Heads-up tourneys and variance

[TNixon]

After doing a bit of digging for information about the variance in heads-up sit-n-gos vs normal 9 or 10 player sit-n-gos, I found an article (through a link on the forums here) titled Basic Bankroll Management, which gives basic guidelines for bankroll requirements for various types of play.

The relevant quote about heads-up-tourneys is:


[ QUOTE ]
Heads up sit and go tournaments are a variance monster all to themselves. Your win rate will have a massive effect on your variance in heads up play, and a player who is only winning 55% of his matches will have huge swings, while a really solid heads up player with a 70% win rate can get away with using numbers about twice the size of the regular SNG numbers in the chart above. The 55% player probably can't have a big enough bankroll no matter what he does; the variance is just too high.

[/ QUOTE ]

I'm having a little bit of trouble understanding why this would be true.

I can certainly see why it would be true that variance would increase heads-up for cash compared to a full ring game, with the reasoning being nearly identical to the reasoning for variance being higher at pot-limit omaha compared to no-limit holdem: because hands are often so close statistically, there are more marginal situations where it is correct to make big bets or calls, while being a slight favorite or underdog.

Similarly, when short-handed, it's less likely that your opponents have premium hands, so you end up putting in more money in marginal situations.

However, when playing a freezeout tournament, isn't your win/cash rate basically the *only* thing that determines variance?

The reasoning given for needing more buyins for a multitable tournament is that with more people, you will reach significant cash less often, so the reccomendation is for a higher bankroll to ride out the dry spells. The "average" player (whatever that really means) has an 11% chance to win a 9 person sit-n-go, but only a 1% chance to win a 100 person tourney. It seems quite obvious that there will be bigger gaps between bigger wins in the MTT situation.

Of course, the fact that you don't have to win to cash in a STT surely makes a difference, with the top 30% getting 2x their buy-in, the top 20% getting 3x, and the top 10% getting 5x, but in a heads-up tournament, the average player has a 50% chance to win. Shouldn't that fact alone *reduce* the variance of a heads-up sit-n-go in comparison to a full single-table tournament, making the dry spells that much shorter?

I have no clue whatsoever know how to calculate it, but I have a hard time imagining that a 20 buyin downswing is more likely to happen on a 50% average over other situations with smaller percentage chances to win bigger pots.

Before this weekend, I would also have said with certainty that my personal experience backed up my gut instinct about heads-up tourneys having less severe downswings, especially at low buyins, because I was having far smaller downswings playing $2-$10 heads-up than I was playing $5 STTs, after working $20 up to almost $200 on heads-up tourneys, playing with about a tenth of my bankroll or less in a single tourney. (if I had $120, for example, I'd play 1 $10 and a couple $5s).

Of course, we're talking about a super-small sample size, and I dropped back down to $20 again on a very long string of losses, but I did clearly identify some very bad tendencies on my part, especially in the $10 tourneys, so at this point, I really have no clue if the downswing was the result of a lot of poor play, combined with a little bit of bad luck, or if that's the normal expected variance playing heads-up?

My gut says that once I stop being stupid in the places I'm being stupid, heads-up tourneys are actually the safest way to slowly increase a bankroll.

Now, I realize completely that even if I'm right about that, $20 isn't really enough to start with (I actually started with $100, but dropped almost instantly to $20 on what was mainly a bad streak of luck on the $5 STTs), and my chances of going broke before getting a stable bankroll are pretty good anyway, but I wanted to get other people's thoughts on the matter.

If it makes a difference, I appear to be significantly better than a 55% winner at the $2 HU tourneys on full-tilt (closer to 70%), and before this weekends big losing streak, which, as I mentioned, appeared to be more me being a donkey and less normal variance, looked to be about a 60%-ish winner on the 5s and 10s.

[Kharlog]

I've read that HUSNGs have lower variance than STTs somewhere in this forum. I started this month and have played 93 HUSNGs winning about 64% of them. That's extremely small sample size and I've been winning almost every day so far (played every day). The losing days haven't been too bad as well. Could be just a good run though...

[ChicagoRy]

Losing 16 buyins at the 5 dollar level or more buyins at the 2 dollar level (unless you continued playing 10s during your downswing) is probably more of a result of a few leaks in your play and not just variance.

