Volunteering and the Strategic Value of Ignorance

concede immediately, and there are two equilibria, each with one individual choosing
q_i (0) = 1, i = 1,2. In the latter case, there are also equilibria in mixed strategies.¹⁰
As players are symmetric, we focus on the (unique) symmetric equilibrium.

Lemma 2 (No individual is informed.)

^a) If T ≤ c/2, in the symmetric equilibrium, q_i (T) = q₂ (T) = 1.

b) If T > c/2, in the symmetric equilibrium, individual i ∈ {1, 2} randomizes his
concession time according to F_i (t) = Φ (t; c, — ∣ + T, 0}.

In the mixed strategy equilibrium (case T > c/2), for any t_j∙ ∈ (0, —c/2 + T), j’s
marginal cost of waiting is one, multiplied by the probability (1 — F_i (t_j∙)) that this
waiting cost has to be paid. The marginal gain of waiting slightly longer is equal to
cF_i' (tj ), i.e. the expected provision cost multiplied by the additional probability that
this cost can be saved. Individual j is indifferent between all t_j∙ ∈ (0, —c/2 + T) if
cost and benefit of increasing t_j∙ (i.e. of waiting slightly longer) are equal. This leads
to F_i (t) = Φ (t; c, — ≤ + T, 0}. The only difference to the standard war of attrition
with complete information is that, due to the time limit, no individual concedes
in (—c/2 + T, T), but instead both choose a concession in T with strictly positive
probability.

In the symmetric equilibrium, no individual concedes immediately with positive
probability (that is, q⁰ = 0). There are asymmetric mixed strategy equilibria where
one of the individuals places a mass point at t = 0, i.e. concedes immediately with
strictly positive probability. Obviously, there can’t be an equilibrium where both
individuals have a mass point at zero, because then waiting an infinitesimally small
amount of time would, at a negligibly higher expected waiting cost, strictly increase
the probability that the rival provides the public good.

The fixed time limit has an important impact on the individuals’ equilibrium
behavior if T > c/2. At the beginning of the game, the individuals are willing to
concede, and they play a mixed strategy for a certain time period (t ∈ (0, —c/2 + T)).

¹⁰For a detailed analysis see Hendricks et al. (1988).

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