Monday, March 14, 2005

Mandelbrot Set

YTD: +27882.59

I remember a while back, Chaos coming up with his theory of strange attractors for tournaments. Without going into too much detail, he was coining a chaos theory term, a branch of physics/maths, to show how certain stack sizes evolve into larger or smaller sizes, somewhat regardless of how the person plays. It's an interesting and useful idea.

Another theory from the same set of disciplines, fractals, also has some interesting ramifications for poker thought. Basically, a fractal can be defined as a system having similar detail at all scales. Have you not noticed that a lot of poker games are like that? Take holdem for example. There are lots of differences between say the 5-10 limit game and the 100-200 game on Stars. But also a huge amount of similarities. Sure, the game plays differently because of a huge increase of aggression, but the fundamentals stay the same. A guy playing 40% of his hands would be just as much a fish in the big game as the little. And a lot of the technical plays, such as blind defense, are also basically the same.

Limit holdem is a fractal.

Looking through Pete B's excellent blog, I realised that PLO is very different. PB is currently exploring low limit PLO and he commented on some hands he played. Looking at them, I realised that I would play them very differently in the games I frequent. This corroborates my old thinking that if you want to learn PLO, don't expect what you learn in the micro-limits to have much value once you move up into meaningful money.

With Pete's permission, here are the hands, with my comments in italics:

Hand 1

PB calls a small raise with 5h 6s 7s 7h and 4 other players take a flop of 3s 4c 2s. UTG calls, as does someone else, the initial raiser folds, and Pete calls. The turn brings a blank, everyone takes off and goes allin, and PB scoops against the other top straights when he hits his small flush on the river.

This is a great example of a backgammon concept that crosses the divide really well - losing one's market. This is that you are NOT afraid that you will get outdrawn, rather that the next card will be so terrifying for your opponent that you won't be able to get any more money out of him. So if he is bluffing, he probably switches off once you flat call anyway; if he has a hand, you need to sweep him in now before he gets nervous about a deadly looking turn. A figure closely approximating zero of online players are capable of passing the "dry" nuts at this point.

Hand 2

PB dealt AAJ8 single-suited under the gun, raises and gets one caller - a conservative player. PB has about $124 against his $30. The flop came 833 rainbow. PB bet 2/3rds of the pot and gets raised the pot. PB puts the foe on a big pair and reraises, foe passes.

I don't like the UTG raise, but that is a stylistic point more than anything. I like the bet on the flop but the reraise is not a good play, at least in my games. First off, he could have a three. But that isn't the real point. The reraise is a variation of my famous "Damned if you do, damned if you don't" play. Ok it's only famous if you have been reading this blog :-( You're damned if he has the 3, cos you are putting in a big chunk of change for 2 outs. But critically, you're kinda damned if he doesn't have the 3 because you are making him pass when he has something like 2 outs. These aren't free cards...he is charging himself for them! If you go limp he may have a rush of blood to the head and bluff off his money. And if you are losing, the result is the same. More upside, same downside.

Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line


Anonymous said...

Dave, could you post a link for Pete's blog? Thanks.

chaos said...

Nice article Dave.

'The damned if you do & don't' argument against, raising or betting inn these situations is very true. Too often I have witnessed myself do such an illogical thing. I've thought about this a bit and tried to consider & correct it, but there is a degree of hard-wiring here.

So why do we do it? Well I think it is the softer aspects of value that rear up to swallow some EV. For one, there is clarity. This probably comes into tournament play morethan cash because in some games or sites, cards don't go on their backs. Though we might crave the uncertainty of gambling, the in-hand uncertainty of having a big hand that might be a mile in front or a long way behind is unpleasant. By reraising all-in the cloud of uncertainty is banished a lot sooner, which is, emotionally, a good thing. I'm sure this is a driver.

Also, in a similar vein, there is once again something decidedly unpleasant about check calling on the turn or river when you might possibly be able to get away from the hand. Perhaps, because if a bet comes again you know the chances of your foe having a monster, have increased. So by raising all-in or betting out on the turn/river you don't have to face such a horrible piece of evidence or quite simply make a tough decision. I've certainly witnessed this breakdown of logic in myself on the river in limit holdem.

There is also the fear of regret, that you might let him improve as well, but you just have to identify these fears and take more time.

While he is undoubtedly charging himself to see the turn, at your decision point you are letting him in for free. But on the risk/reward side of things it's got to be worth letting him hit a 2-outer in the hope he will bluff the turn. Though I suppose you are probably giving him 1 and a bit cards at a two outer by just calling, since he may follow your check with one on the turn.

