Topic 3
Games in Extensive Form
1
A. Perfect Information Games in Extensive Form
.
1
Fold
Raise
2
Fold
(0,0)
2
Raise
(-1,1)
Fold
Raise
(1,-1)
1
Fold
(-1,1)
Raise
(2,-2)
2
A. Perfect Information Games in Extensive Form
In a perfect information extensive form game, each player
knows exactly where they are in the game when they
take a move.
The only thing they don’t know is how future moves will
be played.
3
A. Perfect Information Games in Extensive Form
Strategies:
Definition: A Pure Strategy for a player is an
instruction book on how to play the game.
This instruction book must be complete and tell the player
what to do at every point at which it must make a move.
Because we want to consider what happens when players
make mistakes we must even include apparently
redundant instructions.
4
A. Perfect Information Games in Extensive Form
1’s Instructions
(Fold,Fold)
1
Fold
Raise
2
Fold
(0,0)
2
Raise
(-1,1)
Fold
Raise
(1,-1)
1
Fold
(-1,1)
Raise
(2,-2)
5
A. Perfect Information Games in Extensive Form
1’s Instructions
(Raise,Fold)
1
Fold
Raise
2
Fold
(0,0)
2
Raise
(-1,1)
Fold
Raise
(1,-1)
1
Fold
(-1,1)
Raise
(2,-2)
6
A. Perfect Information Games in Extensive Form
1’s Instructions
(Raise,Raise)
1
Fold
Raise
2
Fold
(0,0)
2
Raise
(-1,1)
Fold
Raise
(1,-1)
1
Fold
(-1,1)
Raise
(2,-2)
7
A. Perfect Information Games in Extensive Form
1’s Instructions
(Fold,Raise)
1
Fold
Raise
2
Fold
(0,0)
2
Raise
(-1,1)
Fold
Raise
(1,-1)
1
Fold
(-1,1)
Raise
(2,-2)
8
A. Perfect Information Games in Extensive Form
Strategies:
In this game player 1 has 4 pure strategies
1. (Raise, Raise)
2. (Raise,Fold)
3. (Fold,Raise)
4. (Fold,Fold)
Player 2 also has 4 pure strategies, but none of her
instructions are ever redundant.
9
A. Perfect Information Games in Extensive Form
2’s Instructions
(Fold,Fold)
1
Fold
Raise
2
Fold
(0,0)
2
Raise
(-1,1)
Fold
Raise
(1,-1)
1
Fold
(-1,1)
Raise
(2,-2)
10
A. Perfect Information Games in Extensive Form
2’s Instructions
(Raise,Fold)
1
Fold
Raise
2
Fold
(0,0)
2
Raise
(-1,1)
Fold
Raise
(1,-1)
1
Fold
(-1,1)
Raise
(2,-2)
11
A. Perfect Information Games in Extensive Form
2’s Instructions
(Raise,Raise)
1
Fold
Raise
2
Fold
(0,0)
2
Raise
(-1,1)
Fold
Raise
(1,-1)
1
Fold
(-1,1)
Raise
(2,-2)
12
B. Imperfect Information Games in Extensive Form
The figures above have the property that everyone knows
everything about the past events in the game – this rules
out:
• players moving simultaneously,
• players knowing something that others players do not.
We need to have a way of representing this. We use
information sets to do this.
13
B. Imperfect Information Games in Extensive Form
An Information set is a collection of nodes with the
property that
1. Each node in the set has the same player’s name.
2. Each node in the set has the same actions available.
Information sets describe nodes that the player cannot
distinguish between.
14
B. Imperfect Information Games in Extensive Form
2 sees 1’s action
1
Raise
Fold
2
Fold
(0,0)
2
Raise
(-1,1)
Fold
Raise
(1,-1)
(0,0)
15
B. Imperfect Information Games in Extensive Form
2 does not see 1’s action
1
Information Set
Raise
Fold
2
Fold
(0,0)
2
Raise
(-1,1)
Fold
Raise
(1,-1)
(0,0)
16
B. Imperfect Information Games in Extensive Form
1. Each node in the set has the same player’s name.
2. Each node in the set has the same actions available.
Both of these are essential, because otherwise the player
would be able to distinguish between nodes in the same
information set by seeing what actions they had and
who they were.
