Extensive Form Games: Perfect Information

ECON2112

Extensive Form Games: Perfect InformationIntroduction

◮ In a normal form game, players take decisions simultaneously.

◮ In an extensive form game, players may observe other players’past decisions.

Extensive Form Games: Perfect InformationIntroduction

◮ A game of perfect information is an extensive form game whereplayers can always observe other players’ past actions

Extensive Form Games: Perfect InformationPerfect Information

Example (Perfect Information)

R

1,3

L

1

b

0,0

a

2,1

2

◮ The open circle is the initial node or root of the game tree.

◮ The black circles and the open circle are called decision nodes.

◮ L, R, a and b are called choices.

◮ 1 and 2 are the names of the players.

◮ We specify a payoff vector in each final node.

Extensive Form Games: Perfect InformationPerfect Information

Example (Perfect Information)

R

1,3

L

1

b

0,0

a

2,1

2

◮ The open circle is the initial node or root of the game tree.

◮ The black circles and the open circle are called decision nodes.

◮ L, R, a and b are called choices.

◮ 1 and 2 are the names of the players.

◮ We specify a payoff vector in each final node.

Extensive Form Games: Perfect InformationPerfect Information

Example (Perfect Information)

R

1,3

L

1

b

0,0

a

2,1

2

◮ The open circle is the initial node or root of the game tree.

◮ The black circles and the open circle are called decision nodes.

◮ L, R, a and b are called choices.

◮ 1 and 2 are the names of the players.

◮ We specify a payoff vector in each final node.

Extensive Form Games: Perfect InformationPerfect Information

Example (Perfect Information)

R

1,3

L

1

b

0,0

a

2,1

2

◮ The open circle is the initial node or root of the game tree.

◮ The black circles and the open circle are called decision nodes.

◮ L, R, a and b are called choices.

◮ 1 and 2 are the names of the players.

◮ We specify a payoff vector in each final node.

Extensive Form Games: Perfect InformationPerfect Information

Example (Perfect Information)

R

1,3

L

1

b

0,0

a

2,1

2

◮ The open circle is the initial node or root of the game tree.

◮ The black circles and the open circle are called decision nodes.

◮ L, R, a and b are called choices.

◮ 1 and 2 are the names of the players.

◮ We specify a payoff vector in each final node.

Extensive Form Games: Perfect InformationPerfect Information

Example (Perfect Information)

RL

1

b

0,0

a

2,1

2

d

1,3

c

4,0

2

◮ L, R, a, b, c, d are called choices (or actions).

◮ What is player 1’s set of pure strategies? S1 = {L,R}

◮ What is player 2’s set of pure strategies? S2 = {ac,ad ,bc,bd}

Extensive Form Games: Perfect InformationPerfect Information

Example (Perfect Information)

RL

1

b

0,0

a

2,1

2

d

1,3

c

4,0

2

◮ L, R, a, b, c, d are called choices (or actions).

◮ What is player 1’s set of pure strategies? S1 = {L,R}

◮ What is player 2’s set of pure strategies? S2 = {ac,ad ,bc,bd}

Extensive Form Games: Perfect InformationPerfect Information

Example (Perfect Information)

RL

1

b

0,0

a

2,1

2

d

1,3

c

4,0

2

◮ L, R, a, b, c, d are called choices (or actions).

◮ What is player 1’s set of pure strategies? S1 = {L,R}

◮ What is player 2’s set of pure strategies? S2 = {ac,ad ,bc,bd}

Extensive Form Games: Perfect InformationPerfect Information

Example (Perfect Information)

RL

1

b

0,0

a

2,1

2

d

1,3

c

4,0

2

◮ L, R, a, b, c, d are called choices (or actions).

◮ What is player 1’s set of pure strategies? S1 = {L,R}

◮ What is player 2’s set of pure strategies? S2 = {ac,ad ,bc,bd}

Extensive Form Games: Perfect InformationPerfect Information

Example (Perfect Information)

RL

1

b

0,0

a

2,1

2

d

1,3

c

2

B

1,4

A

4,0

1

◮ Player 1 set of pure strategies: S1 = {LA,LB,RA,RB}

◮ Player 2 set of pure strategies: S2 = {ac,ad ,bc,bd}.