I started to write something about a recent experience of mine relating to your fractal analogy, but it when I’d finished it had gone on for too long. But I would say that it is true the more short-handed it becomes, there is quite a considerable difference in the way high-limit ring games are played, compared with low-limit games. But in HU you might find it harder to discriminate.

On the strange attractors front, it reminds me that I must dig through your posts to find some of the stuff I’ve written. I am terrible at half writing articles; I love starting them, but seem to be lazy when it comes to finishing them off. I am going to try to complete some of the articles I’ve started as well as write up some of the material that as flowed from threads: if only to spring clean my mind. But at some stage I will put them on a blog. I’m being a little mischievous, as I have had a blog for about 3 months, but it is an anonymous one that I’ve used to keep results, write a few thoughts and about 4 articles and only my girlfriend reads – that I know of.

It isn’t hidden, but it’s not advertised either: I don’t think ‘poker’ is in the list of hobbies. At some stage I’ll either bin it or go live with it, but as I’ve hinted at before I wouldn’t be able to maintain a frequency that folk seem to expect.
Even if I don’t go live with it, I’m sure I’d like a candid opinion on what I’ve written!

Big Dave D said...

Pete's excellent blog can be found at then /users and then /_pjb_ (Pete doesnt want spider traffic :-)

Anonymous said...

From Peter B

Hi Dave, I could write for ages on your points. I agree with most of them. As the cards lie on the hand where I have the nuts with redraws, you are obviously right. I guess that I was scared of a set, so I wanted to get all the money in when I was a big favourite rather than a marginal favourite. But I take your point about "losing your market".

I'm the worst backgammon player in Christendom, but I see what you mean. Are you suggesting that if somehow you find yourself an 80% favourite in a backgammon game against a clever player (with the doubling die to you), you should deliberately make some bad plays to reduce your chance to 60%, in order to get your opponent to accept the double?

I really hadn't considered this before, but it makes a lot of sense. "Keeping your market": An excellent idea for a general bit of poker work.

In limit, of course, you are perpetually trying to contrive situations where your opponent does NOT have the value to call (and you hope that he does anyway).

And as for the AA hand. Yes, you are dead right. I shall remember this one as well.

I tried some of the low-stakes PLO on Stars. A whole raft of publishable hands! Most of which I played terribly. But I ended up $30 to the good after not losing my nerve or allowing the red mist to descend.

Speak later.


Big Dave D said...


That isnt quite what losing one's market means. Imagine you have an edge worth doubling, say a nice lead in a race. If there is no chance of a n immediate change, then you may as well wait until much later on, where your foe will still have to call, but you will still have an edge. In PLO this is analagous to having a dry straight in a big pot, vs a good foe, and waiting for the turn, to see if the draws come.

Losing one's market is to do with volatile positions. Say you have some great chances to hit your foe and you have a semi-strong board,and a slight lead in the race. Although taken seperately these may not be a double, together they are, because if you wait HE MAY HAVE NO TAKE. This is the PLO situation you were in.



Big Dave D said...


I agree with what you say about the dark reasons we often abdicate control in these spots. One of the best parts of my game is my ability to go limp and check-call...I havent got a macho hangup about it at all and I know its one of the best parts of my game.

I'm not sure whether the strange attractor thing was here on the good old days at doesnt spider my blog too well :-(

And in future, don't worry if it seems to it! I rely on your content to raise the tone of mine :)



Anonymous said...

mayeb you want to run that by us again - for those not familiar with backgammon!

chaos said...

Trust me Dave, you do just fine without my distractions. Going limp at will, as you put is a huge strength and is one reason why I have never considered myself to be a natural poker player. (I can do it, but it takes effort!) Thinking about poker and responding to it use two very different parts of our brain. There is , clearly, no emotional-EV in thinking about the right answer whilst sipping a cup of tea. There is emot-EV in almost every decision we make, but for some players it is minimal - they are the lucky ones. Rest assured my MR Hyde is a pretty sick gambler, as those laying me at Cheltenham this can verify.

Fortunately, there are other criteria too that allow us mortals to catch up those emotionally programmed for poker. Awareness, though, is the quickest way to close the gap.


Big Dave D said...


I'm afraid going further into the backgammon examples may took up the rest of the blog! And I'm no expert by any means. I think the poker examples mostly speak for themselves.

