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Kelly Criterion is important in betting, horse racing, investing, money management, and poker cash games, and poker tournaments.
I touched on this slightly in “big moves are not required“.
Basically the idea of the kelly criterion is based on the fact that if you go broke when investing, you have 0 ability to earn in the future. If you lose a significant bet, your ability to earn is hindered significantly as well. If you lost 10% of your bankroll, it will take an 11% gain to make up for your loss. If you lose 20% you’ll need a 25% gain. If you lose 95%, you’ll need a 2000% gain. Because of this, large losses will actually hinder your ability to earn. In a cashgame you can rebuy so it’s irrelevent other than for bankroll management decisions. But tournaments are essentially cash games where you have all your money at risk, and can’t leave until one person takes it all. One major difference is that it is not always winner take all, so you are rewarded for survival more so than overal chip gain. Essentially betting more than the amount defined by the Kelly Criterion decreases the probability of very good results, while still increasing the probability of very bad results. You actually expect to gain MORE in the long run, by betting less than “all in” in a tournament… Unless of course you have a 100% chance of winning. How much you bet depends on the “kelly” value.
It doesn’t matter whether a bet is profitable or not, without money management you will be expected to lose eventually. That’s why the best players in the world are often good at small ball and bankroll management. There are some exceptions of players who gain control and dominate the table early and put the peddle to the medal right off the start. There goal is to gain so many chips that no one can catch them. If they can get lucky, or not get unlucky early, and continue to power away, they can eventually reach a point where everyone at their table will never force them to risk too significant amount of their chips, and from that point on, as long as they continue to play at a high gear, they will not have to take on dramatic risk.
I prefer a combinationm of the two. Understanding that regardless of how you play at some point you MAY have to take on dramatic risk, but until you get to that point, you don’t know exactly when it will be, you can only guess. However, I also think that because the blinds are so much higher later, that there is more bennefit in taking on dramatic risk late, making a move and continuing to go from there. If it were possible to win without ever being all in, why would you want to be all in early. That’s not to say that such a strategy is possible, but I prefer to wait until a “focal point” in a tournament to take that aggressive action, knowing that ultimately coming out ahead at that time period is much better overall. I understand that there may be some pots I miss out on becasue I didn’t have a commanding stack and intimidation, but generally I don’t think that it really comes into play early on at least with my style anyways.
I think that as long as it is possible, you should have limited risk. An experienced player, or someone who has a spredsheet set up to show them how their skill factor and blind structure understands how that will impact their number of big blinds. So it may become neccesary to violate this Kelly criterion before you get so short stacked that you will have to take on that risk anyways.
The kelly criterion teaches you how you can maximize your growth in a tournament. Your long term growth of your chipstack is dependent upon your ability to survive. If for example, you had a 60% chance of winning a hand and it was blind vs blind confrontation and no antes in the pot, the bet would be 1:1. The kelly criterion would indicate that you would not want to bet more than 20% of your bankroll. If you do, it actually will hurt you in rhw long run. A player who risked 20% of their bankroll on a 60/40 would do better than a player who went all in on 60/40 bets. However, a player that risked 20% of their chipstack/bankroll would also do better than a player who risked 5% of their bankroll.The vairence can wipe out your entire ankroll which will hinder your ability to earn entirely.
Poker is much more complicated though because once you put in 20% of your stack, you will see the flop and then have to make another decision based on the flop and your chances of winning, and that’s before you get to receive the results from your first 60/40 bet back, and so it could certainly cause you to put in far more chips than you want since the only way for you to receive winnings is to either have the best hand on the river, or get your opponent to fold. So you ideally would not want to even bet 20% preflop unless you KNEW that your opponent would check it down until the river. Even so, poker is a game of imperfect information, and given that information it’s imperfect probability. It’s like a chess game where not only you can’t see your opponents moves, but when you make a move, there’s still a chance that your chess peice will end up on a different square then what is expected. So I believe aiming for 8% is ideal for preflop in terms of being in position to rereraise if needed without risking much, but in terms of the kelly, I think raising more like 5% is ideal, then getting more money involved dependent upon the strength of your flop, vs your opponents calling range.
