The Prisoner's Dilemma

By Leon Felkins, written 9/5/95

Note: this section was revised on 6/12/98

General Model of the Two player Game

The Prisoner's Dilemma model as presented by Robert Axelrod, Douglas Hofstadter, and others (See References at end), goes as follows:

Two prisoners, lets call them Joe and Sam, are being held for trial. They are being held in separate cells with no means of communication. The prosecutor offers each of them a deal. He also disclosed to each that the deal was made to the other. The deal he offered is this:

Put yourself in Joe's position. If Sam stays mum and you sing, you get zero years. If he stays mum and you stay mum, you will each get 2 years. On the other hand if both of you confess, you both get 4 years. Finally, if he confesses and you don't, you will get 5 years. Whatever Sam does, it is to your advantage to admit your wrong doing. Of course, Sam is also a rational person and he will, therefore, come to the same conclusion. So you both end up confessing which nets a total of 8 man-years in the pokey. The paradox is, if you had both denied the crime, a total of only 4 man-years would be spent behind bars. Wait a minute! Can it really be that rationality leads to an inferior result? Let's look at this one more time. We will use a payoff matrix, a common tool of the game theoreticians. The payoff matrix is usually presented in the following form: 
      ACTION                        PAYOFF
  Joe       Sam               Joe        Sam
 Cooperate  Cooperate         -2 (R)     -2 (R)
 Cooperate  Defect            -5 (S)      0 (T)
 Defect     Cooperate          0 (T)     -5 (S)
 Defect     Defect            -4 (P)     -4 (P)
 
 (The codes represent standard terminology for each action:
 
     R       Reward for mutual cooperation
     S       Sucker's payoff
     T       Temptation to defect
     P       Punishment for mutual defection                 )

The general form of the Prisoner's Dilemma model is that the preference ranking of the four payoffs be, from best to worst, T, R, P, S and that R be greater than the average of T and S. That is, any situation that meets these conditions will be a "Prisoner's Dilemma".

In summary, the Prisoner's Dilemma model postulates a condition in which the rational action of each individual is to not cooperate (that is, to defect), yet, if both parties act rationally, each party's reward is less that it would have been if both acted irrationally and cooperated!

The model can be applied to many real world situations, from genetics to business transactions to international politics.

Iterated "Prisoner's Dilemma" with multiple participants

If the game is played only once there is no incentive for either player to do anything but defect, as discussed above. In fact, if the game is to be played a known number of rounds, there is no better choice than to defect. (Why? Because you both know you will defect on the last move. So, scratch off the last move. That puts you in the same situation for the next to the last move - and so on for all.) But if the game is to be played an indefinite number of times, under certain conditions, cooperation can evolve as the best policy.

Another addition to the game that makes it more realistic is to assume that each player interacts with a multitude of other players. Additionally, it can be assumed that each player remembers the past history of the interactions with each of the other players and that past history is the only information he has.

The Iterated "Prisoner's Dilemma" has been the subject of much study and computer simulation (see references). An interesting and possibly useful result of these studies is that a player's best strategy in this "game" is "Tit for Tat", with the additional proviso that the player be initially cooperative. That is, "I'll start off being nice but from that point on, whatever you do to me, I will do to you on the next interaction". This strategy has been shown to be clearly more productive than "The Golden Rule"!

Note that we are discussing multiple participants in which activities are between pairs of "actors". There is yet another more complex situation in which an individual is interacting with ALL of the other participants at once. This situation, which is more common in the real world, is called the "Many- person-dilemma" or, in my terminology, the "Voter's Paradox". See the companion essay, "Voter's Paradox" at this and other sites.

A new book that provides much greater insight into the Prisoner's Dilemma and the possibility of cooperation is The Origins of Virtue by Matt Ridley (see the references). Matt presents the idea that cooperation may have evolved in spite of the "rationality" of defection. That is, we may be more like ants than we would like to admit!

Author: Leon Felkins
Email: [email protected]

References:
  1. Axelrod, Robert. 1984. The Evolution of Cooperation. New York: Basic Books. A summary of his work is on the internet at this site.
  2. Hofstadter, Douglas R. 1983, "Metamagical Themas: Computer Tournaments of the Prisoner's Dilemma Suggest How Cooperation Evolves". Scientific American 248 (no.5):16-26.
  3. On the Internet: http://pespmc1.vub.ac.be/PRISDIL.html or click here. Author: F.Heylighen. Date: Apr 13, 1995 (modified)
  4. See the PARC papers on Dynamics for some simulation results
  5. For an introduction to Game theory, check out Al Roth's Game Theory and Experimental Economics Page
  6. Related games ("Game of Chicken", etc.) are briefly discussed in "CURRENT RESEARCH IN SOCIAL PSYCHOLOGY"
  7. Ridley, Matt. 1996. The Origins of Virtue. New York: Viking.
  8. A non-technical, easy to read book: Murnighan, J. Keith. 1992. Bargaining Games. New York: William Morrow and Co. (A chapter is on line here)
  9. Several other essays are on the Internet. You can look at the Constitution site for several references or just do a search on "Prisoner's Dilemma".
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