A Simple Explanation of The Prisoner's Dilemma
Introduction - Game Theory and the Prisoner's Dilemma
Game theory is commonly referred to in psychology, economics, political studies and even biology.
It tries to apply mathematical models to situations where multiple parties can either co-operate or be in conflict with each other.
The "game" is what the situation is referred to although it does not have to be a game as such.
It could be going to war, competing for food or running for political office.
One of the common "games" that is discussed is known as the prisoner's dilemma (PD).
This is actually a relatively simple problem to understand on a basic level but if you look at a matrix like the following it might actually not make much sense at first:
| Q | C | |
|---|---|---|
| Q | 1,1 | 3,0 |
| C | 0,3 | 2,2 |
I will try to present this in a simple and easy to understand manner without getting bogged down in the mathematical details.
I don't fully understand the mathematical component myself!
For me it is always easier to visualise situations in my mind and that is what I will try to use in my explanation.
Most of us have never been in prison or involved in criminal enterprises so I have constructed a situation which most of us are likely to be familiar with!
A Simple Example
The School Prank:
Jack and Susan both work together on planting stink bombs throughout the school as a prank. The prank is a great success resulting in the school having to be evacuated and all the children having to be sent home!
Their teacher Mrs Smith suspects the pair but does not have any evidence to prove it. Being a fan of game theory she separates the two of them and offers them both the same deal on their own:
"This is a serious matter. I know you were both involved in this but I can't prove it. If you confess now you can go home and your partner in crime will get 3 detentions. If you both confess you will get 2 detentions. If you both say nothing you will still get 1 detention."
What should Jack or Susan do?
- If neither of them confess then they both only suffer the single detention.
- If Jack confesses and Susan stays quiet, then he goes home and she gets 3 detentions.
- If Susan confesses and Jack stays quiet then she goes home and he gets 3 extra detentions.
- If they both confess then they both get 2 detentions.
We can express this in a table like so:
| Susan Quiet | Susan Confesses | |
|---|---|---|
| Jack Quiet | 1,1 | 3,0 |
| Jack Confesses | 0,3 | 2,2 |
- The first value to the left of the comma expresses the time in detention that Jack gets.
- The second value to the right of each comma expresses the time that Susan gets.
This is the basic convention used in these matrices.
Sometimes the options for each "player" may also be slightly raised or lowered to highlight the distinction.
I have also seen some tables that make them different colours for the same reason but I can't do that using Markdown!
So for example if we look at the top right situation where Jack is quiet but Susan confesses.
The result 3,0 indicates that Jack gets 3 detentions whereas Susan gets zero.
This is how to read these tables/matrices.
This can help us to break down the choices in a more simple visual way.
Jack's Options:
It can sometimes be easier to break things down to the individual viewpoints.
If we consider Jack's choices if he thinks that Susan will stay quiet we can simplify the matrix/table as follows:
| Susan Quiet | |
|---|---|
| Jack Quiet | 1 |
| Jack Confesses | 0 |
If he stays quiet he gets one detention. If he confesses he goes home without any further detention.
Therefore the optimal choice for him is to confess.
What if he thinks Susan will confess?
Let's isolate what happens to him then:
| Susan Confesses | |
|---|---|
| Jack Quiet | 3 |
| Jack Confesses | 2 |
In this case if Jack remains quiet he gets 3 detentions, if he confesses he gets 2. Again the best choice is confession.
This tells us that the best strategy for Jack regardless of what Susan does is to confess.
What about Susan?
If she knows Jack will be quiet:
| Susan Quiet | Susan Confesses | |
|---|---|---|
| Jack Quiet | 1 | 0 |
The situation is exactly the same for Susan as it was for Jack.
The best option is for her to confess as she goes home with 0 detentions.
If she thinks Jack will confess:
| Susan Quiet | Susan Confesses | |
|---|---|---|
| Jack Confess | 3 | 2 |
Again as with the situation with Jack, the optimal solution for her is to confess, because it only gives her 2 detentions vs 3.
The best strategy for Susan, as for Jack is to confess.
Confession is the Dominant Strategy
Since the strategy of confession is better for both parties than staying quiet, confession is said to dominate keeping quiet.
It might seem counterintuitive, since if both kept quiet they would only get 1 detention each.
The point is this is not the optimal strategy from either individual standpoint.
The way I conceptualise it is that dual confession is the equilibrium point if both parties are trying to minimise their punishment.
This becomes all the more important if you don't know for certain how your partner or opponent for that matter may behave.
In most real world situations I think that is actually the case.
Hierarchy of Options is More Important than Absolute Values
It should also be noted that in the Prisoner's dilemma the actual relative values of the punishments are not important.
All that matters is that there is a hierarchy of punishments or losses.
Conclusion
I hope you have found this explanation useful and easy to understand.
I am still learning about this and am by no means any kind of expert - I will apologise in advance for any mistakes or errors!
This is the basic PD but it there are numerous special variants.
