Are There Lucky Players?

Brian Alspach

Poker Digest Vol. 4, No. 24, November 16 - 29, 2001

Last time, I discussed a betting strategy that is fundamentally flawed by a common misunderstanding of what is meant by ``Things even out in the long run''. Let's continue the thread by looking at our perception of luck in the poker context and asking the question: Are there lucky players?

Trying to answer such a question introduces many complicating factors. There are problems involving selective memory. If a certain player has cracked my pocket aces twice with two small cards, it will stick in my memory much more strongly than in the memory of some guy who was watching baseball on the large TV screen while the two hands were taking place.

There are problems involving the small windows through which we view other people. I might say to someone that so-and-so has been really lucky in the last couple of sessions we've played together, only to be told by that someone that he has seen so-and-so get hammered during a couple of recent sessions. It depends when you were observing.

If you were to individually ask many people whether or not there are lucky players, you would receive a wide range of answers. You would conclude there is more psychology involved than anything else. Nevertheless, let's take a mathematical viewpoint.

Do some people have more than their ``fair'' share of luck? I remember reading about an investment scam that is related to the topic.

The operator of the scam chooses a volatile stock. He writes to 32,000 investors, telling half of them the stock is going to go up this week, and telling the opposite to the other half. The following week he writes to the people for whom his previous prediction was correct, telling half of them it will go up the second week, and telling the other half it will go down.

After five weeks, there are 1,000 investors who have received five consecutive correct predictions for a volatile stock. The scammer now offers, at a good price, of course, his predictive abilities for a wide range of stocks. He may get some takers.

The preceding example illustrates how successive positive outcomes is a perfectly natural occurrence for a shrinking pool of people. After several consecutive successful outcomes, people may notice and say how lucky so-and-so is, but forget the large number of people who fell by the wayside. The existence of a small fraction of people who have had more than their share of good fortune is perfectly natural.

Let's now look at the phenomenon of things evening out in the long run. For simplicity, let's discuss two equally likely outcomes such as heads and tails when flipping an unbiased coin. Let H(n) and T(n)denote the numbers of heads and tails, respectively, after n coin flips. What we mean by saying H(n) and T(n) even out in the long run is that the ratio H(n)/T(n) approaches one as n grows larger with probability approaching one.

Let's see just how poor the information about the ratio can be.

Suppose the poker goddess approaches you with the following offer: She knows you have been losing more than your share of heads-up confrontations with your nemesis. She tells you she has functions f(n)and g(n) which control how often the two of you win future battles. She then adds sweetly that the two of you will win equally often in the long run. When you ask her what she means by that, she tells you the ratio f(n)/g(n) approaches one as n goes to infinity, but she will not tell what the functions actually are.

Should you take the offer? As it turns out, she is up to her usual mischief and is using the functions $f(n) = \frac{n}{2}-\sqrt{n}$ and $g(n) = \frac
{n}{2}+\sqrt{n}$, where both are rounded off, when necessary, so that the sum is n. So even though the ratio of f(n) and g(n) approaches one as n increases, the difference is growing larger and larger as n increases. The poor sucker who chooses f(n) as his function will begin to believe the other player is much luckier.

The preceding example demonstrates that knowing something about the behavior of the ratio of two functions of n is a poor basis for judging the difference of the two functions. Of course, the example does not apply to two opponents whose outcomes are subject to the same probability distribution, because f(n) and g(n) are defined so that one is occurring more frequently than the other. However, even when two outcomes are subject to the same probability distribution, our intuition may be challenged.

Let's return to flipping an unbiased coin. If we flip such a coin once a second for a year, most people believe over that many trials that heads and tails are in the lead roughly equally often. Nothing could be further from the truth! The probability that the less fortunate outcome is in the lead less than 54 days is slightly more than .5. The probability that the less fortunate outcome is in the lead less than nine days is .2.

These figures indicate there is a very good chance one of the outcomes will be in the lead most of the time. A person observing these results might suspect there is something wrong with the coin, or that one of the outcomes is lucky. However, this kind of behavior is exactly what is expected from analyzing the mathematics. Before the mathematics was understood, there were some classical false conclusions drawn from experimental data because the scientists of the time could not believe similar behavior could be random.

The proper conclusion to draw from the above is that it is only natural some people will have considerably more positive outcomes than other people, especially when we consider the short time span people live.

So what do we do when we are told so-and-so is a lucky player?

People making such a statement are doing so from a psychological viewpoint. It is a statement about history and is colored by the interaction between the source of the statement and the subject of the statement. What should interest us as poker players is whether or not the source of the statement indicates that so-and-so's luck has any predictive powers.

When you sit down at a table and are told by one or two people, ``Stay away from Joe tonight...he's catching cards you wouldn't believe and if he's in the pot, I'm throwing away my hand no matter what I have'', be ready to take advantage of the situation. The players who told you this may just be blowing smoke, but they may, indeed, be players who let Joe's lucky streak alter their normal game. There also is a chance Joe himself is starting to play marginal hands because he's on a rush. Joe also may be staying in longer than he should because he's caught some good river cards.

Of course, you have to catch some cards to take advantage of the situation. There is nothing you can do if you have crap to play with. Nevertheless, by being aggressive when you enter the hand, you probably will reduce the field if Joe is in there. You may even isolate him. Realize that if you take down Joe a couple of times, the ``Joe's on a rush'' aura will disappear and people will return to normal.

Yes, there are lucky players in the sense that some collection of players will have caught sufficiently more positive outcomes at the right time to make people remember them. No, there are no lucky players in the sense that they have a better chance of having random positive outcomes in the future than anyone else.

Yes, there are players perceived to be lucky and they can use this perception to their advantage. So can you if people think you are lucky.

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