equilibrium, however, Vf is continuous at T = c∕2.
This continuity in T makes the analysis for the selected equilibrium more appealing.
Yet the following proposition holds independently of which equilibrium is selected in
case only one individual decides to learn his provision cost.¹⁴

¹³We still assume that Assumption 1 holds. If T < ¾∕2, then decisions on information are
irrelevant, since both individuals never concede before T. If T > c∏∕2, the war of attrition always
has equilibria where one of the individuals concedes immediately, independent of the decisions on
information.

¹⁴Due to the possible multiplicity of equilibria of the war of attrition in case of T > c∕2, depart-
ing from the analysis of the reduced form game makes the equilibrium analysis more complex in

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