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Bargaining and Rationalist Explanations for War
Unformatted Document Text:  38 Appendix In this appendix, I introduce how to generate the unique SPE in the bargaining with complete information. I first introduce the case of pure bargaining, that is, war is an incredible threat. This is similar to Rubinstein’s original study. Then I apply this logic to the situation of crisis bargaining when war becomes available outside options of bargaining. Let us first look at the case of pure bargaining. Rather giving a formal proof, I only discuss the logic to derive the unique SPE (see Osborne and Rubinstein 1990, 46-48 for the proof). In the situation that all the five Rubinstein assumptions are satisfied, there exists a unique pair of agreements , defined as ) , ( *2 * 1 * x x x ≡ and ) , ( *2 * 1 * y y y ≡ . More precisely, ) 1 , ( * 1 1 * 1 x v y = and ) 1 , ( *2 2 *2 y v x = . To see why, the whole game can be categorized into two groups of subgames: the group of subgames initiated by player 1, denoted as ) 1 ( Γ and the group of subgames initiated by player 2, ) 2 ( Γ . Apparently, ) 1 ( Γ indicates those subgames starting from those even-numbered rounds, denoted by ) 1 ( t Γ , where t is even; 22 and ) 2 ( Γ those subgames starting from those odd-numbered rounds, written as ) 2 ( t Γ , where t is odd. Let us first look at ) 1 ( Γ . Suppose there is an agreement reached in one game of ) 1 ( t Γ in round t. That is, player 1 makes an offer x in round t and player 2 rejects it. The outcome is (x,t). However, to both players, ) , ( t x ) 0 , (x i p , that is, if there is an agreement in round t, both will be better off if such agreement can be made earlier. In short, the Nash equilibrium of all the subgames ) 1 ( Γ must lead to an agreement in ) 1 ( 0 Γ , ) 0 , ( * x . For the same reason, the Nash equilibrium of all the subgames ) 2 ( Γ must lead to an agreement in ) 2 ( 1 Γ , ) 1 , ( * y . Now, let us 22 The whole game can therefore be treated as a special case of .

Authors: Ye, Min.
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38
Appendix
In this appendix, I introduce how to generate the unique SPE in the bargaining with complete
information. I first introduce the case of pure bargaining, that is, war is an incredible threat. This
is similar to Rubinstein’s original study. Then I apply this logic to the situation of crisis
bargaining when war becomes available outside options of bargaining.
Let us first look at the case of pure bargaining. Rather giving a formal proof, I only
discuss the logic to derive the unique SPE (see Osborne and Rubinstein 1990, 46-48 for the
proof). In the situation that all the five Rubinstein assumptions are satisfied, there exists a unique
pair of agreements
, defined as
)
,
(
*
2
*
1
*
x
x
x
and
)
,
(
*
2
*
1
*
y
y
y
. More precisely,
)
1
,
(
*
1
1
*
1
x
v
y =
and
)
1
,
(
*
2
2
*
2
y
v
x =
. To see why, the whole game can be categorized into two groups
of subgames: the group of subgames initiated by player 1, denoted as
)
1
(
Γ
and the group of
subgames initiated by player 2,
)
2
(
Γ
. Apparently,
)
1
(
Γ
indicates those subgames starting from
those even-numbered rounds, denoted by
)
1
(
t
Γ
, where t is even;
22
and
)
2
(
Γ
those subgames
starting from those odd-numbered rounds, written as
)
2
(
t
Γ
, where t is odd.
Let us first look at
)
1
(
Γ
. Suppose there is an agreement reached in one game of
)
1
(
t
Γ
in
round t. That is, player 1 makes an offer x in round t and player 2 rejects it. The outcome is (x,t).
However, to both players,
)
,
(
t
x
)
0
,
(x
i
p
, that is, if there is an agreement in round t, both will be
better off if such agreement can be made earlier. In short, the Nash equilibrium of all the
subgames
)
1
(
Γ
must lead to an agreement in
)
1
(
0
Γ
,
)
0
,
(
*
x
. For the same reason, the Nash
equilibrium of all the subgames
)
2
(
Γ
must lead to an agreement in
)
2
(
1
Γ
,
)
1
,
(
*
y
. Now, let us
22
The whole game can therefore be treated as a special case of
.


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