1. INDUSTRIAL ENGINEERING DEPARTMENT
Introduction to Operations Research III
Game Theory
1. A rural Midwestern county is served by two commercial banks, Farmer’s First Bank and
Rancher’s First Bank. Total deposits in the two banks area approximately equal. The state
has recently passed a law that, for the first time, will allow banks to have branches within the
county. Farmer’s First Bank has decided on full-service branches. It has the capital to build a
maximum of two of these branches. Market studies indicate that each of these branches will
add Php6 million deposits to the bank. These deposits will be taken from Rancher’s First
Bank.
Rancher’s First Bank has decided to expand with automated electronic tellers, rather than
full-service branches. It has the capital to install a maximum of three of these tellers. It is
estimated that each of these installations will add Php4 million in deposits, which will be
taken from Farmer’s First Bank. Let Farmer’s First Bank be player X and Rancher’s First
Bank be player Y. The manager of each bank would like to maximize total deposits.
Formulate this as a two-person, zero-sum game.
2. A game called “Rock, Scissors, and Paper” is played as follows. Two players simultaneously
choose one of three strategies: rock, scissors, and paper. If both players choose the same
strategy, no points are awarded to either player. If one player chooses scissors and the other
player chooses paper, then the player choosing scissors gains 1 point and the player choosing
paper loses 1 point. (This is because “scissors cut paper.”) If scissors and rock are
competing strategies, then the person choosing rock gains1 point and the person choosing
scissors loses 1 point. (This is because “rock breaks scissors.”) finally, since “paper covers
rock,” a person choosing paper would win 1 point while a person choosing rock would lose 1
point. Formulate this as a two-person, zero sum game.
3. Triple River City is divided intro three major sections by the joining of 3 rivers, as shown in
the accompanying figure.
B C
30% 30%
A
40%
Of the city’s residents, 40% live in section A, 30% in section B, and 30% in section C. At
present, Triple River City has no ice skating rinks. Two companies, X and Y, have plans to
build rinks in the city. Company X will build two rinks, one each in two of the town’s three
sections. Company Y will build only one rink. Each company knows that if there are twon
rinks in a given section of town, the two rinks will split that sections’ business. If there is
only one rink is a section of town, that rink will receive all of that section’s business. If there

2. is no rink built in a particular section, the business from that section will be split equally
among the city’s three rinks. Each company would like to locate its rinks or rink in an area
that would maximize its market share. Formulate this situation as a game from Company X’s
point of view.
4. Determine these strategies and the value of the game for the following:
a) b)
Y1 Y2 Y3 Y1 Y2 Y3 Y4
X1 0 4 −1 X1 13 7 12 7
X2 3 −1 − 2 X2 4 3 8 5
X3 9 7 6 X 3 12 7 13 7
5. Use the method of dominance to reduce each of the following games to a 2 x 2 game.
a) b)
Y1 Y2 Y3 Y4 Y1 Y2 Y3
X1 0 4 1 −5 X1 4 8 3
X2 −3 3 2 −4 X2 2 10 2
X3 5 −1 − 3 3 X3 0 3 6
6. Solve each of the following games.
a) b)
Y1 Y3
Y1 Y2 Y3
X1 0 3
X1 40 15 20
X2 −4 0
X2 10 20 30
X3 −2 5
c)
Y1 Y2 Y3
X1 −1 3 2
X2 11 − 1 5
7. Use the simplex method of linear programming to determine the value of the game and the
optimal mixed strategy for player Y.
a) b)
Y1 Y2 Y3
Y1 Y2 Y3 X1 0 4 2
X1 −1 3 6 X2 2 1 4
X2 11 − 1 − 4
X3 3 3 1