probability 413

The Game of Chicken

An interesting variation of the prisoner’s dilemma game

is the game of chicken. It is modeled on the dangerous

driving game in which two drivers head directly toward

one another on a single-lane road. Each driver must

decide at the last minute whether to swerve to the right,

or to not swerve. To bring mathematics into the game,

it is convenient to introduce numerical values for the

outcomes of the game.

Suppose 10 points are assigned to the winner of the

game: she who chooses not to swerve given that the

other driver did. As this is embarrassing to the second

driver, 0 points are awarded to her. If both drivers

decide to swerve, then, as this is only mildly embarrass-

ing to each, assign to each player 5 points. On the other

hand, if both players decide not to swerve, then the out-

come is disastrous and assign –10 points to each driver

in this instance. The outcomes of the game of chicken

can be summarized in the following two payoff tables:

Notice that neither player has a dominant strategy

in the game of chicken: his or her best action does

depend on what choice the other driver is going to

make. Moreover, the game of chicken has two distinct

Nash equilibria: (Swerve, Not Swerve) and (Not

Swerve, Swerve). (It is to neither player’s advantage to

deviate from one of these options.) This suggests that a

compromise (Swerve, Swerve) is not easy to achieve.

Many conflicts in real life, such as labor-manage-

ment disputes and international trade conflicts, have

the flavor of a game of chicken. As we know, the out-

comes of such disputes do indeed vary: one party may

decide to “cave in” so as to avoid a disastrous out-

come, while many times neither party surrenders, and

strikes and wars result.

See also J

OHN

N

ASH

.

probability The principles of probability theory were

first identified by 16th-century Italian mathematician

and physician G

IROLAMO

C

ARDANO

(1501–76), and

later by French mathematicians B

LAISE

P

ASCAL

(1623–62) and P

IERRE DE

F

ERMAT

(1601–65). The key

idea is that if a situation can be described in terms of

possible outcomes, each equally likely, then the proba-

bility of any particular outcome is defined to be the

number 1 divided by the total number of outcomes.

For example, if a die is cast, six outcomes are possible:

{1,2,3,4,5,6}. We usually believe that each outcome is

equally likely, and so we say that the probability of

rolling any one particular outcome, such as a 5 for

example, is 1/6.

More generally, if more than one possible outcome

is deemed acceptable, then we define the probability of

obtaining one of these outcomes to be ratio of the

number of desired outcomes to the total number of

outcomes. For example, there are three ways to roll an

even number when casting a die. Thus we say that the

probability of casting a multiple of 2 is 3/6 = 1/2. The

chances of casting any of the numbers 1, 4, 5, or 6 are

4/6 = 2/3.

Mathematicians call the set of all possible out-

comes of an experiment the

SAMPLE SPACE

, and any

particular set of outcomes (or just a single outcome) an

EVENT

. For example, in casting a die, the sample space

is the set {1,2,3,4,5,6}, and an event could be the sub-

set {2} (rolling a two), for example, or {2,4,6} (rolling

an even number). An event is always a subset of the

sample space.

If Arepresents an event, that is, a set of desirable

outcomes, then the notation P(A) is used to denote the

probability of that event occurring, that is, the proba-

bility that the outcome from one run of the experiment

will belong to the set A. A value P(A) is always

between zero and 1, with value P(A) = 0 indicating

that event Awill never occur and value P(A) = 1 indi-

cating that event Awill always occur. In casting a die,

for example, we have P(even) = , P({1,4,5,6}) = ,

P(a multiple of 7) = 0, and P(a whole number) = 1.

Assigning probabilities to events requires careful

counting. Often the greatest difficulty is identifying

2

–

3

1

–

2