Tuesday, April 10, 2007

You Gotta Know When to Hold 'Em

Canada Bill Jones' Motto: It is morally wrong to allow suckers to keep their money. Supplement: A Smith and Wesson beats four aces.

It has long been known that mathematicians have a bias toward games of chance. In fact, some people become mathematicians to try to find ways to beat the roulette wheel or win at the crap table. Now it seems that a physicist claims to have plumbed the secrets of the poker game, Texas Hold 'Em.

For those of you who don't indulge in losing money at cards, allow me to briefly explain Texas Hold 'Em. However many players are around the table, two players put money in (ante up) prior to the deal. One puts in twice as much (the big blind) as the other (the little blind). This is a little different from other games, where everyone antes the same amount. After the ante, each player is dealt two cards, at which point they can bet or fold. The dealer then lays down three cards (the flop) and players bet again. The dealer then deals out another card (the “turn”) and a final card (the “river”; don't you love my grasp of terminology?). Betting takes place after each card. The winner is the player who can make the best hand using his dealt cards plus any three of the community cards.

The variant that gets everyone excited is No-Limit Hold 'Em, because anyone can bet all their chips at any time. This means that a player can become very wealthy or very broke in an instant.

Thanks to television, Texas Hold 'Em has gotten to be big stuff, with Internet sites popping up all over running tournaments, and even giving people the chance to win a place in a major real tournament. All sorts of books and articles abound with information on how to win and become rich and famous.

Now physicist Clement Sire claims to have applied his mathematical skills to create a model that predicts certain aspects of Hold 'Em.

I'm not sure if I'm more depressed that he wasted his time doing this or that Science News thought it was worthy of reporting.

Here are some of Mr. Sire's insights:

  • When a player has double the average number of chips held by the players at large, he or she will be in the top ten players.
  • The blinds are increased as the tournament goes on. This causes the number of players in the tournament to decrease more rapidly.
  • The number of chips held by the chip leader is a function of the number of the players in a tournament.

Congratulations, Mr. Sire, you have discovered the rules of a zero-sum game.

In a typical tournament, players pay an entry fee, which has nothing to do with how much money the player starts with. The starting “bankroll” is set by the tournament organizers. The players aren't playing for actual money at the table; they play to eliminate other players and win prize money which has no relationship to the value of the chips earned during play.

So, there's a set number of chips to be won. The more players in the tournament, the more the chip leaders will get as they knock out other players. So, of course, chip leaders will have higher totals if there are more guys in the tournament, because there are more chips to be won.

As to the nonsense about the average number of chips, it stands to reason that anyone who is in the top ten will have more than twice the average chips. As players are eliminated, the average chips per player rises, but since the most common way of eliminating players is when one goes “all-in”. This means large numbers of chips can change hands abruptly, but it also means that the balance of chips will tend to be uneven. In fact, without doing an in-depth analysis, I would guess that within a relatively short time in any tournament, the 80-20 rule will come into play; that is, 20 % of the players will have 80% of the chips. But, because of the all-in betting, which 20% is holding those chips can change suddenly.

What Mr. Sire didn't seem to be able to discover is a strategy that might govern winning. Oh, he found that players have tendencies about going all-in, and when they vary from those tendencies, they don't do well. This isn't a big surprise either, because successful players tend to lose when they get out of their normal rhythms.

We are told Mr. Sire is a good card player, which would explain his interest in crunching numbers. But he certainly hasn't discovered any secrets. On the other hand, I don't play cards well, but I have learned some secrets, and I'm glad to share them.

  • Good players are those who can bet intelligently enough to hang around until good cards come their way.
  • Good players pick up on tendencies in their opponents and bet accordingly. Real good players can break their tendencies just often enough to fool the good players. Great players recognize when the real good player is doing that.

  • Impatience will cost any player. Even great players who lose a big hand will suddenly turn into amateurs trying to get back into the game quickly.

  • Luck beats skill any day of the week. The greatest player in the world is going to lose when he holds three aces and the other guy hits a full house on the last card.

I'll be waiting to hear from Science News to report my amazing findings.


Clem said...

Dear John,

I am sorry that you got confused in exposing the results of this work.

The actual result is: when you have twice the average stack you are ranked in the top 10% of the tournament, whatever the number of players (not too small though). In addition, the theory tells you your typical ranking for any value of your stack. This is actually useful when you don't have access to the live standings, as in many Internet poker rooms (Party Poker, for instance).

You'll find more results simply (but correctly) exposed on this non technical review article, which is however more precise than the SA coverage (which mentioned the above result correctly though).


Best regards

The Gog said...

Thanks for your comments, Clem, but I don't think I'm confused. First of all, since a fairly dealt game of poker is a probabilistic event, the "random walk" model you note on your site would, of course, fit well. Also, probabilistic games generally follow the patterns of certain biological activities. While this was no doubt entertaining to verify, it is hardly startling news.

Neither is the "finding" that organizers have figured out empirically that increasing the blind eliminates players more quickly. The same theory is applied to forcing American college football teams to go for a two point conversion after three overtimes.

I have no doubt you are an excellent card player, but I am willing to bet that even you win based more on the principles I elucidate at the end of my piece than on your discoveries.

In any event, I wish you well and hope you'll always fill that inside straight.

Clem said...

Dear John,

on the last account you are perfectly right, although it was never stated that this work is of any help to win at poker. Though, I think that the ability to evaluate one's ranking in a tournament is an interesting and useful result, which I humbly think you dismissed a bit too rapidly in your post. In addition, although completely useless for players, the calculation of the total number of different chip leaders in a tournament is also interesting. Just to know that this quantity obeys a mathematical law should be appealing to the curious man. The fact that the same phenomenon is observed in species evolution (or genes appearance) models is also noteworthy.

Best regards

The Gog said...

From the article: "Sire's model includes functions that reproduce the most basic tasks a poker player must carry out, such as deciding whether to bet strictly on the strength of his or her hand."

That's about winning.

I guess my complaint (to the extent that I'm complaining at all) is with the tone of the Scientific American article, not with your original work as shown in the link you provided above. While the work is interesting from a purely intellectual point of view (and there's nothing wrong with that), I wouldn't say that it "unlocks the secrets" of poker, as the SciAm article does.