Extensive Form Games: Perfect InformationPerfect Information

Example (Perfect Information)

RL

1

b

0,0

a

2,1

2

d

1,3

c

2

B

1,4

A

4,0

1

◮ Player 1 set of pure strategies: S1 = {LA,LB,RA,RB}

◮ Player 2 set of pure strategies: S2 = {ac,ad ,bc,bd}.

Extensive Form Games: Perfect InformationPerfect Information

Example (Perfect Information)

RL

1

b

0,0

a

2,1

212

0,3

12

N

d

2,3

c

4,0

2

◮ We can introduce moves of Nature.

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

◮ Player 2 set of pure strategies: S2 = {ac,ad ,bc,bd}.

Extensive Form Games: Perfect InformationPerfect Information

Example (Perfect Information)

RL

1

b

0,0

a

2,1

212

0,3

12

N

d

2,3

c

4,0

2

◮ We can introduce moves of Nature.

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

◮ Player 2 set of pure strategies: S2 = {ac,ad ,bc,bd}.

Extensive Form Games: Perfect InformationPure and Mixed Strategies

What is a pure strategy in an Extensive Form Game?

In a game of perfect information, a pure strategy is a plan of action forthe entire game, that tells the player what particular choice to take atevery possible juncture of the game where it is his turn to move.

What is the set of Pure Strategies?

The set of all possible plans of action.

What is a Mixed Strategy?

A probability distribution on the set of pure strategies.

Extensive Form Games: Perfect InformationPure and Mixed Strategies

What is a pure strategy in an Extensive Form Game?

In a game of perfect information, a pure strategy is a plan of action forthe entire game, that tells the player what particular choice to take atevery possible juncture of the game where it is his turn to move.

What is the set of Pure Strategies?

The set of all possible plans of action.

What is a Mixed Strategy?

A probability distribution on the set of pure strategies.

Extensive Form Games: Perfect InformationPure and Mixed Strategies

What is a pure strategy in an Extensive Form Game?

In a game of perfect information, a pure strategy is a plan of action forthe entire game, that tells the player what particular choice to take atevery possible juncture of the game where it is his turn to move.

What is the set of Pure Strategies?

The set of all possible plans of action.

What is a Mixed Strategy?

A probability distribution on the set of pure strategies.

Extensive Form Games: Perfect InformationNormal Form Representation

◮ Once we know what is the set of pure strategies for each player,we can obtain the normal form representation of a an extensiveform game

Example

R

1,3

L

1

b

0,0

a

2,1

2

a bL 2,1 0,0R 1,3 1,3

Extensive Form Games: Perfect InformationNormal Form Representation

◮ Once we know what is the set of pure strategies for each player,we can obtain the normal form representation of a an extensiveform game

Example

R

1,3

L

1

b

0,0

a

2,1

2

a bL 2,1 0,0R 1,3 1,3

Extensive Form Games: Perfect InformationNormal Form Representation

◮ Once we know what is the set of pure strategies for each player,we can obtain the normal form representation of a an extensiveform game

Example

R

1,3

L

1

b

0,0

a

2,1

2

a bL 2,1 0,0R 1,3 1,3

Extensive Form Games: Perfect InformationNormal Form Representation

◮ Once we know what is the set of pure strategies for each player,we can obtain the normal form representation of a an extensiveform game

Example

R

1,3

L

1

b

0,0

a

2,1

2

a bL 2,1 0,0R 1,3 1,3

Extensive Form Games: Perfect InformationNormal Form Representation

◮ Once we know what is the set of pure strategies for each player,we can obtain the normal form representation of a an extensiveform game

Example

R

1,3

L

1

b

0,0

a

2,1

2

a bL 2,1 0,0R 1,3 1,3

Extensive Form Games: Perfect InformationNormal Form Representation

Example

RL

1

b

0,0

a

2,1

2

d

1,3

c

4,0

2

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

Extensive Form Games: Perfect InformationNormal Form Representation

Example

RL

1

b

0,0

a

2,1

2

d

1,3

c

4,0

2

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

Extensive Form Games: Perfect InformationNormal Form Representation

Example

RL

1

b

0,0

a

2,1

2

d

1,3

c

4,0

2

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

Extensive Form Games: Perfect InformationNormal Form Representation

Example

RL

1

b

0,0

a

2,1

2

d

1,3

c

4,0

2

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

Extensive Form Games: Perfect InformationNormal Form Representation

Example

RL

1

b

0,0

a

2,1

2

d

1,3

c

4,0

2

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

Extensive Form Games: Perfect InformationNormal Form Representation

Example

RL

1

b

0,0

a

2,1

2

d

1,3

c

4,0

2

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

Extensive Form Games: Perfect InformationNormal From Representation

Example

RL

1

b

0,0

a

2,1

2

d

1,3

c

2

B

1,4

A

4,0

1

ac ad bc bdLA 2,1 2,1 0,0 0,0LB 2,1 2,1 0,0 0,0RA 4,0 1,3 4,0 1,3RB 1,4 1,3 1,4 1,3