If you had AK vs 45s your odds are 60/40. If you put in 20 big blinds when you had 100, and your opponent has 100 and the flop comes K67 with 2 spades you now have about a 45% chance of winning. Suddenly you wish you had some of your chips back, but you can’t change that now. Now had the flop came without any spades, you would have maybe about a 68% of winning. Now you started with 100 and if you hve a 68% chance of winning the kelly criterion will give you a number of 36% of your stack or 36 big blinds total assuming a bet of 1:1. You can look at it this way, or you consider your bankroll is now only 80. 36% of 80 is 28.8 so you can put an additional 28.8. The problem is the result of this is dependent upon the 60/40 you started with. so this would be incorrect to bet 28.8, but correct to bet a toal of 36, or 16 more. So you put in the 16 into a 40 pot and your opponent calls. Now the turn is another king. Now your opponent has 8 cards with 44 unseen cards in the deck left. He has an 18% chance of winning, you have a 82%. You can now put in 64% of your bankroll. To maximize your long term gain, you can put in 64% or 64 big blinds. You have put in 20 and 16 already, so you can put in 28 more. You bet and he calls and the river comes and your opponent hits his straight. Now you fold leaving you with 36 big blinds. This situation comes up again and plays out the same way, only you risk 23 big blinds and win bringing you from 36 to 59. Then you rise to 97 and you’re back. Had you won and your opponent not rivered you, you would have gone from 100 to 164 big blinds. However, because your odds can actually get WORSE as the hand plays out, and because you can’t take money OUT of the pot, you should bet less than kelly. In addition you have to leave room for your opponent to reraise as well.
Real poker never plays out like this. You don’t want to bet as much early because then you can’t build the pot if the flop turns out unfavorable, and players won’t generally just call regardless of how much you bet and do nothing else. You don’t always have a choice of how much to bet. However, you can at least attempt to control the bet size. I would advice aiming fo half the kelly criterion early, then growing the bet as the hand plays out if the cards go into your favor. Of course, that’s the other problem, you don’t KNOW if your opponent has you crushed, or if he’s just drawing. However, a better understanding of the Kelly Criterion and the fact that the best expected value doesn’t always equal the best long term growth of your chipstack is a HUGE HUGE HUGE thing to understand. Everyone trying to maximize their expected value by pushing every small draw just doesn’t know their Kelly Criterion.
Now… what happens if you put pot odds into play?
Lets say you get 2:1 odds. Now what is the ideal “kelly criterion” for a 60/40?
Now you must take (2(.60)-.40)/1=.80 So 80%
However, if you had good pot odds, it may be worth it to take on more risk.
There are several problems with Kelly Criterion as well. It i not calculated based on dependent bets such as those involved in poker. In addition, it doesn’t neccesarily factor time. If an investor would invest in the stock market and bet 10% in 5 investments, the entire stock market could crash bringing them all down, so to avoid this, an investor might invest 10%, then when that is over, reinvest 10%. The problem is, “the long run” is based on an infinite number of trials, rather than a fixed amount of time. In a tournment losing once will also hurt you more because the blinds will rise and it will be increasingly difficult to find situations in which you can safely bet the “kelly”. A loss will cause you to avoid profitable situations for the sake of preserving your tournament life. Losing once does however improve your pot odds on your future decisions in all cases but those when both the small blind and big blind are in with no antes, and in all cases but everyone in the pot when there is antes, so there are some advantages. Poker is a game where current action IS dependent upon it’s previous actions.
If there comes a time when you can gain a substantial lead, this will allow you to control “kelly” and bet amounts where your opponents will be more likely to lose in the long run due to lack of bankroll management, and amounts that still will help maximize your bankroll. So taking on more risk at a certain point may be worth doing. However, in most cases, due to “risk of ruin” Kelly is a poor model, and advises you to bet TOO much. Poker tournament’s payout structure make Kelly criterion even more interesting, because if you ruin after a certain point, you still get paid, while if you ruin before that point you get nothing (the bubble). In my opinion, this would indicate that you should try to mimize your risk of ruin at cost to not being able to fully maximizing your growth before the bubble, while understanding that by gaining control afterwards has advantages. For example gaining large share of the overall chips can allow you to put your opponent in a spot that is over their kelly while still under yours allowing you to win in the long run, even when you have the worst of it. Take for example a situation where you have a million times more chips than your opponent. mno matter his cards, the liklihood of him being able to win so many all ins to be able to beat you is so miniscule that it doesn’t matter how small of a probability of winning you have, you could call his all in with 27 every time if you wanted and still beat him before he gets enough chips to beat you. Although your opponent could have AA everytime, it wouldn’t matter, because YOU forced him to risk all his chips, and eventually that will cause disaster. You would never be able to take it to this extreme, however, you certainly may be able to get enough chips where you force your opponents to either continue to blind down, or call a greater than 20% bet. The odds can be in yourfavor, even when they are not. It’s much better to risk elimination by betting too much when you have a large edge, then allowing others to force you to continously accept a bet over your “kelly” when you only have a slight edge.
This is a strange and wonderful concept, and we will explore more later.
