07.slides_b

Extensive Form Games

ECON2112

Extensive Form GamesReferences

◮ Perfect Information Games (Week 6)◮ Dixit & Skeath’s book: Chapter 3.◮ Osborne’s book: Chapters 5 and 7.

◮ Extensive Form (Week 7)◮ Dixit & Skeath’s book: Chapter 6.◮ Osborne’s book: Chapter 7 and chapter 10 , pp. 313–323.

Extensive Form GamesIntroduction

◮ A perfect information game is an extensive form game whereplayers always know all their opponents past moves.

◮ In an extensive form game, a player may not know all the pasthistory of moves when is his turn to move.


Example (A game of perfect information)

RL

1

b

0,0

a

2,1

2

d

1,3

c

1,3

2

◮ When player 2 has to move, he knows what has been the move ofplayer 1.


Example (An extensive form game without perfect information)

RL

1

b

0,0

a

2,1

b

1,3

a

1,3

2

◮ When player 2 has to move, he does not know what has been themove of player 1.

Extensive Form GamesDefinitions and Terminology

23

13

N

RL

b

3,1,0

a

2,1,2

ba

2

f

1,0,8

e

2,3,1

3

RL

1

d

3,4,1

c

1,2,3

d

3,6,1

c

5,3,4

d

1,2,0

c

2,4,1

3

◮ The game starts at the open circle and continues along thebranches according to Nature’s and players’ moves.

◮ N represents Nature.

◮ Black circles and the open circle (the root of the game tree) aredecision nodes.


23

13

N

RL

b

3,1,0

a

2,1,2

ba

2

f

1,0,8

e

2,3,1

3

RL

1

d

3,4,1

c

1,2,3

d

3,6,1

c

5,3,4

d

1,2,0

c

2,4,1

3

◮ At every ending node there is an assignment of payoffs to players.

◮ Dotted lines are called information sets. An information setcontains the decision nodes that the player whose turn it is tomove cannot tell appart.

◮ Information sets can be singletons.

Extensive Form GamesDefinitions and Terminology 2

313

N

RL

b

3,1,0

a

2,1,2

ba

2

f

1,0,8

e

2,3,1

3

RL

1

d

3,4,1

c

1,2,3

d

3,6,1

c

5,3,4

d

1,2,0

c

2,4,1

3

◮ When player 1 has to move he does not know what has beenNature’s move. But he knows the probabilities.

◮ When player 2 is called to move, he does not know whetherplayer 1 chose L or R.

◮ Player 3 knows in his singleton information which one has beenNature’s move, he also knows that player 1 chose R and thatplayer 2 chose a.


23

13

N

RL

b

3,1,0

a

2,1,2

ba

2

f

1,0,8

e

2,3,1

3

RL

1

d

3,4,1

c

1,2,3

d

3,6,1

c

5,3,4

d

1,2,0

c

2,4,1

3

◮ Information sets are labeled by the player who moves at thatinformation set.

◮ Players have the same set of moves available at every decisionnode inside the same information set.

◮ A pure strategy is a plan of action for the game that tells whichmove to take at every information set where the player moves.


23

13

N

RL

b

3,1,0

a

2,1,2

ba

2

f

1,0,8

e

2,3,1

3

RL

1

d

3,4,1

c

1,2,3

d

3,6,1

c

5,3,4

d

1,2,0

c

2,4,1

3

◮ Player 1 set of pure strategies: S1 = {L,R}.

◮ Player 2 set of pure strategies: S2 = {a,b}.

◮ Player 3 set of pure strategies: S3 = {ec,ed , fc, fd}.


23

13

N

RL

b

3,1,0

a

2,1,2

ba

2

f

1,0,8

e

2,3,1

3

RL

1

d

3,4,1

c

1,2,3

d

3,6,1

c

5,3,4

d

1,2,0

c

2,4,1

3

◮ There is a unique path from the root to each decision node. (Andfrom the root to each ending node.)

Extensive Form GamesInformation Sets, Perfect Recall and Subgames

Information Sets

◮ An information set contains all the information that a playerpossesses when he has to move.

◮ For each information set of a player, this player cannot distinguishbetween decision nodes contained in that information set.

◮ Every decision node inside the same information set must havethe same set of choices available.


Perfect RecallWe only consider extensive form games with perfect recall. Perfectrecall means that players remember all his previous moves andwhatever they knew before.

