[LordMushroom2]

I am a little interested in what we are actually looking for here, finding out which HU gametype has the higher standard deviation per hour when the SnG buy-in is 100 times greater than the big blind in the cash-game.

But I wasn´t too interested and was hoping the guys would find this out for me. This seems far away at the moment, so I will try and help (read: make it worse [img]/images/graemlins/laugh.gif[/img]).

First I wish to dispute that a shortstack will have smaller variance than a big stack if the blinds are the same.

I can understand how one may think it is so (I did it myself not long ago). One would think that a 10 big blind stack would have a smaller variance than a 100 big blind stack because it is limited how big pots it can create. With a 10BB stack the pot can maximum be 20BBs, but if you had 100BBs you could have made the pot bigger because you have more to throw into the pot.

But the thing is there are situations where a small stack would create a bigger pot than a big stack would. When stacks are smaller than 10BBs, for example, it is common to open-raise all-in. If your stack is 9BBs, you are effectively making a 9BB raise! And the opponent is not more scared of calling than he would have been if stacks had been 100BBs and the raise only 3BBs!

Such a huge open-raise would never take place with a big stack. With a big stack you would have bet 3BBs and if the opponent calls one of you will often take it down by making a 4BB bet on the flop winning a 6BB pot instead of the 18BB pot in the shortstack-case.

There are of course situations where a big stack would create a larger pot too, and this roughly cancels out the larger pots of the shortstacks, making both stacks creating roughly the same average pot.

Just look at the 30BB Cap-games on Full Tilt, they have about the same average pot as non-cap-games.

Obviously when stacks get tiny, the average pot must go down as the stacks are getting so small at some point that even if both players were all-in in all hands, the average pot would be smaller than the average pot of a normal game. I don´t know exactly where the stacks start decreasing the average pot, but I would guess around 7BB stacks.

Then here is how I think we should approach our problem:

1) We simplify and assume there is no rake and no fees.

2) We simplify further and assume all players are equally skilled.

3) We assume you can play the same number of hands per minute in an SnG as in a cash-game.

4) Since stacks don´t influence the standard deviation in cash-games as long as stacks are greater than 7BBs, we make it easy for ourselves and assume the cash-game stacks are always 9BBs and either go all-in or fold (easy to relate too and not too unrealistic).

5) Let´s say both players will go all-in in the SB with their 70% best hands and call all-in in the BB with their 40% best hands.

This means the BB will win the SB 100% - 70% = 30% of the time. The SB will win the BB 0,7 * (1-0,4) = 0,42 = 42% of the time. And they will compete for an 18BB pot 0,7 * 0,4 = 0,28 = 28% of the time.

Pokerstove says the SB has a 44,6% chance of winning the 18BB pots and the BB thus has a 55,4% chance of winning. I am ignoring the small chance of a draw because it is so rare.

6) Let´s set a specific monetary size on the games. It is NL100 and a $100+0 SnG.

7) We know that in a $100+0 SnG, you will either win $100 or lose $100 after having played a number of hands. So the "variance" (used very unmathematically here) of it is $100.

8) Run a simulation of a hand in a cash-game where the possible outcomes of a hand are according to the probabilities pointed out in step 5. Note the amount player A (just choose one of them) won/lost.

Then do the same for the next hand and add/subtract what player A won/lost to/from his gain/loss from the last hand.

Then do the same for next hand a summarize the gain/loss from all three hands.

Keep doing this until player A has either won or lost a total of $100, then note the number of hands it took.

9) Repeat step 8 (note the number of hands it took). Do it again and again until you feel you have done it enough to get a fair sample. Then calculate the average number of hands it took.

10) If a particular type of HU SnG lasts more number of hands on average than the number found in step 9, it has less variance.