17
B. Imperfect Information Games in Extensive Form
Information Sets can be used to describe situations where
players move simultaneously:
18
B. Imperfect Information Games in Extensive Form
Simultaneously
1
Information Set
Raise
Fold
2
Fold
(0,0)
2
Raise
(-1,1)
Fold
Raise
(1,-1)
(0,0)
19
C. Nash Equilibrium and Subgame Perfection
At a Nash equilibrium each player’s strategy is a best
response to the other players’ strategies:
20
C. Nash Equilibrium
A Nash Equilibrium.
1
Fold
Raise
2
Fold
(0,0)
2
Raise
(-1,1)
Fold
(1,-1)
Raise
(-2,-2)
21
C. Nash Equilibrium
A Nash Equilibrium.
1
Fold
Raise
2
Fold
(0,0)
2
Raise
(-1,1)
Fold
(1,-1)
Raise
(-2,-2)
Checking that 1’s strategy is a best response
If she folds she gets -1.
22
C. Nash Equilibrium
A Nash Equilibrium.
1
Fold
Raise
2
Fold
(0,0)
2
Raise
(-1,1)
Fold
(1,-1)
Raise
(-2,-2)
Checking that 1’s strategy is a best response
If she raises she gets -2.
23
C. Nash Equilibrium
A Nash Equilibrium.
1
Fold
Raise
2
Fold
(0,0)
2
Raise
(-1,1)
Fold
(1,-1)
Raise
(-2,-2)
Checking that 2’s strategy is a best response
If she changes her action on the right her payoff is unaltered.
24
C. Nash Equilibrium
A Nash Equilibrium.
1
Fold
Raise
2
Fold
(0,0)
2
Raise
(-1,1)
Fold
(1,-1)
Raise
(-2,-2)
Checking that 2’s strategy is a best response
If she changes her action on the right her payoff is unaltered.
25
C. Nash Equilibrium and Subgame Perfection
A Nash Equilibrium.
1
Fold
Raise
2
Fold
(0,0)
2
Raise
(-1,1)
Fold
(1,-1)
Raise
(-2,-2)
Checking that 2’s strategy is a best response
If she changes her action on the left her payoff goes down.
26
C. Nash Equilibrium and Subgame Perfection
A Nash Equilibrium.
1
Fold
Raise
2
Fold
(0,0)
2
Raise
(-1,1)
Fold
(1,-1)
Raise
(-2,-2)
Checking that 2’s strategy is a best response
If she changes her action on the left her payoff goes down.
27
C. Nash Equilibrium
Nash equilibrium alone in an extensive form game is un
satisfactory because it builds in non-credible threats.
At a Nash equilibrium each player assumes the others will
stick to their equilibrium actions when they test their
own action.
They assume other players are committed to playing their
strategy.
This may not be a good assumption in a dynamic model
because players may make threats that are not rational
for them.
28
C. Nash Equilibrium
A Non-Credible Threat.
1
Fold
Raise
2
Fold
(0,0)
2 is making a threat here
that she would not carry
out .
2
Raise
(-1,1)
Fold
(1,-1)
Raise
(-2,-2)
29
C. Nash Equilibrium
A Non-Credible Threat.
1
Fold
Raise
2
Fold
(0,0)
2 is making a threat here
that she would not carry
out .
2
Raise
(-1,1)
Fold
(1,-1)
Raise
(-2,-2)
30
C. Nash Equilibrium
A Non-Credible Threat.
1
Fold
Raise
2
Fold
(0,0)
2 is making a threat here
that she would not carry
out .
2
Raise
(-1,1)
Fold
(1,-1)
Raise
(-2,-2)
Once player 1 realizes this she is better off raising too
31
C. Nash Equilibrium
A Non-Credible Threat.
1
Fold
Raise
2
Fold
(0,0)
2 is making a threat here
that she would not carry
out .
2
Raise
(-1,1)
Fold
(1,-1)
Raise
(-2,-2)
Once player 1 realizes this she is better off raising too
32
C. Nash Equilibrium
This is another Nash equilibrium.(Check)
At this Nash equilibrium no non-credible threats are made.
Can we always find such a Nash equilibrium?
Answer: Yes by using a process called backwards
induction.
This always works in games of perfect information (i.e.
games without information sets).
