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.

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

B 30%

C 30%

A 40%

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)

679213140

3

2

1

321

XXX

YYY

−−−

7137125834712713

3

2

1

4321

XXX

YYYY

5. Use the method of dominance to reduce each of the following games to a 2 x 2 game.

a) b)

331542335140

3

2

1

4321

−−−−−

XXX

YYYY

6302102384

3

2

1

321

XXX

YYY

6. Solve each of the following games.

a) b)

302010201540

2

1

321

XX

YYY

520430

3

2

1

31

−−

XXX

YY

c)

5111231

2

1

321

−−

XX

YYY

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)

4111631

2

1

321

−−−

XX

YYY

133412240

3

2

1

321

XXX

YYY