The Prisoner's Dilemma
The prisoners dilemma have been a rich source
of research material since the 1950's. However, the publication of Axelrod's
book [AXE84] in the 1980's was largely
responsible for bringing this research to the attention to other areas, outside
of game theory, including evolutionary computing, evolutionary biology,
networked computer systems and promoting cooperation between opposing
countries. Although the prisoners dilemma, in the context of game theory, has
been an active research area for almost 60 years [SCO63, SCO62,
SCO60a, SCO60b, SCO59a,
SCO59b] (it can be traced back to von
Neumann and Morgenstern [VON44] and, of
course, John Nash [NAS53, NAS50]), it is still an active research area
with a large number of scientific articles published every year.
The prisoners dilemma has a modern day version in the form of the TV
show "Shafted"  a game show recently screened on terrestrial TV in
the UK (note that this show is not a true prisoners
dilemma as defined by Rapoport [RAP96], but does demonstrate that the ideas
have wider applicability). At the end of the show two contestants have
accumulated a sum of money and they have to decide if to share the money or to
try and get all the money for themselves. Their
decision is made without the knowledge of what the other person has decided to
do. If both contestants cooperate then they share the money. If they both
defect then they both receive nothing. If one cooperates and the other defects,
the one that defected gets all the money and the contestant that cooperated
gets nothing.
In the prisoners dilemma (PD) you have to decide whether to cooperate with an
opponent, or defect. Both you and your opponent make a choice and then your
decisions are revealed. You receive a payoff according to the following matrix
(where the top line is the payoff to the column).

The opponent 

Cooperate 
Defect 

You 
Cooperate 
R=3 R=3 
T=5 S=0 
Defect 
S=0 T=5 
P=1 P=1 
The question arises: what should you do in such a game?
Suppose you think the other player will cooperate. If you cooperate then you
will receive a payoff of 3 for mutual cooperation. If you defect then you will
receive a payoff of 5 for the Temptation to Defect payoff. Therefore, if you
think the other player will cooperate then you should defect, to give you a
payoff of 5.
But what if you think the other player will defect? If you cooperate, then you
get the Sucker payoff of zero. If you defect then you would both receive the
Punishment for Mutual Defection of 1 point. Therefore, if you think the other
player will defect, you should defect as well.
So, you should defect, no matter what option your opponent chooses. Of course,
the same logic holds for your opponent. And, if you both defect you receive a
payoff of 1 each, whereas, the better outcome would have been mutual
cooperation with a payoff of 3. The dilemma each prisoner faces is that mutual
defection seems to be inevitable but the payoff of mutual defection is less
than that could have been achieved by two cooperating players.
Mutual defection is a Nash equilibrium of PD.
Informally, a set of
strategies is a Nash equilibrium if no player can do
better by unilaterally changing his or her strategy. Mutual defection is the unique Nash equilibrium of PD, which means that
it is the only stable solution to this game. In real world scenarios, however, a Nash equilibrium is not necessarily played. Some
conditions to guarantee that the Nash equilibrium is played are: