Poker Is Potentially Profitable People!

First of all, I should warn readers of this blog (if there ever are any) that, as the title of this entry illustrates, I am overly fond of consonance.  But let’s move past the style to the substance.

I got an eye roll from one of my colleagues when I compared playing poker to investing in stocks.  His father was a compulsive gambler, which probably contributed to the reaction.  (I’m sure children of alcoholics are not interested to hear that a glass of wine with dinner is actually good for you.)  However, my colleague’s response reflects widespread ignorance about how poker differs from pure gambling.  An anecdotal illustration of the difference is the cautionary tale of Las Vegas legend Archie Karas. In the early 1990s, Karas won $40 million dollars beating people at poker and pool, games of skill, and then lost it all over a period of three weeks playing high stakes craps and baccarat, games of chance.  But let’s look more precisely at the difference.

Suppose you play roulette and wager $1 on number 13.  The numbers on an American roulette wheel are 1 to 36, along with both a 0 and a 00.  So the odds against you winning are 37:1.  If you do win, you will be paid $35 for every $1 you wagered.  So, on average, for every thirty eight $1 wagers you make, you will lose $37 and win $35 for a net of -$2 over 38 tries.  (Alternatively, your expected value for any one bet is 1/38*$35-37/38*$1=-$1/19.)  All house games are like this, which is why they cannot be beaten.  You will always lose in the long run.

In poker, though, you win or lose almost all your money from other players, not the casino.  The casino makes a profit by taking a “rake” from every pot, usually 10% and capped at a certain amount (e.g., never more than $4 per pot for low stakes).  This makes it sound like the casino doesn’t make very much from poker.  It doesn’t.  Most casinos would rather have slot machines in what are now their poker rooms.  However, customers who play poker are a cranky and insistent bunch, and believe it or not that can make a difference.  In addition, casinos count on the poker players (or their spouses) also stopping by the pit to lose money at games like craps and roulette.  In other words, poker is often what salesmen call a loss leader.

Of course, even if the house is taking very little out of the pot, that doesn’t prove by itself that any one player can win.  But there are two reasons why one can win money.

1. Most players make mathematical errors in their betting.
2. Most players are very predictable.

(1) Suppose it is the middle of a poker hand, with $2000 already in the pot.  (If you are completely ignorant of poker, the “pot” is the money that has already been wagered and will be taken by whoever wins the hand.)  Arnie and Brenda each have $1000 in their “stacks” (the money that each has not yet wagered).  Arnie has a pair of aces, while Brenda only has an inside straight draw.  Brenda will win if she successfully completes her straight, but will lose otherwise.   It is Arnie’s turn to act, and he wagers all of his remaining $1000, so the pot is now $3000.  Brenda has two choices:  she can fold, which means that she gives up any chance of winning the pot, but also risks no more money, or she can call by adding $1000 to the pot, which means that she still has a chance to win the pot, but can also lose the additional $1000.  So if Brenda calls she is risking $1000 to win $3000, giving her odds of 3:1.  Assuming that Brenda will get one more card, the odds are approximately 11:1 against her completing her straight.  So for every 12 times Brenda calls in a situation like this, she will win $3000 one time, and lose a total of -$11,000 for the other times, for an average net of -$8000 over every 12 hands.  (If you prefer, the expected value of her call is 1/12*$3000-11/12*$1000=-$667.)  Although Brenda should definitely fold, it is amazing how often players in similar situations call.

What about Arnie?  If he does not bet, he is giving Brenda a chance to beat him at no cost to her, so he should bet.  If he bets and Brenda folds, he wins $2000 at no cost to himself.  If he bets and Brenda calls, he will lose -$1000 one time out of 12 and win $22,000 the other 11 times, so the situation is immensely profitable for him on average.  However, many players in Archie’s situation either check (do not bet) or bet an amount small enough that Brenda actually does have the right odds to call.

An objection may occur:  “Your description of the situation ignores the fact that Brenda doesn’t know that Arnie has a pair of aces and Arnie doesn’t know that Brenda has an inside straight draw.”  This brings us to the second reason that you make money at poker.

(2) Most players are very predictable.  I’ll illustrate this with the Baluga Whale Theorem (BWT), a generalization well known among savvy poker players.  I should explain that BWT is not a “theorem” in the sense we talk about mathematical theorems.  Rather, it is a generalization about how people tend to play at low stakes.  BWT is also not a recommendation about how people should play.  However, it has implications for what the correct play is.  I’m going to give a very oversimplified version of it here simply to make a point.

