Saturday, 22 September 2012

What Is Game Theory?

Reader's of Schoolonomic already know by now that economics is not an isolated discipline - the non-conclusive list of connections includes history, philosophy, psychology and also mathematics. And while we have written extensively about the first three, so far has been little coverage of the latter.
So to open a new line of explanatory posts aimed at explaining mathematical economic concepts, this post will cover game theory, the mathematic study of decision-making.

Game theory's practical applications include poker, chess and several other forms of classical games.

So What is Game Theory?


Game Theory has at least as many definitions as situations in which it can be used - but very simply put, game theory is an analysis of multiple participants in a situation where wins and losses are pre-determined. For example, when you are dividing your birthday sweets and deciding over the size of the portions for guests (too much for them means less for you, too little for them and they get angry), you are actually conducting game theory. Piece of cake, right? (Pun intended.)
To make it sound more science-ish, mathematicians developed a set of terms they use when they are talking about game theory. These are:

Game - The situation being analysed
Players - A decision-maker in the situation
Strategy - A set of decisions a single player will make in the game
Payoff - The reward of the game for a single player
Information set - The data available at a given point of the game
Equilibrium - The situation where players made all their decisions and an outcome is reached, and payoff is given.

When to use game theory?


As it can be seen above, the concept of game theory is quite broad and hence many problems can be addressed from a game theory-point of view - and it is a very popular one among both mathematicians and economists. It's uses are however also restricted. Firstly, it assumes rationality - that every player will decide based solely on calculations rather than intuitions, or in more simple terms: that everyone is familiar with game theory. Secondly, it assumes payoff maximisation - which means that every player will want to maximise his or her own output, without regard to other players or the player group as a whole.

An example


Let's highlight the meaning of the terms using the example above - you cutting up a cake for you and say, your sister - this is actually a widely known basic problem of game theory, called the fair division problem. The game is to cut up the cake with a knife held by you, yet make the division fair. The players are you and your sister, playing for the obvious payoff, a share of cake. The information set available is the cut in the cake - you can curve it wherever you want, while the equilibrium is the point when you finish cutting. Now one question remains - what should be the strategy?

Solution


The cake size you receive and the one your sister does are reversely proportional to each other - the bigger you get, the smaller does she. In order to mediate between these interests, the most basic solution is the one called 'divide and choose'. In this strategy, you cut up the cake into halves which you think are equal, while your sister chooses the half she wishes to get - this way you'll be inclined to make the sizes as equal as possible, given the fact that your maximum payoff is only half of the cake - if you cut into say, 3/4 and 1/4, your sister will take the bigger part. 

Congratulations! Now you know the concepts of basic game theory. Similar explanatory post's will follow throughout the following weeks, as well as real-life applications on very real-life economic problems. 

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