concede immediately, and there are two equilibria, each with one individual choosing
qi (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, qi (T) = q2 (T) = 1.
b) If T > c/2, in the symmetric equilibrium, individual i ∈ {1, 2} randomizes his
concession time according to Fi (t) = Φ (t; c, — ∣ + T, 0}.
In the mixed strategy equilibrium (case T > c/2), for any tj∙ ∈ (0, —c/2 + T), j’s
marginal cost of waiting is one, multiplied by the probability (1 — Fi (tj∙)) that this
waiting cost has to be paid. The marginal gain of waiting slightly longer is equal to
cFi' (tj ), i.e. the expected provision cost multiplied by the additional probability that
this cost can be saved. Individual j is indifferent between all tj∙ ∈ (0, —c/2 + T) if
cost and benefit of increasing tj∙ (i.e. of waiting slightly longer) are equal. This leads
to Fi (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).
10