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9.1 Games and Strategies

9.1 Games and Strategies. Optimal Strategy for Side Player (R) Select the smallest value in each row Optimal Strategy for Top Player (C) Select the largest value in each column. The saddle point (Pareto point) is where these two values overlap.

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9.1 Games and Strategies

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  1. 9.1 Games and Strategies Optimal Strategy for Side Player (R) • Select the smallest value in each row Optimal Strategy for Top Player (C) • Select the largest value in each column The saddle point (Pareto point) is where these two values overlap. • A game that has a saddle point is called a strictly determined game. • This point is also the “value of the game” • The game is called a fair game if it’s value is 0.

  2. 9.1 Games and Strategies Determine the saddle point if it exists. State whether or not the matrix is strictly determined. State the value of the game.

  3. 9.2 Mixed Strategies When there is no saddle point, a strategy called mixed strategies will be needed to determine the outcome. Mixed Strategies need: • A payoff matrix • R’s probability row matrix or matrices • How could there be multiple?? • C’s probability column matrix or matrices • How could there be multiple?? Suppose the payoff matrix is: • Show there is no saddle point Now, let’s say: • R’s mixed strategies are: • C’s mixed strategy for both columns is: We compare these mixed strategies by using the expected value!

  4. 9.2 Mixed Strategies Given the payoff matrix is: With the following mixed strategies: • R: • C: Each possibility for the game must be explored to find the expected value. Now, expected value is calculated like before! E(X) = sum of each win * probability

  5. 9.2 Mixed Strategies Given the payoff matrix is: With the following mixed strategies: • R: • C: Calculate the expected value of this game. Which strategy is more advantageous for R? C?

  6. 9.1 Games and Strategies • Problems to complete from section 9.1 • Pg. 445 #7, 8, 12 • Problems to complete from section 9.2 • Pg. 451 #1

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