It may make it a little easier to understand if you know that backgammon allows the player to raise the stakes of the game, which his opponent may refuse, and payoff for the original stakes, or accept and play for the doubled stakes of the game. And now only the acceptor of the double can double the stakes in the future.

So as you can imagine, this creates some situations that are poker-like, and also makes the game quite technical mathematically. It also explains why many top poker players were original backgammon is certainly the harder of the two disciplines.



chaos said...

Ok guys, honest opinions. Too academic, not enough poker?


Anonymous said...


I enjoyed your article. Whilst your writing style is certainly not concise, you always have interesting points to put forward. Personally I would have liked you to expand more on this paragraph, maybe at the expense of part of the quiz analogy

"There can be precious little doubt that Limit-Hold'em tournaments order players better than No-Limit (NL). Certainly, skilful Limit-Holdem players will expect to do better than skilful NL players in their respective tournaments. "

Whilst I generally agree with this I certainly dont think it is obvious. It is fine if we consider a poor aggresive player comapred to solid expernienced player. But you talk of sorting the skills of 500 players, meaning you will have some players very close in skill sets. For players close in skills it is not obvious that your statements are true.

Say their win rates at a limit holdem cash game are 1.25 compared to 1.30 BBs per hour or even smaller. It will take a long time for this difference to become apparent in a series of limit holdem tournamnets. How do you know it would take longer with NL?


chaos said...

Well either I'm pissed or the blog is. After I posted there appeared to be about 20 messages (9 repeats from me), but nothing appears on the Blog.

Butch, thanks for your comments - I'll post a reply when this thread has untangled itself.

Big Dave D said...


I cleaned them up 4 u mate...a think Blogspot has been poorly :)

Isn't your article a bit "pearls before swine"? Who the fuck is going to understand it who read PIE...they're all tourney players :)

Although its nice if it is in the print copy cos then you can keep a momento.

I liked the article, fwiw, but i think unfortunatley that NL is the future. Because it orders much worse.

Also, I'm not sure that was always the case either. I just think that the new style of NLHE in tournies has made it so, increasing the penalties for poor decisions making and also the questions in the test have got easier :(



chaos said...

Thanks guys for taking the trouble to read it – not a trivial thing.

In fairness to JS, it wasn’t an article that was particularly useful to anyone, but I quite enjoyed thinking about it. It was an issue that had always lingered in my mind, but I’d never mentally worked through it, as I did then.

Dave: A thousand ty’s that could have been very embarrassing, weren’t you just a little bit tempted to leave them up?

I suppose I was rather defending NL on the skill issue, because it is criticised because poor players often win. As you said, it’s popularity will ensue because of it’s trait of ordering poorly while still holding high levels of skill.

Butch: I’ve only played a handful of limit-holdem tournaments, but I certainly noticed a lot of differences between cash and tournies, so I wouldn’t necessarily expect a smooth transfer of close cash to close tournies players. But I’m being evasive; basically your question refers to the respective discriminating powers of NL and Limit between two close tournament players. The crucial difference is in the variance: a profit of say 1BB an hour could come from high or low volume (in size & number of pots/decisions). Hopefully, this will suffice:

Imagine two games between two closely matched players. In both games player A has a 0.01BB advantage per hand over player B. Suppose that in the limit game the average volume bet per hand by each player is 2BB, in the NL it is 10BB. Assuming an unlimited bankroll is in play for both players, after 10000 hands the best player would ‘expect’ to be 100 BB’s in front. In the NL game he only expects to be ‘5 hands in front’, but in the limit game he’d expect to be ‘25 hands in front’. Consequently, there would be a much higher chance of the better player losing after 10000 hands in NL than in limit since it is easier to close this ‘expected gap’ for the weaker player.

In limit, players play more hands: the larger the sample size the more confidence you have that ‘the truth will out’. Also, there is greater variance in the number of hands played in NL: in a limit tournament it’s almost impossible for one player will play a hundred times as many hands in a tournament than another. Both these factors make it tougher for the NL tournies to express the differences between two closely matched players

My stats skills are basic to say the least. My use of expected hands in front from average hand/volume is probably a bit dodgy. But I think the sense of it stands

Re the multi choice: it probably was over-egged, though in order to show that turning up the gas on difficulty doesn’t necessarily lead to better ordering, I needed to put in an example was needed.