Extensive Form Games: Perfect InformationNormal From Representation

Example

RL

1

b

0,0

a

2,1

2

d

1,3

c

2

B

1,4

A

4,0

1

ac ad bc bdLA 2,1 2,1 0,0 0,0LB 2,1 2,1 0,0 0,0RA 4,0 1,3 4,0 1,3RB 1,4 1,3 1,4 1,3

Extensive Form Games: Perfect InformationNormal From Representation

Example

RL

1

b

0,0

a

2,1

2

d

1,3

c

2

B

1,4

A

4,0

1

ac ad bc bdLA 2,1 2,1 0,0 0,0LB 2,1 2,1 0,0 0,0RA 4,0 1,3 4,0 1,3RB 1,4 1,3 1,4 1,3

Extensive Form Games: Perfect InformationNormal From Representation

Example

RL

1

b

0,0

a

2,1

2

d

1,3

c

2

B

1,4

A

4,0

1

ac ad bc bdLA 2,1 2,1 0,0 0,0LB 2,1 2,1 0,0 0,0RA 4,0 1,3 4,0 1,3RB 1,4 1,3 1,4 1,3

Extensive Form Games: Perfect InformationNormal From Representation

Example

RL

1

b

0,0

a

2,1

2

d

1,3

c

2

B

1,4

A

4,0

1

ac ad bc bdLA 2,1 2,1 0,0 0,0LB 2,1 2,1 0,0 0,0RA 4,0 1,3 4,0 1,3RB 1,4 1,3 1,4 1,3

Extensive Form Games: Perfect InformationNormal From Representation

Example

RL

1

b

0,0

a

2,1

2

d

1,3

c

2

B

1,4

A

4,0

1

ac ad bc bdLA 2,1 2,1 0,0 0,0LB 2,1 2,1 0,0 0,0RA 4,0 1,3 4,0 1,3RB 1,4 1,3 1,4 1,3

Extensive Form Games: Perfect InformationNormal From Representation

Example

RL

1

b

0,0

a

2,1

2

d

1,3

c

2

B

1,4

A

4,0

1

ac ad bc bdLA 2,1 2,1 0,0 0,0LB 2,1 2,1 0,0 0,0RA 4,0 1,3 4,0 1,3RB 1,4 1,3 1,4 1,3

Extensive Form Games: Perfect InformationPerfect Information

Example

RL

1

b

0,0

a

2,1

212

0,3

12

N

d

2,3

c

4,0

2

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

2 1,3 2, 32 1,3

Extensive Form Games: Perfect InformationPerfect Information

Example

RL

1

b

0,0

a

2,1

212

0,3

12

N

d

2,3

c

4,0

2

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

2 1,3 2, 32 1,3

Extensive Form Games: Perfect InformationPerfect Information

Example

RL

1

b

0,0

a

2,1

212

0,3

12

N

d

2,3

c

4,0

2

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

2 1,3 2, 32 1,3

Extensive Form Games: Perfect InformationPerfect Information

Example

RL

1

b

0,0

a

2,1

212

0,3

12

N

d

2,3

c

4,0

2

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

2 1,3 2, 32 1,3

Extensive Form Games: Perfect InformationPerfect Information

Example

RL

1

b

0,0

a

2,1

212

0,3

12

N

d

2,3

c

4,0

2

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

2 1,3 2, 32 1,3

Extensive Form Games: Perfect InformationPerfect Information

Example

RL

1

b

0,0

a

2,1

212

0,3

12

N

d

2,3

c

4,0

2

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

2 1,3 2, 32 1,3

Extensive Form Games: Perfect InformationNormal Form Representation

The normal form representation of an extensive form game isuseful to

◮ Compute Nash equilibria

◮ Find Dominated Strategies.

Extensive Form Games: Perfect InformationNas Equilibrium

Definition (Nash Equilibrium)The Nash equilibrium of a perfect information game (or, moregenerally, of an extensive form game) is the Nash equilibrium of itsNormal form representation.

