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11.4: Chapter Review

  • Page ID
    37946
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    1. Determine whether the games are strictly determined. If the games are strictly determined, find the optimal strategies for each player and the value of the game.
      1. \(\left[\begin{array}{ll}
        2 & 3 \\
        3 & 4
        \end{array}\right]\)
      2. \(\left[\begin{array}{ccc}
        0 & 3 & -1 \\
        1 & 3 & -2 \\
        -1 & 2 & -5
        \end{array}\right]\)
      3. \(\left[\begin{array}{ccc}
        3 & 2 & -1 \\
        5 & 3 & 4
        \end{array}\right]\)
      4. \(\left[\begin{array}{cc}
        4 & 2 \\
        -1 & 3 \\
        4 & 3 \\
        1 & -3
        \end{array}\right]\)
    2. Two players play a game which involves holding out a nickel or a dime simultaneously. If the sum of the coins is more than 10 cents, Player I gets both the coins; otherwise, Player II gets both the coins.
      1. Write a payoff matrix for Player I.
      2. Find the optimal strategies for each player and the value of the game.
    3. Lacy's department store is thinking of having a major sale in the month of February, but does not know if its competitor store Hordstrom's is also planning one. If Lacy's has a sale and Hordstrom's does not, Lacy's sales go up by 30%, but if both stores have a sale simultaneously, Lacy's sales go up by only 5%. On the other hand, if Lacy's does not have a sale and Hordstrom's does, Lacy's loses 5% of its sales to Hordstrom's, and if neither of the stores has a sale, Lacy's experiences no gain in sales.
      1. Write a payoff matrix for Lacy's.
      2. Find the optimal strategies for both stores.
    4. Mr. Halsey has a choice of three investments: Investment A, Investment B, and Investment C. If the economy booms, then Investment A yields 14% return, Investment B returns 8%, and Investment C 11%. If the economy grows moderately, then Investment A yields 12% return, Investment B returns 11%, and Investment C 11%. If the economy experiences a recession, then Investment A yields a 6% return, Investment B returns 9%, and Investment C 10%.
      1. Write a payoff matrix for Mr. Halsey.
      2. What would you advise him?
    5. Mr. Thaggert is trying to decide whether to invest in stocks or in CD's(Certificate of deposit). If he invests in stocks and the interest rates go up, his stock investments go down by 2%, but he gains 1% in his CD's. On the other hand if the interest rates go down, he gains 3% in his stock investments, but he loses 1% in his CD's.
      1. Write a payoff matrix for Mr. Thaggert.
      2. If you were his investment advisor, what strategy would you advise?
    6. Determine the optimal strategies for both the row player and the column player, and find the value of the game.
      1. \(\left[\begin{array}{cc}
        2 & -2 \\
        -2 & 2
        \end{array}\right]\)
      2. \(\left[\begin{array}{cc}
        -2 & 2 \\
        5 & 0
        \end{array}\right]\)
      3. \(\left[\begin{array}{cc}
        3 & 5 \\
        4 & -1
        \end{array}\right]\)
      4. \(\left[\begin{array}{cc}
        -2 & 5 \\
        4 & -3
        \end{array}\right]\)
    7. Find the expected payoff for the given game matrix G if the row player plays strategy R, and the column player plays strategy C.
      1. \(G=\left[\begin{array}{cc}
        3 & 5 \\
        4 & -1
        \end{array}\right] \quad R=\left[\begin{array}{ll}
        1 / 2 & 1 / 2
        \end{array}\right] \quad C=\left[\begin{array}{l}
        1 / 4 \\
        3 / 4
        \end{array}\right] \nonumber\)
      2. \(G=\left[\begin{array}{cc}
        -2 & 5 \\
        4 & -3
        \end{array}\right] \quad R=\left[\begin{array}{ll}
        2 / 3 & 1 / 3
        \end{array}\right] \quad C=\left[\begin{array}{l}
        1 / 3 \\
        2 / 3
        \end{array}\right]\)
    8. A group of thieves are planning to burglarize either Warehouse A or Warehouse B. The owner of the warehouses has the manpower to secure only one of them. If Warehouse A is burglarized the owner will lose $20,000, and if Warehouse B is burglarized the owner will lose $30,000. There is a 40% chance that the thieves will burglarize Warehouse A and 60% chance they will burglarize Warehouse B. There is a 30% chance that the owner will secure Warehouse A and 70% chance he will secure Warehouse B. What is the owner's expected loss?
    9. Two players play a game which involves holding out a nickel or a dime. If the sum of the coins is odd, Player I gets both the coins, and if the sum of the coins is even, Player II gets both the coins. Determine the optimal strategies for both the row player and the column player, and find the expected payoff.
    10. A football quarterback has to choose between a pass play or a run play depending on how the defending team is going to react. If he chooses a pass play and the defending team is expecting a pass, he expects to gain 4 yards, but if the defending team is expecting a run, he gains 20 yards. On the other hand, if he calls a run play and the defending team expects a pass, he gains 7 yards, and if he calls a run play and the defending team expects a run, he loses 2 yards. If you were the quarterback, what would your strategy be?
    11. The Watermans go fishing every weekend either at Eel River or at Snake River. Unfortunately, so do the Nelsons. If both families show up at Eel River, the Watermans can hope to catch only 3 fish, but if the Watermans fish at Eel River and the Nelsons at Snake River, the Watermans can catch as many as 12 fish. On the other hand, if both families fish at Snake river, the Watermans can catch about 5 fish, and if Watermans fish at Snake river while the Nelsons fish at Eel river, the Watermans can catch up to 15 fish. Determine a mixed strategy for the Watermans, and the expected payoff.
    12. Terry knows there is a quiz tomorrow, but does not remember whether it is in his math class or in his biology class. He has time to study for only one subject. If he studies math and there is a quiz in it, he gains 10 points and even if there is no quiz he gains two points for acquiring the extra knowledge which he will apply towards the final exam. If he studies biology and there is a quiz in it, he gains ten points but there
    13. Reduce the payoff matrix by dominance. Find the optimal strategy for each player and the value of the game.
      1. \(\left[\begin{array}{ccc}
        -3 & 1 & 2 \\
        -3 & 5 & 3 \\
        2 & 4 & -1
        \end{array}\right]\)
      2. \(\left[\begin{array}{lll}
        1 & 2 & 3 \\
        4 & 1 & 4 \\
        2 & 3 & 4 \\
        1 & 2 & 2
        \end{array}\right]\)
      3. \(\left[\begin{array}{cccc}
        4 & 3 & 9 & 7 \\
        -7 & -5 & -3 & 5 \\
        -1 & 4 & 5 & 8 \\
        -3 & -5 & 1 & -1
        \end{array}\right]\)
      4. \(\left[\begin{array}{cccc}
        2 & 3 & 1 & 5 \\
        -2 & 2 & 1 & 3
        \end{array}\right]\)
      5. \(\left[\begin{array}{cccc}
        0 & 3 & 2 & 1 \\
        0 & 2 & 1 & -7 \\
        -4 & -9 & 5 & 4 \\
        4 & -7 & 6 & 6
        \end{array}\right]\)
      6. \(\left[\begin{array}{cccc}
        1 & 0 & 2 & 2 \\
        2 & 2 & 0 & 2 \\
        2 & -3 & 0 & 4 \\
        2 & -2 & -3 & 2
        \end{array}\right]\)

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