Game Theory: Minimax, Maximin, and Iterated Removal
Naima Hammoud
March 14, 2017 Last Lecture: expected value principle Colin
A B
Rose A 2 -2 -3 3
B 0 0 3 -3 Rose’s expected Suppose that Rose knows Colin will play ½ A + ½ B payoff if she plays strategy A is -1/2 Rose’s Expectations for playing pure strategies E (A)=1/2 2+1/2 ( 3) = 1/2 Rose ⇥ ⇥ ERose(B)=1/2 0+1/2 (3) = 3/2 Rose’s expected ⇥ ⇥ payoff if she plays strategy B is 3/2 Last Lecture: expected value principle Colin
A B
Rose A 2 -2 -3 3
B 0 0 3 -3 Suppose that Rose knows Colin will play ½ A + ½ B Because 3/2 > -1/2 Rose’s Expectations for playing pure strategies Rose chooses to maximize her payoff ERose(A)=1/2 2+1/2 ( 3) = 1/2 by playing B. That’s ⇥ ⇥ of course only if ERose(B)=1/2 0+1/2 (3) = 3/2 Colin is playing ⇥ ⇥ ½ A + ½ B Last Lecture: expected value principle Colin
A B
Rose A 2 -2 -3 3
B 0 0 3 -3
Rule of thumb: If you know your opponent is playing a mixed strategy and will continue to play it, you should use a strategy that maximizes your expected payoff. Last Lecture
• We saw that in soccer penalty-kick data collected by Ignacio Palacios- Huerta (2003) that kickers and goal-keepers seem to be playing the Nash equilibrium! But is that really the case?
Goalie Left Goalie Right Kicker Left Kicker Right
Nash frequency 0.42 0.58 0.38 0.62
Actual frequency 0.42 0.58 0.4 0.6 Last Lecture
• We saw that in soccer penalty-kick data collected by Ignacio Palacios- Huerta (2003) that kickers and goal-keepers seem to be playing the Nash equilibrium! But is that really the case?
• The player is actually trying to maximize their own gain and minimize the gain of the goal keeper
• It turns out that in zero-sum games, the Nash equilibrium, maximizing your own gain, and minimizing your opponent’s gain actually coincide. Zero-sum Games zero-sum game: A zero-sum game is one in which the sum of the individual payoffs for each outcome is zero.
Example: Matching pennies Colin
The sum of payoffs for Heads Tails this outcome is zero, as is the sum of payoffs for every other outcome. Heads 1 -1 -1 1 Rose Tails -1 1 1 -1 Minimax, Maximin zero-sum game: A zero-sum game is one in which the sum of the individual payoffs for each outcome is zero.
Minimax strategy: minimizing one’s own maximum loss
Maximin strategy: maximize one’s own minimum gain Zero-sum game example
Column Column player 2 player 2 2, 2 0, 0 1, 1 201 Row 4, 4 3, 3 2, 2 Row 4 32 player 1 player 1 21, 1 2, 2 2, 23 21 223 4 5 4 5 Since the payoffs of the column player (shown red) are just the negative of the payoffs of the row player, we can write a matrix only showing payoffs of the row player (on the right). Once we have that, we can find the maximin & minimax. Maximin strategy for player 1: maximize their own minimum gain
Column player 2 A B C minimum gain A 201 0 Row player 1 B 4 32 2 3 C 1 22 If player 1 plays the first strategy4 (strategy A) then their5 minimum gain is 0. Maximin strategy for player 1: maximize their own minimum gain
Column player 2 A B C minimum gain A 201 0 Row player 1 B 4 32 3 2 3 C 1 22 If player 1 plays4 strategy B then their minimum5 gain is -3. Maximin strategy for player 1: maximize their own minimum gain
Column player 2 A B C minimum gain A 201 0 Row player 1 B 4 32 3 2 3 C 1 22 2 If player 1 plays4 strategy C then their minimum5 gain is -2. Maximin strategy for player 1: maximize their own minimum gain Minimax strategy for player 2: minimize their own maximum loss
Column player 2 A B C minimum gain A 201 0 Row player 1 B 4 32 3 2 3 C 1 22 2 maximum loss 4 If player 2 plays strategy A then4 their maximum loss is 4 (their max5 loss is player 1’s max gain) Maximin strategy for player 1: maximize their own minimum gain Minimax strategy for player 2: minimize their own maximum loss
Column player 2 A B C minimum gain A 201 0 Row player 1 B 4 32 3 2 3 C 1 22 2 maximum loss 4 0 If player 2 plays strategy B then4 their maximum loss is 0 (their max5 loss is player 1’s max gain) Maximin strategy for player 1: maximize their own minimum gain Minimax strategy for player 2: minimize their own maximum loss
