be precise, there is a continuum of payoff-equivalent equilibria where i concedes
immediately and j chooses a (sufficiently) high waiting time for each of the two
possible provision costs he could have been informed of (sufficiently high to make it
optimal for i to concede immediately). Given that Assumption 1 holds, by Lemma
1, jH will never provide the public good with strictly positive probability before
T. Thus, there is no further pure strategy equilibrium. To see why, suppose that
i concedes in t' > 0 with probability one. j√s best response is either tjL = 0, or
tjL > t', and i strictly prefers a concession in t'/2 over a concession in t' since in
both cases this doesn’t change his probability of contribution, but strictly reduces
the expected waiting cost.
There can, however, be an additional equilibrium which is in mixed strategies. In
fact, if c/2 < T < c/2 — clnpH, there is a ‘mixed strategy equilibrium’ where i and
jL randomize their concession time. By Assumption 1 and Lemma 1, jH will never
provide the public good before T. Thus, in any equilibrium in mixed strategies, only
i and jL contribute before T with strictly positive probability, and the equilibrium
strategies exhibit similar properties as in the case of complete information.
Contrary to the case where no individual knows his cost, the mixed strategy
equilibrium is uniquely determined by the condition that there is zero probability
that any individual concedes in (—c/2 + T, T) and that therefore jL concedes before
—c/2 + T with probability one (see Appendix). This requires that FjL has a mass
point at zero, and thus i’s payoff in the mixed strategy equilibrium is strictly higher
than V — c, which is i’s payoff from conceding immediately.
The mixed strategy equilibrium characterized in Lemma 3b(ii) has several inter-
esting properties. Whenever pH and/or T are large, this equilibrium does not exist:
as it is likely that j has a high cost and the waiting time until T is costly, waiting
becomes too costly for individual i; thus i prefers to volunteer immediately. When
T → c/2 — clnpH (from below), the probability that individual jɪ, concedes imme-
diately converges to zero, and i’s expected payoff converges to V — c, which is equal
to his payoff in the pure strategy equilibrium. On the other hand, when T → c/2
(from above), the probability that jɪ, concedes immediately converges to one, and the
probability that i concedes before T converges to zero. The equilibrium strategies in
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