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<center><h1>Introduction - Game Theory and the Prisoner's Dilemma</center></h1>
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##### 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!
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<center><h1>A Simple Example</center></h1>
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## The School Prank:

https://i.imgsafe.org/66d550c17e.jpg

>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."
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## What should Jack or Susan do?
https://i.imgsafe.org/66f5a4c799.jpg

1. If neither of them confess then they both only suffer the single detention.
2. If Jack confesses and Susan stays quiet, then he goes home and she gets 3 detentions.  
3. If Susan confesses and Jack stays quiet then she goes home and he gets 3 extra detentions.
4. If they both confess then they both get 2 detentions.
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## 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.
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## 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.  
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## 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.
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## Confession is the Dominant Strategy
https://i.imgsafe.org/66f595ef1a.jpg

##### 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.
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## 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.
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<center><h1>Conclusion</center></h1>
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https://i.imgsafe.org/66d544031d.jpg

##### 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.
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<center><h1>References</center></h1>
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1. Spaniel, William. 2011. Game Theory 101: The Complete Textbook. CreateSpace Independent Publishing Platform.

2. 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.

3. Wikipedia contributors. 2017. “Game Theory.” Wikipedia, The Free Encyclopedia. January 20. https://en.wikipedia.org/w/index.php?title=Game_theory&oldid=761028022.

3. 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.
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