Example (Extensive form that does not satisfy perfect recall)

RL

1

ba

2

dc

2

fe

1

fe

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Subgames

A subgame is a part of an extensive form game that could beconsidered as a separate game.


Example

RL

1

BA

1

b

−2,0

a

2,1

b

−1,4

a

−2,0

2DC

1

d

0,0

c

4,1

d

1,4

c

0,0

2

◮ We say that this game has three subgames.

◮ The entire game is also considered as a subgame.


Example (A game without proper subgames)

RL

1

2

BAB

2,2

A

3,1

1

b

1,4

a

4,1

b

0,2

a

1,2

◮ This extensive form game has only one subgame, which is theentire game.

◮ We say that this game has no proper subgames.

Extensive Form GamesStrategies

◮ We can distinguish the following types of strategies

Definition (Pure Strategy)

In an extensive form game, a pure strategy is a plan of action for theentire game, that tells the player which particular choice to take atevery possible information set.

Definition (Mixed Strategy)

A mixed strategy is a probability distribution over the set of purestrategies.

Definition (Behavioral Strategies)

In an extensive form game, a player’s behavioral strategy specifies atevery information set where he has to move a probability distributionover the choices that follow.

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Extensive Form GamesStrategies

Example

RL

1

2

BAB

2,2

A

3,1

1

b

1,4

a

4,1

b

0,2

a

1,2

◮ La, Lb, Ra and Rb are pure strategies for player 1.

◮13 La+ 1

3 Lb+ 13 Rb is an example of a mixed strategy for player 1.

◮ ( 12 L+ 1

2 R,a) is an example of a behavioral strategy for player 1.

◮ ( 12 L+ 1

2 R,B,a) is an example of a behavioral strategy profile.

Extensive Form GamesNash Equilibrium

Definition (Nash Equilibrium)The Nash equilibrium of an extensive form game is the Nashequilibrium of its normal form representation.

Extensive Form GamesNash equilibrium

Example (Battle of the Sexes with Entry)

InOut

2,5

1

BT

1

R

0,0

L

4,1

R

1,4

L

0,0

2

L ROutT 2,5 2,5OutB 2,5 2,5

InT 4,1 0,0InB 0,0 1,4

◮ For simplicity, when computing Nash equilibria, we can collapsestrategies OutT and OutB into the strategy Out and solve thefollowing game:


ExampleL R

Out 2,5 2,5InT 4,1 0,0InB 0,0 1,4

◮ NE ={

(Out,βL+(1−β)R) : β ∈[

0, 12

]}

∪{(InT ,L)}

◮ However, we are interested in the Nash equilibria of the originalgame.


ExampleInOut

2,5

1

BT

1

R

0,0

L

4,1

R

1,4

L

0,0

2

L ROutT 2,5 2,5OutB 2,5 2,5

InT 4,1 0,0InB 0,0 1,4

◮ The set of Nash equilibria is:

{

(αOutT +(1−α)OutB,βL+(1−β)R) : α ∈ [0,1],β ∈[

0, 12

]}

∪{(InT ,L)}.

Extensive Form GamesBackwards Induction

ExampleInOut

2,5

1

BT

1

R

0,0

L

4,1

R

1,4

L

0,0

2

◮ Is the entire set of Nash equilibria

{


0, 12

]}

∪{(InT ,L)}.

reasonable?

Extensive Form GamesBackwards Induction

Definition (Backwards Induction/Informal Idea)Players should make their choices in a way consistent with deductionsabout other players’ rational behavior in the future.

◮ How can we apply the principle of backwards induction inextensive form games?

◮ To respond to that, we have to make precise what “rationalbehavior in the future” means.

◮ One possibility is to require a Nash equilibrium in every subgame.

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Extensive Form GamesSubgame Perfect Equilibrium

Definition (Subgame Perfect Equilibrium)

A strategy profile is a Subgame Perfect Equilibrium of an extensiveform game if it induces a Nash equilibrium in every subgame.

Subgame Perfect Equilibrium

◮ It is one possible implementation of idea of backwards inductionin extensive form games.

◮ It is equivalent to backwards induction in games of perfectinformation.

◮ It is computed by first resolving the last subgames in the game.

◮ Then we substitute the subgames by the expected payoff inducedby the Nash equilibria of the subgame.

◮ And we keep resolving last subgames in the same way until weget to the root of the game.

