User blog:Wythagoras/Finite Promise Games

Note that I think that we should replace all greater than and lesser thans with greater than or equals and lesser than or equals. This gives a smoother function. I use this definition.

FPCI(1)
The game lasts one round, degrees of the functions are at most 1.

Alice offers him \(x=(w!)!\), beacause Bob hasn't played any integers yet. Note that per definition \(x\) can't be part of the components of the P or Q inversion of \(x\) beacause the components must be strictly between 0 and \(x/2\). Therefore Bob can always keep all his promises, so \(FPCI(1) = 0\).

FPCI(2)
I have thought about it a lot now, but I realized that I need to know something more about these games and what restrictions there are on the polynomials. For example, is P(x,y) = xy allowed? Is a constant polynomial allowed?