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Variance - The Grossness (or Beauty!) of Poker
Variance - The Grossness (or Beauty!) of PokerWhile many players realize that statistical swings are part of poker, very few have a real understanding of variance and how it can affect results and bankroll. But the ability to understand and, more importantly, handle the swings is a key characteristic in winning poker players. Even a rudimentary understanding can keep you off tilt, help you be a better judge of your game and better manage your bankroll. Consider this… Three poker playing friends are each offered 100 separate bets in which they’re a 2 to 1 favorite. They can place a $400 wager on each individual bet. In other words, 65% of the time, they win $400; 35% of the time, they lose $400. Now at the end of the 100 bets, they compare results to see how they’ve done and to their surprise, each has a different bankroll, in some cases, thousands of dollars apart. Three different guys, all making the same wager. Three very different results and likely, three very different opinions of their game.
Why? Variance. While each bet has the same favorable expected outcome, you can easily have situations in which you lose multiple trials in a row. Add the results up for those wagers for each of the three players, and you’ll have very different bankrolls at the end of the session. With smaller edges, say 55/45, the variance is even more dramatic. Note that if you’re a cash game player, you should always want to take positive EV bets offered. For example, if you have QQ and you know your opponent has AKo, you should be willing to call his push since you’re a 57/43 favorite. [1] Unfortunately, the smaller the edge, the wider the swings. The attached spreadsheet will help demonstrate the type of swings you can expect to face even if you’re playing perfect poker and getting your money in as a favorite every time. The spreadsheet allows you to run a scenario where you are a favorite over and over again. You can run just a few scenarios or up to 5,000 scenarios of the same hand and see what type of variance you should expect. Each time you click F9 (the recalc key in Excel) the graph and summary results section changes to reflect new and potential outcomes. For example, let’s assume you are a 68/32 favorite (you have QQ against your opponents AJs) in a situation where you are heads-up pre-flop and you both bet your entire $400 stacks You would expect to win $144 on average ($400*.68 - $400*.32 or 68% of the time you win $400 and 32% of the time you lose $400). Now let’s assume you have this exact same situation 100 times. You should expect to win $14,400… right? Well if you hit F9 a few times on the spreadsheet you’ll see that the actual results may vary considerably. Below is one possible result:  The red line in the above chart (which was taken right out of the spreadsheet) shows that in these 100 scenarios you only won $7,200, not $14,400. Since the results tend to work out closer to your expected win/loss the more scenarios you run, the results will often look closer to the following chart over 100 scenarios:  RUNNING BAD I’m sure you have heard poker players claim they are running bad. Running bad often is the result of playing bad poker, but sometimes it simply means that the player is losing even though they are usually getting their money in the middle as a favorite. If you look at the RESULTS summary box in the spreadsheet it demonstrates what the maximum positive and negative swings are during all scenarios. In other words, it shows what the maximum consecutive winning or losing streak would be. For example, in the above chart, while you would win close to $14,000 over all 100 scenarios, you would lose approximately $2,000 starting around scenario 18. If you run the spreadsheet several times (by pressing the F9 key) you will see that it is not uncommon to see negative swings of as much as $5,000. VARIANCE OVER FEWER SCENARIOS But what happens if you only run 20 scenarios? Here’s a graph which shows the potential results over 20 scenarios:  Over these 20 scenarios when you were a 68% favorite each time you would have lost $800 and would have been down as much as $1,600 at one point. The point here is that you cannot expect the results to approximate the expected outcome if you are only running a few trials/hands. That is why the good players are able to accept the bad beats that are typical in poker – they know that over the long-run they will win if they continue to get their money in the middle as the favorite. RACE SITUATION – PAIR VS. TWO OVERCARDS Now try changing the X% favorite to 55% (roughly, QQ v AKs, for example).[2] The resulting graphs still have an upward trend, but many show repeated losses. Here are a few potential graphs from running the 55%/45% match-ups over 100 scenarios. .jpg) Although in this typical race you would expect to win just a bit more than you would lose, even over 100 scenarios you will see wide swings in your results. INCREASING YOUR LIMITS Now let’s assume you are running well in the $2/4 No Limit game and decide to step up to the $5/$10 No Limit game but are only taking the bankroll you’ve built in the $2/$4 game. To simulate this, change the amount wagered in trials 51 through 100 to $1,000 as the 68% favorite (cell C13 in the spreadsheet). Again, if you click F9 a number of times, the trend is usually up, but every few clicks or so, you should see a sharp downturn. Here are a few examples of these scenarios: .jpg) Add in smaller edges, mistakes (hey, we all make them) and occasional donkey moves and you could see large bankroll swings. Variance is what drives the need to manage your bankroll and attitude. You don’t need to be a statistician to play good poker or manage your bankroll, but you do need to understand that even juicy bets like the 68/32 offered above can put a serious dent in your bankroll and ego. So how do you manage it? First, play within your bankroll. Have at least 20 buy-ins before you attempt to tackle any game, and if you have 20 buy-ins, but you feel uncomfortable in the game, step back down. There’s more shame in playing a weak game in a higher limit than your normal, strong A-game at a level at which you’re more comfortable. Second, if your results seem all over the map, make sure you’ve got a reasonable number of hands to make a good assessment. If the results are for small numbers of hands (hundreds), you’ll see wider swings than if the sample is larger (i.e., thousands of hands). Third, review your hands and make an honest assessment of your play. Was your money in right? If not, you’ll obviously see some downward swings. If your money went in as a favorite and your hand(s) didn’t hold, relax and keep playing within your bankroll. Your results will turn. And finally, ask yourself how you’ve been feeling at the table. Have you been bright, alert and making thoughtful plays, or were you tired, distracted and playing sloppy? If you’re playing within your bankroll, getting your money in right and not tilting, your results will turn; it’s just a matter of playing more hands. If you’re uncomfortable, getting your money in wrong and not playing sharp, take a break, read a book and consider spending time away from the tables, because chances are those results will only get worse! Good luck! |