Game Theory question

I recently started to practise a bit Game Theory, but I wondered....


Is there a function that describes the relation between the chance of winning with a hand (heads up) and the percentage of hands you play if there are 5 community cards to come (like Hold'em) but no player can bet or raise on later rounds(it's just checked down)?


So let:

X=the chance of winning with a hand, and

P=ratio of hands played,


Is there a function X(P)? I would say if P increases X would diminish (so dX/dP<0)


Thanks


Regards

I realise, having thought about it some more, that there isnt probably a function, but it's a domain between extremes, since there is an uncertainty in the random hand your opponent has.


Furthermore there are some other conditions:

-P and X are between 0 and 1

-dX(P)/dP<0

-(P,X)=(0,0) is an a-continous point.

-Probably some other points


To me, the only way of finding such "function" would be through simulation, and let the computer find the appropriate formula for the extremes. Am I right here?


Regards

I think its probably as easy as just setting up a computer to play a few million (or billion) hands heads up all the way to the river, with neither player folding, and then seeing how often each hand won. Your "function" then is a simple table lookup on the win percentage of whatever hand you have been dealt.