Understanding Poker Variance

At the flip of a card, on any given day, any player can beat the best player in the world in poker. Such uncertain outcomes are what draw a lot of players to poker. Even professional players enjoy the intensity and the range of emotion that comes from such risk, as evidenced by the surrounding vernacular; upswing, downswing, busted, running bad. However, most players rarely pause to consider the impact of poker variance on their game, and on the games of their opponents. Even a rudimentary understanding of variance transforms gamblers into thinkers with long-term winning strategies.

Win rate is typically described by an hourly rate (like a real job!) or in big blinds won per hour. Most players recognize their win rate as a tidy number. One that is easy to remember and easy to calculate. However, the true circumstances surrounding win rate are quite murky. Experienced players welcome this uncertainty, this mathematical murkiness, because it attracts recreational players to the game. A sample win rate might be $60 won per hour, or three big blinds won per hour. To simplify and standardize what is to follow, let’s stick with the big blind option, rather than directly referring to money.

On any given date, a player’s winnings are quite different than his expected win rate would indicate. This disparity begs the question of how to create a working definition of what the mathematically inclined call standard deviation. Without giving a crash course in statistics, standard deviation squared equals variance, and it is a term that supplements that neat number you most hear about when discussing a win rate. The ugly truth is that a player’s win rate, if it is correctly calculated, is on average quite far from his winnings on any given day. In a typical live no-limit hold’em session, for example, an aggressive player may win or lose two buy-ins within a few hours. That’s a long way away from what he expects to win on average in those same few hours, which might be an amount around eight big blinds.

An easy way to wrap your mind around this type of poker variance is with an analogy proposed by poker player and mathematician Bill Chen, conveniently dubbed the “Chen Coin Flip.” The Chen Flip puts win rate and variance into a coin flip payoff that normalizes over time to equal the expected win rate. In other words, playing poker for a given amount of time is like flipping for your buy-in with an edge that is proportionate to your win rate. Let’s look at Chen’s example.

When we sit down at a table, every hour we’re essentially doing a coin toss for a rack of chips. Now, if you’re a skilled player, you may have an overlay of half a stack (assuming 100 chips in a rack and 20 in a stack). So if you’re a skilled $3-$6 player, it’s like flipping a coin and getting $120 if you win and losing $100 if you lose, or if you’re a $15-$30 player, it’s $600 if you win, $500 if you lose. Now this is a pretty huge edge when compared to blackjack on a per-hand basis, but we shouldn’t be too surprised at all if we get on a bad streak and lose $3,000. How easy is it to flip tails six times a row? It’s bound to happen if you flip coins all the time — now 10 or 15 in a row is a little unlucky but nothing too phenomenal.

Especially with marginal edges, high-risk environments tend to be emotionally charged. The expected outcome is only a tiny portion of the typical result. Because most people react to the present and not to their expected value, which is a bit of an abstract concept, those feelings of poker elation and rage make perfect sense. To extend the concept a bit, if we make what we believe to be a profitable bluff and our opponent calls us with a backdoor-flush draw or other unlikely nutted-hand combination, then did we make a mistake? Not necessarily. Perhaps, and it’s more likely if our first calculation is correct, we simply ran into the strongest portion of the range of holdings that our opponent plays a certain way. If two percent of the time he has the nut flush, then that should not affect our feeling about our play or the reasoning we made the play.

The Chen Flip also gives downswings their proper context. As Chen points out, it’s not uncommon that a fair coin, when flipped six times, lands all six times on heads. It happens once in every 64 trials. That means for a break-even player that once in every 64 times that you string together six sessions, the results aren’t going to be so pretty. Now the more skill you have, and the larger your edge in a game, the more mitigated your risks are and the less likely a large downswing is to occur. That said, the likelihood is certainly within the realm of a normal possibility for any cash-game player, especially aggressive ones that make plays designed to scrap every bit of value possible out of a game.

Similar methods can be used to justify the results of a skilled tournament player. Even with a solid edge on the field, a fantastic tournament player may easily go without cashing for ten tournaments in a row. Consider for a moment the implications on the bankroll, and you can see the importance of understanding variance in poker. Understanding variance isn’t just a way to make yourself feel warm and fuzzy while you’re sitting at the poker table and winning or losing stacks regularly.

Expected outcomes are quite different from the results. It’s why professionals don’t blink an eye (or lose their minds) when the deck gives them a bad card, and it’s why you can be a better player. By extending the idea of the Chen Flip to all poker play, or to any bet, it’s quite easy to cling to our sanity during a high-variance game. ♠

Reid Young is a successful cash game player and poker coach. He is the founder of TransformPoker.com.