and the equilibrium payoffs are as follows:
1 - δ2
v1
v2
-----——2 [(1 — δ1δ2)(1 + α2λ2)λι + (δ1 — δ2)(α1 — α2λ1λ2)]
(8)
(9)
δ1δ )2
1---χ .''i2∖ [(1 — δ1δ2)(λ1 + α1) + (δ1 — δ2)(α1 — α2λ1λ2)]
— δ1δ2)2λ1
3) If g ≤ α1 ≤ 1, the equilibrium demands are y2 =1 and x1 = xe1 ∈ (0, 1), where
(1 — δ2)[l — δ1δ2 + α2λ2(l — δ1)(1 + δ2)]
(1 — διδ2)
and the equilibrium payoffs are as follows:
v1
v2
(1 — δ2)[λ1(l — δ1δ2 + α2λ2(l — δι)(1 + δ2)) + α1δ1]
(11)
(12)
1 — δ1δ2
δ2[1 — δ1δ2 + α2λ2(1 — δ1)]
1 — δ1δ2
Proof. The indifference conditions between accepting and rejecting an offer are the
following
1 — χ1 + α2λ2l-δδδ2 = δ2 (y2 + α2λ2(1-δ1δδ2)
(1 — y2)λ1 + α11--δδ2 = δ1 (x1λ1 + α1 (1-δ2δ21 ´
(13)
The solution of the system (13) are the demands x1 and y2 defined in (6) and (7)
above. It can be shown that x1 > 0 if and only if α1 <aand x1 < 1 if and only if
α1 >b. Similarly, y2 > 0 if and only if α1 >f and y2 < 1 if and only if α1 <g. It is
straightforward to see that f ≤ b and g ≤ a. Then, there is an intersection between
the interval [b, a] and [f, g] if and only if g > b, that is λ2 ≤ λ2. Under this condition we
can distinguish three sections on the α1 axis. First, when 0 ≤ α1 ≤ b, the equilibrium