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Poker and Variance

Discussion in 'Poker Strategy Articles' started by gavrosh, Nov 17, 2010.

  1. gavrosh

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    The so called "variance" perhaps is the element of poker most talked about by the public but also the least understood. In fact the term is pretty wide and includes several different things. In this article I'll try to explain the aspects of that "variance in poker" from mathematical and statistical point of view, without digging in the theory too much.
    Generally speaking, variance shows deviation from the average (or some Expected Value) - either in a single hand or a huge sample. And as all of you know, variance is something uncontrollable, "preventing" things from being "what they are supposed to be"; something manifested in many ways that puts us to test and often has significant impact on our monthly results or sometimes even our entire poker career.

    Standard deviation

    The term "variance" has several meanings in mathematics as well, one of them being synonymous with "standard deviation". In my opinion the standard deviation is the smallest bit of "the variance in poker" and that's why I start with it.
    The standard deviation (marked with 'sigma', but I'll use just 'sig') is directly connected to the mathematical expectation (E or 'EV' as it is known in poker) and it displays to what degree the result tends to deviate from its expected value. I won't bother you with formulas, it's sufficient to say that sig depends on EV and all the possible outcomes with their respective odds. If you don't fully understand what EV is you should do it before continuing with this article. The standard deviation in poker is measured by money/chips/blinds - just like the EV.
    Here are 3 trivial examples showing spots with similar EV but quite different sigs:

    * We examine all-in preflop AsAd vs AhAc, both hands absolutely equal in strength. EV=0 for both players. Either of them has 2% odds for winning the had by making a flush, 2% for losing it and 96% for split. Great deal of the time the opponents would split the 200bb pot and would be break-even at the end (neglecting the rake). But in rare cases one of the players would take the entire pot, winning 100bb. We can calculate sig in this spot (through a big formula - don't ask :p ) and we get sig = 20bb. Yes, 20 whole blinds deviation in a spot looking like pure split!

    * Now let us see the similar at first look spot: AsKs vs AhKh all-in preflop. Obviously both players have EV=0 again and the stake is the same: 100bb from each player. The difference here is that now it's easier for either player to make his flush: 7% winning odds, 7% losing odds and 86% split odds. We calculate sig = 37,5 bb - almost twice the sig from the example above.

    * The last one is more exotic: 2s2h vs Tc8d all-in preflop, pretty standard don't you think? :D Now both players have 49% of winning and 2% split odds. This makes for the impressive result of sig = 99bb , almost entire buy-in.

    So what does the numbers mean? They show how much the end results deviate from the EV on the average. All of the examples have zero expectation but the 3rd gives way bigger deviations than the 1st.

    And those are some of the classical all-in preflop spots with different EVs:

    * We get АsАd vs АcКh (92% win, 7% loss and 1% split). Our EV = 85bb - a dream spot - but sig = 51,7bb therefore even here we deviate with half buy-in from our EV!

    * АsAd vs KcKd (82% win, 18% loss), EV = 64bb, sig = 77bb. Deviation is still big, even bigger than the EV itself.

    * АА vs KK vs QQ preflop. EV = 100bb, sig = 140bb. Nice hand...

    Note: in all-in preflop spot between 2 players, the one players EV equals the -EV of the other: in AA vs KK, EV(AA) = 64bb, while EV(KK) = -64bb. But the standard deviation is the same for both players!

    Deal it twice!

    This is a mean to "fight variance" well known from the high stakes games where swings could be stunning. The pot is being split in 2 halves and the board is dealt twice, each time for half of the pot. And so, if we deal twice in spot AA vs KK all-in preflop for a total pot of $1000, this equals 2 standard spots AA vs KK for 500$ each. Indeed variance (and here I mean the standard deviation) is reduced: from $385 (or 77bb) to $270 (or 54bb) - it's divided by square root of 2, because it's harder for one of the players to win both boards.
    If we wish to reduce the variance even more we could deal more times for smaller pieces of the pot. If we deal it 3 times for a third of the pot, the standard deviation will drop 1.73 times (it's divided by square root of 3). If we deal 4 times - standard deviation is divided by 2 (this is sqrt 4). Theoretically, if we deal infinite times the variance will disappear (sig will be equal to 0) and therefore we will win part of the pot exactly equal to our EV.

