An Introduction to Game Theory

Game Theory

When the concept of game theory was first dwelled upon, it was a revolutionary hypothesis. Nowadays, not only can it be applied at the poker tables, but it’s generally a part of our everyday lives.

The most common example of game theory is that of the prisoner’s dilemma. This particular game indicates that rational players will often reach an outcome which isn't the most beneficial to either of them, given the options which they are presented.

Two men are arrested by the police, who proceed to interrogate the men separately. The police offer each man the same deal; If one testifies against the other and the other remains silent, then the betrayer will be set free and the other man will receive a one year jail sentence. If both remain silent, both of them are only sentenced to one month in jail each. If they both “rat” each other out, they each receive a three month sentence in jail.

What is the optimal choice to the above problem? If we assume that each man is primarily concerned about reducing his own personal jail time and nothing else, then the problem becomes quite simple. If both men stay silent, then they will each receive one month in jail.

There is one significant problem with the “fairest” outcome suggested above. It encourages one of the men to deviate from the outcome in order to serve their own interests (reducing their personal jail time). Both men are aware of this and recognise that if they choose to stay silent, then their fellow inmate has a strong motivation to start singing like a canary. Ultimately, if both prisoners’ are given these options in a one-off scenario, the optimal strategy is for both inmates to rat one another out. Indeed, the option of testifying against the other inmate dominates the option of staying silent, and in the one-off scenario, it’s always optimal to testify against the other inmate.

The prisoner’s dilemma is an example of a game in which we have perfect information. Both inmates know what sort of jail sentence they may receive given their own choices, dependant on the choice made by the other inmate. Poker presents a much more complicated problem, in that the information we have on our opponent is imperfect – We are unaware of their exact holdings, but we can make educated guesses and pursue certain strategies given their specific betting tendencies and behaviour. Furthermore, unlike the above example of a one-off prisoner’s dilemma game, poker is an infinitely repeated game. Interestingly, if the prisoner’s dilemma game was in fact repeated over and over again, it’s very likely that the inmates would reach a different outcome. Both inmates would be encouraged to stay silent and maximise their long run pay off. However – they still might be tempted to deviate at one point in order to maximise their return in the short run.

Relevance to Poker

Indeed, we can draw many similarities from a repeated game of the prisoner’s dilemma to that of poker. At the poker tables, cooperation with our opposition is generally the strategy that we deploy at first, until we have a better idea of our opponent's thought process and how they act in different situations.

This will result in you starting to call his check raises, even re-raising them. Indeed, your opponent’s perceived deviation from cooperation forces you to change your strategy in order to maximise your own expected value. Your opponent may then adjust to this by bluffing less, forcing you to change your strategy again. Ultimately, this battle is effectively never ending and you are constantly trying to balance how often you cooperate with your opponent, and how often you deceive your opponent.

Future articles surrounding game theory will present real life examples of why “mixing it up” is so beneficial to your bottom line and how approaching the poker tables with game theory in mind can determine whether you will be a winner or loser in the long run.