The Prisoner dilemma

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Grayson

Senior Member
Messages
1,079
The Prisoner dilemma

Game Theory and the "Prisoners? Dilemma"

The nebulous concept called "Game Theory" is intriguing. It holds the promise of inserting rigidity into the fluidity of probability and of improving the odds of winning at poker. It appears to be the perfect way to insert certainty in an inherently uncertain world

Von Neumann was an American-Hungarian Mathematician who, in 1928, set out to develop a strategy for winning at poker. The more he thought about this challenge the more he realized that the principles involved might have far ranging implications for any situation that involved choices based on conflicting human motivations.

Game Theory is a branch of mathematics that tries to logically analyze and evaluate the optimum solution to problems involving conflicting human situations. It aims to insert mathematically defined strategy into problems involving emotions and probability.

Game theory has matured from a poker strategy to an all-encompassing strategy for predicting human behavior in situations involving multiple, conflicting choices. Cold war warriors supposedly used it extensively to determine the effectiveness of nuclear strategy.

What are the mathematical aspects of a Game Theory? This theory can be illustrated by a two-person game known as the Prisoners? Dilemma. This particular dilemma is a fascinating thought experiment with implications far beyond its entertaining qualities:

Two men are suspected of having committed a serious crime together. After they are arrested, they are separated and cannot communicate with each other. The Prosecution lacks sufficient proof to convict them. He visits each of them and makes each of them, separately, an offer:

If you turn State?s evidence and help me convict your friend, you will be freed, but your friend will get 20 years in prison for refusing to cooperate. I am making this same offer to your friend. If you do not take my offer and if your friend cooperates, you are the one who will spend 20 years in prison.

If both of you take me up on my offer and both of you implicate each other independently, you will both get 5 years because you are obviously guilty, but your confessions will save me a trial.

However, if neither one of you cooperates with the police, you will most likely get a sentence of one year on a lesser charge. You will definitely not get off free.

This situation is not really a game because it is played out in prison every day. It is euphemistically referred to as Plea Bargaining. What should each of them do? Cooperate with the Prosecutor, or not? Betray his friend, or not? What are the underlying mechanics?

Here is the dilemma, concisely: Each suspect can only choose between two options: Confess or not confess.

1. He confesses and implicates his friend and he gets:

Either 5 years in jail, if his friend also confesses independently

If he confesses but his friend does not confess, he will get off free, but his friend will get 20 years in jail.

2. He does not confess or implicate his friend and he gets:

Either 1 year on a minor charge, provided his friend holds out, too

Or he gets 20 years if his friend defects and betrays him

The essence of this dilemma lends itself to several possible solutions: Each prisoner has a choice of two options but cannot make a good decision without knowing what the other one will do.

A parallel to the Prisoner's Dilemma occurs in competitive market situations. What will my competitor do if I raise prices? In global confrontations: Should we strike first or should we wait for the enemy to launch an attack? In spousal relationships: Will she still make love to me even if I don?t take out the garbage?

In the case of the prisoners, the mathematical pay-off matrix still leaves everybody in a conundrum: If one of the prisoners betrays his buddy and gives evidence against him, he is rewarded by going free, but only if his friend does not cooperate with the Law. This situation involves grave risks for both prisoners. If both of them decide to testify against each other, both of them will go to prison for six years. Only if both of them have the absolute certainty that the other one will not confess, can they minimize their losses by not confessing at all. In this case, they will both end up with only one year in jail. The catch is, that neither one of them can have this absolute certainty of cooperation because they cannot communicate with each other.

This scenario points out the limitations of mathematics as exemplified in the Game Theory. Human actions are always based on two factors: 1. Emotional motivators, also called impulses, are governed by the evolutionary need of every living organism to always act in what a person considers to be in his best self-interest. This emotion-based process does not necessarily rely on logical sequences. 2. A thin veneer of rationality, the ability of our intellect to make logical decisions, tempers these raw emotions. However, basic human motivators remain anchored deeply in our emotions.