    In the long term...

    And what if we examine some of the examples above in the long term? For instance - we go all in AA vs KK many times for 100bb. We know that in long term we should win approximately as much as our EV. But what happens with the variance? The popular opinion is, that variance drops (is reduced) in the long run, "because we are supposed to get closer to our EV". Well I have good news and bad news for you:
    Bad news is, standard deviation is actually INCREASING in long term. Going into multiple similar spots in fact works exactly the opposite way as the multiple dealing the board. Both mechanisms preserve the EV the same but "dealing it N-times" divides the sig by sqrt(N) while getting into N similar spots multiplies sig by sqrt(N). This is derived from the formula for sig itself. And therefore sig could become huge for huge values of N.
    Good news is, if you are on the winning side (EV+ spot), sig increases slower than the EV and because of that in the long term the standard variation won't be that big of a problem. If we go AA vs KK once we have ЕV = 64bb, sig = 77bb. But if we do it 100 times we have EV = 6400 bb, sig = 770 bb. Make it 10 000 times - EV = 640 000 bb and sig = 7 700 bb; so after 10 000 such spots we will be somewhere between 620k and 660k profit and variances in this interval aren't that painful, are they? Deviation becomes negligible concerning the profits - that of course if there are profits, i.e. if we are EV+ in the spot.

    Binomial distribution

    This term also comes from the Probability Theory and basically displays the odds of given number wins from given number trials with fixed odds of winning a single trial. Again I wont dig into calculations, I'll just show you some results. Binomial distribution could describe all the possible outcomes in almost all casino games, most of all the games with fixed EV like roulette, heads and tails or craps.
    Now lets examine a game (no matter what type) where we have 70% winning odds for every trial and we are going to play 10 such trials. As you already can guess we have expectation of 7 wins out of those 10 attempts (because of the 70% odds). But let's see what are the odds of ending up with 0 wins, 1 win, 2 wins etc. I'll mark the odds of 'n' wins with B(n):

    B(0) = 0,0006 %

    B(1) = 0,01 %

    B(2) = 0,14 %

    B(3) = 0,9 %

    B(4) = 3,67 %

    B(5) = 10,3 %

    B(6) = 20,0 %

    B(7) = 26,7 %

    B(8) = 23,34 %

    B(9) = 12,1 %

    B(10) = 2,8 %

    As we suggested the odds for 7 wins are the best but still only 26,7% ; the next best odds are for 6 and 8 wins and so on. On Figure 1 you can see a table of those odds. The odds for extreme outcomes (10 wins or 10 losses) are pretty small, especially for 10 losses which would be a very bad run. Note how the odds of ending between 5 and 9 wins (7 +/- 2) is whole 92%.
    Now lets play more and increase the trials up to 50:


    The "shape" remains the same and again, the best odds are for the expected outcome - 35 wins. But now it's merely 12%. Imagine how in "long term" the odds of matching our EV becomes next to nothing. On the other hand the odds to be in the fairly tight interval 32-38 wins (35 +/- 3) is 72%, and between 30 and 40% - about 91%. Yes, the variations do increase, but obviously the game is very profitable for us, right?
    Now what does the "shape" of this graph means? When we play a game with positive expectation (like this one) the graph will be drawn to the right i.e. to the bigger values. We have space to move right from the EV - with bigger odds but in tighter interval; or we can move left - with smaller odds but in wider interval. Therefore we are prone to observe often upswings with smaller amplitudes (deviations from the mean) as well as rare downswings but with bigger amplitudes. This phenomenon often has two effects on the players mind:
    1) because of the often small upswings, we are prone to take them for granted and thus - to slightly overestimate our EV.
    2) the rare but frightening big downswings impress us a lot, and even more if we already have our EV overestimated.

