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 the first values a bit nicer. I use this definition.

## FPCI games

### 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 \(FPCI(1) = 1\). (It can't be 0, beacause Friedman said that s≥1.)

### Alternative definition

There seems to be a problem. The current definition says:

- Let FPCI(a) be the minimal N such that Wojtek can win G
_{N}(P,Q,n,N).

Since the numbers that can be selected are between -N and N, it has no effect that he has to accept all double factorial numbers larger than N. Therefore, FPCI(n) = 1 for every n.

I think it is actually better to make this function 2 argument, as in

- Let FPCI(a,s) be the minimal N such that Wojtek can win G
_{N}(P,Q,n,s).

### FPCI(2,s)

\(FPCI(2,1) = 0\)

\(FPCI(2,2) ≥ 1\). If he has to accept 1 and 2, this he'll lose the game for P(x) = x+y and Q(x)=x^{2}+y^{2}.