33
D. Backwards Induction
Example
1
Fold
Raise
2
Fold
(0,0)
2
Raise
(-1,1)
Fold
Raise
(1,-1)
Start at the last move and figure
out what is optimal there
1
Fold
(-1,1)
Raise
(2,-2)
34
D. Backwards Induction
Example
1
Fold
Raise
2
Fold
(0,0)
2
Raise
(-1,1)
Fold
Raise
(1,-1)
Start at the last move and figure
out what is optimal there
1
Fold
(-1,1)
Raise
(2,-2)
35
D. Backwards Induction
Example
1
Fold
Raise
2
Fold
(0,0)
Do all last moves
2
Raise
(-1,1)
Fold
Raise
(1,-1)
1
Fold
(-1,1)
Raise
(2,-2) 36
D. Backwards Induction
Example
1
Fold
Raise
2
Fold
(0,0)
Do all last moves
2
Raise
(-1,1)
Fold
Raise
(1,-1)
1
Fold
(-1,1)
Raise
(2,-2) 37
D. Backwards Induction
Example
1
Fold
Raise
2
Fold
(0,0)
2
Raise
(-1,1)
Fold
Raise
(1,-1)
Now do the second last move.
1
Fold
(-1,1)
Raise
(2,-2) 38
D. Backwards Induction
Example
1
Fold
Raise
2
Fold
(0,0)
2
Raise
(-1,1)
Fold
Raise
(1,-1)
Now do the second last move.
1
Fold
(-1,1)
Raise
(2,-2)
39
D. Backwards Induction
Example:
1
Fold
Raise
2
Fold
(0,0)
2
Raise
(-1,1)
Finally do the first move
Fold
Raise
(1,-1)
1
Fold
(-1,1)
Raise
(2,-2)
40
D. Backwards Induction
Example:
1
Fold
Raise
2
Fold
(0,0)
2
Raise
(-1,1)
Finally do the first move
Fold
Raise
(1,-1)
1
Fold
(-1,1)
Raise
(2,-2)
41
D. Backwards Induction
Notes:
1. This process works for all finite perfect information
games.
2. Proves the existence of a NE in such games (Zermelo’s
Theorem) – Chess Draughts etc.
3. It is a generalization of dynamic programming.
4. BUT backwards induction will not work in games with
information sets.
5. So Selten generalized the idea of Backwards induction
to create Subgame perfection.
42
E. Subgame Perfection
Subgame perfect equilibrium divides the game tree up into
subgames (that is, parts of the tree that can be
considered separately).
It requires that the player’s strategies are a Nash
equilibrium on every subgame.
Again these can be found by working backwards, taking
the last independent game and finding strategies that are
a Nash equilibrium.
Then taking the next last independent game and so on.
Backwards induction always finds a subgame perfect
equilibrium.
43
E. Subgame Perfection
Example
1
Fold
Raise
(1,1)
2
L
R
1
a
(1,2)
1
b
(-1,1)
a
(0,1)
b
(1/2,0)
44
E. Subgame Perfection
A (Bad) NE
1
Fold
Raise
(1,1)
2
L
R
1
a
(2,2)
1
b
(-1,1)
a
(0,1)
b
(1/2,0)
45
E. Subgame Perfection
A (Bad) NE:
1
Fold
Raise
(1,1)
The move of player
2 is not credible
it always prefers
L to R if she gets
to move!
2
L
R
1
a
(2,2)
1
b
(-1,1)
a
(0,1)
b
(1/2,0)
46
E. Subgame Perfection
A (Bad) NE:
1
Fold
Raise
(1,1)
But cannot do
backwards induction
sometimes 1 prefers
‘a’ sometimes ‘b’.
2
L
R
1
a
(2,2)
1
b
(-1,1)
a
(0,1)
b
(1/2,0)
47
E. Subgame Perfection
1
Fold
Raise
(1,1)
2
Where are the subgames?
L
R
1
a
(2,2)
1
b
(-1,1)
a
(0,1)
b
(1/2,0)
48
E. Subgame Perfection
1
Fold
Raise
(1,1)
2
Here
L
R
1
a
(2,2)
1
b
(-1,1)
a
(0,1)
b
(1/2,0)
49
E. Subgame Perfection
1
Fold
Raise
(1,1)
2
And here
L
R
1
a
(2,2)
1
b
(-1,1)
a
(0,1)
b
(1/2,0)
50
E. Subgame Perfection
Let’s look at the
last subgame
and make the
players play a
NE on it.
1
Fold
Raise
(1,1)
2
L
R
1
a
(2,2)
1
b
(-1,1)
a
(0,1)
b
(1/2,0)
51
E. Subgame Perfection
1
Fold
Raise
(1,1)
It has
only 1 NE,
because player
2 always prefers
L to R.