BWT applies to small stakes hold’em poker.  In hold’em, each player is dealt two “hole cards” that only she sees.  There is then a round of betting.  Next three “community cards” (called the “flop”) are dealt face up in the middle of the table.  Each player can use any or all of these cards to make the best poker hand in combination with their two hole cards.  There is another round of betting after the flop.  Then a fourth card is dealt face up (called the “turn”) and there is another round of betting.  Finally, a fifth card is dealt face up (called the “river”) and there is the last round of betting.

Suppose you raise on the first round of betting and are called by one player.  The flop comes, you bet and are called by the same player.  The turn comes, and it is a card that would appear to most players to be a “blank” (a card that is unlikely to connect with anyone’s hole cards).  You bet, and the other player now raises you.  The Baluga Whale Theorem says that the opponent who raises you here, after calling pre-flop and on the flop, probably has a very strong hand.  Why?  Because that is the way that most small stakes players play when they make a big hand on the flop in this situation.  The reason this is so important is that, if you know BWT, you can figure out what kind of hole cards your opponent has, even though you can’t see them directly.  You can save money by folding what would normally be a strong hand in the face of a BWT-type raise on the turn, or if you have an exceptionally strong hand, you know that you can put in a big re-raise and get called.  This is just one example of a common pattern of play.

Since playing predictably allows thoughtful players to determine your hole cards, why not play unpredictably?  Part of the answer is that better plays do play unpredictably to a certain extent.  Good players will sometimes check with a hand you would expect them to bet, or raise a hand you would expect them to fold.  However, the only way to be absolutely unpredictable is to completely randomize one’s play:  check, bet, fold and raise without regard to the value of your hands.  But there is a high cost of complete randomization.  You will frequently wager a lot with weak hands, and wager little with strong hands.  This will be a mathematical error, and against even a moderately skilled player you will lose in the long run (remember our point 1).  Consequently, against a good player you have to partially randomize your play, and do so at points that are likely to be misleading to the particular player you are up against.

How much randomization?  And what kinds of randomizing will mislead your opponent?  There is no fixed rule about this.  Indeed, there cannot be a fixed rule about deceptive play, because there is a counter strategy for every deception.  Perhaps the most basic strategy in hold’em poker is to make a “continuation bet” on the flop, after raising pre-flop, regardless of whether the flop helped your hand.  The continuation bet is a sort of bluff bet, in which you pretend that either the flop made your hand stronger, or you already had such a strong hand that you didn’t need the flop to help you.  A continuation bet will often allow a player to win the pot with a weak hand.

However, since continuation betting is a basic strategy, almost everyone knows it, and many players will respond by calling your flop raise.  If you check on the turn, your opponent will guess that you were simply continuation betting the flop, and will bet the turn himself, winning the pot.  (This technique is called “floating.”)  Of course, many players know about floating, so they will continuation bet on the flop and then bet again on the turn, even if the turn did not help their hand either.  (This is called “double-barreling.”)  Of course, many players know about double-barreling, so…and the process goes on forever.  Truly great players have a skill that goes beyond any rules that allows them to win consistently in the long run.

Long ago, Aristotle recognized that there are skills like this.  One of his favorite examples was sailing a boat.  A boat captain knows many rules of thumb (“red sky at night, sailor’s delight; red sky in the morning, sailor take warning”).  He also knows some mathematical facts (“if the ship’s center of gravity is above the water line, it will begin to capsize”).  However, the complete set of rules underdetermines the correct action in any situation.  This is why sea captains are given both immense discretionary authority but also held very accountable for anything that goes wrong.  Aristotle thought that ethical wisdom, phronesis, is a similar kind of skill that is underdetermined by precise rules.   I think so too, and I think my knowledge of poker has helped me philosophically by giving me another example of a knowledge that is like Aristotelian practical wisdom.  In both poker and in ethics, there are right choices and wrong choices, despite the fact that there is no algorithm for the decision procedure.  And, in both, some people are demonstrably better at making the right choices. (I could go on:  going “on tilt” is one poker manifestation of Aristotelian akrasia, “weakness of will,” in which one knows, in some sense, the right choice but makes the wrong choice.)