Ready? There doesn't need to be another thread about this. Ever.

Game = "round" versus an opponent.
Match = the whole shindig, from posting to profit.
WR1 = Winrate for Game 1 of a 4-man HUSNG
WR2 = Winrate for Game 2 of a 4-man HUSNG
BI = Buy-in for the match, without rake
.0455BI = Rake paid for a turbo

From my post in a previous thread:
2-man: ROI2 = WR1 * 2BI - .0455BI
4-man: ROI4 = WR1 * WR2 * 4BI - .0455BI

So if your WR2 > .50 as it should be, your ROI will be higher in 4-man matches, but who cares? We want the hourly! Since we play only 1 game per 2-man match, and an average of 1 + WR1 games per 4-man match, we have your hourly function being proportional to your profit per game, so:

PPG2 = ROI2 - BI
PPG4 = (ROI4 - BI) / (1 + WR1)

And we're done! Okay okay, so sample numbers... For a constant winrate of of .56 per game, we have:

ROI2 = 1.0745BI
ROI4 = 1.2089BI
PPG2 = .0745BI
PPG4 = .1339BI

So you make almost DOUBLE the hourly playing 4-mans. However, lets be more realistic, and then I'll explain why this is the case.

For a WR1 of .56 and a WR2 of .53, we have:
ROI2 = 1.0745BI
ROI4 = 1.1417BI
PPG2 = .0745BI
PPG4 = .0926BI

STILL higher. Some basic algebra shows that our profit per game is the same when WR2 = (.5)WR1 + .238625. As an example, if we win 56% of our Game 1's, then we only need to win 52% or our Game 2's to make a higher hourly at the 4-man's.

So why does this happen? Think about it like this. In a 4-man HUSNG, you're playing 1 game at your regular stake-level, then whenever you win that game, you take a shot at a higher stake-level, except you get to play it WITHOUT RAKE. However, before you make a switch to 4-man HUSNGs, think about your bankroll! HUSNG bankrolls have the huge benefit of being easily computable using the Kelly Criterion. Use this (props to whoever posted it first): http://hquotes.com/kelly.html

For example, say you win 56% of Game 1's at your current stake level. Say that you estimate you can win 54% of Game 2's in 4-man HUSNGs. Then we have:

2-man BR Requirement:
Win Probability = .56
Win/Loss Ratio = .9130
Kelly Value = .0781 = 7.8% of your roll at a Full-Kelly, or 3.9% at a Half-Kelly (suggested)
So you would need approximately 26 buy-ins.

4-man BR Requirement:
Win Probability = .3024
Win/Loss Ratio = 2.8261 ((4BI - 1.0455BI) / 1.0455BI)
Kelly Value = .0555 = 5.6% of your roll at a Full-Kelly, or 2.8% at a Half-Kelly (suggested).
So you would need approximately 36 buy-ins.

So what am I REALLY telling you?

Say you have 40 buy-ins of your current level, and the next level is double the stakes. Then you don't have enough by the Kelly-Criterion to play higher, but you have enough to play 4-man's at your current level. Your profit per game will be higher, so DO IT.

Say you have 60 buy-ins of your current level, and the next level is double the stakes. Estimate your win-rate, run it through the calculator, smile at the numbers, and MOVE UP. You're giving up equity by playing lower stakes or playing the 4-man's at your current stake level.

Whether you play 4-man's or 2-man's is dependent on your BANKROLL, and NOTHING ELSE (unless you really like seeing a high ROI under your name, or game select like crazy).

I hope I didn't miss anything, and I hope this helps. Maybe people can use the search function and stop asking this.

Wait, did you calculate the waiting time between the two games at a 4-man match in? That must hurt your $/hr.

Nicho did this in his blog with waiting times. From what I understand $ per hour should be better at 4mans assuming the same rake.

I didn't include waiting times. For me it doesn't even matter because I play 3-4 tables and just let it come up when it comes up. Wouldn't be hard though. Estimate average waiting time, compare it to your average length per game, then instead of dividing by (1 + WR1), divide by something slightly larger than it.

Yes, $/hr is better at 4-mans assuming the same stake-level. The main point is that what you play shouldn't be as much based on that logic as it should be based on your bankroll. 4-man HUSNGs are an intermediate step between two different 2-man HUSNG stake-levels (or you can view it the other way around). The $/hr is higher in 4-mans because you are essentially playing a higher stake-level.

Your formula is off which is way you're producing different numbers than mine.

