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# ps6 - MASSACHUSETTS INSTITUTE OF TECHNOLOGY 15.053...

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M ASSACHUSETTS I NSTITUTE OF T ECHNOLOGY 15.053 – Optimization Methods in Management Science (Spring 2007) Problem Set 6 Due April 4 th , 2007 at 4:30 pm. You will need 160 points out of 188 to receive a grade of 5. Problem 1: Rock-Paper-Scissors Play (15 Points, 5 Each) As indicated in lecture, rock-paper-scissors is a classic problem studied in game theory. Part A: Play 10 to 25 rounds of the game with your partner and record your outcome. If you do not have a partner, play with a friend or if you get really desperate you can play with an online opponent at http://www.playrps.com/ . Part B: The winner of the game should answer the following questions: Was your win luck or strategy/skill? If you used a strategy, what was it and why do you think it was successful? Fun Trivia: In Japan there are two variations of R-P-S: Snake, frog , and slug Warrior, tiger, and warrior’s mother There is also an Indonesian variation as well: Elephant, human, and ant. Part C: Write down the payoff matrix for the game. Next week in Recitation we will have a rock, paper, scissors tournament and find out who the ultimate champion is. Practice with all the “friends” you meet over spring break. Page 0 of 9

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Problem 2: Game Theory, LPs and Excel (45 Points) Consider the following payoff matrix. The numbers represent payoffs to the row player R. -2 2 1 5 2 -1 0 -2 1 0 -1 1 Part A: (5 Points) Formulate a linear program that will find an optimal mixed strategy for the row player. Call this R’s problem. Part B: (5 Points) Formulate a linear program that will find an optimal mixed strategy for the column player. Call this C’s problem Part C: (10 Points) Solve both R’s problem and C’s problem in Excel. What are the optimal solutions? Format and turn in your spreadsheet and show your work. Part D: (10 Points) How do the two answers relate to each other? Is this always the case? Explain your answer using duality theory. Part E: (5 Points) What is the payoff to the row player if the row player chooses the optimal strategy and the column player decides to choose column 4? Part F: (10 Points) Is it possible to change one entry such that either the column player or row player has multiple optimal strategies?
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ps6 - MASSACHUSETTS INSTITUTE OF TECHNOLOGY 15.053...

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