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9.1 Strictly Determined Games Game theory is a relatively new branch of mathematics designed to help people who are in conflict situations determine the best course of action out of several possible choices. It has applications in the business world, warfare and political science. The pioneers of game theory are John Von Neumann and Oskar Morgenstern.

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Von Neumann Von Neumann's awareness of results obtained by other mathematicians and the inherent possibilities which they offer is astonishing. Early in his work, a paper by Borel on the minimax property led him to develop... ideas which culminated later in one of his most original creations, the theory of games.Borel In game theory von Neumann proved the minimax theorem. He gradually expanded his work in game theory, and with co-author Oskar Morgenstern, he wrote the classic text Theory of Games and Economic Behaviour (1944).

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Fundamental principle of Game Theory 1. A matrix game is played repeatedly. 2. Player R tries to maximize winnings. 3. Player C tries to minimize losses.

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Two-person zero-sum matrix 1. R chooses (plays) any one of m rows. 2. C chooses (plays) any one of m columns.

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An example Suppose you have $10,000 to invest for a period of 5 years. After some investigation and advice from a financial counselor, you arrive at the following game matrix where you ( R ) are playing against the economy ( C). Each entry in the matrix is the expected payoff after 5 years for an investment of $10,000 in the corresponding row designation with the future state of the economy in the corresponding column section. The economy is regarded as a rational player who can make decisions against the investor – in any case, the investor would like to do the best possible irrespective of what happens to the economy. Find saddle values and optimal strategies for each player.

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Finding the saddle point(s) if they exist 1. R strategy: circle the lowest number in each row (worst case scenario)

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C’s strategy: Put a square around the greatest value of each column. There are two saddle values located in the first row. So R should choose the first row. C (the economy) can either fall or have no change. In either case the gain of R is the corresponding loss to the economy. The value of the game is 5870 and since this value is not equal to zero, the game is considered to be not fair..

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9.4 Linear programming and m x n Games: Simplex Method and the Dual Problem In this section, the process of solving 2 x 2 matrix games will be generalized.

9.4 Linear programming and m x n Games: Simplex Method and the Dual Problem In this section, the process of solving 2 x 2 matrix games will be generalized.

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