The Conclusion: Mathematics is a completely rational process and does not yield to the irrationality and emotionality of human emotions. Therefore, Game Theory provides an illusion of being helpful in resolving conflicting human situations. Human decisions must take into account, not merely various alternatives and the rational connectivity of mathematics and logic, but the unpredictability of human emotions. The only common denominator that applies to human emotions is the irrefutable axiom: Every human being always acts in what he considers to be in his best self-interest, in order to eliminate pain and enhance his feeling of well being.

The human disposition to always engage in a course of action that a person considers to be in his best self-interest, is often misunderstood because it seems to eliminate the concept of altruism and cooperation. Altruism is indeed one of many manipulative human mirages that sounds noble but that does not exist in reality: A person referred to as an altruist merely commits altruism because he enjoys helping others, not because he hates doing so. He is selfishly doing what he enjoys doing. Therefore, what he does to help others actually is done to make himself feel better, to benefit himself.

In many cases, altruism is merely the attempt to compensate for imagined sins or for guilt. The concept of cooperation works a little differently than so-called altruism because both parties clearly imply that their cooperation will benefit all parties involved: They do not try to hide their selfish motivations behind a facade of altruism.

Whatever a person decides to do may not actually be in best interest, but he would not act in a specific manner unless he thought that he was furthering his best self-interest. Any person, who willfully acts in a way that he actually believes to be against his best interest, would be confined in a mental institution in order to protect him from his own actions.

Even a person attempting suicide is acting in what he considers to be his best self-interest because the pain in his life has become so unbearable that he considers the alternative, death, a pleasant escape and thus in his best self-interest.

The above principles of human behavior are essentially based on our inherently irrational emotions, rather than our rational mind, and are thus impossible to quantify by rational mathematical analysis. This perspective may not be palatable to some mathematicians but the question remains: Why does nobody actually utilize the mathematics of Game Theory?

Many mathematicians may go through the calculations provided by Game Theory. However, are the results of their highly touted mathematical concepts of value to anyone? The answer is self-evident because poker players do not pay the slightest attention to Game Theory; neither do the politicians who have control over our nuclear warheads.

All these players have one thing in common: Instead of using Game Theory, they simply use Common Sense, the understanding of human nature and emotions based on their own life-experience. Not even the most intricate matrixes of Game Theory can teach a poker player how to bluff.

Nuclear strategy during the Cold War revolved around the simple, common sense understanding that any nuclear war could only result in MAD, Mutually Assured Destruction. Nobody needs complex mathematical matrixes and formulas to come to this conclusion.

Instead of using Game Theory, we use common sense, based on our understanding of the fact that we will always do what we consider to be in our best self-interest, and that other persons will do precisely the same. This is the reason why all wars will end when all nations achieve nuclear capability. The Mexican Standoff, in the form of nuclear weapons strategy, has been successfully operative during the Cold War and is the only effective deterrent to war.

The medieval monk/philosopher Ockham formulated the maxim of common cense known as Ockham?s Razor: The simplest solution to any problems is the best solution. The simplest solution to human conflicts is not a complex Game Theory, but will always be plain and simple Common Sense, based on our empirical knowledge of how real people really behave.

The knowledge represented by the Game Theory is actually background information. Game Theory is similar to Quantum Mechanics and Relativity: They are spheres of knowledge that we need to be marginally aware of but that do not contribute one iota to our happiness or to a successful life.

Game Theory can be interesting as background knowledge for any person who insists on seeing the largest possible segment of Objective Reality and who would like to obtain the widest possible perspective of human existence. However, abstract mathematics and Game Theory are irrelevant to our achievement of Happiness.

Game Theory is somewhat akin to Economics: Economists generate mostly hot air although they use extremely elaborate mathematical formulas. Very few, if any, economists are millionaires. No economist can predict interest rates or stock prices even from one day to the next. In real life, no poker player or prisoner uses the elaborate mathematics of Game Theory.

If prisoner A could know with absolute certainty, based on his knowledge of the character of prisoner B, that the other prisoner will not confess, than he is best off choosing what is obviously in both his and in their mutual best self-interest: They must refuse to confess and receive one year in jail, the minimum sentence

However, if prisoner A is doubtful in the least whether prisoner B will confess or not, it is in his best self-interest to confess immediately and he thus has a good chance of going free. Even if the other prisoner confesses, too, he will only get six years in jail. This alternative is much more desirable than to rely on the dubious expectation that the other prisoner will not confess. This stance will result in 20 years in jail, the worst of all alternatives.

