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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
First-mover advantage (Coordination Games)
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
Second-mover advantage (Zero-sum Games,
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.