    This busts another common myth: "Only good players run under EV, if yo want to run above EV you should stack-off with garbage and suck-out." There is only a small piece of truth here: if we are losing players (with negative EV), our graph will be drawn to the left (the smaller values) and we will have often small downswings, but rare and big upswings. But don't forget those are just variations and our goal is playing with the biggest possible EV.

    EV difference

    This is the most popular way to "analyse your luck" for a sample (session, month etc.). As the name shows this is the difference between the expected profits: EV or $ Won(EV adjusted) and the money actually won ($ Won). Such difference emerges in almost every spot when players are all-in before the whole board is dealt. In our favorite example AA vs KK all-in preflop for 100bb with our EV=64bb, if we win the hand we win 100bb or 36bb more than EV (signed with minus in the trackers). If we lose, we would be 164bb under the EV (this is signed with plus). Split pot means we are 64bb under EV. And so, this hand could bring EV difference of either -36, +164 or +64 big blinds.

    All-in can occur on streets other than preflop but the same principle applies: if we win - we ran above EV, if we lose - we ran under EV. The only exception is when some of the players is drawing dead (0% odds of winning the hand). In fact all-in on the river is the same thing: one of the players has already won the hand.

    Building graphs of $ WON and EV for the session we see that the $won-line follows the wins and losses during the session and the EV-line will remain parallel to the first line for all hands that didn't went all in before river (i.e. hands that bring no EV difference) and will diverge sharply up/down for every all-in situation. These lines could miss each other by far but could also cross each other regularly. In such a moment of crossing we could say we are with exactly "neutral" luck.

    Long periods of running above/under EV are common sight and we've all seen abundance of graphs on that topic - usually sad ones. As said above, if a player goes all-in with better odds for the most part, he's prone to frequent small ups and not so frequent but bigger big downs. And a graph, displaying a run of 1 buy-in above EV for 5000 hands couldn't impress us, but one with 8 buy-ins below EV looks way more imposing and would eventually find place in the poker forums. The same phenomenon occurs even in trivial situations: if a player's aces hold up he accepts it as something normal, just and logical (although he just ran above EV) even if it happens multiple times; but if his aces get cracked he will whine, not realizing this is balancing (from EV point of view) the times he held up.
    With the help of the EV Diff graphs we can track the effect of the all-in situations as well as their outcome. We could see if we ran good or bad and how much were we supposed to win from these situations. But this is just a small portion of our hands therefore we track only small portion of the variations that shaped our results.

    Sklansky Bucks or " Street by street EV"

    This is another analysing method that at some point reminds of the All-in analysis. Sklansky Bucks (S$) track the odds we have whenever we put money in the pot. If we invest 10bb with 75% odds against one opponent, this means we "own" 75% of the total pot of 20bb, 75% * 20bb = 15bb, therefore we just won 5 theoretical dollars with our preflop actions. But we miss the flop badly and now we have only 20% equity (odds of making better hand by the river) and we decide to call a bet of 15$; we own 20% of the newly-added to the pot 30$, or 20% * 30$ = 6$, but we invested 15$, therefore we lose 9 Sklansky bucks on the flop. The turn is checked by both of us and no S$ are won/lost. The ricer gives us the best hand (i.e. 100% odds), we bet 25$ and get called. This means we own 100% of the 50$ added to the pot on river, and since we invested 25$, we won 50-25 = 25 S$, which in fact matches the money actually won on the river because our hundred per cent odds. Now we sum the Skalnsky bucks street by street and we find out the hand earned us 5-9+25= 21 Sklansky Bucks. the money actually won are 10+15+25 = 50$, so we "ran good" in this hand because we make a bad (from S$ point of view) call on the flop and we get lucky on the river. But what if the opponent folded on the river? Then we would have won only 10+15 = 35$, while our Sklansky bucks would be mere 5-9 = -4 С$, therefore the hand was statistically losing.