Even Mafia members inform on each other, without regard to their blood oath to remain silent. For anyone caught in the Prisoner's Dilemma there can be only one solution: Make the best deal possible with the Law, without regard to anyone else.

Every human being will always act in what he considers to be in his best self-interest. This dictum, not Game Theory, remains the supreme evaluator and predictor for human behavior.

The best way to avoid the Prisoners Dilemma is not to commit a crime or, if you have decided to commit a crime, not to involve any accomplices. Do it yourself.

Edit: Text edit
 

Phoenix

Active Member
Messages
622
The Prisoner dilemma

The math involved in the prisoner dilemma scenario would work as follows.

Expected value if you tell.

Let us say a 50/50 chance of your friend telling.

.50*5 +.50*0 = expected value of 2.5 years in jail if you tell.

Expected value if you do not tell.

.50*1+.50*20= expected value of 10.5 year in jail if you do not tell.

Thus assuming a 50/50 chance it is better if you tell.

Now say you do trust that your partner in crime is 90% likely to not tell on you.

If you tell on this friend

.10*5 +.9*0 = expected value of .5 years in jail.

Expected value if you do not tell

.90*1 + .10*20 = expected value of 2.9 years in jail.

It is still to your advantage to tell on your friend even if he is 90% likely to not tell on you.

now it can be computed just how trustworthy you would need to consider your friend for either alternative to be equally advantageous.
t = chance your friend tells
d = chance your friend doesn't tell
t + d = 1
t*5 +d*0=d*1+t*20

t = 1 - d
5t =20t + d

15t = -d
-t = -1 +d
14t = -1
t = -.07
In this scenario it is impossible for your friend to be trustworthy enough for him to lesson your sentence.

You would need to trust your friend 107% of the time which is impossible.

Now you may modify these results to. Say, for you not telling on a friend is actually worth spending 15 years in jail for. In your own perspective.

Then the results in the 50/50 evaluation would be as follows.

17.5 virtual years if you tell on your friend
10.5 actual years if you don't tell on your friend.

Here, because how much you value not telling on your friend it becomes worth the risk of more time in jail.
 

Grayson

Senior Member
Messages
1,079
The Prisoner dilemma

Originally posted by Phoenix@Jul 6 2004, 12:07 AM
The math involved in the prisoner dilemma scenario would work as follows.

Expected value if you tell.

Let us say a 50/50 chance of your friend telling.

.50*5 +.50*0 = expected value of 2.5 years in jail if you tell.

Expected value if you do not tell.

.50*1+.50*20= expected value of 10.5 year in jail if you do not tell.

Thus assuming a 50/50 chance it is better if you tell.

Now say you do trust that your partner in crime is 90% likely to not tell on you.

If you tell on this friend

.10*5 +.9*0 = expected value of .5 years in jail.

Expected value if you do not tell

.90*1 + .10*20 = expected value of 2.9 years in jail.

It is still to your advantage to tell on your friend even if he is 90% likely to not tell on you.

now it can be computed just how trustworthy you would need to consider your friend for either alternative to be equally advantageous.
t = chance your friend tells
d = chance your friend doesn't tell
t + d = 1
t*5 +d*0=d*1+t*20

t = 1 - d
5t =20t + d

15t = -d
-t = -1 +d
14t = -1
t = -.07
In this scenario it is impossible for your friend to be trustworthy enough for him to lesson your sentence.

You would need to trust your friend 107% of the time which is impossible.

Now you may modify these results to. Say, for you not telling on a friend is actually worth spending 15 years in jail for. In your own perspective.

Then the results in the 50/50 evaluation would be as follows.

17.5 virtual years if you tell on your friend
10.5 actual years if you don't tell on your friend.

Here, because how much you value not telling on your friend it becomes worth the risk of more time in jail.

I knew that, honest. I said that, see, look up. :D

The logic of it makes more sense than the maths to me. I do understand the math of it, but find it easier to describe mathematical situations with language rather than numbers for the most part. Used to drive my math teacher potty.

Anyway, thanks for that Phoenix.
 


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