While the maths student in me barely has a pulse nowadays, I think its spirit lives on in my writing. I often feel the need to justify or qualify the statements I make. I have been quick to criticise flawed logic in others, so when I don’t do it isn’t a surprise if someone jumps on me if what I say isn’t self-evident. But, having said that, you were right to pull me up on the sorting issue. It was something that was intuitive to me, as it was to you, but I hadn’t worked through a proper explanation. (Though it still isn’t proper, but hopefully sufficient)


chaos said...

I was in a mad rush to get out this afternoon, consequently that last paragraph didn't come across too well. Essentially, I agree, Butch, that my style needs to be more concise. I certainly wasn't criticising you for pulling me up the sorting issue: it's just, through doing maths i guess, one is used to getting pulled up for making unproven statements.



Anonymous said...

Hi Chaos,

Thanks for your response. It is probably the academic in me that made me ask you about it (I am just finishing up my physics phd.)

A few of my thoughts on this idea (these are off the cuff comments and I have nothing to back these up ;)

Generally I would expect the better player to have a larger edge in a hand of NL as compared to limit. So just because limit players play more hands doesnt mean the better player has more chance of being ahead. We might need to compare the chance of being ahead in 25 52-48% shots in limit with 5 65-35% shots in NL for example.

By using tournaments to sort people we presumabley comapre prize money won over time. Inevitabley the money won will come down to a few key hands at the business end of the tourney, so maybe there aren't that many more 'key' hands in a limit tourney than a NL one.

As I said before, I agree with your sentiment, just interested in the thoughts behind it. I enjoying thinking about these sort of things but I never get around to doing any analysis as I always decide I should play cards and make some money instead.

On a vaguely related note, I've never found a satisactory discussion of the concept of variance in NL compared to limit, as there are so many variables to consider. All I can say for sure is that in my experience online NL cash games offer a lower variance to limit ones for the same win rate.

Don't feel you have to reply just thought I'd share my thoughts.


chaos said...

Hi Butch, your comments are much appreciated and interesting. Below are some thoughts relating to some of your comments, but I suppose only modelling or simulation would provide reasonably conclusive answers.

The example served only to show that different types of games would have different success at separating close players even if the difference in the edge per hand were the same in both flavours of the game. My assumption that the average volume/pot in the limit game was less may be incorrect in very tight aggressive games, but the variance of that volume would certainly be much higher in NL. This will, naturally, handicap the game’s ability to order correctly. Which game is likely to provide the greater edge for two equivalent matches I can’t say, but that is another issue.

I don’t know whether a player drawn, say, from the top 10 % will have a greater edge per hand against a player in the top 30% in NL over limit. But even if that were true then it is still possible to construct a greater edge and variance for the NL player over the limit player that leaves the limit player more likely to win the race or the HU tournament than the NL player. In a series of such tournaments of the limit player have a higher win rate, but a lower hourly rate. With a similar distribution, but in a cash game with an unlimited bankroll, one would expect that the NL player would be expected to win more after say 10,000 hands (depending on the edge difference), but the limit player would more be likely to be winning.

Re key hands: I feel there would be more key ‘money hands’ in limit than NL, but that is based on only a little experience & gut feeling. Even if that weren’t true it maybe that a good limit player is more likely to find himself in these situations than the NL player.

Cash NL variance vs Limit variance: Well I don’t know as I’ve only played a little NL cash, but player-type certainly would make a difference. Thinking about it could easily be true; I suppose in limit you know when you enter a hand that the potential damage is limited, but in NL it’s bounded only by your stack. That fact could lure us into playing more hands than in NL and potentially increase the hourly variance above NL. Of course in NL you fold a lot more when you catch a flop than in limit and more of the right plays are to fold.

Analysis(generally): My poker-maths is very poor. It was probably the first thing I thought about when I got seriously interested in poker. But I was putting the cart before the horse in trying to find solutions to marginal situations - what point is there knowing what the right decision is if you don’t enforce it? Much of the analysis people do is, for me, icing on the cake, because there are more fundamental problems that they ignore. Players are always looking to improve and so focus on finding the answers to questions they haven’t solved, rather than maximising the utility from what they do understand. To me is the hardest and most important most important task, if I can reduce the number of times that I think ‘Yeah I knew that was the wrong play’ and push it to the back of the mind, then I’m making progress.

Anonymous said...

Hi Again

Interesting stuff, I think you have convinced me ;). We're probably as close to a natural end to this debate as we will get.

I have no argumnets what so ever with your last paragraph. Practicing what you preach is certainly the biggest weakness in my game and I suspect it is for most people.


Anonymous said...

Poker is a game of skill and strategy. No one could count on luck.