### Document 7432513

```Combining Sequential
and Simultaneous
Moves
Simultaneous-move games in tree
from


Moves are simultaneous because players cannot
observe opponents’ decisions before making moves.
EX: 2 telecom companies, both having invested \$10
billion in fiberoptic network, are engaging in a price war.
CrossTalk
High
Low
GlobalDialog
High
Low
2, 2
-10, 6
6, -10
-2, -2
G’s information set
C
High
Low



G
G
High
Low
High
Low
(2, 2)
(-10, 6)
(6, -10)
(-2, -2)
C moves before G, without knowing G’s moves.
G moves after C, also uncertain with C’s moves.
An Information set for a player contains all the nodes
such that when the player is at the information set, he
cannot distinguish which node he has reached.
A strategy is a complete plan of action,
specifying the move that a player would
make at each information set at whose
nodes the rules of the game specify that is
it her turn to move.
 Games with imperfect information are
games where the player’s information sets
are not singletons (unique nodes).


Battle of Sexes
Starbucks
Sally
Banyan
Harry
Harry
Starbucks 1, 2
Banyan
0, 0
Starbucks 0, 0
Banyan
2, 1
Harry
Sally
Starbucks
Banyan
Starbucks
1, 2
0, 0
Banyan
0, 0
2, 1

Two farmers decide at the
beginning of the season
what crop to plant. If the
season is dry only type I
crop will grow. If the
season is wet only type II
will grow. Suppose that
the probability of a dry
season is 40% and 60%
for the wet weather. The
following table describes
the Farmers‘ payoffs.
Dry
Crop 1 Crop 2
Crop 1
2, 3
5, 0
Crop 2
0, 5
0, 0
Wet
Crop 1 Crop 2
Crop 1
0, 0
0, 5
Crop 2
5, 0
3, 2
A
Dry 40%
Nature
Wet 60%
A
1
2
1
2
1
2
B
B
B
B
1
2
1
2
1
2
2, 3
5, 0
0, 5
0, 0
0, 0
0, 5
5, 0
3, 2



When A and B both choose Crop 1, with a 40% chance
(Dry) that A, B will get 2 and 3 each, and a 60% chance
(Wet) that A, B will get both 0.
A’s expected payoff: 40%x2+60%x0=0.8.
B’s expected payoff: 40%x3+60%x0=1.2.
1
2
1
0.8, 1.2
2, 3
2
3, 2
1.8, 1.2
Combining Sequential and Simultaneous Moves I

GlobalDialog has invested \$10 billion. Crosstalk is
wondering if it should invest as well. Once his decision
is made and revealed to G. Both will be engaged in a
price competition.
G
C
I
C
NI
G
High
Low
High
Low
High
2, 2
-10, 6
Low
6, -10
-2, -2
0, 14
0, 6
Subgames
G
C
NI
High
Low
I
G
C
C
High
Low
High
Low
High
Low
0, 14
0, 6
2, 2
6, -10
-10, 6
-2, -2
★

Subgame (Morrow, J.D.: Game Theory for
Political Scientists)




It has a single initial node that is the only member of
that node's information set (i.e. the initial node is in
a singleton information set).
It contains all the nodes that are successors of the
initial node.
It contains all the nodes that are successors of any
node it contains.
If a node in a particular information set is in the
subgame then all members of that information set
belong to the subgame.

Subgame-Perfect Equilibrium
A configuration of strategies (complete
plans of action) such that their
continuation in any subgame remains
optimal (part of a rollback equilibrium),
whether that subgame is on- or offequilibrium. This ensures credibility of the
strategies.
C has two information sets. At one, he’s
choosing I/NI, and at the other he’s
choosing H/L. He has 4 strategies, IH, IL,
NH, NL, with the first element denoting his
move at the first information set and the
2nd element at the 2nd information set.
 By contrast, G has two information sets
(both singletons) as well and 4 strategies,
HH, HL, LH, and LL.

HH
HL
LH
LL
IH
2, 2
2, 2
-10, 6
-10, 6
IL
6, -10
6, -10
-2, -2
-2, -2
NH
0, 14
0, 6
0, 14
0, 6
NL
0, 14
0, 6
0, 14
0, 6
(NH, LH) and (NL, LH) are both NE.
 (NL, LH) is the only subgame-perfect Nash
equilibrium because it requires C to
choose an optimal move at the 2nd
information set even it is off the equilibrium
path.

Combining Sequential and Simultaneous Moves II

C and G are both deciding simultaneously if
he/she should invest \$10 billion.
G
I
C
I
N
H
G
L
0,
14
6
N
,0
0, 0
C
H
14
L
6
G
C
H
L
H
2, 2
-10, 6
L
6, -10
-2, -2
G
C
I
N
I
-2, -2
0, 14
N
14, 0
0, 0
One should be aware that this is a simplified
payoff table requiring optimal moves at every
subgame, and hence the equilibrium is the
subgame-perfect equilibrium, not just a N.E.
Changing the Orders of Moves in a
Game


Games with all players having dominant
strategies
Games with NOT all players having dominant
strategies
FED
Low interest High interest
rate
rate
CONGRESS
Budget balance
3, 4
1, 3
Budget deficit
4, 1
2, 2

F moves first
Congress
Low
Balance
4, 3
Deficit
1, 4
Balance
3, 1
Deficit
2, 2
Fed
High
Congress

C moves first
Congress
Fed
Balance
Low
High
1, 3
Low
4, 1
High
2, 2
Deficit
Fed
3, 4

SALLY
Starbucks
Banyan
Starbucks
2, 1
0, 0
Banyan
0, 0
1, 2
HARRY

H first
Harry
Sally
Starbucks
Banyan
Sally
Starbucks
2, 1
Banyan
0, 0
Starbucks
0, 0
Banyan
1, 2

S first
Sally
Harry
Starbucks
Banyan
Starbucks
2, 1
Banyan
0, 0
Starbucks
0, 0
Harry Banyan
1, 2

but not necessary)
Navratilova
DL
CC
DL
50
80
CC
90
20
Evert

E first
Evert
Nav.
DL
CC
Nav.
DL
50, 50
CC
80, 20
DL
90, 10
CC
20, 80

N first
Nav.
Evert
DL
CC
Evert
DL
50, 50
CC
10, 90
DL
20, 80
CC
80, 20
Homework
1.
2.
Exercise 3 and 4
Consider the example of farmers but now change the
probability of dry weather to 80%.
(a) Use a payoff table to demonstrate the game.
(b) Find the N.E. of the game.
(c) Suppose now farmer B is able to observe A’ move
but not the weather before choosing the crop she’ll
grow. Describe the game with a game tree.
(d) Continue on c, use a strategic form to represent the
game.
(e) Find the N.E. in pure strategies.
```