Pondering more Problems
Enriching the Alice-Bob story
Bob
Go to Movies
Bob
Go shoot pool
Alice
Go to A
Go to B
Alice
Go to A
2
3
Alice
Go to B
0
0
Go to A
1
1
Go to A
Go to B
Go to B
3
2
2.5
1
Alice knows that Bob is shooting pool before she
decides which movie to go to.
2.5
0
How many strategies are possible for
Alice?
A)
B)
C)
D)
E)
1
2
3
4
8
How many strategies are possible for
Bob?
A)
B)
C)
D)
E)
1
2
3
4
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
First find the subgames.
1) Starts at node where Alice knows that Bob is shooting pool.
2) 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.
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 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.
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) 7
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) 7
First step: Look at proper subgames
What do we know so far?
A strategy for player 1 takes the form
w/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.
The circled subgame in strategic form
Player 2
Player 1
c1
d1
c2
d2
2,3
2,1
1,5
4,2
How many pure strategy Nash equilibria does this game have?
A) 0
B) 1 C) 2
D) 3
E) 4
Mixed strategy N.E?
Player 2
Player 1
c1
d1
c2
d2
2,3
2,1
1,5
4,2
Does this game have any mixed strategy Nash Equilibria?
A) Yes B) No
Hint: Note that there is a strictly dominant strategy for Player 2
If Player 2 uses that strategy, would is a mixed strategy a
best response for Player1?
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 ?/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
Player 2
a2
b2
a1
4,2
1,2
b1
2,1
3,2
Player 1
How many pure strategy Nash equilibria does this game have?
A) 0 B) 1 C) 2 D) 3 E) 4
Pure strategy N.E
• Two Nash equilibrium profiles, a1 for Player 1, a2 for Player 2
• b1 for Player 1, b2 for 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
?/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.
And
Player 1’s strategy is b1/d1/c1/d1 and Player 2’s is b2/d2.
Are we there yet?
Player 2
Player 1
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 mix
a2 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 2
Player 1
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=3p1
• the expected payoff to Player 1 from playing b1 is 2p+3(1p)=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.
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.
And then:
Equilibrium strategies: Player 1, a1,d1,d1,d1,d1,d1
Player 2, b2,b2
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
Player 1
Player 2
c1
d1
a2
3,4
1,4
b2
4,2
2,5
Nash equilibrium profiles in this subgame are
1) Player 2 plays a2, Player 1 plays c1
2) Player 2 plays b2, Player 1 plays d1
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?
Problem 8, Chapter 9
Trimming the game tree
Subgame between Players 2 and 3
a2
b2
a3 b3
2,1 2,0
0,2 3,4
Two Nash equilibria, at a2,a3 and b2,b3
Finishing the job
The case of a2,a3 equilibrium in subgame between 2 and 3
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
The case of b2,b3 in subgame between 2 and 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
1/3
2/3
1/3
a2
b2
a3
2,1
0,2
2/3
b3
2,0
3,4
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
Happy weekend…and keep those
decision trees trimmed