[0,1] game and tournament play?

[Bozeman]

The tournament nature only comes in if you try to maximize ROI instead of $/hand or you have more players.

[Aisthesis]

Well, in the ultra-simple game, B can actually pick his ROI, as long as it's strictly less than 100%. But the closer he wants to get to 100%, the longer it will take him to play a hand. That's why I figured EV/hand was the best way to go.

But of course, one big shortcoming here is that there's no blind pressure on B at all in the ultra-simple game. For the moment, I'd just like to use the ultra-simple game to get some kind of grasp on how to structure the calculation.

Striving for EV/hand, I can't even really prove that the cash game variation is best. All I see at this point is 3 variables, call them x, x' and x" where x is the call threshold in medium stack, x' the call threshold in small stack and x" the call threshold in the big stack.

Is there any real reason why they have to be identical for the sake of maximizing EV/hand in the tournament structure?

They'll definitely give you some kind of functions that tell what the average number of hands played will be and what the ROI is. And from there you can figure EV/hand pretty easily. I think it would actually be possible to write an explicit equation for average number of hands played as well as ROI, given B's choices for x, x' and x". But by the time you converted all that to EV/hand and tried to maximize, I fear one would end up with some absolutely horrible derivatives to find, if one did it that way.

I'm no doubt making it too complicated here.

On the ultra-simple game, I think I could figure it for multiple players if I had an airtight approach on just 2.

Anyhow, my problem at the moment is that the ultra-simple game doesn't really seem all that simple in a tournament setting. Maybe rotating blinds would actually make it easier in some way.

Or, one could start by having both players put in $1, starting stacks of maybe $2 each. Then, rotate who is first to act but allow both players to raise (and fold to an opponent's raise). If you set up the tournament that way, then you could just consider maximizing ROI without worrying about how long it would last.

Maybe something like that would be a start? If that presents an easy solution for 2 players, it shouldn't be that hard for a 3-player with winner-take-all, etc.

I really do think this [0,1] game should have some tournament potential, but I have trouble figuring out a really simple starting point from which to progressively add some complexity. Any suggestions (or solution to the one just described)?

[Aisthesis]

Just as an example (for the ultra-simple game):

All it takes for the tournament to end is for either player to win twice in a row. If the result of the first two played hands is 1 win and 1 loss for B (or A), then we're back to the beginning state.

So, if we define x as the medium stack call threshold for B, x' as the short stack call threshold for B, and x" as the big stack call threshold for B, then B's ROI is simply:
(1/4)*(2x + x' + x" + x*x" - x*x')

That's actually not too bad. It's summing the infinite series for average number of hands played that is scaring me. I'll have to try it (picking specific numbers, it always takes on a rather easy form) when I get a chance. If the values resulting from variables x, x' and x" (all between 0 and 1) will take on a coherent pattern, it might work out into something doable.

My apologies for resorting to calculus again on this one, but I can't see how to get a grip on it otherwise.

In any case if we define r as 2 times ROI (buy-in was $2), and n as average number of hands until the tournament is decided, then r/n will be EV/hand.