Game Theory: Minimax, Maximin, and Iterated Removal Naima Hammoud

Game Theory: Minimax, Maximin, and Iterated Removal Naima Hammoud March 14, 17

Last Lecture: expected value principle Colin A B Rose A - - B - Suppose that Rose knows Colin will play ½ A + ½ B Rose s Expectations for playing pure strategies E Rose (A) =1/ +1/ ( ) = 1/ E Rose (B) =1/ +1/ () = / Rose s expected payoff if she plays strategy A is -1/ Rose s expected payoff if she plays strategy B is /

Last Lecture: expected value principle Colin A B Rose A - - B - Suppose that Rose knows Colin will play ½ A + ½ B Rose s Expectations for playing pure strategies E Rose (A) =1/ +1/ ( ) = 1/ E Rose (B) =1/ +1/ () = / Because / > -1/ Rose 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 principle Colin A B Rose A - - B - 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 () 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.4.58.8.6 Actual frequency.4.58.4.6

Last Lecture We saw that in soccer penalty-kick data collected by Ignacio Palacios- Huerta () 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 Rose Heads Tails Heads 1-1 -1 1 Tails -1 1 1-1 The sum of payoffs for this outcome is zero, as is the sum of payoffs for 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

Zero-sum game example Column player,, 1, 1 44, 4,, 1, 1,, 5 Column player 1 44 1 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 : maximize their own minimum gain A Column player A B C 1 minimum gain B 44 5 C 1 If plays the first strategy (strategy A) then their minimum gain is.

Maximin strategy for : maximize their own minimum gain A B Column player A B C 1 44 5 minimum gain C 1 If plays strategy B then their minimum gain is -.

Maximin strategy for : maximize their own minimum gain A B Column player A B C 1 44 5 minimum gain C 1 If plays strategy C then their minimum gain is -.

Maximin strategy for : maximize their own minimum gain Minimax strategy for player : minimize their own maximum loss A B Column player A B C 1 44 5 minimum gain C 1 maximum loss 4 If player plays strategy A then their maximum loss is 4 (their max loss is s max gain)

Maximin strategy for : maximize their own minimum gain Minimax strategy for player : minimize their own maximum loss A B Column player A B C 1 44 5 minimum gain C 1 maximum loss 4 If player plays strategy B then their maximum loss is (their max loss is s max gain)

Maximin strategy for : maximize their own minimum gain Minimax strategy for player : minimize their own maximum loss A B Column player A B C 1 44 5 minimum gain C 1 maximum loss 4 If player plays strategy C then their maximum loss is (their max loss is s max gain)

Maximin strategy for : maximize their own minimum gain Minimax strategy for player : minimize their own maximum loss A B Column player A B C 1 44 5 minimum gain maximin C 1 maximum loss 4 minimax 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 : maximize their own minimum gain Minimax strategy for player : minimize their own maximum loss A B Column player A B C 1 44 5 saddle point minimum gain maximin C 1 maximum loss 4 minimax If a saddle point exists, it should always be played. Here plays A and player plays B

Maximin strategy for : maximize their own minimum gain Minimax strategy for player : minimize their own maximum loss A B Column player A B C 1 44 5 saddle point minimum gain maximin C 1 maximum loss 4 minimax A saddle point is a Nash equilibrium

More examples player 1 6 1 7 41 15 1 1 minimax None of the row minima equals any of the column maxima, so no saddle points maximin player 1 1 6 1 7 41 15 1 1 1 The highlighted entry is the saddle point, and both 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 Left Center Right Up 1 Middle 1 1 1 1 5 Down 1 4 1

Column player Left Center Right Up 1 Middle 1 1 1 1 5 Down 1 4 1 Column player will never play Right because it is strictly dominated by Center. The payoffs of player playing Right are (,, 1), which are dominated by (1, 1, ) from playing Center. Therefore we can remove Right.

Column player Left Center Right Up 1 Middle 1 1 1 1 5 Down 1 4 1 player will never play Middle because it is strictly dominated by Up. Payoffs of Middle are (1, 1) which are dominated by (, ) from Up.

Column player Left Center Up 1 Down 1 4 The new game matrix is now smaller.

Column player Left Center Up 1 Down 1 4 Column player will never play Left because it is strictly dominated by Center. Payoff of (, 1) from Left versus (1, ) from Center.

Column player Center Up 1 Down 4 Now row player is better off playing Down than Up, because the payoff is 4 instead of. So (4, ) is a unique Nash equilibrium

FAITH TELEVISION S NEW HIT GAMESHOW You have observed the host to be 99.98% accurate in the last 1, games. If he predicted that the contestant chooses only Box #, he rewards their faith with the million dollars.! Do you take both boxes or only Box #? Box 1 $1 Box $1 million or nothing 4

THE MATRIX FOR NEWCOMB S PROBLEM HOST Predicts that you select both boxes Predicts that you select Box # CONTESTANT You select both $1, $1,1, You select Box # $ $1,,

TWO ARGUMENTS Argument 1: Have faith and take Box # In your observations of the last 1, 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 $1. If you have faith (in the host, in your observations), and select Box #, you will win the million dollars. Argument : Take both boxes What does it matter what the host predicted? Either there is one million dollars in Box # 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 $1 or $1,1,. This is better than $ or $1,,.