Pondering more Problems - PowerPoint Presentation

Slide1

Pondering more Problems

Slide2

Enriching the Alice-Bob story

Go to A

Go to B

Go to A

Alice

Alice

Go to B

Go to A

Go to B

2

3

0

0

1

1

3

2

2.5 1

Go shoot pool

Alice knows that Bob is shooting pool before she

decides which movie to go to.

Bob

Bob

Go to Movies

Alice

Go to A

Go to B

2.5

0Slide3

How many strategies are possible for Alice?

1

2

3

4

8Slide4

How many strategies are possible for Bob?

1

2

3

4

8Slide5

The strategies

Alices’s strategies take the form x/y where x is the movie she chooses if she knows Bob is going to the movies and y is the movie she chooses if she knows he is shooting pool.

Bob’s strategies take the form x/y where x is Movie or Pool and y is Movie A or Movie B.Slide6

Subgame perfect strategies

First find the subgames

.

Starts at node where Alice knows that Bob is shooting pool.

Starts at node where Bob has decided to go to movies.

Subgame

perfection requires that Alice goes to movie A if Bob shoots pool.

The subgame where Bob goes to the movies has 2 pure strategy Nash equilibria. Alice goes to A and Bob goes to A. Alice goes to B and Bob goes to B.

This subgame also has one mixed strategy Nash equilibrium.

Slide7

Pure strategy SPNE

These are Alice goes to Movie B if Bob goes to the Movies and she goes to Movie A if he shoots pool. Bob goes to the movies and chooses Movie B.

Alice goes to Movie A if Bob goes to the Movies and to Movie A if Bob shoots pool. Bob chooses pool and if he went to the movies he would go to A.Slide8

What about mixed strategy equilibrium if Bob goes to movies.

In subgame where Bob goes to movies, if Alice goes to A with probability p, Bob will be willing to mix his strategies if his expected payoff is the same for each movie.

For Bob

Payoff from Movie A is 2p+0(1-p)=2p.

Payoff from Movie B is p+3(1-p)=3-2p

He will mix if p=3/4. Slide9

Payoff in mixed strategy equilibrium

In mixed strategy Nash equilibrium, Alice goes to A with probability ¾ and Bob goes to B with probability ¾. Expected payoff to Bob from either action is

3/4x 2=3/2.

The mixed strategy

subgame

equilbrium for the movies gives him a lower payoff than shooting pool.Slide10

Mixed strategy SPNE

So this game has one more SPNE. This one in which Bob goes to shoot pool. If Alice were to find out that Bob is going to the movies, she would go to Movie A with probability ¾ and Bob would go to Movie B with probability ¾. If Alice finds out that Bob is going to shoot pool, she would go to Movie A for sure. Slide11

Chapter 9, Problem 16

Let’s find the SPNE(s)

How

many proper

subgames

does this game have?

A) 0 B) 2 C) 3 D) 4 E) 7Slide12

Chapter 9, Problem 16

Let’s find the SPNE(s)

How many information sets does Player 1 have?

A) 0 B) 2 C) 3 D) 4 E) 7Slide13

First step: Look at proper

subgames

Slide14

What do we know so far?

A strategy for player 1 takes the formw/x/y/z where entries specify what is done in each of player 1’s information sets (reading from top down and left to right)

From what we know so far, in a SPNE Nash equilibrium, Player 1’s strategy must be of the

Form ?/?/c1/d1

Let’s look into the second question mark.Slide15

The circled

subgame in strategic form

c2

d2

c1

2,3

1,5

d1

2,1

4,2

Player 2

Player 1How many pure strategy Nash equilibria

does this game have?A) 0 B) 1 C) 2 D) 3 E) 4 Slide16

Mixed strategy N.E?

c2

d2

c1

2,3

1,5

d1

2,1

4,2

Player 2

Player 1

Does this game have any mixed strategy Nash Equilibria?

Yes B) NoHint: Note that there is a strictly dominant strategy for Player 2If Player 2 uses that strategy, would is a mixed strategy a best response for Player1?Slide17

What do we know now?

We already knew that in a SPNE Nash equilibrium, Player 1’s strategy must be of the

form ?/?/c1/d1

Since the only Nash equilibrium for the circled

subgame

is Player 1 plays d1, Player 2 plays d2, we now know that in SPNE, Player 1’s strategy is of the form ?/d1/c1/d1 and Player 2’s strategy is of the form ?/d2Slide18

A simplified game

Given the restrictions of

subgame

perfection, we have only

two remaining question marks. What does Player 1 do at his

First node? What does Player 2 do at her first node?Slide19

Strategic form of simplified game

a2

b2

a1

4,2

1,2

b1

2,1

3,2

How many pure strategy Nash equilibria does this game have?

A) 0 B) 1 C) 2 D) 3 E) 4

Player 1Player 2Slide20

Pure strategy N.E

Two Nash equilibrium profiles, a1 for Player 1, a2 for Player 2

b1 for Player 1, b2 for Player 2.Slide21

And now what do we know?

Earlier we figured out that in SPNE, Player 1’s strategy is of the form ?/d1/c1/d1 and Player 2’s strategy is of the form ?/d2

Now we see that there are two pure strategy Nash

equilibria

, corresponding to filling in the ?’s with the Nash

equilibria that we just found. So these equilibria

are Player 1’s strategy is a1/d1/c1/d1 and Player 2’s is a2/d2.AndPlayer 1’s strategy is b1/d1/c1/d1 and Player 2’s is b2/d2.Slide22

Are we there yet?

a2

b2

a1

4,2

1,2

b1

2,1

3,2

Not quite. We need to see whether the reduced game that we studied has any mixed strategy Nash

equilibria. Note that b2 weakly dominates a2. Player 2 would be willing to mixa2 and b2 only if she is sure that Player 1 plays a1.

