which again is equal to V — (1 — pL) exp (—1/2 + T/c) c. Hence, i is indifferent to all
t E (0, —c/2 + T] U {T}. Any ti E (—c/2 + T, T) leads to a lower payoff.
Now turn to j and suppose that i follows Fi. The equilibrium strategy of jH
follows from Lemma 1. For jL, a concession in t E [0, —c/2 + T] yields an expected
payoff of
I (v — x) —e cl dx + e cl (v — cL — t)
= V — Cl-
Jo cl
Hence, jL is indeed indifferent to all t E [0, —c/2 + T]. For all t > —c/2 + t, jLs
expected payoff is strictly lower. The ex ante expected payoffs in the second row of
(6) and (7) follow directly from these calculations.
A.4 Proof of Lemma 4
By Assumption 1 and Lemma 1, FiH (t) = 0 for all t < T and FiH (t) = 1 otherwise,
i = 1, 2. Thus, for iL, a concession in T is strictly preferred to any t E (—cL/2 + T, T).
Suppose that jL follows FjL. For any ti E [0,t), iL’s payoff is
Pl / (v — x)
Jo
----e cl dx + e cl
PlCl
(v — ti — cL ) = v — cL-
By choosing ti = T, iL gets
Pl
[t (v — x)
Jo
PlCl
cl dx +
cL (v
CL)
Cl + exp
+CL)∙
If
— ^2~ + T < — CL ln(1 — pL) , (13)
c = — ɪ + T, and iL is indifferent between all ti E [0,t) U {T}: (13) implies that
FiL (t) < 1 and iL waits until T with strictly positive probability. If (13) is violated,
30
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