412 prisoner’s dilemma

The height of a prism is defined to be the distance

between the two parallel planes that contain the bases

of the figures. The

VOLUME

Vof a prism is given by the

area of its base multiplied by the figure’s height:

V= area of base ×height

This follows from C

AVALIERI

’

S PRINCIPLE

.

A prismatoid is a polyhedron whose vertices lie in

one or the other of two parallel planes. The two bases

of the figure are not required to be congruent, nor even

have the same number of vertices. All lateral faces in a

prismatoid are either triangular or quadrilateral. If the

number of vertices of each base polygon is the same

and each lateral face is a quadrilateral, then the prisma-

toid is called a prismoid.

prisoner’s dilemma In

GAME THEORY

, any two-per-

son variable-sum game of partial conflict that mimics the

following classic scenario is called a prisoner’s dilemma:

Two prisoners, held in separate rooms incom-

municado, must choose to either confess or

deny involvement in a team crime. If both con-

fess, then each will be sentenced to two years

of hard labor. If both deny involvement, then

each will be sentenced to four years of hard

labor. However, if one prisoner denies and the

other confesses, then the denial carries just one

year of hard labor, and the confession six.

The following tables show the expected payoffs for

each prisoner Xand Yfor each of the four possible

outcomes of the game:

Each prisoner can argue as follows:

I have no indication as to what my partner will

do. If he is to choose option C, to confess, then

it is to my advantage to choose option D, to

deny. If he is to choose option D, then, again,

it is to my advantage to choose D. Either way,

I should choose option D.

The choice of Dis thus a dominant strategy for

each prisoner, and it is likely that both prisoners will

deny involvement in the crime. Moreover, it is worth

noting that neither player is tempted to deviate from

this choice in an attempt to trick his opponent in the

game: the risk of being the only confessor inhibits this.

Thus the outcome (D,D) is a stable outcome for the

game, and both prisoners will likely each be sentenced

to four years of hard labor. (Any outcome to a game,

such as (D,D) for the prisoner’s dilemma, is called a

“Nash equilibrium” for the game if no player can bene-

fit by departing unilaterally from it.)

The prisoner’s dilemma provides a

PARADOX

:

In the game of prisoner’s dilemma, each player

has a dominant strategy that, when used,

yields an outcome to the game that is less ben-

eficial than if both were to deviate from the

dominant strategy.

This phenomenon is also seen in the predicament of an

arms race between two nations: mutual disarmament is

of benefit to both nations, but the fear of an opposing

nation choosing to defect from such an agreement

inhibits cooperation.

Elements of the prisoner’s dilemma can be extended

to games involving more than two players. For exam-

ple, a teacher asks each of his students to write on a

piece of paper his or her name and either the word

cooperate or the word defect. The students know that

candy pieces will be distributed among the class

according to the following rules:

If each student chooses to cooperate, then each

will receive 10 pieces.

If two or more students defect, then all

will be punished. Those that cooperate will

receive only five pieces of candy, and the defec-

tors shall receive none.

If, on the other hand, there is a single bold

student willing to be the lone defector, then

that defector will receive 80 candy pieces, and

all other students none.