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If we disregard deals, players accidentally mucking winning hands, and time-limited tournaments, in order to win a poker tournament, you must win the very last hand of the tournament. In order to win the last hand, exactly one of these statements must be true:

  1. You were the underdog, and you won the hand. (You got lucky.)
  2. You were the favorite to win the hand, and you did. (You did not get unlucky.)
  3. You were even up to win the hand, and you won it. (You got very lucky!)

(If you're wondering why I said very lucky for number 3- it's because I think you'd have to have the same hand as your opponent to be even up- which means the only way you can win is by making a flush. The most common example that you'll never forget is when you have AA, you're up against AA and you win/lose with/to a flush.)

To summuraize, to win a tournament, on the last hand you either have to get lucky, or not get unlucky.

That statement can be extrapolated to the rest of the preceeding hands in the tournament too, i.e., the above logic can be applied to every hand you win in the entire tournament.

Multiple times after winning a tournament (small ones, mostly 10-100 people), I remember thinking that I won because I never got unlucky- no one pulled a bad beat on me. Sometimes I recall winning a tournament because I got lucky a few times, and sometimes it was a combination of the two. Sure you'll have a few hands here and there where you catch a bad beat and still end up winning, especially in large tournaments, but in the smaller tournaments, most of the time a meaningful hand where you get unlucky will send you to the rail, or lead to it shortly after, particularly closer to the end of the tournament.

So this means that to win a tournament, you will have traversed a course of mostly getting lucky and not getting unlucky. Occaisionally, you will also get unlucky, and many times you will also not get lucky. (Not get lucky means you were the underdog and didn't win.) We now have 4 "main" ways your chip count can significantly change per hand throughout the course of the tournament:

  1. Got lucky. (Was underdog, won hand.)
  2. Got unlucky. (Was favorite, lost hand.)
  3. Didn't get lucky. (Was underdog, lost hand.)
  4. Didn't get unlucky. (Was favorite, won hand.)

I would like to know if we can quantity the approximate expected ratio of those occurrences throughout the course of a tournament, separated by wins and loses, for someone who wins the tournament. I'm not sure if this can be done mathematically without real data. Ideally we could datamine tracking software to get these numbers. For example, perhaps if you just won a tournament, and looked back at those types of hands, you might see this:

Hands won:

  • Didn't get unlucky: 90%
  • Got lucky: 10%

Hands lost:

  • Didn't get lucky: 90%
  • Got unlucky: 10%

Why does this matter? If we can get real numbers, it helps assign the skill vs luck ratio of poker. This number may be important for certain governments in how they determine whether to define poker and/or poker tournaments as pure gambling, vs a game of skill.

Update: Imorin pointed out many reasons why my above method for attempting to quantify luck may not work, and I agree. I still believe there may be a way to achieve this. Here is what prompted me to ask this in the first place. I believe that almost everyone who wins a tournament can think of at least one hand in the tournament where they got lucky, and had they not gotten lucky, they would have either busted out, or very likely not have won the tournament. A few times I remember winning single table tournments thinking afterwad, "I won that because I never got unlucky. No one sucked out on me the entire tournament." But I've never won a multi-table tournament where there wasn't at least one hand where I got lucky, and had I not, I probably wouldn't have won it.

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The common conditions/rules of being independent from luck in the tournament:

1) You are playing tournament with deep stacks and reasonable blind level lengths. It means turbo tournaments with 5 minutes per level contain enough luck-dependent situations. Not playing "turbos" will allow to avoid rapid short stack preflop all-in situation.

2) You don't ignore your opponent's push diapason, nor his image, nor any other aspects of tournament play. In this case typically the worst situation you will face: AQ vs KJ, where you are up to 60% ahead of your opponent.

3) Tending to have chip leader stack (or near it) at least on your table. In this case preflop allin 50/50 will not be fatal for you.

4) And finally: try to win without showdown as much as you can, which is a poker basics.

Taking into account these factors will reduce the influence of luck in the tournament to a minimum.


Update: (Answering the comment from @TTT below the post).

To measure the "luck" component of tournament play you can do the following:

  1. Get the hands history of all tournaments where you got into big prizes. Top1-Top3... Let's approximate it with all MTT where you got on the final table.
  2. Get the product of all your chances on "risky for life" show-downs during each tournaments from (1). That means any show-down hands where you was risking to be beaten (even AA vs A2 pre-flop, when all your chips are in pot) get into this statistics.
  3. Get the average of this products. In case of at least 10-20 such tournaments in the statistics you will get the average probability for you to get on the final_table/top_3/ITM etc.
  4. This number could be surprisingly low, but you can increase it using points from the first part of my post. Because it is very dependent from your playing style.

p.s. I am not sure that calculating this number for another players (data mining) could be enough helpful, because it will present other players' winning chance and reflect their playing style. But at least you will see what this numbers approximately could be.

p.p.s. It would be really interesting to compare your "lucky" number with another players and get your playing style closer to the best of them. But this is something related to using another players' hands history... what is not fair :)