    The S$ measure how profitable is every single action in every hand we play. That's why the S$ graph is more fluctuating than the EV graph. At first this type of analysis seems a pretty good decision providing a lot of information, but things aren't that good. Bluffs, semi-bluffs, fold-equity, pot odds, implied odds, reversed odds are amongst the most important elements of poker and S$ couldn't take into account any of that. For instance, if we bluff someone that would fold almost anything but in this case we get called by the nuts, S$ would conclude we played the hand awfully since we put a lot of money with 0% equity. Or if we draw with excellent odds and we miss. Well, the correct S$ interpretation is something difficult and they are rarely used nowadays.

    The big picture

    Thus far we examined only isolated cases, concerning small part of the different possible situations in poker. But not all spots are like them, so not only the board dealt brings variances in our game. Variances come from lot of things: good/bad table selection, playing style, setups we fall into, tilt (both ours and opponent's), table flow, positions, opponents, metagaming and what not - even our promotional bonuses could bring variance. Some of these factors - like the EV diff - are neutral (decrease their impact and change their direction) in long term, but if we have steady tilt issues this means steady harms on our profits; and vice-versa - solid table selection could greatly improve our situation.

    The combined long-term effect of all these factors determines whether and how winning players are we. Of course the main factors are the quality of our game and the strength of our opponents, but none of the other things is to be underestimated. The measure of our strength in cash games is called Winrate and it shows how many bb (big blinds) we win on the average per 100 hands : for example wr=4bb/100hands. Sometimes BB (big bets) are used instead of bb (big blinds) with the rate being 1BB = 2bb. So this parameter measures our "speed" of winning money and it "foretells" our expected winnings for the next, say, 50 000 hands. If we play 50 000 hands with winrate=4bb/100hands, we expect to win (4/100)*50 000 = 2000bb = 20 BI. Now let's examine this measure in details:


    Firs of all, I should make note that there is a difference between our real winrate and the one we see in the tracker. The real WR is formed in long term by the force of the factors above and is more or less an abstract variable: we can't calculate it precisely. The Winrate shown by trackers is simply the money won divided by the hands played and it fluctuates all the time (even for large samples) - it jumps after a "good" session and drops after a "bad" one, therefore this WR is subject to variations. This statistically found winrate will approximate the real one in long term and this is in fact the only way to "find out" our real winrate: by taking the winrate from a huge sample of hands (hundred of thousands). Even then, we will get only the WR for this playing style against these opponents and all the other circumstances under which the sample was played. Note that the evolution in our (as well as the opponent's) game leads to "changes in motion" in the real winrate, making it's determination even harder.

    Although difficult to figure out and difficult to improve, in my opinion the real winrate is the most important stat for a player. This is our "edge" and positive WR means winning player, the bigger the winrate - the better the player. There are "rakeback pros" as well - they have WR close to zero (slightly more or less) and they get profits from their bonuses/rakeback only.
    There is similarity between Winrate and single-hand EV: both tell us how much we win in the long run. EV shows how much you win on the average from an isolated spot while WR shows the average profit from ANY hand. There is resemblance, but the scales are very different: EV for AA vs KK is 64bb or 6400bb /100 hands (in this form); but Winrate is a lot more modest, even for the best players - you can rarely see a two-digit winrate, and a good player could be glad even with winrate of 4bb/100hands.

    Winrate (the real one) is a linear function: if we play 100 hands with expected profit of 5bb, we expect 10bb for 200 hands, 15bb for 300 hands and so on:


    The results will fluctuate around this line, but how?

    Standard Deviation - part 2

    Now let's expand the term "Standard deviation" we already discussed. Earlier in the article we described the variances in a single hand with given EV. And if the long-term analog of EV was Winrate, than what should be the long-term analog of "sig"? Well it has the same name: Standard deviation, it is marked as SD in trackers and much like "sig" it measures the degree to what the results tend to vary from the average state (in this case - from the winrate line). The crucial difference between the two parameters is the unit of measurement. While "sig" for a single hand is measured in just in big blinds, the SD is measured in bb/100 hands, in order to take account of the variance increase in long run. If we have SD = 50bb/100hands, for 200 hands this makes sqrt(2)*50 = 70.71 bb/200 hands, for 400 hands we have SD = sqrt(4)*50 = 100bb/400 hands. Even though SD increases in long term, it has decreasing influence over the end results.

