1

Topic 3

Games in Extensive Form

2

A. Perfect Information Games in Extensive Form

. 1

RaiseFold

1

22

Fold Fold

Fold Raise

Raise

(0,0) (-1,1)

Raise

(1,-1)

(-1,1) (2,-2)

3

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.

4

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.

5

A. Perfect Information Games in Extensive Form

1’s Instructions(Fold,Fold)

1

RaiseFold

1

22

Fold Fold Raise

(0,0) (-1,1)

Raise

(1,-1)

(-1,1) (2,-2)

RaiseFold

6

A. Perfect Information Games in Extensive Form

1’s Instructions(Raise,Fold)

1

RaiseFold

1

22

Fold Fold Raise

(0,0) (-1,1)

Raise

(1,-1)

(-1,1) (2,-2)

RaiseFold

7

A. Perfect Information Games in Extensive Form

1’s Instructions(Raise,Raise)

1

RaiseFold

1

22

Fold Fold Raise

(0,0) (-1,1)

Raise

(1,-1)

(-1,1) (2,-2)

RaiseFold

8

A. Perfect Information Games in Extensive Form

1’s Instructions(Fold,Raise)

1

RaiseFold

1

22

Fold Fold Raise

(0,0) (-1,1)

Raise

(1,-1)

(-1,1) (2,-2)

RaiseFold

9

A. Perfect Information Games in Extensive Form

Strategies:In this game player 1 has 4 pure strategies1. (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.

10

A. Perfect Information Games in Extensive Form

2’s Instructions(Fold,Fold)

1

RaiseFold

1

22

Fold Fold Raise

(0,0) (-1,1)

Raise

(1,-1)

(-1,1) (2,-2)

RaiseFold

11

A. Perfect Information Games in Extensive Form

2’s Instructions(Raise,Fold)

1

RaiseFold

1

22

Fold Fold Raise

(0,0) (-1,1)

Raise

(1,-1)

(-1,1) (2,-2)

RaiseFold

12

A. Perfect Information Games in Extensive Form

2’s Instructions(Raise,Raise)

1

RaiseFold

1

22

Fold Fold Raise

(0,0) (-1,1)

Raise

(1,-1)

(-1,1) (2,-2)

RaiseFold

13

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.

14

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.

15

B. Imperfect Information Games in Extensive Form

2 sees 1’s action 1

RaiseFold

22

Fold Fold Raise

(0,0) (-1,1)

Raise

(1,-1)

(0,0)

16

B. Imperfect Information Games in Extensive Form

2 does not see 1’s action1

RaiseFold

22

Fold Fold Raise

(0,0) (-1,1)

Raise

(1,-1)

(0,0)

Information Set

17

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.

18

B. Imperfect Information Games in Extensive Form

Information Sets can be used to describe situations where players move simultaneously:

19

B. Imperfect Information Games in Extensive Form

Simultaneously 1

RaiseFold

22

Fold Fold Raise

(0,0) (-1,1)

Raise

(1,-1)

(0,0)

Information Set

20

C. Nash Equilibrium and Subgame Perfection

At a Nash equilibrium each player’s strategy is a best response to the other players’ strategies:

21

C. Nash Equilibrium

A Nash Equilibrium. 1

RaiseFold

22

Fold Fold Raise

(0,0) (-1,1)

Raise

(1,-1)(-2,-2)

22

C. Nash Equilibrium

A Nash Equilibrium.

Checking that 1’s strategy is a best responseIf she folds she gets -1.

1

RaiseFold

22

Fold Fold Raise

(0,0) (-1,1)

Raise

(1,-1)(-2,-2)

23

C. Nash Equilibrium

A Nash Equilibrium.

Checking that 1’s strategy is a best responseIf she raises she gets -2.

1

RaiseFold

22

Fold Fold Raise

(0,0) (-1,1)

Raise

(1,-1)(-2,-2)

24

C. Nash Equilibrium

A Nash Equilibrium.

Checking that 2’s strategy is a best responseIf she changes her action on the right her payoff is

unaltered.

1

RaiseFold

22

Fold Fold Raise

(0,0) (-1,1)

Raise

(1,-1)(-2,-2)

25

C. Nash Equilibrium

A Nash Equilibrium.

