Is variance smaller when buy-in in with a smaller buy-in?

[LordMushroom2]

I am arguing with a guy over whether or not playing with a minimum buy-in gives you less variance (BB/100) than playing with a full buy-in.

If anyone could explain why the variance is or isn´t smaller with a smaller buy-in, preferably mathematically, that would be great.

Also it would be nice to hear what standard deviation people who have played with a shortstack experienced.

[soah]

it's pretty rare when having a shorter stack will cause you to play a significantly larger pot than what you'd play with a deep stack, but there are tons of times when having a shorter stack will cause you to play a much smaller pot than what you'd play with a deep stack

thus, standard deviation is smaller

[chicheebee]

i don't know about "significantly" but i thin there are plenty of situations where shorter stacks do play bigger pots than full stacks due to becoming pot committed

[xorbie]

im not convinced that standard deviation is smaller at all, these guys are constantly getting 20bb in pf and actually often play bigger pots w/ hands because of that (how often do you get 20bb in w/ 66 or AT?).

[lucky_mf]

[ QUOTE ]
it's pretty rare when having a shorter stack will cause you to play a significantly larger pot than what you'd play with a deep stack, but there are tons of times when having a shorter stack will cause you to play a much smaller pot than what you'd play with a deep stack

thus, standard deviation is smaller

[/ QUOTE ]

I disagree with this. I think it depends on what is bigger versus smaller. If you are playing 200bb+ deep (with your oppenents) there are plenty of instances in where turn would get checked for pot control, whereas if the same hands were played with 100bb there would be turn bets. You are also a lot less likely to see someone do something like push over a flop bet with a flush draw playing 200bb+. These differences would lead to increased variance in a shorter (100bb) buy-in game versus a deep (200bb+) game under normal conditions (i.e. not very many huge pots in the deep game).

Now if small is 20bb-40bb and big is 100bb you might be right. But even in very short games the money is going to be going in pf a lot with flips. As probably know the variance of a transformation of a binomial random variable (the transformation being winnings) is maximized when the probability of success is 0.50.

Lucky

[soah]

I'm not saying that there aren't some hands where the shortstack would play a bigger pot than if both players were deep, but the difference between sometimes getting 20bb in the pot instead of 10bb pales in comparison to the times that they get 20bb in instead of 100bb.

[ogdundar]

I don't think this can be answered qualitatively... You need statistics on it.

My immediate thought is that shortstacked poker increases variance though...