Column player 2 A B C minimum gain A 201 0 Row player 1 B 4 32 3 2 3 C 1 22 2 maximum loss 4 0 2 If player 2 plays strategy C then4 their maximum loss is 2 (their max5 loss is player 1’s max gain) Maximin strategy for player 1: maximize their own minimum gain Minimax strategy for player 2: minimize their own maximum loss Column player 2 A B C minimum gain A 201 0 maximin Row player 1 B 4 32 3 2 3 C 1 22 2 maximum loss 4 0 2 4 minimax 5 Take the maximum of the minimum gains, i.e. the maximum of row minima (maximin), and the minimum of the maximum losses, i.e. the minimum of column maxima (minimax). If they are equal, you have a saddle point. Maximin strategy for player 1: maximize their own minimum gain Minimax strategy for player 2: minimize their own maximum loss Column saddle point player 2 A B C minimum gain A 201 0 maximin Row player 1 B 4 32 3 2 3 C 1 22 2 maximum loss 4 0 2 4 minimax 5
If a saddle point exists, it should always be played. Here player 1 plays A and player 2 plays B Maximin strategy for player 1: maximize their own minimum gain Minimax strategy for player 2: minimize their own maximum loss Column saddle point player 2 A B C minimum gain A 201 0 maximin Row player 1 B 4 32 3 2 3 C 1 22 2 maximum loss 4 0 2 4 minimax 5
A saddle point is a Nash equilibrium More examples
player 2 player 2
32100 31100 01200 01200 player 1 2 3 player 1 10210 210213 0 631227 1 maximin 631227 1 6 7 6 7 43 2 2 25minimax 43 1 2 25 None of the row minima The highlighted entry is the equals any of the column saddle point, and both maxima, so no saddle points players will play it. Dominated strategies: iterated removal
Dominated strategy: There is some other strategy that does better than it.
• A dominated strategy will never be played, so we can remove it from the game
• We can iterate until we get to to the dominant strategy
• This is called iterated removal of dominated strategies iterated removal example
Column player 2
Left Center Right Up 3 0 2 1 0 0 Row player 1 Middle 1 1 1 1 5 0
Down 0 1 4 2 0 1 Column player 2
Left Center Right Up 3 0 2 1 0 0 Row player 1 Middle 1 1 1 1 5 0
Down 0 1 4 2 0 1
Column player will never play Right because it is strictly dominated by Center. The payoffs of player 2 playing Right are (0, 0, 1), which are dominated by (1, 1, 2) from playing Center. Therefore we can remove Right. Column player 2
Left Center Right Up 3 0 2 1 0 0 Row player 1 Middle 1 1 1 1 5 0
Down 0 1 4 2 0 1
Row player will never play Middle because it is strictly dominated by Up. Payoffs of Middle are (1, 1) which are dominated by (3, 2) from Up. Column player 2
Left Center Up 3 0 2 1 Row player 1 Down 0 1 4 2
The new game matrix is now smaller. Column player 2
Left Center Up 3 0 2 1 Row player 1 Down 0 1 4 2
Column player will never play Left because it is strictly dominated by Center. Payoff of (0, 1) from Left versus (1, 2) from Center. Column player 2
Center
Row Up 2 1 player 1 Down 4 2
Now row player is better off playing Down than Up, because the payoff is 4 instead of 2.
So (4, 2) is a unique Nash equilibrium “FAITH” — TELEVISION’S NEW HIT GAMESHOW
You have observed the host to be 99.98% accurate in the last 10,000 games. If he predicted that the contestant chooses only Box #2, he rewards their faith with the million dollars. ! Do you take both boxes or only Box #2?
Box 2 Box 1 $1 million $1000 or nothing
4 THE MATRIX FOR NEWCOMB’S PROBLEM
HOST
Predicts that you Predicts that you select both boxes select Box #2
You select both $1,000 $1,001,000 CONTESTANT You select Box #2 $0 $1,000,000 TWO ARGUMENTS
Argument 1: Have faith and take Box #2 In your observations of the last 10,000 games, the host has been shown to possess 99.98% accuracy in predicting the contestants choice. If you select both boxes, you will almost certainly get only $1000. If you have faith (in the host, in your observations), and select Box #2, you will win the million dollars.
Argument 2: Take both boxes What does it matter what the host predicted? Either there is one million dollars in Box #2 or there isn’t. The host’s prediction does not change the contents of the box here and now. By opening both boxes, you get either $1000 or $1,001,000. This is better than $0 or $1,000,000. ! Take both boxes.