To be sure you should post some big pots that you lost (say, pots over 1k) and the corresponding action previously before some of these big pots (where needed) and let the forum determine if you have some tendencies that might need fixing.

[TNixon]

[ QUOTE ]
Losing 16 buyins at the 5 dollar level or more buyins at the 2 dollar level (unless you continued playing 10s during your downswing) is probably more of a result of a few leaks in your play and not just variance.

[/ QUOTE ]
Oh, I know for a fact there was a big poor play factor in the big downswing. There was a little bit of bad luck, too, but mostly just bad play. I clearly have some tendencies that need fixing, but they're not particularly difficult to identify. I just didn't bother until it was way too late. Most of them were big all-in calls where I had no business whatsoever calling. And I did continue playing the 10s much longer than I should have before stepping down as well. [img]/images/graemlins/smile.gif[/img]

I'm just trying to resolve my (admittedly limited) experience and gut instinct with an article claiming HUSNGs have a much higher variance than normal ones.

The question I *really* want answered is what's the cheapest, lowest-variance way to build up a bankroll, outside of grinding away at limit cash tables, which I simply don't have the patience for. So far, (minus a day's worth of complete stupidity), heads-up tourneys seem the best of all the things I've tried, but the one article I was able to find on the issue claims otherwise, and a forum search here is actually what pointed me to that article in the first place.

[img]/images/graemlins/smile.gif[/img]

[ChicagoRy]

Link please.

[omgwtfnoway]

[ QUOTE ]
However, when playing a freezeout tournament, isn't your win/cash rate basically the *only* thing that determines variance?]
this is absolutely correct

consider this analogy for thinking about hu sngs and 6handed or 9handed sngs:
for hu sngs there are two outcomes, you win or lose. flipping a coin is a decent but not great analogy (i'm assuming that you have a decent edge)
for sngs or mtts with more players roll d&d style die with 50+ sides.
the point is that you're simply going to need a lot more trials with the die before you even get all the possible outcomes much less all the outcomes in their expected proportions. hence, there is far less variance in husngs than in other sngs and tournies.

chicagory - if i remember correctly that's from an article written by pokerfox for the pocketfives forum. if you look up their article archive and search for his first articles you'll find it pretty easily. i think it was the first article he wrote for p5.

[jay_shark]

Here is the link if you're interested . If you feel the need to ask questions , please do so and I'll reply back .

http://forumserver.twoplustwo.com/showfl...part=1&vc=1

[TNixon]

That's perfect. Wonder why that post didn't show up in my searching. Figured it should have shown up when I was searching this forum for "variance" at least. [img]/images/graemlins/smile.gif[/img]

Your figures, along with the results from a quick-n-dirty simulator I threw together, show pretty much what I expected to see, that 10 buyins gives you a fair chance of going bust, but not so much that I'm afraid of busting out my last $20 before catching back up a little.

10-25 buyins, based on how much ROR you're willing to accept, sounds *much* more reasonable than the "absolute minimum of 40 buyins for a 70% winrate" that other article reccommended.

[img]/images/graemlins/smile.gif[/img]

In fact, according to my simulator, even a 55% winrate gives a decent chance (71%) of surviving through a string of 250 games, although on average you only have 28 buyins at the end of that 250 games when you do make it through.

I've only run stats on games from one of the two computers I mainly play on, but I'm about 68% on the $2s, and 63% on the $5 (over an admittedly small sample, 50 $2 games and 90 $5 games). I'll have to check the other computer to be sure, but I'm fairly positive it's the $10s where I was getting clobbered. (48% over 64 games on this one, which would lead me to believe that I'm adjusting *extremely* poorly to the players at the higher buyin)

[TNixon]

More interesting results from my simulator:

I added the concept of stepping up and stepping down, added in the actual levels from Full-Tilt, just to get a feel for what a correct step-up and step-down strategy might be, and the results are extremely interesting.

