Page 202 - Probability Demystified

CHAPTER 11 Game Theory 191

3
Hence, Player A should play his black card 10 of the time and his white
7
card 10 of the time. His expected gain, no matter what Player B does, when
p ¼ 3 is
10

3 3
5p 2ð1 pÞ¼ 5 21
10 10

15 7
¼ 2
10 10
15 14
¼
10 10
1
¼ or $0:10
10
On average, Player A will win $0.10 per game no matter what Player B does.
Now Player B decides she better figure her expected loss no matter what
Player A does. Using similar reasoning, the table will look like this when the
probability that Player B plays her black card is s, and her white card with
probability 1 s.

Player B’s Card:

Player’s A Card: Black White

White $5s $2(1 s) $5s $2(1 s)

Black $2s $1(1 s) $2(s) þ $1(1 s)

Solving for s when both expressions are equal, we get:

5s 2ð1 sÞ¼ 2ðsÞþ 1ð1 sÞ
5s 2 þ 2s ¼ 2s þ 1 s
7s 2 ¼ 3s þ 1
7s þ 3s 2 ¼ 3s þ 3s þ 1
10s 2 ¼ 1
10s 2 þ 2 ¼ 1 þ 2
10s ¼ 3
10s 3
¼
10 10
3
s ¼
10