Enriching the AliceBob 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 25 1 Go shoot pool ID: 571961
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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
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 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. 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
Does this game have any mixed strategy Nash equilbria? A) Yes B) No Slide16
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 ?/d2Slide17
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?Slide18
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 1Player 2Slide19
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.Slide20
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 2Slide21
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 2Slide22
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.Slide23
Problem
7,
Chapter
9
How many strategies are possible for Player 1?
A) 10 B) 12 C) 16 D) 32 E) 64Slide24
How many proper
subgames
does this game have?
A) 2 B) 4 C) 5 D) 6 E) 7Slide25
Sawing branches from
subtrees
: Easiest casesSlide26
Sawing branches from
subtrees
: Using dominanceSlide27
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.Slide28
Suppose 1 chooses c1 at lower left node.
What would 2 do then at 2’s left node?Slide29
Now what?Slide30
One SPNE
Player 1’s strategy is a1, c1, d1, d1,d1. Player 2’s strategy is a2,b2Slide31
If 1 takes the other road at lower left node.Slide32
Equilibrium strategies: Player 1, a1,d1,d1,d1,d1,d1
Player 2, b2,b2
And then: Slide33
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.
Slide34
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 d1Slide35
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?Slide36
Problem 10, page 287Slide37
Some tree-trimming
a3
b3
a2
2,1
2,0
b2
0,2
3,4
Subgame
between 2 and 3Slide38
a3
b3
a2
2,1
2,0
b2
0,2
3,4
How many pure strategy Nash equilibrium does
this subgame have?
A) 0 B) 1 C) 2 D) 3 Slide39
Truncated
game with a2, a3 the Nash equilibrium in subgame between 2 and 3
2
4
2
3
3
2
1
3
Player 1
a1
b1Slide40
Truncated
game if b2, b3 in subgame between 2 and 3
2
4
2
3
0
1
1
1
Player 1
a1
b1Slide41
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
3,2,1,3Slide42
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 2,4,2,3Slide43
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=2.66Slide44
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/a4