2 man ROI =~ (2*WR - 1.0455)/1.0455

So a 60% win rate translates to an ROI=~14.77%

4 man ROI =~ (4*WR1*WR2 - 1.0455)/1.0455

A 60% first round and 58% second round translates to an ROI =~ 33.14%

Since the first round of a 4 man is not skewed towards better players and there is no reason to suggest so , we have that WR1(4 man) = WR(2man). A good estimate is that your second round match may drop your win % by a percentage or 2 .

So if a 4 man lasts about twice as long as a regular sng , then you're better off playing the 4 man with an adequate bankroll for it . Given the above probability distribution for first and second place , we would be indifferent to playing either game if the length of the 4 man is ~ 2.24 (33.14/14.77 ~ 2.24) times as long as the regular .

However , your standard deviation for the 4 man tournament is almost double which raises your bankroll requirements by about a factor of 2. This means that a player with 30 buy-ins for a regular sng , may need about 60 buy-ins for the 4 man tournament.

If you want ROI as a percentage with respect to your initial investment, your formula is right. The ROI I outlined above is different. It simply calculates how much you get back if you put in 1.0455BI. Subtract 1.0455BI to get your profit per game. Or, subtract 1.0455BI then divide by 1.0455BI to get your ROI as a percentage, as you've suggested.

The rest of what I wrote is still correct, and my slight error further emphasizes the idea that equivalent stake 4-man games are more profitable, but that what you play should be determined by your bankroll and not something else. The Kelly Criterion is gold here.

Jesus nope I was right the first time.

What are you arguing jayshark. We don't care about ROI.

I don't think so.

You need to think this over again because you're clearly confused.

Your standard deviation for a typical sng is about 1 unit/sng. This is true for a wide variety of win rates which makes the approximation a good one.

For a 4 man tournament , your standard deviation may be computed the following way:

Var(x)=E(x^2)-E(x)^2 where E(x) is your win rate(or mean) for a random variable x . For this particular case , we regard the variable x as +1-1/22 for when we win and -(1 +1/22) for when we lose.

var(x) = (3-1/22)^2*p^2 + (1-1/22)^2*(1-p^2) - [(3-1/22)*p^2 - (1+1/22)*(1-p^2)]^2

var(x) ~ 3.5707 for when p=0.6 .

The standard deviation is just the square root of this number and so
s.d(x)=1.889 which is almost double the standard deviation for the singles.

In fact , your variance is not going to deviate too much for p not close to 0 or 1 which is certainly the case here .

I think it is probably more than (or at least close to) that. So I don't think it would be a good idea to play the 4-mans unless you are a multi-tabling wizard like skates
and even if you are, it may worth it to give up some $/hr for avoiding a ****load of variance especially if you are prone to tilt. Also if you need double the bankroll for it then you are better of IMHO with just moving up to the next level for less variance and more hourly instead of grinding the 4-mans.

With the flippping dialema, at what point is it no longer profitable to play 4-mans?

I don't see how flipping ever changes anything in the 4-mans. Are you talking about something I'm unfamiliar with? I assume you mean two players chopping whenever they are matched in a 4-man.

In general, chopping is 0EV in a 4-man. If you meet in round 1, then you essentially play 2 buyins and 2 rakes for the finals, which is the same you would have paid if you just played a stake level up anyway. If you meet in round 2, you just keep the money you earned in round 1, which is the same as you would have if you played a 2-man. Really, chopping only hurts you since it costs time. It is financially neutral, however.

I am no bueno at the math stuff, but would it be better to play the 4 mans on full tilt even tho the rake is more?

What about having to wait longer for people to sit with you in 4 mans rather than 2 mans?

That and the waiting time in between two matches really pushes me towards "hourly is higher in 2 mans."

I don't have numbers but any numbers that leave out those two VERY important facts are really not anywhere near correct.

At the lower levels the "waiting longer for a game to start" is less of a factor obviously, but if you look at the 100-500 level 4 mans, good luck getting 2-4 of those going at once. Especially in the 500s where Croixdawg and Checkdeznuz (sp?) are constantly sitting, you're not going to make more money playing those two.

True True. And I do think this is more applicable to lower levels than higher levels for those reasons. I can sit in a $230 4-man at 11pm and play other games for 40 minutes before it fires off.

I play mostly 4-mans for the reasons that have been mentioned. Something else to think about, though: a really profitable situation in the 2mans happens when you find one really exploitable/tilty/stupid villain who is happy to keep replaying you. Since there's no replay-button type thing in the 4mans, you give up these goldmines.