Checking that 2’s strategy is a best responseIf she changes her action on the right her payoff is

unaltered.

1

RaiseFold

22

Fold Fold Raise

(0,0) (-1,1)

Raise

(1,-1)(-2,-2)

26

C. Nash Equilibrium and Subgame Perfection

A Nash Equilibrium.

Checking that 2’s strategy is a best responseIf she changes her action on the left her payoff goes

down.

1

RaiseFold

22

Fold Fold Raise

(0,0) (-1,1)

Raise

(1,-1)(-2,-2)

27

C. Nash Equilibrium and Subgame Perfection

A Nash Equilibrium.

Checking that 2’s strategy is a best responseIf she changes her action on the left her payoff goes

down.

1

RaiseFold

22

Fold Fold Raise

(0,0) (-1,1)

Raise

(1,-1)(-2,-2)

28

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.

29

C. Nash Equilibrium

A Non-Credible Threat.1

RaiseFold

22

Fold Fold Raise

(0,0) (-1,1)

Raise

(1,-1)(-2,-2)

2 is making a threat herethat she would not carry out .

30

C. Nash Equilibrium

A Non-Credible Threat.1

RaiseFold

22

Fold Fold Raise

(0,0) (-1,1)

Raise

(1,-1)(-2,-2)

2 is making a threat herethat she would not carry out .

31

C. Nash Equilibrium

A Non-Credible Threat.

Once player 1 realizes this she is better off raising too

1

RaiseFold

22

Fold Fold Raise

(0,0) (-1,1)

Raise

(1,-1)(-2,-2)

2 is making a threat herethat she would not carry out .

32

C. Nash Equilibrium

A Non-Credible Threat.

Once player 1 realizes this she is better off raising too

1

RaiseFold

22

Fold Fold Raise

(0,0) (-1,1)

Raise

(1,-1) (-2,-2)

2 is making a threat herethat she would not carry out .

33

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).

34

D. Backwards Induction

Example

Start at the last move and figureout what is optimal there

1

RaiseFold

1

22

Fold Fold Raise

(0,0) (-1,1)

Raise

(1,-1)

(-1,1) (2,-2)

RaiseFold

35

D. Backwards Induction

Example

Start at the last move and figureout what is optimal there

1

RaiseFold

1

22

Fold Fold Raise

(0,0) (-1,1)

Raise

(1,-1)

(-1,1) (2,-2)

RaiseFold

36

D. Backwards Induction

Example

Do all last moves

1

RaiseFold

1

22

Fold Fold Raise

(0,0)(-1,1)

Raise

(1,-1)

(-1,1) (2,-2)

RaiseFold

37

D. Backwards Induction

Example

Do all last moves

1

RaiseFold

1

22

Fold Fold Raise

(0,0) (-1,1)

Raise

(1,-1)

(-1,1) (2,-2)

RaiseFold

38

D. Backwards Induction

Example

Now do the second last move.

1

RaiseFold

1

22

Fold Fold Raise

(0,0) (-1,1)

Raise

(1,-1)

(-1,1) (2,-2)

RaiseFold

39

D. Backwards Induction

Example

Now do the second last move.

1

RaiseFold

1

22

Fold Fold Raise

(0,0) (-1,1)

Raise

(1,-1)

(-1,1) (2,-2)

RaiseFold

40

D. Backwards Induction

Example:

Finally do the first move

1

RaiseFold

1

22

Fold Fold Raise

(0,0) (-1,1)

Raise

(1,-1)

(-1,1) (2,-2)

RaiseFold

41

D. Backwards Induction

Example:

Finally do the first move

1

RaiseFold

1

22

Fold Fold Raise

(0,0) (-1,1)

Raise

(1,-1)

(-1,1) (2,-2)

RaiseFold

42

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.

43

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.

44

1

E. Subgame Perfection

Example 1

RaiseFold

2

L R

(1,1)

(1,2) (-1,1) (0,1)

ba

1

b a

(1/2,0)

45

1

E. Subgame Perfection

A (Bad) NE 1

RaiseFold

2

L R

(1,1)

(2,2) (-1,1) (0,1)

ba

1

b a

(1/2,0)

46

1

E. Subgame Perfection

A (Bad) NE:

The move of player2 is not credibleit always prefersL to R if she getsto move!

