In this episode, the main mathematical concepts derive from a field called game theory. This active area of research is one of the main facets of modern economics. The goal of game theory is to take some social situation with some notion of winning and come up with a formal, mathematical way of discussing it. Thus the parties involved in the situation are called "players" and the set of rules we define is called "the game." Usually the discussion focuses around what the best strategy for playing the game is, i.e. how best to go about winning. Sometimes, though, determining the strategy is difficult, and mathematicians will focus on determining whether an optimal strategy exists. Taking chess as an example, is it possible for black to always win or regardless of what black does can white always force a draw? Since a general game could be almost anything, mathematicians usually focus on classes of games with a few aspects in common. Most real life games can be categorized pretty easily.
In the 1960s and 1970s, Monty Hall hosted a game show called Let's Make a Deal. Though the following problem was never a game on the show exactly as stated, it is similar in format to games that appeared.
Suppose there are three closed doors before you. Behind one door is some prize of great value, but nothing lies behind the other two doors. You are given the opportunity to select one door. The host of the show then opens up one of the other two doors - always one with nothing behind it. You are given the choice to keep your original guess or to switch.
Let's go through how to solve this problem. We'll consider two different cases, one where our first guess is correct and one where our first guess is incorrect.
On our first selection, suppose we unknowingly choose the door with the prize behind it. This happens exactly 1/3 of the time. The remaining two doors are both losers, so after the host opens one door the one remaining is a loser. Then the strategy of never switching wins. The strategy of always switching loses.
Supposing on the other hand that on our first selection we chose incorrectly. This happens exactly 2/3 of the time. The door the host opens is the other losing door, so the remaining door is the winning door. If we stick with our initial choice, we lose. If we change, though, we win.
The strategy of always staying with your first guess loses 2/3 of the time. The strategy of always switching wins 2/3 of the time! Our gut instinct (usually) is that it shouldn't matter whether we switch or not. However, this perfectly exemplifies why rationality is a better problem solving tool than common sense: it is always better to switch.
Charlie mentions the Prisoner's dilemma. This is one of the most common basic problems in Game Theory. In this problem, there are two prisoners, 1 and 2, who are guilty of some crime which carries a sentence of 10 years. The police only have enough evidence to convict the two prisoners on lesser crimes which carry only a sentence of one year, so officers will put the prisoners in separate rooms and offer them only 5 years in prison if they will swear under oath that the other prisoner commited the crime. The two prisoners each have a choice of remaining loyal or betraying their partner.
|The prisoner's dilemma is most easily understood by putting it into what is called normal form. In a two player game, the normal form is given by a matrix. Each column corresponds to a choice by one party, and each row corresponds to a choice by the second party. The entry where a particular row and column intersect is the outcome of each party taking those two actions. To the right is the normal form for the prisoner's dilemma. Each entry gives the sentence of prisoner 1 and then prisoner 2.||
The Nash Equilibrium is named after John Nash - the subject of the book and movie A Beautiful Mind. In 1951, Nash published an article in which he proved that any finite game with any finite number of players - i.e. a game with a finite number of possible actions on the part of any player - has an equilibrium with the properties stated in the previous paragraph.
In any event, the result of the prisoner's dilemma is that the stable equilbrium of Nash is not the optimal one for both players - they each will end up with five-year sentences instead of one-year terms. One could think instead of an analogous situation of two-equally sized companies manufacturing some product. Naturally, they are both trying to maximize their profit. One company might undercut the other to increase their sales. However, the other company will likely attempt the same strategy. This results in both companies actually reducing their profit. Using more careful analysis, one can use this idea to actually determine when two companies are acting as a duopoly - that is, working together.
Tit for tat is a surprisingly successful strategy for prisoner's dilemma type problems with multiple rounds of competition. In this strategy, the player will follow the following rules each round against a competitor X:
Suppose that there are now 5 prisoners. Each prisoner will play the prisoner's dilemma game against each other prisoner a total of six times. Suppose that 3 of the prisoners are greedy and that the other two take the tit for tat strategy. Call the greedy players Gs, and the tit for tat players Ts.