Page 203 - Probability Demystified

192 CHAPTER 11 Game Theory

So Player B should play her black card 3 of the time and her white card 7
10 10
of the time. Player B’s payout when s ¼ 3 is
10

3 7
$5s $2ð1 sÞ¼ 5 $2
10 10
15 14
¼
10 10
1
¼ or $0:10
10
Hence the maximum amount that Player B will lose on average is $0.10
per game no matter what Player A does.
When both players use their strategy, the results can be shown by
combining the two tree diagrams and calculating Player A’s expected gain
as shown in Figure 11-3.

Fig. 11-3.

Hence, Player A’s expected gain is

9 21 21 49
$5 $2 $2 þ $1
100 100 100 100

45 42 42 49 10
þ ¼ ¼ $0:10
100 100 100 100 100

The number $0.10 is called the value of the game. If the value of the game
is 0, then the game is said to be fair.