Pondering more Problems

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

23 0

011

32

2.5 1

Go shoot pool

Alice knows that Bob is shooting pool before shedecides which movie to go to.

Bob

BobGo to Movies

Alice

Go to A Go to B

2.50

How many strategies are possible for Alice?

A) 1B) 2C) 3D) 4E) 8

How many strategies are possible for Bob?

A) 1B) 2C) 3D) 4E) 8

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.

Subgame perfect strategies

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.

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.

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.

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.

Mixed strategy SPNE

So this game has one more SPNE. This is a mixed strategy equilibrium 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.

Chapter 9, Problem 16Let’s find the SPNE(s)

How many proper subgames does this game have? A) 0 B) 2 C) 3 D) 4 E) 7

Chapter 9, Problem 16Let’s find the SPNE(s)

How many information sets does Player 1 have? A) 0 B) 2 C) 3 D) 4 E) 7

First step: Look at proper subgames

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 theForm ?/?/c1/d1Let’s look into the second question mark.

The circled subgame in strategic form

c2 d2

c1 2,3 1,5d1 2,1 4,2

Player 2

Player 1

How many pure strategy Nash equilibria does this game have?

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

Does this game have any mixed strategy Nash equilbria? A) Yes B) No

What do we know now?

We already knew that in a SPNE Nash equilibrium, Player 1’s strategy must be of theform ?/?/c1/d1Since 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 ?/d2

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?

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

Two Nash equilibrium profiles, a1 for Player 1, a2 for Player 2B1 for Player 1, B2 for Player 2.

Player 1

Player 2

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 ?/d2Now 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.

Are we there now?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 1 plays a2 with probability p. When would player 1 be sure to play a1?

Player 1

Player 2

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 2

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.

Problem 7, Chapter 9

How many strategies are possible for Player 1?A) 10 B) 12 C) 16 D) 32 E) 64

How many proper subgames does this game have?A) 2 B) 4 C) 5 D) 6 E) 7

Sawing branches from subtrees: Easiest cases

Sawing branches from subtrees: Using dominance

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.

Suppose 1 chooses c1 at lower left node. What would 2 do then at 2’s left node?

Now what?

One SPNE

Player 1’s strategy is a1, c1, d1, d1,d1. Player 2’s strategy is a2,b2

If 1 takes the other road at lower left node.

Equilibrium strategies: Player 1, a1,d1,d1,d1,d1,d1 Player 2, b2,b2

And then:

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.

Another look:Strategic form of subgame after 1 plays a1

c1 d1

a2 3,4 1,4

b2 4,2 2,5Player 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 d1

Also some mixed strategies

• At the node where 1 has gone a1 and 2 has gone 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?

Problem 10, page 287

Some tree-trimming

a3 b3a2 2,1 2,0

b2 0,2 3,4

Subgame between 2 and 3

a3 b3a2 2,1 2,0b2 0,2 3,4

How many pure strategy Nash equilibrium does this subgame have?

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

Truncated game with a2, a3 the Nash equilibrium in subgame between 2 and 3

2423

3213

Player 1

a1b1

Truncated game if b2, b3 in subgame between 2 and 3

2423

0111

Player 1

a1b1

One SPNE

• Player 1 uses b1• Player 2 uses b2/a2• Player 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

3,2,1,3

Another SPNE

• Player 1 chooses a1• Player 2 chooses b2/b2• Player 3 chooses b3 • Player 4 chooses a4/b4/b4/a4

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

Mixed strategy eq in 2,3 game

a3 b3a2 2,1 2,0b2 0,2 3,4

2/3

1/3

1/3 2/3

Payoff to Player 1 from this equilibrium:

3(2/9)+4(4/9)+2(1/9)+0=2.66

A third SPNE

• Players 1 plays b1• Player 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/a4