Extensive Form Games: Perfect InformationNash Equilibrium

Typically, not every Nash equilibrium of a perfect information game isreasonable.

Example

R

1,3

L

1

b

0,0

a

2,1

2 a bL 2,1 0,0R 1,3 1,3

◮ The game has two Nash equilibria (L,a) and (R,b).

◮ Under (R,b), player 2 is threating player 1 with playing b if heplays L.

◮ But, is this a credible threat?

Extensive Form Games: Perfect InformationNash Equilibrium

Typically, not every Nash equilibrium of a perfect information game isreasonable.

Example

R

1,3

L

1

b

0,0

a

2,1

2 a bL 2,1 0,0R 1,3 1,3

◮ The game has two Nash equilibria (L,a) and (R,b).

◮ Under (R,b), player 2 is threating player 1 with playing b if heplays L.

◮ But, is this a credible threat?

Extensive Form Games: Perfect InformationNash Equilibrium

Typically, not every Nash equilibrium of a perfect information game isreasonable.

Example

R

1,3

L

1

b

0,0

a

2,1

2 a bL 2,1 0,0R 1,3 1,3

◮ The game has two Nash equilibria (L,a) and (R,b).

◮ Under (R,b), player 2 is threating player 1 with playing b if heplays L.

◮ But, is this a credible threat?

Extensive Form Games: Perfect InformationNash Equilibrium

Example

R

1,3

L

1

b

0,0

a

2,1

2 a bL 2,1 0,0R 1,3 1,3

◮ If player 2 finds himself in a situation where he has to choosebetween a and b, he will always choose a.

◮ Player 1 should foresee this, and play L instead of R.

◮ (In any case, note that b is a dominated strategy.)

Extensive Form Games: Perfect InformationBackwards Induction

◮ This idea can be generalized as the Principle of BackwardsInduction.

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

◮ The implementation of the Backwards Induction principle togames of perfect information involves a computational method.

Extensive Form Games: Perfect InformationBackwards Induction. Computation

Backwards Induction. Computation

◮ First we resolve the last moves in the game.

◮ Then we substitute decision nodes by the expected payoff thatarises from a rational move at such decision node.

◮ And we keep resolving last moves in the same way until we get tothe root of the game tree.

Extensive Form Games: Perfect InformationRollback equilibrium

Example

RL

1

b

0,0

a

2,1

2

d

1,3

c

2

B

1,4

A

4,0

1

◮ The optimal moves of the Rollback equilibrium are A,a,d and L.

◮ This corresponds to the pure strategy profile (LA,ad).

◮ However, we can also give the result as the behavioral strategyprofile (A,a,d ,L).

Extensive Form Games: Perfect InformationRollback equilibrium

Example

RL

1

b

0,0

a

2,1

2

d

1,3

c

4,0

2

◮ The optimal moves of the Rollback equilibrium are A,a,d and L.

◮ This corresponds to the pure strategy profile (LA,ad).

◮ However, we can also give the result as the behavioral strategyprofile (A,a,d ,L).

Extensive Form Games: Perfect InformationRollback equilibrium

Example

RL

2,1

1

d

1,3

c

4,0

2

◮ The optimal moves of the Rollback equilibrium are A,a,d and L.

◮ This corresponds to the pure strategy profile (LA,ad).

◮ However, we can also give the result as the behavioral strategyprofile (A,a,d ,L).

Extensive Form Games: Perfect InformationRollback equilibrium

Example

R

1,3

L

2,1

1

◮ The optimal moves of the Rollback equilibrium are A,a,d and L.

◮ This corresponds to the pure strategy profile (LA,ad).

◮ However, we can also give the result as the behavioral strategyprofile (A,a,d ,L).

Extensive Form Games: Perfect InformationRollback equilibrium

Example

R

1,3

L

2,1

1

◮ The optimal moves of the Rollback equilibrium are A,a,d and L.

◮ This corresponds to the pure strategy profile (LA,ad).

◮ However, we can also give the result as the behavioral strategyprofile (A,a,d ,L).

Extensive Form Games: Perfect InformationRollback equilibrium

Example

R

1,3

L

2,1

1

◮ The optimal moves of the Rollback equilibrium are A,a,d and L.

◮ This corresponds to the pure strategy profile (LA,ad).

◮ However, we can also give the result as the behavioral strategyprofile (A,a,d ,L).