In all the following simulator runs, I start with $20, which is enough for 9 attempts at full-tilt's $2 HUSNGs. I allow stepping up through the $50 HUSNGs, and assume a fixed win rate (which I know is absolutely wrong, since your win rate is likely to be smaller at higher levels, especially when you first step up). But the win-rate I've been running with is a pretty small 58%, which I don't feel is unreachable for most people who care enough to read this forum. [img]/images/graemlins/smile.gif[/img]

Each run is 10k attempts at a string of 250 games, with a survival rate and a finish bankroll rate, which results in an equity calculation. (percentage survival multiplied by the average ending bankroll)

The step-up and step-down multipliers are how many buyins at the next and previous level you have to have before stepping up and stepping down. So, if you're currently at $5 + $0.25 with $80, and a stepup and stepdown multiplier of 10, you'll move up when you have $105, and down when you have $21.50.

<font class="small">Code:</font><hr /><pre>
Trying 10000 times over 250 games with $20.00. WR: 0.58 SU: 5.00 SD: 10.00
Survived 6955 times with 426.75 avg remaining
296.804880 equity
Trying 10000 times over 250 games with $20.00. WR: 0.58 SU: 8.00 SD: 10.00
Survived 7769 times with 240.48 avg remaining
186.830201 equity
Trying 10000 times over 250 games with $20.00. WR: 0.58 SU: 10.00 SD: 10.00
Survived 8027 times with 161.66 avg remaining
129.764105 equity
Trying 10000 times over 250 games with $20.00. WR: 0.58 SU: 13.00 SD: 10.00
Survived 8128 times with 102.65 avg remaining
83.437791 equity
Trying 10000 times over 250 games with $20.00. WR: 0.58 SU: 15.00 SD: 10.00
Survived 8246 times with 85.94 avg remaining
70.865849 equity
</pre><hr />

So, fixing the stepdown value at 10, and varying the stepup, larger stepup values (waiting longer to stepup) give you a higher chance of surviving, (12% higher going from a stepup of 5 to 15), but at a HUGE opportunity cost. Clearly, you'd be far better off stepping up early rather than late, and simply redepositing if you went broke.

Now the other end, varying the stepdown:
<font class="small">Code:</font><hr /><pre>
Trying 10000 times over 250 games with $20.00. WR: 0.58 SU: 5.00 SD: 5.00
Survived 5382 times with 633.26 avg remaining
340.821326 equity
Trying 10000 times over 250 games with $20.00. WR: 0.58 SU: 5.00 SD: 8.00
Survived 6537 times with 490.59 avg remaining
320.700775 equity
Trying 10000 times over 250 games with $20.00. WR: 0.58 SU: 5.00 SD: 10.00
Survived 7007 times with 422.73 avg remaining
296.204438 equity
Trying 10000 times over 250 games with $20.00. WR: 0.58 SU: 5.00 SD: 13.00
Survived 7537 times with 306.03 avg remaining
230.654902 equity
Trying 10000 times over 250 games with $20.00. WR: 0.58 SU: 5.00 SD: 15.00
Survived 7883 times with 236.34 avg remaining
186.310536 equity
</pre><hr />

While the previous results indicated that stepping up earlier was better, because of the very slow increase in survivability compared to the decrease in equity, the results here are almost the opposite, in a couple respects.

Stepping down late (waiting until you only have 5 buyins at the previous level is pretty late) gives the highest equity, but with a pretty significant failure rate. When stepping down that late, 47% of the time you'll go broke over the 250 games, and 53% of the time, you'll end up with an average $640 bankroll.

Stepping down sooner increases your survivability, just like stepping up later does, but the overall equity cost seems to be much smaller. While you pay a heavy price for waiting longer to step up, it appears the price for stepping down sooner is much less. To me, it seems worth sacrificing a small amount of equity for significant increases in survival rate.

In fact, it's very encouraging that the strategy I was already using (step up early and step down later) appears to be ideal for me, given my level of acceptable risk. In fact, SU:5 SD:10 is *very* close to what I was already doing anyway without actually attaching explicit numbers onto the transitions, and a 70% survival rate over 250 games seems perfectly acceptable to me, especially given the fact that it will leave me with around 20x my original startup on average. And if I have to redeposit, well, it will have to be more than $20 anyway, so I can re-figure a strategy from there.

[img]/images/graemlins/smile.gif[/img]

However, it's quite obvious that stepping up with 5 buyins and down with only 10 is not a good overall long-term strategy. When running with 5000 games instead of 250, waiting longer to step up and stepping down sooner *increases* your equity up to a point, rather than decreasing it, since your survivability has such a big influence over so many games. From there, it's just a matter of finding the right balance between stepping up and stepping down. Interestingly enough, it still appears to be better to step up and step down fairly early. Right around SU:10 and SD:20 seems to be a maximum, at least given a winrate of 58%.