1

RaiseFold

2

L R

(1,1)

(2,2) (-1,1) (0,1)

ba

1

b a

(1/2,0)

47

1

E. Subgame Perfection

A (Bad) NE:

But cannot dobackwards inductionsometimes 1 prefers‘a’ sometimes ‘b’.

1

RaiseFold

2

L R

(1,1)

(2,2) (-1,1) (0,1)

ba

1

b a

(1/2,0)

48

1

E. Subgame Perfection

Where are the subgames?

1

RaiseFold

2

L R

(1,1)

(2,2) (-1,1) (0,1)

ba

1

b a

(1/2,0)

49

1

E. Subgame Perfection

Here

1

RaiseFold

2

L R

(1,1)

(2,2) (-1,1) (0,1)

ba

1

b a

(1/2,0)

50

1

E. Subgame Perfection

And here

1

RaiseFold

2

L R

(1,1)

(2,2) (-1,1) (0,1)

ba

1

b a

(1/2,0)

51

1

E. Subgame Perfection

Let’s look at thelast subgameand make the players play aNE on it.

1

RaiseFold

2

L R

(1,1)

(2,2) (-1,1) (0,1)

ba

1

b a

(1/2,0)

52

1

E. Subgame Perfection

It hasonly 1 NE,because player2 always prefersL to R.

1

RaiseFold

2

L R

(1,1)

(2,2) (-1,1) (0,1)

ba

1

b a

(1/2,0)

53

1

E. Subgame Perfection

Oncewe have solved thislast game it is easyto figure outwhat player 1 willdo.

1

RaiseFold

2

L R

(1,1)

(2,2) (-1,1) (0,1)

ba

1

b a

(1/2,0)

54

1

E. Subgame Perfection

Oncewe have solved thislast game it is easyto figure outwhat player 1 willdo.

1

RaiseFold

2

L R

(1,1)

(2,2) (-1,1) (0,1)

ba

1

b a

(1/2,0)

55

1

E. Subgame Perfection

This is thesubgame perfectequilibrium.

1

RaiseFold

2

L R

(1,1)

(2,2) (-1,1) (0,1)

ba

1

b a

(1/2,0)

56

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.

57

E. Subgame Perfection

1

YX

2

b

1

L R

(1,3)(0,0) (0,0)

a

1

b a

(2,2)

1

L R

(3,1) (0,0) (0,0)

a

1

b a

(2,2)

2

b

IF 1 plays right the players then play a coordination gamewhich has 3 NE’s.

58

E. Subgame Perfection

1

YX

2

b

1

L R

(1,3)(0,0) (0,0)

a

1

b a

(2,2)

1

L R

(3,1) (0,0) (0,0)

a

1

b a

(2,2)

2

b

IF 1 plays right the players then play a coordination gamewhich has 3 NE’s.

59

E. Subgame Perfection

1

YX

2

b

1

L R

(1,3)(0,0) (0,0)

a

1

b a

(2,2)

1

L R

(3,1) (0,0) (0,0)

a

1

b a

(2,2)

2

b

IF 1 plays right the players then play a coordination gamewhich has 3 NE’s.

60

E. Subgame Perfection

1

YX

2

b

1

L R

(1,3)(0,0) (0,0)

a

1

b a

(2,2)

1

L R

(3,1) (0,0) (0,0)

a

1

b a

(2,2)

2

b

IF 1 plays X the players then play the battleof the sexes which also has 3 NE’s.

61

E. Subgame Perfection

1

YX

2

b

1

L R

(1,3)(0,0) (0,0)

a

1

b a

(2,2)

1

L R

(3,1) (0,0) (0,0)

a

1

b a

(2,2)

2

b

IF 1 plays X the players then play the battleof the sexes which also has 3 NE’s.

62

E. Subgame Perfection

1

YX

2

b

1

L R

(1,3)(0,0) (0,0)

a

1

b a

(2,2)

1

L R

(3,1) (0,0) (0,0)

a

1

b a

(2,2)

2

b

IF 1 plays X the players then play the battleof the sexes which also has 3 NE’s.

