Page 209 - Probability Demystified

198 CHAPTER 11 Game Theory

Player B

Player A Rock Tree

Pistol 0.5 0.2

Rifle 0.3 0.8

3. Person A has two cards, an ace (one) and a three. Person B has two
cards, a two and a four. Each person plays one card. If the sum of the
cards is 3 or 7, Person B pays Person A $3 or $7 respectively, but if
the sum of the cards is 5, Person A pays Person B $5. Construct a
payoff table, determine the optimal strategy for each player, and the
value of the game. Is the game fair?
4. A street vendor without a license has a choice to open on Main Street
or Railroad Avenue. The city inspector can only visit one location per
day. If he catches the vendor, the vendor must pay a $50 fine; other-
wise, the vendor can make $100 at Main Street or $75 at Railroad
Avenue. Construct the payoff table, determine the optimal strategy
for both locations, and find the value of the game.
5. Find the optimal strategy for Player B in the last example (basketball).

ANSWERS
1. Let p be the probability of running and 1 p ¼ the probability of
passing. Then
2p þ 10ð1 pÞ¼ 5p 6ð1 pÞ

2p þ 10 10p ¼ 5p 6 þ 6p

8p þ 10 ¼ 11p 6

8p 11p þ 10 ¼ 11p 11p 6

19p þ 10 ¼ 6

19p þ 10 10 ¼ 6 10