Problem Set 5: Zero-Sum Games

Prepared by:

Joseph Malkevitch
Department of Mathematics
York College (CUNY)
Jamaica, New York 11451


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1. For each of the zero-sum matrix games below (named U, V, and W respectively), where payoffs are shown from the point of view of the row player:

a. Determine the value of the game using pure strategies if there is such a value. (If optimal play with pure strategies is possible what are these pure strategies?)

b. Determine if the game has a saddle point and if so indicate which cell is the saddle point and the value of the game.

c. What is the result of using dominant strategy analysis for the games?

d. Determine the value of the game using optimal mixed strategies for the two players if there is no optimal pure strategy solution.

e. Which, if any of these games is fair? Do you see a pattern?

Game U

  Column I Column II
Row 1 11 -7
Row 2 -8 4

Gave V


Column I

Column II
Row 1 6 -3
Row 2 -4 2

Game W

  Column I Column II
Row 1 -5 9
Row 2 2 -4

2. Because their is no pure strategy way of optimally playing the game below (payoffs shown from Row's point of view) each of the players separately decides that flipping a coin is the optimal way to find a mix of playing the rows by Row and of playing the columns by Column.

  Column I Column II
Row 1 8 -4
Row 2 -4 2

How does this decision compare for the payoffs of the players with the earnings that are available to the players by optimal play?

2. Find the value, if possible using dominant strategy analysis, looking for a saddle point, and/or using mixed strategy analysis for the two zero-sum games below with payoffs from Row's point of view. Do games X and Y have a value?

Game X

Row/Column I II III
1 6 9 4
2 7 8 10
3 5 10 2

Game Y

Row/Column I II III
1 -3 17 -15
2 1 4 3
3 -2 -4 14