63

E. Subgame Perfection

1

YX

2

b

1

L R

(1,3)(0,0) (0,0)

a

1

b a

(2,2)

1

L R

(3,1) (0,0) (0,0)

a

1

b a

(2,2)

2

b

IF 1 plays X the players then play the battleof the sexes which also has 3 NE’s.

64

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…

65

E. Subgame Perfection

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….

1

RM

22

a a b

(2,2) (0,1)

b

(3,1)

(1,3)

(0,0)

L

66

F. Perfect Bayesian Equilibrium

To remedy the problem with the game above we need another notion of equilibrium…

Perfect Bayesian equilibrium.

67

F. Perfect Bayesian Equilibrium

If you are goingto do somethinglike Backwardsinduction in thisgame, then youare going to needto decide what action player 2 will take.

1

RM

22

a a b

(2,2)(0,1)

b

(3,1)

(1,3)

(0,0)

L

68

F. Perfect Bayesian Equilibrium

If you are goingto do somethinglike Backwardsinduction in thisgame, then youare going to needto decide what action player 2 will take.

Must give player 2 beliefs

1

RM

22

a a b

(2,2)(0,1)

b

(3,1)

(1,3)

(0,0)

L

p 1-p

69

F. Perfect Bayesian Equilibrium

Can now evaluate player 2’s expected payoff from each action

Action a = 2p+1(1-p)=1+p

1

RM

22

a a b

(2,2)(0,1)

b

(3,1)

(1,3)

(0,0)

L

p 1-p

70

F. Perfect Bayesian Equilibrium

Can now evaluate player 2’s expected payoff from each action

Action b = 1p+0(1-p)=p

1

RM

22

a a b

(2,2) (0,1)

b

(3,1)

(1,3)

(0,0)

L

p 1-p

71

F. Perfect Bayesian Equilibrium

Action a is always better than action, so player 2 should

play a.

1

RM

22

a a b

(2,2) (0,1)

b

(3,1)

(1,3)

(0,0)

L

p 1-p

72

F. Perfect Bayesian Equilibrium

Hence 1 should play R.

1

RM

22

a a b

(2,2) (0,1)

b

(3,1)

(1,3)

(0,0)

L

p 1-p

73

F. Perfect Bayesian Equilibrium

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.

1

RM

22

a a b

(2,2) (0,1)

b

(3,1)

(1,3)

(0,0)

L

p 1-p

74

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).

75

F. Perfect Bayesian Equilibrium

A SignallingGame

0

22

(0,0) (-1,1)(1,2) (0,0)

11

2 2

1/2 1/2

T TW W

a a a ab b b b

(1,2)(0,0) (0,0)(-1,1)

76

F. Perfect Bayesian Equilibrium

A SignallingGame

We must give player 2 two lots of beliefs (sometimes these

are called assessments).

0

22

(0,0) (-1,1)(1,2) (0,0)

11

2 2

1/2 1/2

T TW W

a a a ab b b b

(1,2)(0,0) (0,0)(-1,1)

1-pp 1-ee

77

F. Perfect Bayesian Equilibrium

A SignallingGame

2 plays a if: >1/3p . 2 plays a if: >2/3e .

0

22

(0,0) (-1,1)(1,2) (0,0)

11

2 2

1/2 1/2

T TW W

a a a ab b b b

(1,2)(0,0) (0,0)(-1,1)

1-pp 1-ee

78

aaa

F. Perfect Bayesian Equilibrium

A SignallingGame

Both types of 1 play T

0

22

(0,0) (-1,1)(1,2) (0,0)

11

2 2

1/2 1/2

T TW W

ab b b b

(1,2)(0,0) (0,0)(-1,1)

1-pp 1-ee

79

aaa

F. Perfect Bayesian Equilibrium

A SignallingGame

Both types of 1 play TSo rational expectations says =1/2p ,but e is arbitrary – must have >2/3e .

0

22

(0,0) (-1,1)(1,2) (0,0)

11

2 2

1/2 1/2

T TW W

ab b b b

(1,2)(0,0) (0,0)(-1,1)

1-pp 1-ee

80

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.