Pondering more Problems. Enriching the Alice-Bob story Go to AGo to B Go to A Alice Go to B Go to A...
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Transcript of Pondering more Problems. Enriching the Alice-Bob story Go to AGo to B Go to A Alice Go to B Go to A...
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
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
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.
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.
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?
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
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