Take No Prisoners!

Have you ever watched an episode of one of the many crime dramas on television (for example, Law and Order) and watched how the police interview (perhaps interrogate is a better word) criminal suspects?  It's particularly interesting when there is more than one suspect.  The police usually separate the two suspects and try to convince one suspect to testify against the other suspect.  It's really a brilliant move, and there is a classic problem in game theory that explains how it works.

The so-called Prisoner's Dilemma was first described by Merrill Flood and Melvin Dresher at the RAND Corporation in 1950.  The RAND Corporation was studying game theory in order to develop a better global nuclear strategy.  However, it was the Canadian mathematician Albert Tucker who first structured the rewards for the game in terms of prison sentences and named it "Prisoner's Dilemma".  William Poundstone further popularized the problem for a lay audience when he wrote his book, The Prisoner's Dilemma in 1993.  Poundstone explained the problem as follows: 

"Two members of a criminal gang are arrested and imprisoned. Each prisoner is in solitary confinement with no means of speaking to or exchanging messages with the other. The police admit they don't have enough evidence to convict the pair on the principal charge. They plan to sentence both to a year in prison on a lesser charge. Simultaneously, the police offer each prisoner a Faustian bargain."

A "Faustian Bargain" is also known as a "Deal with the Devil" or "Mephistophelian bargain" (after the classic German legendary figure Faust) and refers to the fact that in the game, the bargain doesn't end well for the individual making it.  Basically, the police ask each prisoner to either betray the other prisoner (testify against the other prisoner - this has also been called "defect" in the literature) and go free or remain silent (this has also been called "cooperate" in the literature) and serve a 1 year prison sentence on a lesser charge.  As outlined above, the possible outcomes to the game are as follows:

• A and B each betray the other (Defect/Testify) - each of them serves 3 years in prison

• A betrays B, but B remains silent - A is set free and B serves the maximum 5 years in prison

• A remains silent, B betrays A - A serves the maximum 5 years in prison and B is set free

• A and B both remain silent - each of them serve only one year in prison on the lesser charge

Shown as a 2x2 matrix, the model looks like this:

As you can see, the best option for both prisoners is to remain silent (i.e. "cooperate" with each other) and spend only 1 year in prison.  Remember though that the prisoners are acting independently.  How can they trust each other not to betray each other?  If only one betrays the other, he or she goes free (and the other goes to prison for five years).  However, if both of them betray each other, they go to prison for three years.  Since three years of prison is better than five years, they end up betraying each other rather than taking the risk of staying silent (which would have been the better option, hence, the dilemma).

Now, here is where the mathematician John Nash comes in.  You may have heard about Nash already, as his story was told in the 2001 movie, "A Beautiful Mind" starring Russell Crowe and Jennifer Connelly.  Nash made several early contributions to game theory, but one of the major concepts bears his name.  It's called the Nash Equilibrium and basically describes the scenario in which no player in a non-cooperative game (like Prisoner's Dilemma) has anything to gain by changing only their strategy.  The Nash Equilibrium in the Prisoner's Dilemma is the scenario in which both prisoners defect (testify against each other).  Importantly, the Nash Equilibrium doesn't always mean that the most optimal strategy is chosen.  Again, in the Prisoner's Dilemma, the best strategy would have been for both prisoners to cooperate.  Unfortunately however, even though mutual cooperation leads to the best possible outcome for both prisoners, if one prisoner chooses cooperation and the other does not, that prisoner's outcome is worse.

The Prisoner's Dilemma illustrates an important principle - when both players pursue their own self-interest, they end up worse off compared to if they had just cooperated.  As it turns out, there are a number of real world examples of the Prisoner's Dilemma problem in politics, business, and even health care.  For example, three investigators published an article several years ago entitled "The diffusion of medical technology: A 'Prisoner's Dilemma' trap?"  More recently, Dr. Niccie McKay suggested that the Prisoner's Dilemma is a potential barrier to collaboration and cooperation between health care organizations (see "The Prisoner's Dilemma: An Obstacle to Cooperation in Health Care Markets").

One can easily imagine setting up a Prisoner's Dilemma situation between two competitor hospitals, say Hospital A and Hospital B.  The strategic decision could involve whether to invest in an important clinical program (heart surgery program, transplant program, etc) or a specific health-care technology (robotic surgery or MRI).  The costs of either the clinical program or technology are significant, but so are the potential pay-offs in terms of new patients.  However, the pay-off are only significant if one hospital invests in the program - there are not enough patients in the region to support two similar programs.  The hospitals are faced with an impossible choice (a Faustian bargain if you will).  The risk of losing patients to their competitor is too great, so each hospital really has no choice but to invest in the new clinical program or technology!

The Prisoner's Dilemma is one of the most well-known problems in game theory.  We will keep on this line for a few more posts, before ending out the year, as I usually do, with my Top Ten list of posts and the 2023 Reading List.