Mathematical Modeling: Practive VI: 2x2 Games

prepared by:

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

email:

malkevitch@york.cuny.edu

web page:

http://york.cuny.edu/~malk/

Matrix Games

1. The matrix below shows an example of a game (payoffs in dollars) involving two players, row and column.:

 Column I Column II Row 1 (6, 16) (-3, 10) Row 2 (-5, 9) (1, 3)

a. If Row plays Row 2 and Column plays Column II what is the payoff to Row and what is the payoff to column?

b. If Row plays Row 2 and Column plays Column I what is the payoff to Row and what is the payoff to column?

c. If Row plays Row 1 and Column plays Column II what is the payoff to Row and what is the payoff to column?

d. If Row plays Row 1 and Column plays Column I what is the payoff to Row and what is the payoff to column?

2. Does Row have a dominating strategy in the game in Exercise 1?

3. Does Column have a dominating strategy in the game in Exercise 1?

4. Is the game in Exercise 1 a zero-sum game?

5. How would you advise Row to play the game in Exercise 1?

6. How would you advise Column to play the game in Exercise 1?

7. Given the game below, would you rather be Row or Column?

 Column I Column II Row 1 (64, -64) (-8, 8) Row 2 (-8, 8) (1, -1)

8. Given the game below, would you rather be Row or Column?

 Column I Column II Row 1 (14, -14) (-8, 8) Row 2 (-10, 10) (4, -4)

9. Given the game below, would you rather be Row or Column?

 Column I Column II Row 1 (10, -10) (-6, 6) Row 2 (-10, 10) (4, -4)

10. Given a row or a column of a matrix game, one can determine for that row or column the minimum entry in the payoff for Row or Column.

a. For the game in Exercise 9 find the row minimum for Row 2.

b. For the game in Exercise 9 find the column minimum for Column II.

11. Given a row or a column of a matrix game, one can determine for that row or column the maximum entry in the payoff for Row or Column.

a. For the game in Exercise 9 find the row maximum for Row 1.

b. For the game in Exercise 9 find the column maximum for Column II.

12. (Technical and somewhat involved) Can you work out the conditions under which a 2x2 zero-sum matrix game which has no dominating row or column is "fair" in the sense that if the game is played many times, in the long run the earnings of each or the players is zero?