The beauty of game theory, pt. 1: An introduction to Prisoner’s Dilemma
Game theory is a mathematical discipline that is almost unheard of in the British a level curriculum, especially in non-private colleges. It is also a discipline that is immensely important in a number of different fields, such as economics and political science. “So why, Cameron-” I hear you saying, “-is it shunned so by our teaching staff? For decision maths is a module available for any student taking Further Pure AS to take!”. Well, guys, i hate to break it to you, but decision maths is incredibly dense. It’s also difficult to teach, which might go some way to explaining it’s lack of inclusion in the majority of people’s further maths AS level.
However, as indicated by the title, I do find a certain beauty in certain aspects of this certainly entrancing subject. Prisoner’s dilemma is one of those aspects. Part one, in fact.
And thus we begin our journey through the looking glass, into the wonderful world of “the study of mathematical models of conflict and cooperation between intelligent rational decision-makers”(1).
But first, an introduction. Prisoner’s dilemma, hereby referred to as PD, as dictated by Albert W. Tucker in 1992:
– Two members of a criminal gang are arrested and imprisoned. Each prisoner is in solitary confinement with no means of speaking to or exchanging messages with the other. The police admit they don’t have enough evidence to convict the pair on the principal charge. They plan to sentence both to a year in prison on a lesser charge. Simultaneously, the police offer each prisoner a Faustian bargain. If he testifies against his partner, he will go free while the partner will get three years in prison on the main charge. Oh, yes, there is a catch … If both prisoners testify against each other, both will be sentenced to two years in jail. –
The most obviously optimal solution, and the solution to which most humans subjected to this dilemma are biased towards, is to stay quiet; the total number of years served by both prisoners combined is minimised to 2, and thereby each criminal gets home to his lovely wife in only a year.
And this is where game theory comes in. If the first prisoner (prisoner alpha) chooses to stay silent, then the prisoner who breaks the trend (prisoner Omega) and decides to betray the other gains a greater reward: no prison sentence, and since he has no allegiance to the other criminal, doesn’t bare guilt about sentencing him to an additional 2 years in the slammer.
If prisoner Alpha is assumed to betray Omega, then prisoner Omega still loses time off his sentence by betraying. And, perhaps a bonus, forces that no good son-of-a-bitch who betrayed him into a longer sentence! As you can see, by systematically proceeding through each possible outcome in a small thought experiment, we have quickly determined that in the classic version of the game, co-operation is dominated by betrayal.
What I really love about this game is the demonstration of the intrinsic egocentricity of the purely rational being. By being selfish, prisoners Alpha and Omega have effectively guaranteed a sentence of 2 years for themselves, when by simply considering the overall system they could have reasoned that the optimal solution would be to both stay silent! Alas, you can never trust a purely rational being. They’re weird and they smell and oft live with their parents at 45.
Perhaps the most obvious real world example of PD is the Cold War nuclear arms race; in a simplified model, the opposing alliances NATO the Warsaw Pact had the option to either arm or disarm. Disarming while their enemy continued to arm would leave them highly vulnerable to attack, while both alliances maintaining an arsenal of arms would guarantee peace. Maintaining a nuclear arsenal costs both alliances greatly, and clearly the optimal solution is for both sides to disarm, guaranteeing peace and avoiding the costs of war. If your opponent chooses to disarm and you continue to arm, nuclear superiority is achieved.
In conclusion, game theory is too cool 4 school.
Cameron G. Mason
EDIT: I have since had my attention drawn to the fact that game theory is NOT as sparsely covered in British A level curriculum as i once thought! Prisoner’s dilemma and oligopolies are covered in A level economics.
For further reading: http://en.wikipedia.org/wiki/Prisoner’s_dilemma
Soon to come: The beauty of game theory, pt. 2