This story leads us into our discussion of the prisoner's dilemma.
The prisoner's dilemma is an example of a game analyzed in game theory that shows why two completely rational individuals (or group of individuals) might not cooperate, even if it appears that it is in their best interests to do so.
Here's how it works:
Two members of a gang are arrested and imprisoned. They are each put into solitary confinement and have no way of communicating with each other. Each prisoner has two options. They can either remain silent and cooperate with the other prisoner, or they can betray the other prisoner by testifying against them.
The options are as follows for prisoner A and prisoner B:
• A and B both remain silent so both of them will serve 1 year in prison
• A and B both betray each other so both of them will serve 2 years in prison
• A betrays B but B remains silent so A will be set free but B will serve 3 years in prison
• B betrays A but A remains silent so B will be set free but A will serve 3 years in prison
http://static1.businessinsider.com/image/5756c8da9105841d008c7255-1200/prisoner%27s-dilemma.png
It should be noted that both prisoners understand the game. Even though they are both in the same gang, they have no loyalty to each other and will have no opportunity to reward or punish the other. Since each prisoner is assumed to be rationally self-interested, each prisoner would betray the other because betraying offers a greater reward than cooperating.
We can see this is true through the following scenarios:
• If B cooperates then A should betray because going free is better than serving 1 year.
• If B betrays then A should betray because serving 2 years is better than serving 3 years.
Either way, A should betray B. The same reasoning shows that B should always betray A.
Hence, the only possibility for all purely rational self-interested prisoners would be for them to betray each other and each would then serve 2 years.
Since betrayal always leads to a better payoff for both prisoners, it is considered a dominant strategy. Well, what's a dominant strategy then? Strategic dominance is simply when one strategy is better than another strategy for one person, no matter how their opponent plays. Since both players have a dominant strategy, the prisoner's dilemma has only one unique Nash equilibrium. However, there are other non-equilibrium outcomes that would be better for both players.
In the prisoner's dilemma, the Nash equilibrium occurs when both players betray each other. Again, this is the only nash equilibrium because it is the only outcome from which each player has nothing to gain by changing their strategy. The reason why mutual cooperation is not a nash equilibrium is because each prisoner could improve their outcome by breaking this mutual cooperation.
You may be asking, well what's the dilemma then?
The dilemma is that pursuing individual reward and acting rationally will logically lead both prisoners to betray each other. If both of them remained silent and cooperated then they would get a better reward (less jail time). However, this is not the rational outcome because the choice to cooperate is irrational.
Here are some questions to consider for discussion:
Do you think that the prisoners would act any differently if the prison times were vastly different like the ones below?
• A and B both remain silent so both of them will serve 5 years in prison
• A and B both betray each other so both of them will serve 20 years in prison
• A betrays B but B remains silent so A will be set free but B will serve life in prison
• B betrays A but A remains silent so B will be set free but A will serve life in prison
The idea of life in prison seems pretty tough but do you still think the prisoners will betray each other and risk serving 20 years in prison versus 5?
Can anyone think of any other real life examples of the prisoner's dilemma?
That's about all I have for now on this topic. If anyone has any further questions or needs clarification on anything please let me know.
Sources:
Packel, Edward W. The mathematics of games and gambling. Washington: Mathematical Association of America, 2008
"Prisoner's dilemma." Wikipedia. March 26, 2017. Accessed April 05, 2017. https://en.wikipedia.org/wiki/Prisoner%27s_dilemma.
"Nash equilibrium." Wikipedia. April 05, 2017. Accessed April 05, 2017. https://en.wikipedia.org/wiki/Nash_equilibrium#Prisoner.27s_dilemma.
"Strategic dominance." Wikipedia. March 26, 2017. Accessed April 05, 2017. https://en.wikipedia.org/wiki/Strategic_dominance.
You mentioned the example of increasing the punishment for both prisoner's betraying each other above, to try and get them to cooperate. Did you find any examples in your research where mathematicians/scientists actually studied this? At what point did people stop betraying each other and start cooperating? Or did they never cooperate?
ReplyDeleteI never actually found any information on whether increased prison time would change their individual decisions. This was more of just an opinion question. However, I still think that no matter how much you increase the prison times, a pair of rational individual thinkers will always choose to betray each other since they don't know what the other person will do
DeleteI would have to agree with Zach. There has definitely been studies done on this, but the majority of them have been psychology related...such as the Stanford Prison Experiment. This experiment in particular involved a lot of betrayal...but it was from both the prison guards, and the prisoners.
DeleteThere is no exact answer to this question because it involves individual preferences. Some people value future time more than others... And what if a person enjoys their stay in jail?
ReplyDeleteThis question has to do with a group of rational thinkers. All purely rational self-interested prisoners would betray each other because betraying offers a greater reward than cooperating. No rational thinker would prefer to stay in jail longer
DeleteThere was an example I read online about Prisoner's Dilemma applying to women wearing makeup which I thought was interesting. Society would likely be better off if no women wore make-up as a great deal of time and money could be saved if make-up didn't exist. Now obviously if nobody wore makeup then there would be a great temptation for one women or a small group of women to wear makeup. Once that group of women grew in size to the tipping point, the women who were still not wearing makeup wouldn't be perceived as beautiful. This leads to the situation we are faced with today where most women wear makeup - even though it isn't beneficial for all women, it is the rational decision to make.
ReplyDeletehttps://www.quora.com/What-is-a-good-real-world-example-of-the-prisoner-s-dilemma-in-recent-history
Has anyone ever gathered research on multiple cases where the Prisoner's Dilemma can be applied?
ReplyDeleteFor Example: an analyst studies all cases in NY between 1980-present where 2 prisoners are arrested and put to trial (like your first example). Let's assume the analyst looks at cases where the jail time isn't as high, say less than 5 years worst case scenario and determines the percentage for all four cases and compares and contrasts them?
Has there been any research like this conducted where we can see the percentages of the outcomes?
http://www.businessinsider.com/prisoners-dilemma-in-real-life-2013-7
DeleteHere's a good article from business insider where two economists studied the dilemma on students and prisoners. The payoffs weren't years of jail time, but money for students, and value of coffee and cigarettes for prisoners. Only 37% of students cooperated while 56% of prisoners cooperated. On a pair basis, only 13% of student pairs managed to get the best mutual outcome and cooperate, while 30% of prisoner pairs did.
Like I said, the payoffs are not as serious as jail time but the results show that prisoners aren't necessarily as calculating and un-trusting as you might expect. The theory that economics tells us does not line up exactly with real behavior.
My Art History SYE class is a good real-life example of prisoner's dilemma. There are 8 people in the class. Everyone gets 1 skip, and there are penalties beyond that. But, if half or more of the class skips, then the professor just cancels class and the fact that people skipped on that day doesn't count as their 1 skip or for any penalties. So, skipping class can lead to class being canceled or to penalties.
ReplyDelete