Suppose that Player 2 plays a2 with probability p. When would player 1 be sure to play a1?

Player 1Player 2Slide23

a2

b2

a1

4,2

1,2

b1

2,1

3,2

If player 2 plays a2 with probability p, then:

the expected payoff to Player 1 from playing a1 is 4p+1-p=3p-1

the expected payoff to Player 1 from playing b1 is 2p+3(1-p)=3+p.Playing a1 for sure is a best response for Player 1 if 3p-1≥3+p, which implies that p≥1/2.

So this game has many mixed strategy equilibria. In all of them,Player 2 plays a2 with probability p≥1/2 and Player 1 plays a1 for sure. In these equilibria

, payoff to Player 1 is 3p-1 and payoff to Player 2 is 2.

Player 1

Player 2Slide24

Finishing it up

Recall that we found two pure strategy Nash

equilibria

, corresponding to filling in the first ?’s

These

equilibria were Player 1’s strategy is a1/d1/c1/d1 and Player 2’s is a2/d2.

Player 1’s strategy is b1/d1/c1/d1 and Player 2’s is b2/d2.Now we also have a bunch of Nash equilibria

in which Player 1 plays a1/d1/c1/d1 and Player 2 plays a strategy x/b2 where her first move strategy x is to play a1 with probability p≥1/2 and b1 with probability 1-p.Slide25

Problem

7,

Chapter

9

How many strategies are possible for Player 1?

A) 10 B) 12 C) 16 D) 32 E) 64Slide26

How many proper

subgames

does this game have?

A) 2 B) 4 C) 5 D) 6 E) 7Slide27

Sawing branches from

subtrees

: Easiest casesSlide28

Sawing branches from

subtrees

: Using dominanceSlide29

Handling ties

In the subgame on the lower left, there are two Nash equilibrium plays for Player 1.

One is c1. The other is d1.

Let’s explore each possibility.Slide30

Suppose 1 chooses c1 at lower left node.

What would 2 do then at 2’s left node?Slide31

Now what?Slide32

One SPNE

Player 1’s strategy is a1, c1, d1, d1,d1. Player 2’s strategy is a2,b2Slide33

If 1 takes the other road at lower left node.Slide34

Equilibrium strategies: Player 1, a1,d1,d1,d1,d1,d1

Player 2, b2,b2

And then: Slide35

Conclusions so far:

There are two pure strategy

Subgame

Perfect Nash

Equilibria

.In one of them,

Player 1’s strategy is a1, c1, d1, d1,d1. Player 2’s strategy is a2,b2. In this case, the course of play is 1 plays a1, 2 plays a2, and 3 plays c1. Payoffs are 4 for player 1 and 3 for player 2.

Another SPNE is a1/d1/d1/d1/d1 for Player 1 and b2/b2 for Player 2. Then the course of play is 1 plays a1, 2 plays b2 and 3 plays d1. Payoffs are 5 for player 1 and 2 for player 2.

Slide36

Another look:

Strategic form of subgame after 1 plays a1

c1

d1

a2

3,4

1,4

b2

4,2

2,5

Player 2

Player 1

Nash equilibrium profiles in this subgame are 1) Player 2 plays a2, Player 1 plays c12) Player 2 plays b2, Player 1 plays d1Slide37

Also some mixed strategies

At the node where 1 has gone a1 and 2 has gone a2, Player 1 is indifferent between c1 and d1, he could also use a mixed strategy.

If

prob

of c1 is p, the expected payoff to 2 from going a2 is

3p+1-p=1+2p and the expected payoff from going b2 is 2. What happens with p<1/2? p>1/2? p=1/2?Slide38

Problem

8, Chapter 9Slide39

Trimming the game treeSlide40

a3

b3

a2

2,1

2,0

b2

0,2

3,4

Subgame

between Players 2 and 3

Two Nash equilibria, at a2,a3 and b2,b3Slide41

Finishing the jobSlide42

The case of a2,a3 equilibrium in

subgame

between 2 and 3Slide43

One SPNE

Player 1 uses b1Player 2 uses b2/a2Player 3 uses a3

Player 4 uses a4/b4/b4/a4

Course of play is then 1 chooses b1, 2 chooses a2, 3 chooses a3, 4 chooses a4. Payoffs are

3,2,1,3Slide44

The case of b2,b3 in

subgame

between 2 and 3Slide45

Another SPNE

Player 1 chooses a1Player 2 chooses b2/b2Player 3 chooses b3 Player 4 chooses a4/b4/b4/a4

Course of play is now 1 chooses a1, 2 chooses b2. Payoffs 2,4,2,3Slide46

Mixed strategy eq

in 2,3 game

a3

b3

a2

2,1

2,0

b2

0,2

3,4

2/3

1/31/3

2/3

Payoff to Player 1 from this equilibrium:3(2/9)+4(4/9)+2(1/9)+0=2.66Slide47

A third SPNE

Players 1 plays b1Player 2 plays b2 if 1 plays a1 and plays a mixed strategy of a2 with probability 2/3 and b2 probability 1/3 if 1 plays b1. Player 3 plays a mixed strategy a2 with probability 1/3 and b2 with probability 2/3.

Player 4 plays a4/b4/b4/a4Slide48

Happy weekend

…and keep those decision trees trimmed