As the time limit approaches, it becomes less costly to wait until the end, and
thus there is a point in time after which the individuals are inactive (for all t ∈
(—c/2 + T, T)) because they prefer the random selection at T. Finally, they put the
remaining probability mass on a concession at T.
From Lemma 2, we can compute the individuals’ expected payoff in the symmetric
equilibrium, which is equal to
V — c/2 — T
V — c
if T ≤ c/2
if T > c/2 ,
(3)
One individual knows his cost of provision. Suppose that only individual j
has become informed about his provision cost, while i = j remained uninformed. j’s
strategy is now contingent on his type (denoted by jL or jH), and i’s optimal strategy
is to choose his concession time as if his cost was c. Recall that we still assume that
Assumption 1 holds.
Lemma 3 (One individual is informed.)
a) If T ≤ c/2, in equilibrium, qi (T) = qjH (T) = 1, and qjL (0) = 1. (If T = c/2,
there is an additional equilibrium where qi (0) = 1 and qjL (T) = 1.)
b) If c/2 <T < c/2 — cln pH,
(i) there is a pure strategy equilibrium where qi (0) = 1;
(ii) there is a mixed strategy equilibrium where Fi (t) = Φ (t; cL, — ⅜ + T, θ},
Fj l (t) = J;$ (t; c, — 2 + t, 1 — (1 — Pl) e 2 + 2), and qjH (T) = 1∙
c) If T > c/2 — clnpH, in equilibrium, qi (0) = 1.
If T < c/2, both i and jH prefer a random selection at T to any concession before
T, and this makes it optimal for jL to concede immediately. Since there is positive
probability that the time limit T is reached, the equilibrium strategies of i and jH
are uniquely pinned down.
If T > c/2, the structure of the equilibrium reverses, and there is a ‘pure strategy
equilibrium’ where i concedes immediately and both jL and jH wait until T. To
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