Page 201 - Probability Demystified

190 CHAPTER 11 Game Theory

play my white card and hope Player B plays her white card, and I will win $1.
But she might realize this and play her black card! What should I do?’’
In this case, Player A decides that he should play his black card some of
the time and his white card some of the time. But how often should he play
his black card?
This is where probability theory can be used to solve Player A’s dilemma.
Let p ¼ the probability of playing a black card on each turn; then 1 p ¼ the
probability of playing a white card on each turn. If Player B plays her black
card, Player A’s expected profit is $5 p $2(1 p). If Player B plays her white
card, Player A’s expected profit is $2p þ $1(1 p), as shown in the table.

Player B’s Card

Player A’s Card Black White

Black $5p $2p

White $2(1 p) $1(1 p)

$5p $2(1 p) $2p þ $1(1 p)

Now in order to plan a strategy so that Player B cannot outthink Player A,
the two expressions should be equal. Hence,
5p 2ð1 pÞ¼ 2p þ 1ð1 pÞ

Using algebra, we can solve for p:
5p 2ð1 pÞ¼ 2p þ 1ð1 pÞ
5p 2 þ 2p ¼ 2p þ 1 p

7p 2 ¼ 3p þ 1
7p þ 3p 2 ¼ 3p þ 3p þ 1
10p 2 ¼ 1

10p 2 þ 2 ¼ 1 þ 2
10p ¼ 3

10p 3
¼
10 10
3
p ¼
10