    Another important difference between these two is the way we calculate them. Sig derives from a mathematical formula using EV and odds, the result from that formula is exact. But SD is determined by statistical methods (the trackers do this job) much like the "not real" winrate, and just like that it's subject to variations. So pretty much the variations themselves could also vary :D Still, we have no better way to determine our SD but we need serious sample of hands - hundreds of thousands.

    SD differs for different playing styles and game types (NL/PL/FL; fullring/shorthanded/heads-up...). ABC player on 6max NLHE has SD about 80bb/100hands (or 40BB/100hands). The nits/passive players have smaller SD, and the maniacs - a bigger one. But don't forget: our goal isn't minimal SD, it's maximal WR, so a lower-variable style doesn't mean better style!

    Distribution of money won

    With WR=5bb/100hands, SD=80bb/100hands, at some point we could predict the results after 10 000 hands (this is hundred times 100): the expected profit is 10 000*(5bb/100) = 100*5bb = 500bb; the deviation for 10 000 hands is sqrt(100) = 10 times bigger or 800bb/10000hands. Now what are the odds to win about 500bb, 510bb, 490bb and so on? To answer that question we must know the type of distribution the profit has. The two dominant opinions on the Internet are: Binomial distribution (the one we already examined) and Normal distribution (a very common type in statistical researches) - these two have different properties. Both opinions are wrong, but are close enough to the truth, so it's safe to assume them both (with small mistake) because "Poker is a game of small edges." Remember the shape of the binomial distribution with 70% odds? It was heavily drawn to the right because of our huge edge. But Winrate 5bb/100hands gives us way smaller edge and the graph of our distribution will be just slightly drawn to the right. This is what the distribution looks like at the moment of the 10 000-th hand:


    In any given moment (hand) the odds of wining the respective amount of money are ordered that way, with the top of the mountain matching the point of the winrate line - i.e. the biggest odds are to reach our expectation. The odds for bigger/smaller profits decrease gradually.

    Now according to the properties of the Normal distribution the odds of being within "one standard deviation" from the average value is 67%, the odds for "two standard deviations" - 95%, and for "3 standard deviations" - more than 99%. Well, our distribution isn't exactly "Normal" but the numbers are close enough. Here are some examples:

    * In 100 hands with WR = 5bb/100hands and SD = 80bb/100hands we have expectance (average value) of 5bb profit, with 67% odds of winning between 5bb - 80bb = -75bb and 5bb + 80bb = +85bb. Also we have 95% odds of winning between -155bb and +165bb; and almost certainly (99,7%) between -235bb and +245bb.

    * In 10 000 hands SD becomes 800bb, the expected profit is 500bb. This gives us 67% odds to be between -300 and +1300 blinds; 95% to be between -1100 and +2100 blinds; 99% to be between -1900 and + 2900 blinds.

    * In 1 000 000 hands SD becomes 8 000 bb, expected profit is 50 000 blinds. We are 67% sure between +42 000 and +58 000 bb; 95% between +34 000 and +64 000 bb; 99.7% between +26 000 and +72 000 blinds. Finally, we reached the "guaranteed" profits!

    If we insist on 100% certainty, the results are between the theoretically worse and best possible scenario: to lose/win whole buy-in in every single hand. Obviously this is useles info and in practice such case couldn't occur.


    Confidence intervals

    Those were just three points (moments) of the graph. But with the so called "Confidence intervals" we can give "frames" to the possible outcomes from given number of hands. Confidence interval of 95% shows us the possible points we could reach at any given moment with probability of 95%. As we already know this is no more than "2 standard deviations" above or under the expected profit. Since both the expected profits and SD increase the more we play (with SD increasing parabolicly) this confidence interval will be a parabola around the winrate line:


    Note that we don't have 95% probability for the WHOLE line of winnings to be inside the parabola, but 95% for any of it's points (i.e. at the moment of any hand), so the line of winnings will occasionally breach the frames. Confidence interval of 99.7% would be breached a lot less, but will be much wider, and Confidence interval of 67% will be tighter but breached frequently, so 95% is a good decision.

