Explanation Given Information: The provided Given Information is: The three-finger Morra game is a game in which two players simultaneously show one, two or three fingers at each round. The outcome depends on a predetermined set of rules. If they both finger A and B differ by 1 , then A loose one point. If they both finger A and B differ by more than 1 , then A round is draw. If they show the same number of finger, A wins an amount equal to the sum of fingers. Formula used: Solve the matrix game as: Step 1) Reduce the payoff matrix by dominance. Step 2) Add a fixed number k to each of the entries so that they all become nonnegative and no column is all zero. Step 3) Write 1 s to the right of and below the matrix, and then write down the associated standard maximization problem. Solve this primal problem using the simplex method. Step 4) Find the optical strategies and the expected value as follow: Column strategy: 1. Express the solution to the primal problem as a column vector . 2. Normalize by dividing each entry of the solution vector by p (which is also the sum of all the entries). 3. Insert zeros in position corresponding to the columns deleted during reduction by dominance. Row strategy: 1. Express the solution to the dual problem as a rown vector. 2. Normalize by dividing each entry by p , which will once again be the sum of all the entries. 3. Insert zeros in positions corresponding to the rows deleted during reduction by dominance. Value of the game: e = 1 p − k Calculation: Consider the given Information is: The three-finger Morra game is a game in which two players simultaneously show one, two or three fingers at each round. The outcome depends on a predetermined set of rules. If they both finger A and B differ by 1 , then A loose one point. If they both finger A and B differ by more than 1 , then A round is draw. If they show the same number of finger, A wins an amount equal to the sum of fingers. Add k = 1 to each entry and put 1 s to the right and be below: Represent this above problem by the matrix as: [ 3 0 1 0 5 0 1 1 0 1 7 1 1 1 1 0 ] Now, write down the dual, which is the LP problem corresponding to the new matrix as: Maximize p = x + y + z Subject to 3 x + z ≤ 1 , 5 y ≤ 1 , x + 7 y ≤ 1 and x ≥ 0 , y ≥ 0 , z ≥ 0 Solve the dual problem using the standard simplex method as: x y z s t u p s 3 0 1 1 0 0 0 1 t 0 5 0 0 1 0 0 1 u 1 0 7 0 0 1 0 1 p − 1 − 1 − 1 0 0 0 1 0 Apply row operation 3 R 3 − R 1 and 3 R 4 + R 1 x y z s t u p x 3 0 1 1 0 0 0 1 t 0 5 0 0 1 0 0 1 u 0 0 20 − 1 0 3 0 2 p 0 − 3 − 2 1 0 0 3 0 Apply row operation 5 R<

Finite Mathematics and Application Calculus (Looseleaf) - Text Only

7th Edition

ISBN: 9781337291484

Author: Waner

Publisher: Cengage

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Chapter 6.5, Problem 43E

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To calculate: the optical strategy of each player and the expected value of the game.

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Chapter 6.5, Problem 42E

Chapter 6.5, Problem 47E

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Finite Mathematics and Application Calculus (Looseleaf) - Text Only

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the optical strategy of each player and the expected value of the game.

Solution Summary: The author explains the three-finger Morra game, where two players simultaneously show one, two or three fingers at each round.

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