Assignment: Practice 1 (Spring, 2013) (Zero-Sum Games)

Prepared by:

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

email:

malkevitch@york.cuny.edu

web page:

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

1. The matrix below shows the payoffs of a zero-sum game from Row's point of view.

.
 Row/Column I II III 1 3 2 -1 2 9 -3 -4 3 0 5 7

Write down a version of the game matrix where the payoff entries in the cells have the form (a, b) where a is Row's payoff and b is Column's payoff.

2. For each of the games below:

a. Indicate if any of the rows dominate other rows

b. Indicate if any of the columns dominate other columns

c. Does the game have a value in "pure strategies?" If the game has a value, what is this value?

d. If there are dominating rows and/or columns write down the matrix that results after carrying out a dominating row and column analysis.

e. If after completing dominating row and column analysis you obtain a 2x2 game, find the optimal mixed strategy for each player. (This means designing a spinner for each of the players and determining the "value" of the game that occurs when these spinners are used to play the game optimally.)

f. If the game has a "saddle point," what cell of the matrix corresponds to the saddle point?

Game I

 Row/Column I II III 1 (2, -2) (6, -6) (-5, 5) 2 (-3, 3) (4, -4) (-6, 6) 3 (3, -3) (-3, 3) (2, -2)

Game II

 Row/Column I II III 1 (-2, 2) (-1. 1) (-2, 2) 2 (-1, 1) (5, -5) (3, -3) 3 (5, -5) (1, -1) (3, -3)

Game III

 Row/Column I II III 1 (1, -1) (-2, +2) (8, -8) 2 (4, -4) (3, -3) (7, -7) 3 (-2, 2) (0, 0) (-4, 4)