game theory:minimax, maximin, and iterated...

Naima Hammoud

March 14, 2017

Game Theory: Minimax, Maximin, and Iterated Removal

Last Lecture: expected value principle

Suppose that Rose knows Colin will play ½ A + ½ B

Rose’s Expectations for playing pure strategies E

Rose

(A) = 1/2⇥ 2 + 1/2⇥ (�3) = �1/2

A B

A 2 -2 -3 3

B 0 0 3 -3

Colin

Rose

ERose

(B) = 1/2⇥ 0 + 1/2⇥ (3) = 3/2

Rose’s expected payoff if she plays strategy A is -1/2

Rose’s expected payoff if she plays strategy B is 3/2

Suppose that Rose knows Colin will play ½ A + ½ B

Rose’s Expectations for playing pure strategies E

Rose

(A) = 1/2⇥ 2 + 1/2⇥ (�3) = �1/2

ERose

(B) = 1/2⇥ 0 + 1/2⇥ (3) = 3/2

Because 3/2 > -1/2Rose chooses to maximize her payoff by playing B. That’s of course only if Colin is playing ½ A + ½ B

Last Lecture: expected value principleColin

Rose

A B

A 2 -2 -3 3

B 0 0 3 -3

Last Lecture: expected value principleColin

Rose

A B

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

Heads Tails

Heads 1 -1 -1 1

Tails -1 1 1 -1

Colin

Rose

The sum of payoffs forthis outcome is zero,as is the sum of payoffsfor every other outcome.

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

2

42,�2 0, 0 1,�14,�4 �3, 3 2,�21,�1 �2, 2 2,�2

3

5

Column player 2

Row player 1

2

42 0 14 �3 21 �2 2

3

5

Column player 2

Row player 1

Zero-sum game example

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 therow player (on the right). Once we have that, we can find the maximin & minimax.

Maximin strategy for player 1: maximize their own minimum gain

minimum gain

0

If player 1 plays the first strategy (strategy A) then their minimum gain is 0.

Column player 2

Row player 1

A B CABC

2

42 0 14 �3 21 �2 2

3

5

minimum gain

0

�3

Column player 2

Row player 1

A B CABC

2

42 0 14 �3 21 �2 2

3

5

If player 1 plays strategy B then their minimum gain is -3.


minimum gain

0

�3�2

Column player 2

Row player 1

A B CABC

2

42 0 14 �3 21 �2 2

3

5

If player 1 plays strategy C then their minimum gain is -2.


minimum gain

0

�3�2

maximum loss 4If player 2 plays strategy A then their maximum loss is 4 (their max loss is player 1’s max gain)

Column player 2

Row player 1

A B CABC

2

42 0 14 �3 21 �2 2

3

5

Maximin strategy for player 1: maximize their own minimum gainMinimax strategy for player 2: minimize their own maximum loss

minimum gain

0

�3�2

0

Column player 2

Row player 1

A B CABC

2

42 0 14 �3 21 �2 2

3

5

maximum loss 4If player 2 plays strategy B then their maximum loss is 0 (their max loss is player 1’s max gain)


minimum gain

0

�3�2

2maximum loss 4If player 2 plays strategy C then their maximum loss is 2 (their max loss is player 1’s max gain)

0

Column player 2

Row player 1

A B CABC

2

42 0 14 �3 21 �2 2

3

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.

minimum gain

�3�2

0

2maximum loss 4 0

Column player 2

Row player 1

A B CABC

2

42 0 14 �3 21 �2 2

3

5


maximin

minimax

If a saddle point exists, it should always be played. Here player 1 plays A and player 2 plays B


minimum gain

�3�2

0

2maximum loss 4 0

Column player 2

Row player 1

A B CABC

2

42 0 14 �3 21 �2 2

3

5maximin

minimax

saddle point

A saddle point is a Nash equilibrium


minimum gain

�3�2

0

2maximum loss 4 0

Column player 2

Row player 1

A B CABC

2

42 0 14 �3 21 �2 2

3

5maximin

minimax

saddle point

player 2

player 1

More examples

The highlighted entry is the saddle point, and both players will play it.

2

664

3 2 1 00 1 2 01 0 2 13 1 2 2

3

775

0001

3 2 2 2

2

664

3 1 1 00 1 2 01 0 2 13 1 2 2

3

775

0001

3 2 21

None of the row minima equals any of the column maxima, so no saddle points

player 2

player 1

maximin

minimax

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

Left Center Right

Up 3 0 2 1 0 0

Middle 1 1 1 1 5 0

Down 0 1 4 2 0 1

Column player 2

Row player 1

iterated removal example

Left Center Right

Up 3 0 2 1 0 0

Middle 1 1 1 1 5 0

Down 0 1 4 2 0 1

Column player 2

Row player 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.

Left Center Right

Up 3 0 2 1 0 0

Middle 1 1 1 1 5 0

Down 0 1 4 2 0 1

Column player 2

Row player 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.

Left Center

Up 3 0 2 1

Down 0 1 4 2

Column player 2

Row player 1

The new game matrix is now smaller.

Left Center

Up 3 0 2 1

Down 0 1 4 2

Column player 2

Row player 1

Column player will never play Left because it is strictly dominated by Center. Payoff of (0, 1) from Left versus (1, 2) from Center.

Center

Up 2 1

Down 4 2

Column player 2

Row player 1

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

4

Box 1 $1000

Box 2 $1 million or nothing

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?

THE MATRIX FOR NEWCOMB’S PROBLEM

Predicts that you select both boxes

Predicts that you select Box #2

You select both $1,000 $1,001,000

You select Box #2 $0 $1,000,000

HOST

CONTESTANT

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.

game theory:minimax, maximin, and iterated...

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