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Game Theory, Part 1 Game theory applies to more than just games. Corporations use it to influence business decisions, and militaries use it to guide their strategies. In fact, game theory grew in popularity and acceptance during WWII.

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Game Theory, Part 1 In essence, the goal of game theory is to determine the best strategy for a participant in any sort of competition to use.

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Game Theory, Part 1 How do we determine the best strategy for each competitor? Begin with constructing a payoff matrix.

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Game Theory, Part 1 How do we determine the best strategy for each competitor? Begin with constructing a payoff matrix. The payoff matrix shows the outcome for each possible combination of strategies by each competitor.

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Game Theory, Part 1 We will consider only competitions that are limited to two participants. The rows of the payoff matrix will represent the actions of one competitor, the columns will represent the options for the other competitor.

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Game Theory, Part 1 Very important: The values in the payoff matrix represent the payoff for the row competitor.

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Game Theory, Part 1 Very important: The values in the payoff matrix represent the payoff for the row competitor. Those values represent exactly the opposite payoff for the column competitor.

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Game Theory, Part 1 Example: Sol and Tina play a game where they each choose to display either heads or tails on a coin. They reveal their coins at the same time, and have the following payoff matrix. H T Tina H T Sol

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Game Theory, Part 1 Remember that the values in the matrix show the payoff for Sol. H T H T Tina Sol If Sol and Tina each play heads, Sol wins 3 pennies. This means, of course, that Tina loses 3 pennies.

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Game Theory, Part 1 Remember that the values in the matrix show the payoff for Sol. H T H T Tina Sol If Sol plays heads and Tina plays tails, then Sol loses 1 penny; Tina gains 1 penny—exactly opposite of Sol.

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Game Theory, Part 1 Remember that the values in the matrix show the payoff for Sol. H T H T Tina Sol What strategy should each player follow?

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Game Theory, Part 1 One method of choosing a strategy is to try to minimize the potential damage or loss. In other words, choose the least of all evils.

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Game Theory, Part 1 Consider the worst-case scenario for each player, for each option. H T H T Tina Sol For Sol, look at the rows. If Sol plays heads, the possible outcomes are to win 3 pennies, or to lose 1 penny.

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Game Theory, Part 1 Consider the worst-case scenario for each player, for each option. H T H T Tina Sol If Sol plays heads, the worst that can happen is that he will lose 1 penny. This is the row minimum for row 1.

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Game Theory, Part 1 Consider the worst-case scenario for each player, for each option. H T H T Tina Sol In order to minimize Sol’s losses, he should choose the maximum value of all the row minimums. -2

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Game Theory, Part 1 Consider the worst-case scenario for each player, for each option. H T H T Tina Sol This value is called the maximin—the maximum of the row minimums—which in this case is –1. -2

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Game Theory, Part 1 Consider the worst-case scenario for each player, for each option. H T H T Tina Sol Now consider Tina. Remember that the values in the payoff matrix are exactly the opposite for her. In other words, large positive numbers mean she loses money. -2

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Game Theory, Part 1 Consider the worst-case scenario for each player, for each option. H T H T Tina Sol Because of this, we find the worst-case scenario for her by searching for the largest numbers in each column. 3 -2

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Game Theory, Part 1 Consider the worst-case scenario for each player, for each option. H T H T Tina Sol If Tina plays heads, the worst thing that can happen is that she’ll lose 3 pennies, as opposed to possibly winning 1 penny. 3 -2

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Game Theory, Part 1 Consider the worst-case scenario for each player, for each option. H T H T Tina Sol If she plays tails, the worst thing that can happen is that she’ll win 1 penny, rather than 2. 3 -2

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Game Theory, Part 1 Consider the worst-case scenario for each player, for each option. H T H T Tina Sol In order to minimize her losses, we want to choose the minimum of the column maximums, called the minimax. 3 -2

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Game Theory, Part 1 Consider the worst-case scenario for each player, for each option. H T H T Tina Sol Notice that Sol’s maximin is the same as Tina’s minimax. This suggests that the outcome of the game will be the same every time—Tina will win 1 penny. 3 -2

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Game Theory, Part 1 Consider the worst-case scenario for each player, for each option. H T H T Tina Sol This is an example of a strictly determined game, where the maximin and the minimax are the same. 3 -2

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Game Theory, Part 1 Consider the worst-case scenario for each player, for each option. H T H T Tina Sol That value for the maximin and minimax is called the saddle point of the game. It shows the outcome each game for the row player. 3 -2

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Game Theory, Part 1 Consider another game where each player displays a card with the letter A, B, C, or D. This game has the following payoff matrix: A B C D A B C D Sol Tina

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Game Theory, Part 1 A B C D A B C D Sol Tina Sol took a close look at his options, represented in the rows, and noticed something interesting.

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Game Theory, Part 1 A B C D A B C D Sol Tina Regardless of what Tina does, row A never provides a better outcome for Sol than row D.

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Game Theory, Part 1 A B C D A B C D Sol Tina Row D is said to dominate row A. Because it can never outdo row D, Sol can simply eliminate A as an option.

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Game Theory, Part 1 A B C D A B C D Sol Tina Likewise, Tina can compare her options in the columns of the payoff matrix. Remember, though, that she wants the smallest (most negative) numbers, as those represent more money won.

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Game Theory, Part 1 A B C D A B C D Sol Tina Column C always outperforms column A, regardless of what Sol does. So Tina can eliminate column A.

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Game Theory, Part 1 A B C D A B C D Sol Tina Now that Sol and Tina have both eliminated A as an option, we really only need to worry about a 3x3 payoff matrix. Evaluate the maximin and minimax.

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Game Theory, Part 1 A B C D A B C D Sol Tina The maximin is –2, which occurs for strategy D. -3 -5 -2

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Game Theory, Part 1 A B C D A B C D Sol Tina This suggests that Sol should play D, and Tina should play D. In such an event, Sol will win 2 pennies. -3 -5 -2 342

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Game Theory, Part 1 A B C D A B C D Sol Tina How long do you think Tina will play this game? -3 -5 -2 342

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