Page 199 - Probability Demystified

188 CHAPTER 11 Game Theory

payoff. When a game consists of the loser paying the winner, it is called a zero
sum game. This means that the sum of the payoffs is zero. For example, if a
person loses a game and that person pays the winner $5, the loser’s payoff
is $5 and the winner’s payoff is þ$5. Hence the sum of the payoff is
$5 þ $5 ¼ $0.
Consider a simple game in which there are only two players and each
player can make only a finite number of moves. Both players make a move
simultaneously and the outcome or payoff is determined by the pair of
moves. An example of such a game is called, ‘‘rock-paper-scissors.’’ Here
each player places one hand behind his or her back, and at a given signal,
brings his or her hand out with either a fist, symbolizing ‘‘rock,’’ two fingers
out, symbolizing ‘‘scissors,’’ or all five fingers out symbolizing ‘‘paper.’’ In
this game, scissors cut paper, so scissors win. A rock breaks scissors, so the
rock wins, and paper covers rock, so paper wins. Rock–rock, scissors–
scissors, and paper–paper are ties and neither person wins. Now suppose
there are two players, say Player A and Player B, and they decide to play for
$1. The game can be symbolized by a rectangular array of numbers called
a payoff table, where the rows represent Player A’s moves and the columns
represent Player B’s moves. If Player A wins, he gets $1 from Player B. If
Player B wins, Player A pays him $1, represented by $1. The payoff table
for the game is

Player B’s Moves:

Player A’s Moves: Rock Paper Scissors

Paper 0 $1 $1

Rock $1 0 $1

Scissors $1 $1 0

This game can also be represented by a tree diagram, as shown in
Figure 11-1.
Now consider a second game. Each player has two cards. One card is
black on one side, and the other card is white on one side. The backs of all
four cards are the same, so when a card is placed face down on a table,
the color on the opposite side cannot be seen until it is turned over. Both
players select a card and place it on the table face down; then they turn the
cards over. If the result is two black cards, Player A wins $5. If the result
is two white cards, Player A wins $1. If the results are one black card and