2
L
R
1
a
(2,2)
1
b
(-1,1)
a
(0,1)
b
(1/2,0)
52
E. Subgame Perfection
1
Fold
Raise
(1,1)
Once
we have solved this
last game it is easy
to figure out
what player 1 will
do.
2
L
R
1
a
(2,2)
1
b
(-1,1)
a
(0,1)
b
(1/2,0)
53
E. Subgame Perfection
1
Fold
Raise
(1,1)
Once
we have solved this
last game it is easy
to figure out
what player 1 will
do.
2
L
R
1
a
(2,2)
1
b
(-1,1)
a
(0,1)
b
(1/2,0)
54
E. Subgame Perfection
1
Fold
This is the
subgame perfect
equilibrium.
Raise
(1,1)
2
L
R
1
a
(2,2)
1
b
(-1,1)
a
(0,1)
b
(1/2,0)
55
E. Subgame Perfection
Backwards induction will usually give you a unique
solution to a game (it will not be unique if there are ties
in the players’ payoffs).
As the subgames have many Nash equilibria, however, a
game can have many subgame perfect equibria.
Example.
56
E. Subgame Perfection
IF 1 plays right the
players then play a
coordination game
which has 3 NE’s.
1
X
Y
2
2
L
R
1
a
(3,1)
L
1
b
(0,0)
a
(0,0)
R
1
b
(1,3)
a
(2,2)
1
b
(0,0)
a
(0,0)
b
(2,2)
57
E. Subgame Perfection
IF 1 plays right the
players then play a
coordination game
which has 3 NE’s.
1
X
Y
2
2
L
R
1
a
(3,1)
L
1
b
(0,0)
a
(0,0)
R
1
b
(1,3)
a
(2,2)
1
b
(0,0)
a
(0,0)
b
(2,2)
58
E. Subgame Perfection
IF 1 plays right the
players then play a
coordination game
which has 3 NE’s.
1
X
Y
2
2
L
R
1
a
(3,1)
L
1
b
(0,0)
a
(0,0)
R
1
b
(1,3)
a
(2,2)
1
b
(0,0)
a
(0,0)
b
(2,2)
59
E. Subgame Perfection
IF 1 plays X the players
then play the battle
of the sexes which also
has 3 NE’s.
1
X
Y
2
2
L
R
1
a
(3,1)
L
1
b
(0,0)
a
(0,0)
R
1
b
(1,3)
a
(2,2)
1
b
(0,0)
a
(0,0)
b
(2,2)
60
E. Subgame Perfection
IF 1 plays X the players
then play the battle
of the sexes which also
has 3 NE’s.
1
X
Y
2
2
L
R
1
a
(3,1)
L
1
b
(0,0)
a
(0,0)
R
1
b
(1,3)
a
(2,2)
1
b
(0,0)
a
(0,0)
b
(2,2)
61
E. Subgame Perfection
IF 1 plays X the players
then play the battle
of the sexes which also
has 3 NE’s.
1
X
Y
2
2
L
R
1
a
(3,1)
L
1
b
(0,0)
a
(0,0)
R
1
b
(1,3)
a
(2,2)
1
b
(0,0)
a
(0,0)
b
(2,2)
62
E. Subgame Perfection
IF 1 plays X the players
then play the battle
of the sexes which also
has 3 NE’s.
1
X
Y
2
2
L
R
1
a
(3,1)
L
1
b
(0,0)
a
(0,0)
R
1
b
(1,3)
a
(2,2)
1
b
(0,0)
a
(0,0)
b
(2,2)
63
E. Subgame Perfection
In a finite game,as a Nash equilibrium always exists so
does a subgame perfect equilibrium.
But again there are certain games for which subgame
perfection is not going to work and we are going to need
to rule out non-credible actions.
Here is an example…
64
E. Subgame Perfection
1
(1,3)
L
R
M
2
a
(2,2)
2
b
a
(0,1)
(3,1)
b
(0,0)
This game has no subgames, but so the above NE is also a
subgame perfect equilibrium, but it appears that 2 is
making a non-credible threat….
65
F. Perfect Bayesian Equilibrium
To remedy the problem with the game above we need
another notion of equilibrium…
Perfect Bayesian equilibrium.
66
F. Perfect Bayesian Equilibrium
1
If you are going
L
to do something
like Backwards
(1,3)
induction in this
a
game, then you
are going to need
(2,2)
to decide what
action player 2 will take.
