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Alex Rousso

06/06/2012

# Luck Horizon

A number of times I’ve wanted to write about “the long run” in poker. Each time I try to articulate the concept, and especially poker players’ general lack of appreciation of it, a clunky phrase comes out. I’ve said things like ‘the long run is longer than you think’ and ‘variance in some forms of poker is higher than in others’.

Take the analysis any further, and I fear I might lose my reader before the end of the sentence. Consider for example the thought that: ‘there’s no specific number of hands or tourneys you can play before you reach the long run, it’s just that the longer you play, the less likely your “in practice” results are to be deviating significantly from your “in theory” results.’ That, as the Bishop said to the Ice Cream Lady, is one hell of a mouthful.

Poker players are wiser and more knowledgeable about the game nowadays, and it’s high time we had a concept which makes it easier to talk in these terms more freely. I’d like to introduce the term “Luck Horizon”, which will be reasonably easy to define and, although hard to quantify mathematically, I hope will be simple to understand.

The use of the word “horizon” in this way usually means a limit, possibly a limit beyond which something does not apply any more. In business, a company’s “cashflow horizon” is the time in the future where, assuming zero sales from that point on, it would run out of money. In astrophysics, the “event horizon” is the point near enough to a black hole that its gravitational pull is so strong that even light cannot escape. At that event horizon, time stands still (i.e. events can no longer occur).

Your Luck Horizon in poker is that notional point in the future where the luck factor would no longer apply to your lifetime results. In other words, at the Luck Horizon, your actual results (i.e. how much you have made/lost from poker) would be exactly as they “ought” to be (e.g. -2.1BB/100 in cash games, an ROI of 7.2% in SNGs, etc., etc.).

In practice, the Luck Horizon can never be reached. There will always be a luck element in poker and however infinitesimal that factor is, it will have an effect on your averages. Imagine you played the WSOP main event each year for ever. Let’s also imagine a point after, say, a million years where it just so happened your results were perfectly in tune with your “real” ROI at that point. The next ME you played, if you busted, you would be running below expectation. Frustrating, huh?

So we actually need to define the Luck Horizon as the point in the future where the luck factor in your results diminishes to within a tolerable limit. There’s a useful corollary to this term, as it also explodes a mythical question about how much luck there is in poker.

In my seminars I’ve been asked quite a few times to answer that eternal question, and people are always expecting some actual figure in return, as in “poker is 70% skill and 30% luck”. The answer, boringly, is that it depends on what game you play (tournaments/cash, stakes, PLO/NLHE, etc.) and how long you play it for. Play one hand against Phil Ivey heads up and the luck factor will be very high. Play 100,000 hands and it will be much lower.

Using the concept of the Luck Horizon, we can see that the correct way to look at this is not considering how much luck is involved as a percentage, but how far away your luck horizon is in this situation. Heads up against Ivey, I would suggest (with all due respect) that it’s not that far away. Entering the Main Event every year I would suggest it is very far away.

We can elucidate further by considering what factors should affect luck horizon. First and foremost is obviously your skill level in the game you’re playing. Just to be clear here: if you have no edge, you have infinite variance – your results graph will just be a random walk.

The second factor is the limit within which you would be comfortable that luck isn’t a major influencer of your results. Defining those limits is a mathematical proposition beyond the scope of this article. As, indeed, is demonstrating how to calculate your variance from your results. However, both are not too difficult – especially with the help of your tracker, an excel spreadsheet and a friendly mathematician. I may be asking too much, but it shouldn’t be too hard to create a website into which you upload your results and it does the whole thing for you (anyone?)

To eradicate the luck factor altogether would mean playing forever. I know you sometimes feel you want to do this (e.g. just having gotten off the plane to Vegas), but you will have to stop eventually, I promise. So how do should we define how small a factor luck is in our results before we’re comfortable?

A couple of concepts might help frame this. The first is “risk of ruin”, which is a factor in bankroll management. Put simply, the limits to how much luck you can handle are – among other things – down to your chances going broke. It’s all very well playing lots of tournaments you have a big edge in, but if they are really expensive (e.g. the Main Event), you only have to get unlucky a few times before you’re too broke to play it again.

Another factor is time scale. The Main Event is another example here (it only happens once a year, so you’ll be lucky if you can play 50 of them – unfortunately not enough for you to reach the long run). However, it might also be a game which breaks often online – you see lobbies full of heads up players sat down at tables waiting, spider-like, for their prey to flutter by. In short, some variants of poker are quite easy to get a consistent game where you have a decent edge. In others, it may not be easy to reach the long run no matter how good you are.

So to summarise, the factors at play in working out your luck horizon are as follows:

- your edge in the game you are playing,
- the limits you will tolerate that your results graph is down to skill and not luck; these are mediated by:
- 1. your risk of ruin for a certain game, and
- 2. the time scale involved to get a big enough sample

Hopefully this concept will be useful for players trying to establish the luck factor in whatever poker variants they play.

I’ve been particularly keen on finding a new term which brings to mind (here’s that clunky phrase again) just how long it takes to reach the long run in poker. If we can actually quantify that with some mathematical analysis to obtain statements of the following ilk:

- with an ROI of 50% and an unlimited bankroll, your luck horizon playing the Main Event every year is 150,000 years.
- with a win rate of 2BBs/100, a bankroll of 100 BIs, and playing 30,000 hands a month online, your luck horizon is 18 months.
- with a 200,000 hand sample displaying a 4BB/100 win rate, the probability that this win rate is accurate to within a tolerance of +/-10% is 96%.

... and so on.

If we can get some actual figures involved (number of months, years, etc.), it makes cross-comparison a lot easier. Certainly, it’s much more preferable to saying “it’s all variance” and “I’m on quite a large downswing at the moment”. The very facet of luck factor being defined in terms of a time scale and being explicitly qualified by win rates, tolerance limits etc. will, I hope, help poker players to make better decisions about their poker careers.

Blogger “Lildavefish” recently pointed out that many players are beginning to hide behind the concept of variance, almost as if it’s an excuse for bad results, and using it rather nebulously as a synonym for luck. It isn’t. Variance is a mathematical term which defines the amount which individual events/points on a graph (of your results) deviate from their average.

So let’s say this explicitly: luck will affect the variance in your results, but so will being bad at the game you’re playing and so will your selection of the games you play. One of your jobs as a poker player is to reverse engineer from the sample of results you have to an understanding of whether you are money making or not at the games you are playing. I hope the introduction of the term “luck horizon” – and, if possible, the mathematical analysis which will follow from it – will help in those endeavours.

*This article first appeared in **Bluff Europe magazine**.*

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