GAME THEORY Day 5 Minimax and Maximin Step

Minimax and Maximin Step 1. Write down the minimum entry in each row. Which

Mixed Strategy Yesterday, we discussed dominance and saddle points. Games where ONE strategy can

Expected Value To analyze the effect of one or both players using mixed strategies,

Expected Value Principle If you know that your opponent is playing a given mixed

Colin’s Point of View How can Colin choose a mixed strategy such that Rose

What about 3 x 3? Colin has strategies A, B, and C with probabilities

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GAME THEORY Day 5

Minimax and Maximin Step 1. Write down the minimum entry in each row. Which one is the largest? Maximin Step 2. Write down the maximum entry in each column. Which one is the smallest? Minimax RoseColi n A B C D A 4 3 2 5 B -10 2 0 -1 C 7 5 2 3 8 7 -4 8 -5 2 D 0 Column Maximum Row Minimum 2 -10 Maximin 2 5 5 Saddle Points Minimax

Mixed Strategy Yesterday, we discussed dominance and saddle points. Games where ONE strategy can be played with certainty to receive the best outcome are pure strategy games Contrastingly, when there is no saddle point (or multiple saddle points) where neither player would want to play a single strategy with certainty, it is called a mixed strategy game Example. RoseColi n A B A 2 -3 B 0 3 Row Minimum: -3 and 0 Maximin 0 Column Maximum: 2 and 3 Minimax 2

Expected Value To analyze the effect of one or both players using mixed strategies, we can use the concept of expected value. Let’s say in our example Colin will flip a coin to decide between strategy A and B. Colin A and Colin B have probability of ½ each Example. RoseColi n A B A 2 -3 B 0 3 Expected Value. Rose A: ½ (2) + ½ (-3) = ½ Rose B: ½ (0) + ½ (3) = 3/2 Thus, if Colin is playing the mixed strategy ½ A, ½ B, then Rose should play Rose B.

Expected Value Principle If you know that your opponent is playing a given mixed strategy, and will continue to play it regardless of what you do, you should play your strategy which has the largest expected value.

Colin’s Point of View How can Colin choose a mixed strategy such that Rose cannot take advantage of it! Let Colin play a mixed strategy of probability x for strategy A and (1 -x) for B where 0<x<1. Thus, Colin can should play mixed strategy ¾ A, ¼ B RoseColi n A B A 2 -3 B 0 3 Expected Value. Rose A: x(2) + (1 -x)(-3) = -3 + 5 x Rose B: x (0) + (1 -x)(3) = 3 - 3 x What can we do with this information? -3 + 5 x = 3 – 3 x So x = ¾ What would Rose’s average payoff be? ¾ Rose A: ¾ (2) + ¼ (-3) = ¾ Rose B: ¾ (0) + ¼ (3) = ¾

What about 3 x 3? Colin has strategies A, B, and C with probabilities (x, y, 1 – x – y) Solve using a system of two equations in two unknowns. Rose A: Rose B: Rose C: RoseCol in A B C A 1 2 2 B 2 1 2 C 2 2 0 (2/5, ½)