Prisoner’s Dilemmas and White-Collar Prisoners, Part 1

One of the more popular kinds of technical economics is known as “game theory,” a mathematical treatment of people’s interactions as guided by their self-interest. Although this branch of mathematical economics has long been associated with John Forbes Nash, Jr., its originators were a mathematician and an economist. The mathematician was John von Neumann, who was also responsible for the basic conceptual design of the computer. The economist was Oskar Morganstern, who is known mainly for being the collaborator with John Von Neumann. The book the wrote together is called Theory of Games and Economic Behavior.

This book discussed many games, but its heart was an analysis of what was called the “two-person zero-sum game,” a phrase which is fairly self-explanatory. The example used in the book was a game between Sherlock Holmes and Professor Moriarty, which has long faded into the obscurity of history of economics. A game invented in 1950, the “Prisoner’s Dilemma,” is the one that’s survived the early years, in large part because it presents a counter-intuitive situation and taps into our suspicions about how other people will behave when our fate is in their hands.

The vintage Prisoner’s Dilemma game works like this: two prisoners are charged with serious crimes, and each are offered plea bargains. If one prisoner rats out the other, then that prisoner will walk; the other prisoner will do serious time…if that other fellow doesn’t plea bargain too. If both plea bargain, then both will spend a long time behind bars, but not as long as a prisoner who keeps his/her mouth shut while the other rolls. If both prisoners keep their mouths shut, then both walk away doing minor time. To be specific, the “payoff matrix,” detailing what each prisoner would be dinged with, depending upon what action each decides to pursue, is the following for the original version of the game:

- If both prisoners “co-operate” by remaining silent, each gets a 6-month sentence.
- If prisoner A “defects” by taking the plea bargain and B “co-operates,” then A goes free and B gets ten years in jail.
- If prisoner A “co-operates” and prisoner B “defects,” then A gets 10 years in jail and B goes free.
- If both prisoners “defect,” then they both get a 5-year sentence.

Like many mathematical treatments of human behavior, this situation is somewhat artificial. If A and B will both walk through staying silent, then there is no way that either will take the plea bargain, unless a positive reward is offered (and at least one of them doesn’t see it as carrying enough of a taint to make the reward not worth it.) Given this restrictive setting, though, a logical analysis of the game concludes that both A and B will defect, because it’s the best choice for them under uncertainty – even though the outcome of both decisions make them both worse off than the optimum result (both keeping their mouths shut) would yield. Hence, the name “Prisoner’s Dilemma.”

The reason for the dilemma is, each prisoner doesn’t know how the other will behave. All they have to go on is how they’ll fare in each situation:

- If a prisoner defects, then the two outcomes are: freedom or 5 years.
- If a prisoner co-operates, then the two outcomes are: 6 months or 10 years.

If 50-50 probabilities are assigned to the other fellow’s actions, then defection carries an average sentence of (0 + 5) / 2 = 2.5 years, whereas co-operation carries an average sentence of (0.5 + 10) / 2 = 5.5 years. It’s obvious that defection – rolling – carries the lower average, so any prisoner who wants to minimize his/her time in jail will go for the defection option. The same advantage to defection holds for all probability levels, thus making it a “dominating” strategy because the average jail sentence for a defecting prisoner is lower no matter what the odds of the other prisoner’s choice are.

And yet, this reasoning implies that both will defect, thus leading to a certain outcome that’s not the best for both.

Since this analysis is counter-intuitive, the Prisoner’s Dilemma game has been exhaustively analyzed, although (as far as I know) no-one has explored the implication that the sub-optimal outcome, combined with the out-of-reach optimal outcome, implies an entrepreneurial niche to break the sub-optimality. Since each prisoner will get 5 years from acting rationally if and only if the other prisoner’s behavior is unknown, but will get 6 months by acting rationally if the other prisoner is sure to co-operate, each prisoner stands to gain 4.5 years off his/her sentence if there were some way to secure verifiable co-operation in advance. This 4.5-year gain might very well be worth paying for in money, effort, and/or different kind of risk.

Also, one or both of the prisoners may put together a counter-game with, say, this kind of payoff matrix:

- Any prisoner who co-operates will be killed (yes, murdered) once outside of the can.

This kind of retaliation, of course, sets up a different prisoner's-dilemma situation once the murder is discovered.


[The second and final part is here.]