Discussion Questions and Problems

Discussion Questions

  1. M4-1 What is a two-person, zero-sum game?

  2. M4-2 How do you compute the value of the game?

  3. M4-3 What is a pure strategy?

  4. M4-4 Explain the concept of dominance. How is it used?

  5. M4-5 How is a saddle point found in a game?

  6. M4-6 How do you determine whether a game is a pure strategy game or a mixed strategy game?

  7. M4-7 What is a mixed game, and how is it solved?

Problems

  1. M4-8 Determine the strategies for X and Y, given the ­following game. What is the value of the game?

    Y1 Y2
    X1 2 −4
    X2 6 10

  2. M4-9 What is the value of the following game and the strategies for A and B?

    B1 B2
    A1 19 20
    A2 5 −4

  3. M4-10 Determine each player’s strategy and the value of the game, given the following table:

    Y1 Y2
    X1 86 42
    X2 36 106

  4. M4-11 What is the value of the following game?

    S1 S2
    R1 21 116
    R2 89 3

  5. M4-12 Player A has a $1 bill and a $20 bill, and player B has a $5 bill and a $10 bill. Each player will select a bill from the other player without knowing what bill the other player selected. If the total of the bills selected is odd, player A gets both of the two bills that were selected, but if the total is even, player B gets both bills.

    1. Develop a payoff table for this game. (Place the sum of both bills in each cell.)

    2. What is the best strategy for each player?

    3. What is the value of the game? Which player would you like to be?

  6. M4-13 Resolve Problem M4-12. If the total of the bills is even, player A gets both of the bills selected, but if the total is odd, player B gets both bills.

  7. M4-14 Solve the following game:

    Y1 Y2
    X1 −5 −10
    X2 12 8
    X3 4 12
    X4 −40 −5

  8. M4-15 Shoe Town and Fancy Foot are both vying for more share of the market. If Shoe Town does no advertising, it will not lose any share of the market if Fancy Foot does nothing. It will lose 2% of the market if Fancy Foot invests $10,000 in advertising, and it will lose 5% of the market if Fancy Foot invests $20,000 in advertising. On the other hand, if Shoe Town invests $15,000 in advertising, it will gain 3% of the market if Fancy Foot does nothing; it will gain 1% of the market if Fancy Foot invests $10,000 in advertising; and it will lose 1% if Fancy Foot invests $20,000 in advertising.

    1. Develop a payoff table for this problem.

    2. Determine the various strategies using the computer.

    3. How would you determine the value of the game?

  9. M4-16 Assume that a 1% increase in the market means a profit of $1,000. Resolve Problem M4-15, using monetary value instead of market share.

  10. M4-17 Solve for the optimal strategies and the value of the following game:

    A table shows the strategies of “A” and “B.”

  11. M4-18 For the following two-person, zero-sum game, are there any dominated strategies? If so, eliminate any dominated strategy and find the value of the game.

    PLAYER Y’s STRATEGIES
    PLAYER X’s STRATEGIES Y1 Y2 Y3
    X1 4 5 10
    X2 3 4 2
    X3 8 6 9

  12. M4-19 Refer to Problem M4-18. There is a saddle point in this game, making it a pure strategy game. Ignore this and solve it as a mixed strategy game. What special condition in the solution indicates that this should not have been solved as a mixed strategy game?

  13. M4-20 Petroleum Research, Inc. (A), and Extraction International, Inc. (B), have each developed a new ­extraction procedure that will remove metal and other contaminants from used automotive engine oil. The equipment is expensive, and the extraction process is complex, but the approach provides an economical way to recycle used engine oil. Both companies have developed unique technical procedures. Both companies also believe that advertising and promotion are critical to their success. Petroleum Research, with the help of an advertising firm, has developed 15 possible strategies. Extraction International has developed 5 possible advertising strategies. The economic outcome in millions of dollars is shown in the following table. What strategy do you recommend for Petroleum Research? How much money can it expect from its approach?

    A table shows the strategies of “A” and “B.”

Bibliography

  • Bierman, H., and L. Fernandez. Game Theory with Economic Applications, 2nd ed. New York: Addison-Wesley, 1998.

  • Bowen, Kenneth Credson, with contributions by Janet I. Harris. Research Games: An Approach to the Study of Decision Process. New York: ­Halstead Press, 1978.

  • Brandenburger, A. and B. Nalebuff. “The Right Game: Use Game Theory to Shape Strategy,” Harvard Business Review (July–August 1995): 57—71.

  • Bushko, David, and Michael Raynor. “Consulting’s Future, Game Theory, and Storytelling,” Journal of Management Consulting 9, 4 (November 1997): 3–6.

  • Davis, M. Game Theory: A Nontechnical Introduction. New York: Basic Books, Inc., 1970.

  • Dixit, A. K., and Susan Skeath. Games of Strategy. New York; W.W. Norton and Co., 1999.

  • Dutta, Prajit. Strategies and Games: Theory and Practice. Cambridge, MA: MIT Press, 1999.

  • Fudenberg, D., and D. K. Levine. The Theory of Learning in Games. ­Cambridge, MA: MIT Press, 1998.

  • Koselka, Rita. “Playing Poker with Craig McCaw,” Forbes (July 3, 1995): 62–64.

  • Lucas, W. “An Overview of the Mathematical Theory of Games,” ­Management Science 8, 5, Part II (January 1972): 3–19.

  • Luce, R. D., and H. Raiffa. Games and Decisions. New York: John Wiley & Sons, Inc., 1957.

  • Shubik, M. The Uses and Methods of Game Theory. New York: American ­Elsevier Publishing Company, 1957.

  • Sinha, Arunava. “The Value Addition Game,” Business Today (February 7, 1998): 143.

  • von Neumann, J., and O. Morgenstern. Theory of Games and Economic ­Behavior. Princeton, NJ: Princeton University Press, 1944.

  • Yuan, L. L. “Property Acquisition and Negotiation Styles: An Asian Case Study,” Real Estate Finance 15, 1 (1998): 72–78.

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