Monte Hall Problem
Problem 6, chapter 2
Let’s Draw a Game Tree…
Monte
Door 1
Door 2
Door 3
Contestant
Door 1
Door 2 Door 3 Door 1 Door 2 Door 3
Door 1 Door 2
Door 3
Door 3
Adding more branches
Monte
Door 1
Door 3
Door 2
Contestant
Door 1
M
2
3
Door 2 Door 3 Door 1 Door 2 Door 3
M M
M
M
M
3
2
3
1
3
1
Door 1 Door 2
M
M
2
1
Door 3
M
1
2
Contestant’s final choice
Monte
Door 1
Door 3
Door 2
Contestant
Door 1
M
2
Stay
3
Door 2 Door 3 Door 1 Door 2 Door 3
M M
M
M
M
3
2
3
1
3
1
Door 1 Door 2
M
M
2
1
Door 3
M
1
Stay
Switch
Switch
This shows 3 of Contestant’s six information sets. What are the others?
2
Information sets in Monte Hall game
• In last move, contestant knows which door he chose
and which Monte opened. The door that Monte
opened is neither the one that contestant opened nor
the one with the good prize.
• Six things contestant might see.
?/1/ 2 (C opened 1, Monte opened 2)
?/1/ 3 (C opened 1, Monte opened 3)
?/2/ 1 (C opened 2, Monte opened 1)
?/ 2/ 3 (C opened 2, Monte opened 3)
?/3/ 1 (C opened 3, Monte opened 1)
?/3/ 2 (C opened 3, Monte opened 2)
Information sets
•
•
•
•
•
•
{1/1/2, 3/1/2} (If C opened 1 and M opened 2)
{1/2/3, 2/2/3} (If C opened 1 and M opened 2)
{1/3/2, 3/3/2} … etc
{2/1/3, 1/1/3}
{2/2/1, 3/2/1}
{2/3/1, 3/3/1}
To stay or to switch: that is the question.
• If your initial choice was right, then you win if
you stay and lose if you switch.
• If your initial choice was wrong, you lose if you
stay and you win if you switch.
• What is the probability that your initial choice
was right?
• What is probability you win if you stay?
• What is probability you win if you switch?
Check your (Mis)understanding?
Number 3.5 page 79
Answer Key claims that:
• For player 1 a strictly dominates c
• For player 2, y strictly dominates w and x.
These claims are correct.
The key claims that
• if player 1 knows that player 2 will not use
w or x, then a dominates b for player 1.
This claim is not true!
Moral of the Story?
• Question authority.
• Even expensive, well-written textbooks make
mistakes.
• So do instructors.
So maybe that’s why…
Dominant strategies
Clicker Question
Player 2
Strategy A
Strategy A
Player 1
Strategy B
1,3
5, 3
2, 4
7, 2
Strategy B
A ) Strategy A strictly dominates Strategy B for both Players.
B) Strategy B strictly dominates A for Player 1. Strategy A weakly dominates B for Player 2.
C) Strategy B strictly dominates A for Player 1. Strategy A strictly dominates B for Player 2.
D) Strategy B strictly dominates Strategy A for both players.
E) No strategy in this game is strictly dominated
Strict and Weak Dominance
• Strategy A strictly dominates strategy B for a
player if that player gets a higher payoff from
doing A than from doing B no matter what
the other player(s) do.
• Strategy A weakly dominates strategy B for a
player if A gives that player at least as high a
payoff as B, no matter what the other
player(s) do and for some actions of the
others gives a higher payoff.
Clicker Question
Player 2
Strategy A
Strategy A
Player 1
Strategy B
2,2
0, 3
3, 0
1,1
Strategy B
A ) Strategy A strictly dominates Strategy B for both Players.
B) Strategy B strictly dominates A for Player 1. Strategy A weakly dominates B for Player 2.
C) Strategy B strictly dominates A for Player 1. Strategy A strictly dominates B for Player 2.
D) Strategy B strictly dominates Strategy A for both players.
E) No strategy in this game is strictly dominated
Game Theory Doctrine (A tautology)
• A rational player who understands the payoffs
of a game and who tries to maximize his own
payoff will
A) never use a strictly dominated strategy.
B) will always use a strictly dominant strategy if
one exists.
Dominant strategies?
Player 2
Strategy A
Player 1
Strategy A
Strategy B
Strategy B
1 0 , 10
0, 11
1 1, 0
1, 1
Does either strategy strictly dominate the other for
Player 1?
Does either strategy strictly dominate the other for
Player 2?
What is the predicted outcome?
What are games like this called?
How about this one?
Player 2
Strategy A
Player 1
Strategy B
Strategy A
1 0 , 10
0, 10
Strategy B
1 0, 0
1, 1
Does either strategy strictly dominate the other
for Player 1?
Does either strategy weakly dominate the other
for Player 1?
How about player 2?
