[0,1] game and tournament play?
 

Does anyone have a simple example in which the [0,1] is applied to a tournament situation?

The simplest one I can think of may be so simple as to be completely trivial (I haven't managed to think it through completely yet):

A and B both buy in for $2 to a tournament which B is going to win most of the time. The rules are this: A always posts $1 blind, and B can call or fold. Initial stacks are $2, and last man standing wins.

B's best play here may well be to just maximize EV with just calling [1/2,1] every time. But he can obviously increase his probability of winning by playing tighter than that. The tighter he plays, the more often he'll win (he just needs to avoid losing both his first and his second hand), approaching 100% if he has the patience. But he has to play a lot of hands to win A's $2.

So, assuming that if B loses, he can just buy-in again under the same circumstances, let's define his "best play" as the hand selection which will maximize his winnings per tournament hand played. That will almost have to be the same way he plays in a "cash game"--namely calling on [1/2,1]. That's really going to have to give B a value of .25 for each tournament hand played (although he can obviously improve on his ROI% by playing tighter).

Or is the identity between best tournament strategy and best cash game strategy less clear than it seems on first glance? If they're the same in this case, does anyone have a (simple, please!!) tournament example where there is a clear difference?

I will say that in my ultra-simple example it could be advisable for B to tighten up on the second hand he plays if he has lost the first one. It seems less likely that he would want to loosen up on the second hand if he has won the first one (now he has $3 left to A's $1) ... but maybe that's also something to be considered.

Re: [0,1] game and tournament play?
 

In this extremely simplistic tournament, just assuming that B keeps the obvious calling criteria of 1/2 for small, medium, or big stack, I come up with some strange results that then turn out to be much less strange:

First, the average number of hands played in this tournament is 32/5 = 6.4 (at least the way I did it, this required summing an infinite series that wasn't nearly as bad as it might seem).

Moreover, B wins the tournament exactly 90% of the time!! Basically, whenever he calls he will on average win 3/4 of the time. And it takes 2 wins in a row or 2 losses in a row for the tournament to end. The probability of B winning twice in a row is hence (3/4)(3/4) = 9/16, and the probability of A winning twice in a row is (1/4)(1/4) = 1/16. So for every time A wins, B wins 9 for a 90% win-rate.

I at first thought that this would mean that B was beating his results in the cash game version. But this isn't the case. His ROI is 2*(9/10) - 2*(1/10) = 8/5. Each tourney lasts 32/5 hands. So his average winnings per hand are (8/5)*(5/32) = 1/4. Same as in the cash game.

So, it seems quite likely that B's tournament play here will be identical with his cash game play. But I still don't really see any objective reason why B might not want to have different calling criteria here in the small, medium, and big stacks in order to maximize EV/hand in tournament play.

Re: [0,1] game and tournament play?
 

If you want to see tourny strategy come out, assume the prize pool is something like $100 and the 4 chips issued are just tourny chips.

PairTheBoard

Re: [0,1] game and tournament play?
 

That should just multiply winnings and hence EV/hand by a factor of 25 but won't influence hand selection.

I've noticed on Jerrod's board that they've discussed some aspects of prize structure. But I haven't seen any precise models.

Re: [0,1] game and tournament play?
 

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

Re: [0,1] game and tournament play?
 

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)?

Re: [0,1] game and tournament play?
 

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.