Simple Question... the mathematics of Game Theory...

[Xavier]

I am getting to study game theory in more and more depth, but I also seem to be coming across more and more mathematical models.

The thing is, up until now I had only been studying some of the underlying principles (like alternate strategies, optimal strategies etc.). But obviously as I dig more deep, I realise I'm gonna have to learn some math in order to keep up and fully appreciate what's going on (it is pretty much all mathematical).

But the thing is, I'm not prepared to learn a lot of stuff that I don't need or use for the purposes of studying game theory (e.g. geometry). So I wanna ask to anyone who has studied the mathematical aspects of game theory (or has any relevant advice):

Exactly what kind of mathematics must I learn in order to get to grips with game theory? For example, how much statistics, probability theory, algebra do I need?

Bear in mind I don't wanna learn any excess mathematics, just whatever I need to apply to game theory.

Thanks.

[Tarokun]

Game Theory: A Nontechnical Introduction

This book covers it pretty well, the basics at least. I don't remember there being much more than basic Algebra in it, either.

The kind of mathematics required for game theory depends on how far into game theory you want to study. The harder stuff probably goes into advanced calculus or statistics, though I wouldn't know because I haven't gone that far into it. Knowing summations is more or less a prerequisite, though.

[js_africanus]

Real analysis and combinatorics, IIRC.

http://dmoz.org/Science/Math/Combina...l_Game_Theory/

http://dmoz.org/Science/Social_Scien...s/Game_Theory/

http://dmoz.org/Science/Math/Combinatorics/

Here is an online textbook in real analysis. I'm not sure if the book is complete yet. http://www.shu.edu/html/teaching/math/reals/reals.html

Basically, I don't think you can realistically be "excessively-mathed" for game theory (or economics generally).

If you want to know more about those two areas and the best ways, short of formal study in school, you might try starting a new thread with a title to that effect.

[Hostile Dialect]

I remember doing a fairly in-depth project on game theory in 10th grade (two years ago) for my geometry class. Obviously, I didn't have any mathematical background beyond geometry.

Can't offer much more than that, except to say that game theory is fascinating, and you're going to have a lot of fun, especially if you're a math person (I've found that I'm not, but that was still one of the most interesting research projects I've ever done).

[Mathochist]

Originally Posted by Xavier

Bear in mind I don't wanna learn any excess mathematics, just whatever I need to apply to game theory.

Apply to what? You can probably get away without knowing more than you already do, since "real-world" situations are never so cut-and-dried as abstract games, and the results of game theory only "really" apply as guidelines and rough sketches of actual behavior. Also, they tend to assume rational opponents.

Anyhow, what would be the horrible problem with learning more math than you absolutely, positively have to know?

[ultrafilter]

Game theory is a mathematically sophisticated area. There'll always be something that you can do without more math, but there will always be things you can't do as well.

[js_africanus]

Originally Posted by Mathochist

Also, they tend to assume rational opponents.

Assuming irrational players doesn't make it any easier.

Anyhow, what would be the horrible problem with learning more math than you absolutely, positively have to know?

Yeah. What he said.

[dakravel]

Assuming irrational players doesn't make it any easier.

Wouldn't assuming rational players be easier than assuming irrational players? You can reason what a rational player is most likely to do, but an irrational player might as well be a roulette wheel, since you can't be sure of what he will choose in any case. Or am I missing something?

[js_africanus]

Originally Posted by dakravel

Wouldn't assuming rational players be easier than assuming irrational players? ...am I missing something?

Yes and no. Irrational doesn't mean that they can snap or do something completely nuts at any moment. (Indeed, that may be perfectly rational!)

Here is an example from that text. If we imagine a road on a circle so that all the homes are at points on a circle. We have two types of people, red & blue. None of them are segregationsits: Any player would prefer to live between one of each type rather than two of one type. At each "turn" two players are chosen randomly and are allowed to switch if it makes them better off (or if it makes one better off and the other not worse off). If we allow players to make a mistake with some probability, i.e. they switch even though it doesn't make them better off (maybe even worse off), we will find that our neighborhood ends up being completely segregated with blues occupying on half of the circle and reds occupying the other.*

So, you are correct (I think) because to allow for mistakes, the modeller must add some more mathematics & complications (e.g. knowing how to work w/ Markov chains). Your error is that an irrational player isn't going to go pell-mell all over the place doing god-know-what. For example, in basketball there are times where it is optimal to foul another player; however, players may not always choose to foul optimally. Working a strategy for players who foul optimally may be easier than when they don't; but, they are still playing within a well defined game and suboptimal fouling doesn't really mean that they're like a roulette wheel.


*I'm sure I'm getting some detail wrong. But I'm sure you get the point.

[zoid]

It can get very deep.

At the U. I attended, Game Theory was a graduate level course. I have NO idea how I passed - I don't remember a thing.

[ultrafilter]

To summarize js_africanus's point, an irrational player is not one who is completely unpredictable. An irrational player is one who does not always make the optimal choice given the information he has available.