Print Page - The Poker Player's Prisoners Dilemma

The mathematical concept called "Game Theory" was something I studied when at University. At the risk of stopping you reading now, it holds the promise of inserting rigidity into the fluidity of probability and was originally developed to study the improving the odds of winning at poker. Don't worry, it gets simpler.

John 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 realised that the principles involved might have far ranging implications for any situation that involved choices based on conflicting human motivations. Von Neumann went on to invent the digital computer, played a key role in the development of the atom bomb and became a defender of a movement to bomb the Russians before they could bomb us (and game theory became huge in Cold War politics). In addition to discovering game theory of course. All because he wanted to win at poker, allowing myself modest dramatic licence.

Now, Game Theory is a huge branch of mathematics that tries to logically analyse and evaluate the optimum solution to problems involving conflicting human situations. It aims to insert mathematically defined strategy into problems involving emotions and probability and has matured from an obscure academics' desire to beat his mates at cards to an all-encompassing strategy for predicting human behavior in situations involving multiple, conflicting choices.

One of the best known applications of Game Theory is shown by a two-person game known as the Prisoners’ Dilemma. It goes as follows:

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 Prosecutor 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 co-operate. 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 co-operates 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 the judicial system every day world wide. It is euphemistically referred to as Plea Bargaining. What should each of them do? Co-operate 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.

In the case of the prisoners, the conundrum is clear:

If one of the prisoners betrays his accomplice and gives evidence against him, he is rewarded by going free, but only if his friend does not co-operate with the lawyer. 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 minimise 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 co-operation 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.

So in essence 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 (or to use the layman's term "a complete waste of time") 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 axiom that we have come to commonly assume: 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.

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 in poker because poker players do not pay the slightest attention to Game Theory.All Poker 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 experience allied to the evidence of the cards in front of them, betting patterns, tells, an analysis of pot odds if necessary and their propensity to gamble . Not even the most intricate matrices of Game Theory can teach a poker player how to bluff.

The medieval monk/philosopher Ockham formulated the maxim of common sense known as Ockham’s Razor: The simplest solution to any problem 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 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 is somewhat akin to Economics: Economists generate mostly hot air (spoken with inside knowledge) 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. If they did, they'd all be lying on a beach in the Caribbean perfectly punting the markets. In real life, no poker player or prisoner uses the elaborate mathematics of Game Theory.

For poker players caught in the Prisoner's Dilemma there can be only one solution: Make the best deal possible with the lawyer, without regard to anyone else. Just remember, 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.

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Poker Forums => The Rail => Topic started by: TightEnd on October 23, 2007, 02:51:38 PM