2 take any σ ne γ and suppose that there is someg a

2

Take any σ ∈ NE ( γ ), and suppose that there is some G ∈ a ( γ ) such that γ |
G
6∈ NE ( G ) (if
not, then σ is a SPNE and we are done). Now pick any ˜ σ ∈ NE ( γ ) and create a strategy
profile σ
0
such that
σ
0
|
h
=
(
˜ σ if h=G
σ |
h
if h 6
3. Consider the following 3 player game. Player 1 announces a number a
1
∈ [0 , 1]. After this,
player 2 observes a
1
and then announces a number a
2
∈ [0 , 1]. If a
1
+ a
2
≤ 1, then the
game ends, and the players’ payoffs are given by
u
1
( a
1
, a
2
) = a
1
,
u
2
( a
1
, a
2
) = a
2
,
u
3
( a
1
, a
2
) = 1 - a
1
- a
2
.
If a
1
+ a
2
> 1, however, player 3 is allowed to choose either player 1 or player 2 as a
coalition partner. If player 3 chooses player 1, then the payoffs are
u
1
( a
1
, a
2
) = a
1
,
u
2
( a
1
, a
2
) = 0 ,
u
3
( a
1
, a
2
) = 1 - a
1
.
If player 3 chooses player 2 as a partner, the payoffs are
u
1
( a
1
, a
2
) = 0 ,
u
2
( a
1
, a
2
) = a
2
,
u
3
( a
1
, a
2
) = 1 - a
2
.
Describe the set of subgame perfect Nash equilibria to this game.