R
M
2
2
b
a
(0,1)
(3,1)
b
(0,0)
67
F. Perfect Bayesian Equilibrium
1
If you are going
L
to do something
like Backwards
(1,3)
induction in this
a
game, then you
are going to need
(2,2)
to decide what
action player 2 will take.
R
M
2
p
1-p
b
a
(0,1)
(3,1)
2
b
(0,0)
Must give player 2 beliefs
68
F. Perfect Bayesian Equilibrium
1
L
(1,3)
2
a
(2,2)
R
M
p
1-p
b
a
(0,1)
(3,1)
2
b
(0,0)
Can now evaluate player 2’s expected payoff from each
action
Action a = 2p+1(1-p)=1+p
69
F. Perfect Bayesian Equilibrium
1
L
(1,3)
2
a
(2,2)
R
M
p
1-p
b
a
(0,1)
(3,1)
2
b
(0,0)
Can now evaluate player 2’s expected payoff from each
action
Action b = 1p+0(1-p)=p
70
F. Perfect Bayesian Equilibrium
1
L
(1,3)
2
a
(2,2)
R
M
p
1-p
b
a
(0,1)
(3,1)
2
b
(0,0)
Action a is always better than action, so player 2 should
play a.
71
F. Perfect Bayesian Equilibrium
1
L
(1,3)
2
a
(2,2)
R
M
p
1-p
b
a
(0,1)
(3,1)
2
b
(0,0)
Hence 1 should play R.
72
F. Perfect Bayesian Equilibrium
1
L
(1,3)
2
a
(2,2)
R
M
p
1-p
b
a
(0,1)
(3,1)
2
b
(0,0)
At an equilibrium we want 2’s beliefs to be consistent with
1’s actions (rational expectations).
So at an equilibrium we will have p=0.
73
F. Perfect Bayesian quilibrium
To find Perfect Bayesian equilibria we equip players with
beliefs at each information set at which they must take a
move.
We require their actions at that set to be optimal given their
beliefs this is called “Sequential Rationality”.
We then work backwards through the tree. Given the
sequentially rational actions we find for the players we
then check that beliefs are consistent given these actions.
(Rational expectations).
74
F. Perfect Bayesian Equilibrium
A Signalling
Game
1/2
W
2
b
1/2
1
1
T
a
0
a
2
b
(1,2) (0,0) (0,0) (-1,1)
T
W
2
a
2
b
a
b
(-1,1) (0,0) (0,0) (1,2)
75
F. Perfect Bayesian Equilibrium
A Signalling
Game
1/2
W
b
1
T
W
1-p
2
a
1/2
1
T
p
0
a
2
b
(1,2) (0,0) (0,0) (-1,1)
e
1-e
2
a
2
b
a
b
(-1,1) (0,0) (0,0) (1,2)
We must give player 2 two lots of beliefs (sometimes these
are called assessments).
76
F. Perfect Bayesian Equilibrium
A Signalling
Game
1/2
W
b
a
2
b
(1,2) (0,0) (0,0) (-1,1)
2 plays a if:
2 plays a if:
1
T
W
1-p
2
a
1/2
1
T
p
0
e
1-e
2
a
2
b
a
b
(-1,1) (0,0) (0,0) (1,2)
p>1/3.
e>2/3.
77
F. Perfect Bayesian Equilibrium
A Signalling
Game
1/2
W
b
1
T
W
1-p
2
a
1/2
1
T
p
0
a
2
b
(1,2) (0,0) (0,0) (-1,1)
e
1-e
2
a
2
b
a
b
(-1,1) (0,0) (0,0) (1,2)
Both types of 1 play T
78
F. Perfect Bayesian Equilibrium
A Signalling
Game
1/2
W
b
1
T
W
1-p
2
a
1/2
1
T
p
0
a
2
b
(1,2) (0,0) (0,0) (-1,1)
e
1-e
2
a
2
b
a
b
(-1,1) (0,0) (0,0) (1,2)
Both types of 1 play T
So rational expectations says p=1/2,
but e is arbitrary – must have e>2/3.
79
F. Perfect Bayesian Equilibrium
At a Perfect Bayesian Equilibrium (PBE) we can choose
beliefs at unused information sets in any way we want.
Two interesting types of PBE in signalling games:
1. Pooling equilibria: All types choose the same, identical
action.
2. Separating equilibria: No two types choose the same
action.
80