According to a paper by Daniel et al there are (at least) 4 main types of PD games:
"the public/private benefits and costs distinction gives rise to four classes of PD games under the names of provision, commons, altruism, and selfish."
Things get complicated fast and frankly the mathematics is beyond my understanding - at least right now.
The most interesting thing for me when it comes to the PD is the numerous different examples and fields where it can crop up.
The binary choice represented by cooperation/betrayal can come up in situations as diverse as war and peace between two nations (or superpowers) or decisions about advertising within a corporation.
Where have you seen the Prisoner's Dilemma at work yourself? Let me know in the comments.
References
Spaniel, William. 2011. Game Theory 101: The Complete Textbook. CreateSpace Independent Publishing Platform.
Wikipedia contributors. 2017. “Prisoner’s Dilemma.” Wikipedia, The Free Encyclopedia. January 19. https://en.wikipedia.org/w/index.php?title=Prisoner%27s_dilemma&oldid=760941605.
Wikipedia contributors. 2017. “Game Theory.” Wikipedia, The Free Encyclopedia. January 20. https://en.wikipedia.org/w/index.php?title=Game_theory&oldid=761028022.
Daniel, G., M. Arce, and Todd Sandler. 2005. “The Dilemma of the Prisoners’ Dilemmas.” Kyklos International Review of Social Sciences 58 (1). Wiley Subscription Services, Inc., A Wiley Company. doi:10.1111/j.0023-5962.2005.00275.x.
Thank you for reading
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I should read more about that as well. I am also puzzled with the maths behind.
The maths behind it is still beyond me! I'm hoping to find some kind of simple guide for it.
think of the punnet square used to describe the chances of a baby having blue eyes ;-)
Yes it is very similar! What I find hard to get my head around is the statistical analysis and the pure maths style expressions which are used to describe them - the matrices - which are really just tables are not a problem.
Please share it as soon as you get it!
I will on my side try to write down some explanation about the other problem (gold and special relativity) we chatted a bit earlier today.
Will do I just hope I'm not too far away from my pure maths days to understand it. Most of the papers I have read just go straight to the complex stuff! I'm no John Nash lol!
^^
In addition to payback by the other party as mentioned by @nxtblg , one might also consider the ability of the one attempting to create the dilemma to enforce it.
In the above given example, if one child knows the following, they might well decide to remain silent even if they know the other will surely try to oust them:
In real life having enough knowledge (and gumption) can make staying silent the best choice in many cases by creating a scenario where the perceived cost/pain of enforcement is too high for the ones presenting the dilemma.
Please, don't do that again! Tonight I dreamed of calculating all these probabilities! :-)
Lol:)
What we play here on Steemit would be considered an iterative game. You might want to check out Axelrod and iterative prisoners dilemma (IPD).
Thanks will do:)
So what happened who confessesed? Lol I think I understood that :)
Lol they both did!
Ahh I was on the edge of my seat lol
:)
I don't know how you do it but overtime i come back online I see another awesome article :)
Thanks you are very kind:)
Funny: the example you give will only be accepted at face value by folks with less than a full ration of common sense. A more normal kind of fellow would add, "and the defector gets punished by the co-operator on the schoolyard."
That's the trouble with the standard prisoner's-dilemma game: it's sequestered from the processes of everyday life.
Not only in that way. Another criticism - taken up by the game-theory community when computers were added! - is that it's unrealistic because it's one-shot. Add computers to run what's called an "Iterated Prisoner's Dilemma", where players repeatedly match up against others for many rounds of the game, and you get this more common-sensical finding: the ideal strategy over the long term is Tit-For-Tat. Your first move should be co-operate, and your next move should be the same as the move that you yourself faced in the last round. If you got screwed over in Round 1, you should screw over in Round 2; if you got co-operation in Round 1, you should co-operate in Round 2; and so on...
I can't prove it, but I'm sure that the band Jefferson Airplane poked a hole in the standard PD model in a song that ribbed John Nash:
From the lyrics the chorus:
[Italicized part is the poke-through of the standard PD model]
It is, but it there is generally a bit of a disconnect (on both sides) between idealized mathematical models of useful things and the useful things themselves. These models are useful exercises and can teach us things but can't (or at least shouldn't) be mapped directly to practice.
It isn't quite that simple, there are other strategies that do better in different models such as forgiving tit-for-tat but again these are still models.
Thanks for your comments.
I never thought about that! I think that is the issue with studying human behaviour you can only model it so far in my opinion. I'm sure the experts will disagree but I can claim ignorance on that for now:)
Great song! Never heard it before.
Thanks! Funny: I didn't clue in until I read Sylvia Nasar's bio of John Nash, the original A Beautiful Mind She included two anecdotes from his grad-school days that sealed my hunch shut:
I really must look into Go - have heard so much about it lately but I know nothing about it! I think an AI defeated the no.1 player not too long ago.
You're right.
"In a Huge Breakthrough, Google’s AI Beats a Top Player at the Game of Go"
Yes thought so:)