Please comment and correct any mistakes or misinformation you find.

What does "HUHU" stand for?

"HU" refers to anything heads-up. In a 6-max table the pot is heads-up (HU) on flop if there are two players in the pot. HUHU abbreviation makes it explicit that we're talking about heads-up tables.

Is the rake beatable?

Depends on your winrate. The lowest beatable limits at PokerStars and Full Tilt are 2/4 and 3/6 although some people have claimed beating 1/2 for decent sample size. This is for rake cap of 50c. Playing 2/4 or 3/6 with $1 rake cap is suicide. (Unless you have a really good rakeback deal.)

The gross rake (not including rakebacks or bonuses) you pay with 50c rake cap is something like 3.5-4BB/100 at 2/4, closer to 3BB/100 at 3/6 and approximately 2BB/100 at 5/10. So that should give you an idea of what it takes to beat the rake. Just be sure to remember that if you have a real marginal winrate, it means that you will have huge swings, which are exaggerated by the higher general variance in HUHU.

The options for players not rolled for 2/4 or preferably 3/6 are to find a site with a real low rake or really generous rakeback deals. Another option is to play limit HU SNGs, which play essentially the same as cash HU. There are some differences when the stacks get extremely short, but a winning Limit HUHU cash player will most certainly be a winning Limit HU SNG player too.

What's a good winrate?

Anything positive is a good start. Anything beyond 5+BB/100 is probably unsustainable. (And if you really are beyond that, you probably won't need this FAQ.) But the point of HUHU is to play the weakest player on the table all the time. So you get to make a lot decisions against a worse player than you, which should generally yield a higher winrate than you have at 6-max (assuming you have to play with other good players). Needless to say, game selection is probably the biggest contributing factor to your winrate.

Does the winrate converge faster because you play so many hands?

No, it does not. Convergence depends on the variance and it's higher in HUHU. The fact that you play so many more hands is irrelevant.

What you can say, though, is that because your winrate can be so much higher in HUHU, the likelihood of being up in a given decent sized sample of hands is generally higher in HUHU.

How big is the variance of HUHU?

Variance in absolute terms is somewhat higher compared to 6-max limit hold'em. But not that much. Typical standard deviations per 100 hands in 6-max are around 18-20BB and in HUHU they are in the order of 25-30BB. So there's a difference, but it's not that huge. Don't get me wrong, it's still a big difference, especially in the short term. The reason it feels significantly higher is that since you play nearly every hand, the swings are packed into shorter periods of time.

What alleviates this fluctuation is higher winrates compared to 6-max. So if you have an expectation of 4BB/100, this will somewhat trump the effects of the variance and you will experience lower midterm and longterm swings than with say 2BB/100 expectation in 6-max.

But sometimes people exaggerate this effect too. Like it would be almost impossible to lose money in HUHU if you're a great player. This is not true. Even if you're a big winner and only play against real bad opposition, it's still possible to experience swings of 100-200BB and sometimes even more. Those who claim the opposite have just ran well.

However, if you take a sample of for example 100k hands, and assuming that you're a big winner, it becomes virtually impossible to be losing. Though the sample is typically still going to include pretty sick swings.

What sort of bankroll do I need for limit HUHU?

This is mostly a function of your personal preferences. Something of a starting point for a recreational player could be 500BB or even lower than that. (Of course, that doesn't tell you much if you are planning to move up and down in stakes.) It's basically a question of how eager you are to drop down in stakes if you happen to run badly.

If you are not willing to drop down, then take the worst downswing you can imagine or estimate, multiply that by two (because you might start tilting and play badly during the downswing), reduce this number from your bankroll and it should still leave you with a decent bankroll. It's not like it would be a good idea to continue to play at your current limit after you lose 400BB of your 500BB bankroll.

However, if you are looking to move up and down in stakes, especially if you're looking to escape the higher rake low stakes games, you can be way more aggressive with bankroll management. The only real limitation is that you don't want to go too much over "Kelly betting". Kelly criterion, or Kelly betting in general, is a concept from sports betting that gives you the betting size that optimizes the growth of your bankroll.

