Wednesday, October 12, 2011

Game Theory: Part V: Non-Zero-Sum Games: The Simple Prisoner's Dilemma

Sometimes, we have the option of cooperating with someone else or refusing to cooperate. Sometimes, others have the option of cooperating with us or refusing to cooperate. Elimination of dominated strategies is not always the best option.

The following material is flagged Green Level. It is intended to reflect material that the author believes to be a matter of consensus among experts in the field. This belief may be incorrect, however; and as the author is not an expert and does not have an expert fact-checking the article, errors may creep in.
Let's look at another non-zero-sum game called the Prisoner's Dilemma. The standard setup goes something like this: There are two people who were captured by the police while trying to rob a bank. The police have enough evidence to get them convicted of unlicensed possession of firearms (a crime carrying a sentence of one year in this world), but offer them a deal: each prisoner can turn state's evidence and go free, but if they do, the other will be imprisoned for twenty years. However, if both try this, they will both be imprisoned for five years.



Cooperate

Defect
Cooperate (-1,-1) (-20, 0)
Defect(0,-20)(-5, -5)
As you can see, each prisoner does better by snitching (the option labeled "Defect"), but both do better if the other stays silent (the option labeled "Cooperate"). The best solution for each prisoner is to defect while their partner cooperates, and the worst is to cooperate while their partner defects. It is a simple matter to find that the Nash equilibrium (remember that from last time?) is that both prisoners defect.
But the Nash equilibrium is not the best outcome.There are outcomes, called the Pareto optimals, which are outcomes that cannot be moved from without leaving at least one player worse off. In a way, you could think of it as sort of a dominant outcome. As you can see, there are three Pareto optimals, none of which are the Nash equilibrium (the optimals being the outcomes with at least one cooperation).
Incidentally, the Prisoner's Dilemma is something that shows up absolutely everywhere, from the actual prison system to economics to psychology. To give one example, take taxes. No one actually likes paying them, but most people recognize that if no one paid them, the government wouldn't be able to do anything. The best situation for any person is to not pay taxes while everyone else does, but (again, to most people; Libertarians have something of a tendency to disagree with this) everyone paying taxes is a preferable outcome to no one paying taxes, since (yet again, any Libertarians out there will disagree) a small part of one's paycheck/inheritance/dividends/whatever is preferable to a government that is unable to provide roads, police, defense, and so forth.
So how do we solve this? How do we get the prisoners to cooperate?
(Keep your hands down, those of you who already know an answer.)
For one solution, tune in next week. Or maybe the week after that. Same Bat-time, same Bat-channel.
<<Non-Zero-Sum Games: Chicken | Game Theory | Cyclic Games: The Iterated Prisoner's Dilemma I: Reciprocal Relationships>>

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