Game Theory Explained

Mathematics certainly has great uses in many applied fields such as Physics, Chemistry, but also Economics, Computer Science and many others. Basically we can find Mathematics everywhere without even noticing it. One of its application which has captured my interest, is Game Theory.

Game Theory is the study of mathematical models of decisions between rational subjects, with each of them having an objective to reach, which can lead to cooperation or conflict with the other individuals. Its applications are countless: Economics, Political Science, Psychology, Logic and of course games like Backgammon and Poker. Game theory can be divided in two branches: Cooperative Game Theory in which analysed situations involve individuals interacting trying to achieve a common goal, and Non-cooperative Game Theory in which each individual interacts with the others trying to achieve his personal goal.

A very interesting example is the Prisoner’s Dilemma, which surprisingly can be found an example in which two rational subjects might decide not to cooperate even if their cooperation would lead to a benefit for both. I think this is a great example showing how in the real world, a little bit more generosity from each person could lead to huge benefits for everyone! Anyways, the rules are the following:

Two criminals of the same gang get arrested and are given two possibilities: betray the “colleague” (defect) or stay silent (cooperate). Their decisions will results in:

- If they both betray each other, they will both serve 2 years in prison
- If they both stay silent, they will both serve only 1 year in prison
- If one of them betrays but the other stays silent, the former will be set free immediately while the latter will get 3 years of prison.

Also, it is assumed that both cannot communicate and there will be no opportunities in the future for each of them to either punish or reward the other. Let’s call the prisoners A and B, and let’s verify the outcome of the 4 different combinations of decisions:

- If A betrays and B stays silent, A gets 0 years and B gets 3 years
- If A betrays and B betrays, A gets 2 years and B gets 2 years
- If A stays silent and B betrays, A gets 3 years and B gets 0 years
- If A stays silent and B stays silent, A gets 1 year and B gets 1 years.

Surprisingly, even if the cleverest decision seems to be the one having both of them staying silent, analysing the different combinations it is clear that, whatever the other prisoner’s decision is, each of them always benefits from betraying the other! Thus if they both make the correct-for-themselves decision, they both take 2 years of prison instead of 1.

This was just one easy example of Game Theory’s applications. Analysing real-life situations using mathematical models is a great way to study Maths and enjoy it the most!