    On this page you can experiment with different WR, SD, confidence intervals and numbers of hands: http://www.castrovalva.com/~la/win.htm (choose Total winnings). Possible runs are being simulated there. Note that the unit of measure there is BB.

    The confidence intervals can answer a frequently asked question: "for how long could I be losing?". Since the initial value of the SD is huge compared to the WR, we could be losing pretty often in small samples of hands. But the variance increases slower than the expected profits and from some point on the effects of SD can no longer keep a wining player "in the red". The moment comes after the parabola cross the zero-line. For instance with WR=5bb/100hands, SD=80bb/100hands, after 100 000 it's near impossible to be behind (check in in the simulator). The bigger the winrate and the smaller the SD - the sooner the "guaranteed" profits come.

    This graph gives us another, even more important knowledge. The lowest point of the parabola represents the worse possible scenario. In the example above it's possible to lose whole 12 buy-ins during the first 30 000 hands, despite our good WR. And that's why a sufficiently big bankroll is a necessity. If we bankrupt during the first 30 000 hands due to a bad run, we won't reap the rewards that await us after that. So the bankroll should be big enough to survive the short term punches. If the WR was smaller or the SD was bigger, even worse scenarios would be possible - try it out on the simulator. Also, here we used 95% confidence interval, if we want more security we need even bigger bankroll. It's not a coincidence one of the first tips the new players get is to employ good bankroll management and the experienced players always keep bankroll of at least 40, 50 or even 100 Buy-ins.

    But let's not be so pessimistic, the probability of running under the WR line is the same as to be above the line: 50%. Obviously a good run won't harm our bank but bears the risk of overestimating our skills and losing idea where (on the graph) are we. Then the inevitable "return" to the average values could be even more painful. After all, the points closer to the average are easiest to reach (with biggest odds) and runs close to the WR line are considered "normal", while heavily deviating runs are considered "extreme" and are rare.

    What's the picture for a losing player - one with negative WR? His line is going down and it's a matter of time for him to bust regardless the size of his bankroll. SD has exactly the opposite effect on him - it could keep him on the winning side only for a limited period of time or bust him even faster than expected.
    And what about the break-even players? They have WR=0 with horizontal line. Thanks to the variance they could go up and down, and with even greater amplitudes in long term because of the increasing SD. And that's why they will eventually bust in the long run, even though their expectation isn't negative (i.e. they aren't losing players) ! Recall how the lowest points of the parabola give us idea of how big bankroll we need. With WR=0 there is no lowest point, the parabola drops infinitely. Therefore there is no adequate bankroll for a break-even player. No matter how big, theoretically the player would bust. But it's uncertain if he will reach that moment in his lifetime...

    Withdraws and variance

    The the most entertaining in my opinion conspiracy theory in poker is "The Withdrawal Curse": it's been said, that after a withdrawal the software brings a downswing upon the player. Indeed, most of the players have experienced a dangerous down after an withdrawal, causing them to drop limits or even bust. The explanation is: not only the bankroll should be conformable to the variations, but so should be the "withdrawal policy". If a player cashes out ALL of his winnings in practice he becomes a break-even player since his bankroll couldn't possibly increase, and we just saw what happens with the break-even players. Thus, part of the winnings should always be kept in the bankroll as a measure to fight variance, doubly so if we intend to move up the limits. There are whole books dedicated to this topic and it definitely deserves the attention of all who play for a living.


    We come to conclusions we more or less already know:

    * Poker is a game of small edges and large variances - we should control them as much as possible.
    * The most important thing in poker is to play with maximal edge: by the best play we are capable of, by good selection of opponents and by minimal tilt. These things provide us with maximal Winrate.
    * Bankroll management is one of the most important skills "beyond" the poker table and bad BRM could ruin even excellent players.

    I hope this article helped you understand the logics and reasons for the listed, and I hope it busted some of the popular myths in the poker community. I finish with a key sentence from a good book: "Understand and accept the realities of poker!"

    Author: Vihren Lindrov
    Last edited: Nov 17, 2010

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