University:
Massachusetts Institute of Technology
Course:
15.501/516 | Accounting
Academic year:

2023
Views:

336

Pages:

2
Author:

gemelgalvanises

Introduction to Optimization 1. Find the value and optimal strategy for the game in the table below. ⎡ 2 2⎤ ⎢1 3 ⎥ ⎣ ⎦ 2. Find the value and optimal strategies for the two-person zero-sum game in the table below. ⎡4 ⎢6 ⎢ ⎢5 ⎢ ⎣6 3. 5 7 7 6 5 6 5 5 8⎤ 9 ⎥⎥ 4⎥ ⎥ 5⎦ Mad Max wants to travel from New York to Dallas by the shortest possible route. He may travel over the routes shown in the table below. Unfortunately, the Wicked Witch can block one road leading out of Atlanta and one road leading out of Nashville. Mad Max will not know which roads have been blocked until he arrives at Atlanta or Nashville. Should Mad Max start toward Atlanta or Nashville? Route New York - Atlanta Length of Route (Miles) 800 New York – Nashville 900 Nashville – St. Louis 400 Nashville – New Orleans 200 Atlanta – St. Louis 300 Atlanta – New Orleans 600 St Louis – Dallas 500 New Orleans – Dallas 300 4. Consider the two-person zero-sum game in the table below. ⎡ 1 2 − 1 − 1⎤ ⎢− 1 1 − 1⎥ 2 ⎥ ⎢ ⎢⎣− 1 − 1 1 ⎥⎦ a. b. c. 5. Write down each player’s LP. We are told that player 1’s optimal strategy has x1 > 0, x2 > 0, x3 > 0. Find the value of the game and each player’s optimal strategies. Suppose the column plays the non-optimal strategy (½, ½, 0). Show how the row player can earn an expected reward that exceeds the value of the game. This problem is from last year’s midterm. After publishing his latest book, Bill earned $500,000, which he wants to invest in stocks. Bill believes the investment will be affected by whoever wins the Presidential election in November. He has narrowed down his investment to three stocks, labeled C, D and E. In this election, the candidates are denoted as B, K, and N. The performance of each stock under each candidate is show in the table below: Stocks C D E B 1.5 1.1 0.9 Candidates K 1.1 1.3 1.2 N 0.7 1.2 1.4 For example, the table says that if Bill invests $1,000 in stock D, and candidate K is elected, then Bill’s position in stock D will be worth $1,300 after one year. Bill is by nature a pessimist, and assumes that no matter how he invests his money, the candidate least favorable to his investments will win. So, he wants to invest conservatively, so as to maximize his payment over the next year in the worst possible case. We will call his objective the maximin objective, since his goal is to maximize the payment in the worst case. a) If Bill were required to invest all his money in one stock, which stock would he invest in? b) Let pC, pD, and pE denote the proportion of the money invested in stocks C, D, and E. Formulate an LP to compute the optimal maximin strategy for Bill. It must be in the form of a linear program. c) Suppose that the pessimistic Bill decides that candidate N really has no chance whatsoever, and so eliminates that column from consideration. Under this scenario Stock E is dominated by Stock D. Determine what proportion of his $500,000 Bill should allocate to Stocks C and D for his optimal maximin portfolio. Please show your work.