Volunteering and the Strategic Value of Ignorance

Proposition 1 Consider the game of information acquisition and suppose that As-
sumption 1 holds. There exists a threshold T > c^∕2 such that

(i) if T < T, there are two asymmetric equilibria where exactly one individual ac-
quires information and one symmetric equilibrium where both individuals randomize
their information decision;

(ii) if T > T, it is strictly dominant to acquire information.

If both individuals remained uninformed, this would cause a high inefficiency
in the volunteering game and lead to the lowest expected payoffs. Therefore, it
is beneficial for at least one individual to find out about his provision cost even
if information acquisition leads to a higher ex ante probability of being the one
who concedes first. As a consequence, there is never an equilibrium where both
individuals decide not to learn their cost of provision. If, however, T is sufficiently
small and only individual j acquires information, then j concedes immediately with
high probability, and i prefers to remain uninformed. Being uninformed constitutes
a strategic advantage in the volunteering game, being a commitment not to volunteer
too early. This, in turn, induces the rival to concede immediately, which outweighs
i’s waiting cost in case j has a high provision cost. For a higher T, this waiting
cost increases, and, in the case of the mixed strategy equilibrium in (N, I), the
probability that j concedes immediately decreases. There exists a threshold T such
that, for T > T, i is better off if he finds out about his provision cost as well. If
the value of information V_i¹ is negative for all (⅛∕2, c∕2), the location of T depends
on which equilibrium is selected in case (N, I). In both cases, the threshold T is
uniquely determined such that V_i¹ is negative for all T < T and positive for all
_ ~
T >T.

Corollary 1 (i) If in case (N, I) the pure strategy equilibrium is selected, T ≤ c∕2.
(ii) If in case (N, I) the mixed strategy equilibrium is selected and p∏ is small, T is
strictly larger than c∕2. Then, there may be no equilibrium where both individuals
acquire information with probability one for all T fulfilling Assumption 1.

this case. Then, players can condition their strategies in the war of attrition on the information
acquisition. As in our analysis for the pure strategy equilibrium in case (N, I), this can support
information acquisition of both players in equilibrium if T > c∕2.

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