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Your suggestions are for limiting the amount of luck in a tournament. However, my question is not about how to limit it, but how to quantify how much luck you can expect to have had when you win a tournament. –  TTT Mar 30 '13 at 2:27
    
Reasonable comment, I have expanded my answer.. –  IharS Apr 1 '13 at 12:37
    
Your comment regarding other player's data possibly not being relevant is a good one. I think this calculation would also need to include the probability of winning based on the playing style. For example, if you play 100 10 person tourneys, you might expect to win 10 on average. If you win 30/100, the probabilities above would likely be drastically different (and more relevant) than if you won 3/100, which would be less significant. I originally intended this question to be based on data from statistically solid and winning players. –  TTT Apr 5 '13 at 16:28
    
Your question was about luck component. When you are saying about winning 30/100 instead of expected 10/100 -- that shows that your skill is better than average. Not luck. –  IharS Apr 6 '13 at 10:43
    
I'm curious about your use of the word diapason to mean (presumably) range. I've only ever heard the term used in a musical context. Do you commonly hear this word used in poker strategy/analysis? –  Robusto Apr 6 '13 at 16:00
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Your partition (underdog/favorite, hand won/lost) won't reveals many things as you will have to work on average. This will give you the performance of your hand range (supposed constant) versus and average hand range met in tournaments. A long term moving average will give you the evolution of your handrange performance. The deviation will give you the effect of luck + focus in the final classement. (assuming some normalization, which may be hard to do.). This could be an interesting approach but the effect of focus is impossible to determine. The better thing to do is to buil a quantitative model for luck...

Some remarks:

I think the way you measure luck is not adapted to reality. First, 49%vs51% is not very different from 51%vs49% in practice, but in your model it is. Then, 49%vs51% is very different from 1%vs99% in practice, but in your model it isn't. So I think an interesting number would be the average of the difference of percentage of chance of loss/win to 50%.

Let's say you have 75% chance to loose:

  • you win: +25% (it's luck)
  • you loose: -25% (it's not luck)

Let's say you have 75% chance to win:

  • you loose: -25% (it's unluck)
  • you win: +25% (understandable as non unluck)

It will give you on average the difference in number of hand you've won to what you are supposed to win (50%).

Is this a good definition of luck ? No, because it reflects just a number of hands. Usually we are more interested in chips. You can easily understand it by looking at some extreme case again. Doubling your stack with 1% chance is way more lucky than stealing 1BB with 1% chance. Once again we have to quantify explicitly to solve comparable situtation (is stealing 1BB with 1% chance luckyer than winning 10 BB with 10% chance ?)

The simple solution, used in cash game is the expected value. The difference of probability to 50% is multiplied by the number of chips in the pot. It give you the amount of chips you can expect to win. You could then compare the number of chips you win/loose after the hand to what you are supposed to have (expected value) to deduce if you have been lucky or not.

Is this a good definition for luck ? Yes for cash game, no for tournament. In cash game your stack is directly linked to your gains, and the result of a hand won't influence the result of following hands (assuming you have enough money).

In a tournament this not possible to suppose these 2 things. Your gain/placement is not proportionnal to the amount you win in a hand. Let's say you double your stack (+1000 chips) in your very first hand, with expected value it will count as the same thing as stealing a blind after an hour or two of game, and nothing to the big blind at the last table. But in term of real gain (or classement) this is absolutely different. You can take this into account by dividing your gain by a reference value. The reference value can be your stack at the moment or the average stack at the moment or the ratio between your contribution and the size of the pot or a combination of the three, depending on what you think is important in a tournament.

This is a simple model, as you don't take into account the effects of a hand on another (we can talk about dynamics of the tournament), you are simply working hand by hand. Take this exemple: you are lucky, you double in the very first hand, you are now comfortable enough to call another all-in, the second all-in is a sure gain, 90% win for you, you win. There is no luck in this second win taken apart of the first one, but without the luck in the first all-in the second wouldn't have happened. It would be very difficult to take it into account. (I guess you have to use recurrence on conditionnal probabilty to guess the effect of a hand on another, but have no clue of how to do it).

Another question is how to calculate your chance of gain. This is a big question and can lead to philosophical considerations. Let's say you have seen the flop before going all-in, what % will you use ? hand / hand, hand + flop / hand + flop ? what % will you use after seeing turn, river ? With all-in the answers are easy to guess. but without all-in ? What % do you have to use ? Maybe you can use different % between each round of bet depending on what you have seen.

"What you have seen" this is where it gets philosophical. You have seen cards, of course, but you have also seen rounds of bets and previous hands. You have more information and it could change some things. We talked about luck in the sense of the diffenrence between what you expect and what you get. But now we have luck in the sense of the difference between your interpretation of information and reality. Some intepret it as skill. But in my opinion (mainly on internet as you can not get tells from your opponent) there is a part of luck in interpretation of informations. And this luck seems impossible to quantify.

Poker is such a complex game that it is very hard to build quantitative model for it.

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Great answer. You pose a lot of questions and provide a lot of scenarios which make my original suggestion for how to quantify luck less useful. I will edit the question with my reasoning behind the original question. –  TTT Apr 12 '13 at 20:56
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