But while it's not a good long-term strategy, it certainly does seem to be a good way to get a leg-up for pretty cheap, with a fairly minimal risk of failure, given that I'm starting with $20 in all the above scenarios.

[img]/images/graemlins/smile.gif[/img]

Oh, and just in case anybody's curious, leaving all the factors the same except for the initial buy-in, and making that $50 instead of $20, which is a much more reasonable number to start with, gives very similar trends, but *much* increased survivability rates, and bigger final bankrolls. I ran with $20 because that's about what I have in full-tilt right now, but here are the 250 game runs starting with $50:

Varying step-up:
<font class="small">Code:</font><hr /><pre>
Trying 10000 times over 250 games with $50.00. WR: 0.58 SU: 5.00 SD: 10.00
Survived 8720 times with 653.82 avg remaining
570.132111 equity
Trying 10000 times over 250 games with $50.00. WR: 0.58 SU: 8.00 SD: 10.00
Survived 9339 times with 498.47 avg remaining
465.520792 equity
Trying 10000 times over 250 games with $50.00. WR: 0.58 SU: 10.00 SD: 10.00
Survived 9523 times with 385.32 avg remaining
366.936872 equity
Trying 10000 times over 250 games with $50.00. WR: 0.58 SU: 13.00 SD: 10.00
Survived 9817 times with 212.85 avg remaining
208.956978 equity
Trying 10000 times over 250 games with $50.00. WR: 0.58 SU: 15.00 SD: 10.00
Survived 9877 times with 158.25 avg remaining
156.300737 equity</pre><hr />
And varying step-down:
<font class="small">Code:</font><hr /><pre>
Trying 10000 times over 250 games with $50.00. WR: 0.58 SU: 5.00 SD: 5.00
Survived 6904 times with 903.74 avg remaining
623.943522 equity
Trying 10000 times over 250 games with $50.00. WR: 0.58 SU: 5.00 SD: 8.00
Survived 8301 times with 731.22 avg remaining
606.986914 equity
Trying 10000 times over 250 games with $50.00. WR: 0.58 SU: 5.00 SD: 10.00
Survived 8780 times with 648.21 avg remaining
569.124702 equity
Trying 10000 times over 250 games with $50.00. WR: 0.58 SU: 5.00 SD: 13.00
Survived 9335 times with 516.80 avg remaining
482.430339 equity
Trying 10000 times over 250 games with $50.00. WR: 0.58 SU: 5.00 SD: 15.00
Survived 9545 times with 440.76 avg remaining
420.704468 equity
</pre><hr />

But over 5000 games, the difference in final bankroll is very small, but the survivability difference is quite large:
<font class="small">Code:</font><hr /><pre>
Trying 10000 times over 5000 games with $20.00. WR: 0.58 SU: 10.00 SD: 20.00
Survived 7823 times with 24260.49 avg remaining
18978.979982 equity
Trying 10000 times over 5000 games with $50.00. WR: 0.58 SU: 10.00 SD: 20.00
Survived 9598 times with 25040.24 avg remaining
24033.626326 equity
</pre><hr />

Amazing what a difference that $30 makes. Obviously in the long term, most failures happen pretty early.

Anyway, enough geeking out for one day. Thanks for the help everyone.

[img]/images/graemlins/smile.gif[/img]

[SkyReVo]

Nice post TNixon but there's one thing I don't understand:

Trying 10000 times over 250 games with $20.00. WR: 0.58 SU: 5.00 SD: 10.00
Survived 6955 times with 426.75 avg remaining
296.804880 equity

You say you'll go up a limit if you got 5 Buy-ins for it and go down with 10 Buy-Ins. Let's say you got 55$ which is 5x $11 so you step up to them. But it also is 10x Buy-in for the lower limit (5.50) which forces you (if we use your rules) to step down after losing just one SNG?

Also I wanted to ask whether you could do that with
10000 times / 5k games / $180. WR: 0.60 SU: 10 SD: 15 for me?
Would be pretty happy about it.