Question
Are there more Zero Sum or non Zero Sum games?
Answer
I believe both classes would be equal in size. The argument hinges on Euclidean spaces of all finite powers being basically the same infinity and equal to the real line.
Take the simplest case, the normal form (i.e. a square matrix representation ) of a 2-player n-move game, assume the game is general, non-symmetric etc. Most general case.
You've got 2*nxn=2n^2 real numbers in the payoff matrices for the two players., P and Q.
Your game description lives in R^(2n^2) Euclidean space, i.e finite dim.
Some subspace of that is the zero-sum manifold (I think it should be a manifold... getting rusty on my terminology and definitions here). Since there will be one constraint per matrix cell of the form P(i,j)+Q(i,j)=0, you've got n^2 constraints. So the manifold should be of dimension R^(2n^2-n^2) = R^(n^2) dimension.
So basically zero-sum is a lower-dimension manifold in a higher-dimensional space, but both dimensions are finite.
If I remember how to count my infinities, all finite-dimensional Euclidean spaces are the same as the real line R. You only get your next infinity when you go to infinite dimensions. (R^w).
Don't want to bother LaTeXing this though...partly because I've forgotten how to typeset things like Aleph Null.
This is all based on very rusty memories of stuff I studied 12 years ago, so if some working mathematician has a better answer, or there are mistakes in my argument, please point 'em out.
Take the simplest case, the normal form (i.e. a square matrix representation ) of a 2-player n-move game, assume the game is general, non-symmetric etc. Most general case.
You've got 2*nxn=2n^2 real numbers in the payoff matrices for the two players., P and Q.
Your game description lives in R^(2n^2) Euclidean space, i.e finite dim.
Some subspace of that is the zero-sum manifold (I think it should be a manifold... getting rusty on my terminology and definitions here). Since there will be one constraint per matrix cell of the form P(i,j)+Q(i,j)=0, you've got n^2 constraints. So the manifold should be of dimension R^(2n^2-n^2) = R^(n^2) dimension.
So basically zero-sum is a lower-dimension manifold in a higher-dimensional space, but both dimensions are finite.
If I remember how to count my infinities, all finite-dimensional Euclidean spaces are the same as the real line R. You only get your next infinity when you go to infinite dimensions. (R^w).
Don't want to bother LaTeXing this though...partly because I've forgotten how to typeset things like Aleph Null.
This is all based on very rusty memories of stuff I studied 12 years ago, so if some working mathematician has a better answer, or there are mistakes in my argument, please point 'em out.