There are numerous limitations for using Kelly betting in poker, but what it will tell you is the rough threshold at which the risk of losing money becomes too high for the growth of your bankroll. Even if you have a controlled stop-loss for a shot in the higher limit, it still might be the case that you are better off not taking the shot. This is because dropping too much from your
current bankroll affects negatively on the future growth of your bankroll.

So let's say you're trying to decide whether to play at 5/10 or 10/20. Further, let's assume that you have a rough idea what you expect your winrate and standard deviation to be at both stakes. Now, you want to know which limit you should choose to maximize the growth of your bankroll. Chen and Ankenman derive this sort of formula in Mathematics of Poker. It comes down to something like this:

cutoff bankroll = (std dev in higher limit ^ 2 - std dev in lower limit ^ 2) / (2 * winrate in higher limit - 2 * winrate in lower limit)

For example, let's say your winrate is 3BB/100 at both limits and standard deviation is 25BB/100. Converting the numbers into dollars and plugging them into the formula ((500^2 - 250^2) / (2 * 60 - 2 * 30)) gives you the cutoff bankroll at around $3125. In other words, to maximize the growth of your bankroll you are better off choosing the 5/10 game if you're below $3125 and 10/20 if over.

Of course, the parameters are difficult to estimate, but this sort of calculation should give you an idea of where the Kelly betting goes. One thing to note though is that if you over-estimate your winrate at the higher limit you will bet over the full Kelly, which hurts the growth of your bankroll more than betting under the full Kelly. So it's better to be a bit conservative with your estimations even if you want to maximize the growth.

Where are the fish?

Oh, yeah, there's this network that has effectively no rake and gets constant flow of massive fish, but wait.. I don't want to tell you where that is.

This one you have to figure out yourself. Although if you see from the lobby that 20 players are waiting for a player in a given limit, it should tell you something.

Should I play "hit-and-runners"?

There isn't any particular advantage for "hit-and-runners" in limit HUHU so of course you should play them. Furthermore, if you know someone is a "hit-and-runner" and he keeps coming keep after "running", then, by definition, he's not a "hit-and-runner" anymore. It's irrelevant whether he leaves after winning a couple of hands.

Which books cover limit HUHU?

Basically none. There are some chapters about super short-handed play in Winning short-handed strategies by Borer, Mak, Tanenbaum, but I don't know how useful the information there is.

There are other sources of information and it should be no secret at this point that several video coaching sites offer pretty solid knowledge on the subject.

What are GTO and Nash equilibrium?

GTO stands for 'game theory optimal' and is a shorter term for Nash equilibrium strategy. Formally defined, two players' strategies are in Nash equilibrium when neither player can improve their expectation by unilaterally changing their strategy. Or, in other words, GTO is a strategy that cannot be exploited in any way. It contains the perfect balance of value-bets, semi-bluffs and bluffs. The maximum expectation (EV) the opponent can obtain against this strategy is exactly zero.

The full solution to limit HUHU is not known currently, but it's possible to infer some properties of it. The book Mathematics of Poker has more to say on this subject.

What's proper ethics with other regular players?

The jury is still out there. Well, to be honest, there isn't a jury out there, which is a bit of a shame. I guess a good starting point would be a goal to make it pleasurable for the customers to come and enjoy a bit of HUHU fun, but I don't know if that's a good place to start ethics discussions.

Generally speaking, playing your button and then quitting is considered unethical. Some people are so annoyed by hit-and-running in general that they think it's unacceptable to play like less than 20 hands, or like 20 hours or whatever. Some people would like to get a signed notice about the possibility of you quitting in the next few hours him mailed and posted in three copies and with a legally binding clause forcing you to pay massive compensations if you break this agreement. Go figure.

Of course, the only thing that really matters whether you get the action you want. So if you want to continue playing the guy and he insists of you not "hit-and-running", then you probably should comply. Other than that, it doesn't really matter. Other will disagree.

Something i would want to add/suggest in regards to winrates and swings:

i think 5BB/100 is a sustainable winrate at the lower limits. i have a 5.4BB/100 winrate at 2/4 over 18k hands and that includes multiple 50BB downswings over the course of a couple hundred hands.