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ExampleInOut

2,5

1

BT

1

R

0,0

L

4,1

R

1,4

L

0,0

2

◮ The last subgame has three Nash equilibria, (T ,L), (B,R) and(

45 T + 1

5 B, 15 L+ 4

5 R)

.

◮ If players use (T ,L), we obtain the SPE (In,T ,L).

◮ If players use (B,R), we obtain the SPE (Out,B,R)

◮ If players use(

45 T + 1

5 B, 15 L+ 4

5 R)

, we obtain the SPE(

Out, 45 T + 1

5 B, 15 L+ 4

5 R)


ExampleIn

4,1

Out

2,5

1


45 T + 1

5 B, 15 L+ 4

5 R)

.



◮ If players use(

45 T + 1

5 B, 15 L+ 4

5 R)


Out, 45 T + 1

5 B, 15 L+ 4

5 R)


ExampleIn

1,4

Out

2,5

1


45 T + 1

5 B, 15 L+ 4

5 R)

.



◮ If players use(

45 T + 1

5 B, 15 L+ 4

5 R)


Out, 45 T + 1

5 B, 15 L+ 4

5 R)


ExampleIn

45 ,

45

Out

2,5

1


45 T + 1

5 B, 15 L+ 4

5 R)

.



◮ If players use(

45 T + 1

5 B, 15 L+ 4

5 R)


Out, 45 T + 1

5 B, 15 L+ 4

5 R)


◮ Remember that in game “Battle of the Sexes with entry” the set ofNash equilibria was

{


0, 12

]}

∪{(InT ,L)}.

◮ We have just seen that the set of subgame perfect equilibria is

{

(In,T ,L),(Out,B,R),(

Out, 45 T + 1

5 B, 15 L+ 4

5 R)}

◮ Which in terms of mixed strategies is:

{

(InT ,L),(OutB,R),(

45 OutT + 1

5 OutB, 15 L+ 4

5 R)}

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Example

RL

1

BA

1

b

−2,0

a

2,1

b

−1,4

a

−2,0

2DC

1

d

0,0

c

4,1

d

1,4

c

0,0

2

◮ The subgame in the left has three Nash equilibria

◮ The subgame in the right has three Nash equilibria

◮ Therefore, there are nine subgame perfect equilibria.


Example

R

c dC 4,1 0,0D 0,0 1,4

L

a bA 2,1 −2,0B −2,0 −1,4

1

◮ The subgame in the left has three Nash equilibria

◮ The subgame in the right has three Nash equilibria

◮ Therefore, there are nine subgame perfect equilibria.

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Sticky Note

The third being mixed?


Subgame Perfect Equilibrium. Properties.

◮ Every subgame perfect equilibrium is a Nash equilibrium.

◮ Not every Nash equilibrium is subgame perfect.

◮ However, subgame perfections has no bite if the extensive formgame does not have proper subgames.



RL

1

2

BAB

2,2

A

3,1

1

b

1,4

a

4,1

b

0,2

a

1,2

◮ If a game has no proper subgames, then every Nash equilibriumis subgame perfect.

◮ But, can we still use the idea of backwards induction to findequilibria that are not self-enforcing?



RL

1

2

BAB

2,2

A

3,1

1

b

1,4

a

4,1

b

0,2

a

1,2

◮ Player 1 will always choose a over b.

◮ Given this, player 2 will always choose B.

◮ Given this, player 1 will choose R

◮ Only (R,B,a) is reasonable. (It is also the only strategy profilethat survives iterated deletion of strictly dominated strategies.)

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Example

RL

1

ba

2,2,4

2

d

0,2,3

c

2

f

0,1,1

e

1,3,2

3

f

0,0,1

e

3,3,0

◮ There are no proper subgames.

◮ Every Nash Equilibrium is subgame perfect.

◮ Is the Nash equilibrium (L,bd ,e) reasonable?

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ExampleThe normal form representation is:

ac ad bc bdL 2,2,4 2,2,4 1,3,2 1,3,2R 3,3,0 0,2,3 3,3,0 0,2,3

e

ac ad bc bdL 2,2,4 2,2,4 0,1,1 0,1,1R 0,0,1 0,2,3 0,0,1 0,2,3

f◮ Note that no strategy is dominated.


Example

RL

1

ba

2,2,4

2

d

0,2,3

c

2

f

0,1,1

e

1,3,2

3

f

0,0,1

e

3,3,0

◮ Is the Nash equilibrium (L,bd ,e) reasonable?

◮ Can we use a backwards induction argument to discard(L,bd ,e)?

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