Variance and Happiness

Most serious poker players have a love-hate relationship with the inherent variance in poker outcomes. In our most frustrated moments, we may resent the inevitable variance due to the randomness of the cards, and what it does to our fragile bankrolls. We may curse the bad luck that gave the biggest pot of the night to the guy who called three bets cold with undercards and a backdoor straight draw, and who took the chips directly to the craps table. If there were no variability, solid players would win all the time, and thinking about it over coffee, that seems pretty fair. But, of course, variability is an important part of what keeps losing players interested. There's a limit to how bad a spot even bad poker players will put up with. They may know they're not likely to win, but many of them certainly wouldn't play if they knew there was no chance whatsoever. We can reasonably argue that with poker, as with many casino games, variance in outcomes is the primary attraction for players who know, deep in their hearts, that they need to get lucky to win. Although the end result for poor players is hardly different from a steady losing rate, the prospect of occasional, or even frequent, winning sessions is important to keeping the games healthy. Nobody tries to get lucky at a game with no luck.

There are lots of ways to look at this graphically, but here's one of my favorite. Look at a graph of your bankroll over time. The numbers are completely fabricated, but if you're a solid but unexceptional winning player, it might look a bit like this:

It features a steady rise, punctuated with numerous small setbacks. If you're a conscientious but still losing player, it might look more like this:

The end result isn't great, but in fact after this short sequence of results, this player wouldn't even have good reason to believe he was a poor player. Both winning and losing players have lots of winning and losing sessions, masking the overall trend somewhat. Note especially how often the losing player has a winning session (everywhere the line goes up, even a little). Rather than complaining about the stress on our bankrolls, we might be thankful for the opportunity to play a game in which the losers are liable to be happy so often. Although there are many games like this, most of them benefit the casinos and not other players.

Another way of visualizing this is in terms of the distribution of results. We can plot a distribution of results with different variances. Losing players will always tend to have more of their results to the left of zero (losing sessions). But variability affects how consistent this relationship is. With low variability, the loser always loses. In the chart below, you can see an entire normal distribution that sits below zero (although an outcome above zero is possible, the probability is negligible):

If we assume the same losing rate (-100 dollars, bets, or whatever per unit time), but a much higher (and more realistic) variance, we might see something like this:

Since the average result is the same, the long-term expectation will be identical. But if you came to the casino looking for more recreation than profit (and we hope so, for players whose expectation is below zero), the latter game would be a lot more fun.

Poor players do pay for this privilege. Intuitively, it seems like the highly variable game would be somewhat dangerous to losing players – that is, that there would also be a higher chance of a much larger loss. But there's more than one way to pay for the variance. Players on a budget will simply quit earlier if things aren't going well. So, all they're paying for this high variability is a greater risk of having to leave early. This has little effect on the skilled player glued to his seat, as long as there's a waiting list.

It seems like there should be some kind of conservation of happiness. The happier the losing players get, the more miserable the winning players will be. And it feels that way sometimes, because losing a huge pot to a long-shot draw is … well, not as much fun as winning. But losing players tend to be more results-oriented, while winning players, at least the most levelheaded winners, tend to focus on playing well. It may take an especially horrendous losing session for a winning player to become unhappy due only to results. So, although winning players may grudgingly think variance is adding to their stress, it probably contributes more to the happiness of losing players than to the misery of winning players. And although it's easy to view gamblers at the poker table disdainfully, the health of the game depends on it providing value for everyone who sits down at the table.