Prisoner’s Dilemma: Part 1

Two prisoners are locked up in separate cellblocks and accused of a crime.  The police give each prisoner a chance to confess.  If one confesses and the other keeps quiet, then the confessor will be set free and the other will face 20 years in prison.  If both prisoners confess, then each will face 10 years in prison.  If both players keep quiet, each will only face 5 years in jail.  What will they do?

This is the prisoner’s dilemma, one of the most studied games in game theory.  Above, we see a classic example of it with some arbitrary values.  There are many versions of it, but we’ll start with a technical definition (P,Q,R,S are variables):

Any version of the prisoner’s dilemma must have these properties:

– 2 players have only two strategies, A and B

– when both play strategy A, they get a negative outcome, P

– when players both play strategy B, they get negative outcome S worse than outcome P (S < P)

– when they play different strategies,

• the one who plays strategy A gets an outcome Q that is significantly worse than outcomes S or P (Q < S and Q < P)

• the one who plays strategy B gets an outcome R that is significantly better than outcomes S or P (R > S and R > P)

– there is NO communication between the two players

Those conditions must be satisfied for a game to be the prisoner’s dilemma (though it is easy to identify once you learn it).  There are variations too when communication is allowed, or values are changed.

Our matrix for the example:

Let’s do some analysis!  Prisoner 2 can either confess or stay silent.  If prisoner 2 confesses, prisoner 1 is better off confessing (10 years) than staying silent (20 years).  Likewise, if prisoner 2 stays silent, prisoner 1 is better off confessing (5 years) than staying silent (10 years).  Similarly, when we focus on prisoner 2’s strategies, we arrive at the same conclusion: he is better off confessing.

We conclude that no matter what the other prisoner does, a prisoner ALWAYS has a better outcome if he confesses.  Their best interests, it seems, lie in confessing every time.

So if both prisoners follow their best interests and do confess, they end up each receiving 10 years in prison.  However, is this really the best outcome?  If both players had stayed silent, they would have only received 5 years in jail.

Here’s our prisoner’s dilemma!  Individually looking at each prisoner, we see that their best interests lie in confessing.  However, if both do indeed follow their best interests, we see that they end up at 10 years, an outcome that is not the best outcome at all!  We have a case where individuals following their best interest ultimately create a position worse for everyone.

Price War Application:

Perhaps the greatest part of the prisoner’s dilemma is its diverse applications.  In economics especially, the prisoner’s dilemma refutes Adam Smith’s idea that following individuals’ best interests will better society.

In microeconomics, prisoner’s dilemma can be used to analyze competition between two companies.  Let’s say we have two soft beverage companies, CocaCola and PepsiCola.  We focus on their most popular and virtually identical product, cola drink.  Assume several things: their product is indistinguishable (customer preference only relies on pricing), the market size does not oscillate (the total number of customers between them does not change) and they are the only companies which sell this product.  In addition, their production costs and the price they charge are identical.  In other words, they are identical companies.

At any time, a company can advertise its product.  If one company does so without the other, then it takes customers from the other company, making one company better off at the expense of the other. If both companies decide to advertise, they still split the market evenly, but are worse off because of advertising fees.  If neither advertises, they still split the market evenly, but get to pocket the cash they would spend on advertising.  If we assign values to these strategies, we get a matrix:

This too is the prisoner’s dilemma!  Ever wonder why these companies spend so much during the Super Bowl?  Both are caught in the prisoner’s dilemma, and are unwilling to stop advertising unless the other does so first.  Price wars between two rival companies like these often happen, and in the end, no one’s a winner!

More Applications

In almost every field, you can find in some way an application of the prisoner’s dilemma.  It is everywhere!

In law, plea bargaining can be described as a prisoner’s dilemma.  When there are two suspects, there is an incentive for them to both confess rather than to stay silent, even if neither committed the crime.

In psychology, drug addiction can be explained by the prisoner’s dilemma.  A drug user has the option of relapsing today, or tomorrow, if not at all.  An analysis of the scenario shows that a drug user is caught in a position where relapsing today becomes his best interest, therefore making him addicted as the same scenario is played the next day.

Even in Las Vegas, a prisoner’s dilemma can be played among the players when there is a poker tournament.  The casino sometimes offers players additional chips for money that isn’t placed in the pool.  Every player tries to buy the chips, making no player better off and leaving the casino with some nice profits.

The prisoner’s dilemma also makes an appearance in movies, such as the Dark Knight, and in TV shows as well.  Here’s one of Dilbert.

And that is your little introduction to the prisoner’s dilemma!  Part I features the classic dilemma, price wars, and other applications of the prisoner’s dilemma.  In Part 2, we’ll feature some more applications, iterated prisoner’s dilemma, tragedy of commons and the good ol’ tit for tat.  Stay tuned…

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