Page 207 - Probability Demystified

196 CHAPTER 11 Game Theory

EXAMPLE: Player A and Player B decided to play one-on-one basketball.
Player A can take either a long shot or a lay-up shot. Player B can defend
against either one. The payoff table shows the probabilities of a successful
shot for each situation. Find the optimal strategy for each player and the
value of the game.

Player B (defense)

Player A (offense) Long shot Lay-up shot

Long shot .1 .4

Lay-up shot .7 .2

SOLUTION:
Let p be the probability of shooting a long shot and 1 p the probability of
shooting a lay-up shot. Then the probability of making a shot against a long
shot defense is 0.1p þ 0.7(1 p) and against a lay-up defense is 0.4p þ
0.2(1 p). Equating and solving for p we get

0:1p þ 0:7ð1 pÞ¼ 0:4p þ 0:2ð1 pÞ
0:1p þ 0:7 0:7p ¼ 0:4p þ 0:2 0:2p

0:6p þ 0:7 ¼ 0:2p þ 0:2

0:6p þ 0:7 0:2p ¼ 0:2p þ 0:2 0:2p

0:8p þ 0:7 ¼ 0:2

0:8p þ 0:7 0:7 ¼ 0:2 0:7

0:8p ¼ 0:5

0:8p 0:5
¼
0:8 0:8
5
p ¼
8