Extensive Form Games: Perfect InformationRollback equilibrium

Example

R

1,3

L

2,1

1

◮ The optimal moves of the Rollback equilibrium are A,a,d and L.

◮ This corresponds to the pure strategy profile (LA,ad).

◮ However, we can also give the result as the behavioral strategyprofile (A,a,d ,L).

Extensive Form Games: Perfect InformationRollback equilibrium

Example

RL

1

b

0,0

a

2,1

212

0,3

12

N

d

2,3

c

4,0

2

◮ Moves that conform with Backwards Induction: d ,a,L.

◮ Expressed as a pure strategy profile: (L,ad).

◮ Expressed as a behavioral strategy profile: (L,a,d) .

Extensive Form Games: Perfect InformationRollback equilibrium

Example

RL

1

b

0,0

a

2,1

212

0,3

12

2,3

N

◮ Moves that conform with Backwards Induction: d ,a,L.

◮ Expressed as a pure strategy profile: (L,ad).

◮ Expressed as a behavioral strategy profile: (L,a,d) .

Extensive Form Games: Perfect InformationRollback equilibrium

Example

RL

2,1

1

12

0,3

12

2,3

N

◮ Moves that conform with Backwards Induction: d ,a,L.

◮ Expressed as a pure strategy profile: (L,ad).

◮ Expressed as a behavioral strategy profile: (L,a,d) .

Extensive Form Games: Perfect InformationRollback equilibrium

Example

R

1,3

L

2,1

1

◮ Moves that conform with Backwards Induction: d ,a,L.

◮ Expressed as a pure strategy profile: (L,ad).

◮ Expressed as a behavioral strategy profile: (L,a,d) .

Extensive Form Games: Perfect InformationRollback equilibrium

Example

R

1,3

L

2,1

1

◮ Moves that conform with Backwards Induction: d ,a,L.

◮ Expressed as a pure strategy profile: (L,ad).

◮ Expressed as a behavioral strategy profile: (L,a,d) .

Extensive Form Games: Perfect InformationRollback equilibrium

Example

R

1,3

L

2,1

1

◮ Moves that conform with Backwards Induction: d ,a,L.

◮ Expressed as a pure strategy profile: (L,ad).

◮ Expressed as a behavioral strategy profile: (L,a,d) .

Extensive Form Games: Perfect InformationRollback equilibrium

Example

R

1,3

L

2,1

1

◮ Moves that conform with Backwards Induction: d ,a,L.

◮ Expressed as a pure strategy profile: (L,ad).

◮ Expressed as a behavioral strategy profile: (L,a,d) .

Extensive Form Games: Perfect InformationBehavioral Strategies

Definition (Behavioral Strategies)

In a game of perfect information, a player’s behavioral strategyspecifies at every decision node where it is his turn to move aprobability distribution over the choices that follow.

Definition (Behavioral Strategy Profile)

In a game of perfect information, a behavioral strategy profile specifiesone behavioral strategy for each player.

Extensive Form Games: Perfect InformationBehavioral Strategies

◮ Compare:

Definition (Pure strategy)

In a game of perfect information, a pure strategy is a plan of action forthe entire game, that tells the player what particular choice to take atevery possible juncture of the game where it is his turn to move.

Definition (Mixed Strategy)

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

Definition (Behavioral Strategies)

In a game of perfect information, a player’s behavioral strategyspecifies at every decision node where it is his turn to move aprobability distribution over the choices that follow.

Extensive Form Games: Perfect InformationBehavioral Strategies

Example

RL

1

b

0,0

a

2,1

2

d

1,3

c

2

B

1,4

A

4,0

1

ac ad bc bdLA 2,1 2,1 0,0 0,0LB 2,1 2,1 0,0 0,0RA 4,0 1,3 4,0 1,3RB 1,4 1,3 1,4 1,3

◮ (LA,ac) corresponds to (L,a,A,c).

◮ ( 12 LA+ 1

2 LB, 13 ac+ 2

3 ad) c.t. (L, 12 A+ 1

2 B,a, 13 c+ 2

3 d).

Extensive Form Games: Perfect InformationBehavioral Strategies

Theorem (Kuhn)(In any extensive form game with perfect recall,) every probabilitydistributions on ending nodes induced by a mixed strategy profile canalso be induced by some behavioral strategy profile.

◮ The consequence is that, although mixed strategies may lookmore general, there is no loss of generality in working only withbehavioral strategies.