Danzasmack (not that he is the authority on winrates) suggests that 5+BB/100 is sustainable if you game select well.

Also, swings can be really big. I went on a 50BB downswing in 92 hands against one guy where he won 88% at showdown.

I have also won 70BB in less than 120 hands playing tilty fish.

On a stylistic point... and this is obviously just my opinion, but i feel like some of your writing for the FAQ is a little bit conversational, where most of the other FAQs are less so. something to consider?

I''ll take another look and see if i think of anything else. also, i might try to find some good essential reading if there is any.

I don't think 18k hands sample is remoutly enough to give you any indication of sustainability of your WR. I kinda doubt if 180k is enough either.

I also think that 50bb or so up or down is normal order of business in huhu and don't even worth mentioning.

The conversational tone was intentional, but apparently not very well received.

I'd delete the article if it was still possible.

I would not.

For one, "What's proper ethics with other regular players?" part is smart, funny and and a lot of food for thought for someone who still struggles with "i hate those f***ing hit&runners" issues.

this

Very good effort Jarno.

Just needs a little polish.

I like this FAQ. Conversational tone is fine for me. I trust the "essential reading" would be added to this?

noobish question: what does "solving" the game mean? I heard Bryce say that hulhe has 10^18 game states, and it would take ~100,000 weeks to solve it if you shut down google and let it work 24/7.

Solving would mean coming up with a strategy that is guaranteed at least zero EV against any other strategy. In other words, a strategy that couldn't be exploited in any way. It would not be optimal, in the traditional sense of the word, because it wouldn't exploit the mistakes of the opponent. But it would be impossible to beat.

Any article with formulas is an article I like

I wanted to clarify something in this paragraph. You mentioned that betting more than full Kelly hurts the growth rate of your bankroll more than betting under full Kelly. This isn't always true since betting at 1.5 Kelly is the same as betting at 0.5 Kelly in terms of growth rates. Also, betting at 1.5 Kelly is better than betting at 0.4 Kelly, etc. Betting anything higher than k = 2 is a losing proposition and the player is better off not gambling.

In general, the growth rate of your bankroll in terms of your standard deviation and win-rate is (k - k^2/2)*(wr/s.d)^2 . It is easy to check that at k = 2 , your growth rate is 0 and therefore you are indifferent to gambling.
Also check that at k = 1.5 and k = 0.5 you get identical growth rates.

This is assuming you're playing in a no rake environment. In the real world, a 0 EV strategy would be an overall money loser due to rake. For instance, two players playing a close approximation of the game theory optimal strategy in an average 5/10 HULHE game would have an approximate winrate of -2BB/100. However, you would still expect to have an edge over most opponents when utilizing a game theory optimal strategy due to the fact that most players employ a strategy which deviates heavily from that optimal strategy.

Once computing power reaches a high enough level or an innovative algorithm is devised to make a close enough approximation to a HULHE game theory optimal strategy (which may have already been achieved by some bots such as Polaris), we can expect to lose to the rake against the majority of opponents and hope that a random fish stumbles onto one of our tables often enough to make a slight profit.

How can this be right?

cutoff bankroll = (std dev in higher limit ^ 2 - std dev in lower limit ^ 2) / (2 * winrate in higher limit - 2 * winrate in lower limit)

Imagine that the std dev. is the same at both higher and lower limits. Then you're cutoff bankroll is 0? That doesn't make sense. I think what you want is the average of the kelly criterion for both stakes. So, you first figure out for the higher limit: Std dev^2/winrate^2, then do the same for the lower limit and then average them. It looks similar to the formula you gave, but it's different in that you divide and then subtract. There's a big difference.

You use dollars, not big bets. Then you'll get your answer.

That still doesn't seem right to me, but if you do that, I guess I see what you're saying.

Basically say winrate and sd are the same in every limit, let's say 15bb/100 and 1bb/100. Then if you are deciding between .50/1 and 1/2 the break even would be (900-225)/(2-1)=675. If you're deciding between 5/10 and 10/20 it would be (90,000-22,500)/(20-10)=67,500/10= 6,750. That's what you expect the answer to be, so I suppose that is right. Thanks for the clarification.