Would a rational player ever play a
weakly dominated strategy?
• Only if he is absolutely certain that the other
player won’t play some strategies.
• Which strategies?
Clicker Question
Player 2
Strategy A
Player 1
Strategy B
Strategy A
1 0 , 10
0, 10
Strategy B
1 0, 0
1, 1
If I were playing this game just once with a stranger
whom I would never meet again, I would:
A) Play Strategy A
B) Play Strategy B
Rousseau’s Stag Hunt
Player 2
Stag
Stag
Player 1
Hare
2 , 2
1, 0
Hare
0, 1
1 , 1
Are any strategies weakly dominated?
Are any strategies strictly dominated?
How would you play?
Clicker Question
Player 2
Stag
Stag
Player 1
Hare
2 , 2
1, 0
Hare
0, 1
1 , 1
If you were playing Rosseau’s stag hunt with a
stranger, whom you will never meet again,
which strategy would you play?
A) Stag
B) Hare
Gaming Pigs
(Iterated dominance)
Are there dominated strategies for Big Pig?
How about Little Pig?
How would you “solve” this game?
What went on in the pigpen
The Entry Game
Challenger
Challenge
Stay out
0
1
Incumbent
Give in
1
0
Challenger’s payoff
Incumbent’s payoff
Fight
-1
-1
Challenger’s payoff
Incumbent’s payoff
Strategic Form of Entry Game
Incumbent
Give in
Stay out
Challenger
Enter
Fight
0,1
0,1
1,0
-1,-1
Dominance in Entry Game?
• No dominant strategy for Challenger. Which is
better depends on what incumbent will do.
• Give-in is weakly dominant for Incumbent.
• If Challenger believes that Incumbent is rational
and that incumbent thinks there is a chance that
Challenger will enter, then Challenger believes
that Incumbent will give in.
• In this case, predicted outcome is Challenger
enters and incumbent gives in.
Kidnapping with imperfect information
Strategic Form
Dominated strategies?
• Guy doesn’t have any dominated strategies
• But for Vivica, Don’t Pay dominates Pay.
• What does iterated elimination of dominated
strategies tell us?
• If Guy knows that Vivica is rational, he knows
she won’t pay ransom.
• If Guy knows that Vivica won’t pay ransom, he
is better off not kidnapping.
Kidnapping with Perfect Information
Kidnapping with complete information
Vivica
Guy
Pay Ransom
Don’t Pay Ransom
Kidnap-- Kill if R, Kill if NR
4,1
2,2
Kidnap—Release if R, Kill if NR
5,3
2,2
Kidnap— Kill if R, Release if NR
4,1
1,4
Kidnap—Release if R, Release if NR
5,3
1,4
Don’t Kidnap– Kill, Kill
3,5
3,5
Don’t Kidnap—Release, Kill
3,5
3,5
Don’t kidnap--Kill, Release
3,5
3,5
Don’t kidnap—Release, release
3,5
3,5
Are any strategies strictly dominated for either player?
Dominated strategies?
• Neither strategy dominates for Vivica
• For Guy, Kidnap—Release if Ransom, Kill if No
ransom weakly dominates all other strategies
that start with Kidnap.
• So if Vivica believe that Guy is rational, then she
believes that if Guy Kidnaps, he will kill if no
ransom and release if ransom.
• So Vivica would pay ransom
• So Guy would Kidnap and release after receiving
ransom.
Does Player 1 have a dominated strategy?
Hint: Compare b and d.
Iterated Elimination of Dominated Strategies-Stage 1
Does Player 2 have a dominated strategy?
Hint: Compare y and z.
The game After first round of Elimination
If each knows the other won’t play a
dominated strategy, we have a
smaller game.
Reduced Game after one iteration.
This is the game if each knows that the other is rational
and each knows that the other knows that the other is rational.
Are there any dominated strategies?
Reduced Game after 2 rounds of iterated elimination of
strictly dominated strategies. (Note that x couldn’t have
been eliminated in the first round.)
Reduced Game after 3 rounds of iterated elimination.
a is eliminated. This couldn’t have been done in earlier rounds.
Are there any strictly dominated strategies in this game?
We have eliminated 12 of 16 strategies, but to get any further,
We’re going to need more tools.
Iterated elimination and Common Knowledge
Strategy a dominates c for Player 1. Strategy y dominates w and x for
Player 2. Rational players won’t use these strategies. If each knows other
is rational, then Player 2 know s that 1 won’t play c and 1 knows that 2
won’t play w or x.
If both are rational and believe other is rational,
Player 1 knows that 2 won’t play x or y, so Player 1 can eliminate b.
Player 2 knows that Player 1 won’t play c, so Player 2 can eliminate y.
If Player 1 knows that Player 2 knows that Player 1 is rational, then
Player 1 knows Player 2 will Play z. What will Player 1 do?
See you on Thursday
…And Steer Clear of Dominated Strategies