One more thing, another interesting thing about the kelly criterion is that by betting LESS, you risk not fully being able to maximize your gain. By betting more, you risk not having a gain and your risk of ruin goes up along with your risk of being crippled and a loss will hinder your ability to gain. Since the Kelly Criterion may result in having some risk of ruin anyways, in my opinion it is even more reason for erroring on the side of caution if possible. you can’t win more chips than all of them anyways. Once the tournament is done, you don’t get to carry them over to your next tournament. The point I’m trying to illustrate is, if you can stil win without increasing your risk of ruin, why bother. I think it is better to attempt to make up for lack of growth later on, when a win and chip position can be used to gain greater chip position, and when having that chip position will result in a win. Remember Kelly Criterion is reserved for INDEPENDENT bets, however a poker tournament is DEPENDENT. Your gains will put you in a better position to win the tournament at a certain time. early on chip position means very little because there’s so many around you that will get more chips than you and there’s higher varience of other players that can outchip you. The wins of a small pot are smaller, blind steals are smaller, and taking on a 50% risk of ruin for a .05% gain in the total share of all chips seems silly when you can survive and continue to win small pots and later on take a 60% risk to gain 5% of all the chips in play. If you could take a 50% chance to ge .01 % of all th chips in play now, or a 50% chance to get 5% of all chips later, which would you take?
Note: Even people who research Kelly will advize betting half that amount. The reason is that with the Kelly amount you have about a 1/3rd chance of halfing your bankroll before doubling it, or in this case your tournament stack. Now when you consider that you have to face escalating blinds if at all possible you want to error on the side of reducing risk so that you aren’t forced in a spot where you will have to take an all in. This isn’t neccesarily true later in the tournament when not gaining fast enough will result in you being in an “all or nothing” situation anyways, and when a larger variance swinging to the positive side will allow you to take control of a large enough share of chips that you control the table.
Early on with 100 players, you have 1% of chips and a double up will bring you to 2%. However, later on with 30 players left, the average stack has 3.3% of chips. Still even later, with 20 players left players have 5%. Not to mention the blinds and antes make up a larger percentage as well. You could double up from 5% to 10% of all chips, steal a few blind then suddenly have 12% of all chips. You will be safe from having to risk your chips, where as had you only taken a smaller amount avoiding risk, you might get up to 6%, then the blinds raise and your are forced to take a bet higher than “Kelly” as those with the 12% of all chips can reraise you without taking on the same kind of risk themselves.
Taking Max ev (expected value) is simply not truly “max EV”
I believe because poker has multiple streets of betting that you will generally waant to have a 5% risk if possible.
All the math book authors like David Skalansky recognize that “utility” is important, they just aren’t sure how to model it, and aren’t sure how to measure it. Because they can’t quantify it and because it varies in so many tournaments, they do without it. It makes sense that a math guy is going to want tangible EXACT numbers with an exact system so he can make sure that all formulas match exactly.
Snyder in his bookPoker tournament formula 2 basically shows you the errors in this. He likens poker to a game of monopoly. The more properties (chips) you gain, the less are available for your opponents. It’s worth mortgaging houses to make sure you secure enough properties so that your opponent cannot.
Now I think that some day there will be a much better model of tournament play that will factor in utility, but until then you can only attempt to edtimate this.
Kelly Criterion is essentially a utility function to maximize your own growth. Since Skylansky proved that any time you get your oponent to make a mistake, you gain, we can assume that this will generally hold true with utility as well. So it may be wise with a large chipstack to deny your shorter stacked opponents the propper “kelly” to safely make a call, provided this doesn’t violate your “kelly” unless you are desperate, in which case all in may be your only move.
Kelly supporters say to bet a fraction of kelly for a number of reasons. First, reducing volitility and providing more stability, reducing risk of ruin while also giving yourself a “hedge” against your numbers being wrong. If you bet the full kelly and estimate you have a 52% chance of winning and you really only have a 51% chance of winning, you will have too great of a chance of going broke.
Check out a kelly criterion calculator. The site calculates the numbers under “results” which is below the calculator. The page will refresh and make reccomendations.
Your gambling bankroll should be your chipstack in a tournament. Your odds offered are the pot odds you get. This is interesting because in poker your odds change based on your bet. The more you bet, the closer to 1:1 odds you get. If you are deciding whether or not to call, then you can calculate the odds. Say there’s 200 in the pot, your opponent bets 100. You have 100 to call to win 300. Although your bet is 100/400 total, your pot odds are 300:100 or 3:1. If the result of the Kelly is that you can bet more, you can raise, but that doesn’t neccesarily mean that you should.
Here are some rules of thumb:
When calling a raise, generally only call if the the bet is 1/2 the kelly
When making a raise, generally only bet a maximum of 1/4th kelly, or even 1/8th. The reason is, if someone reraises you, you don’t want to have to accept a bet over the kelly amount. The faster the structure, the higher and closer to the kelly you should target.
The higher the blinds are in proportion to your stack, the closer to kelly you should be willing to go. This is because of the lack of choice.
The faster the blinds escalate, the closer to the kelly you should be willing to go. In a fast blind structure, if you do not take risks and win it will result in you taking a far worse bet in terms of “kelly” later as you will be forced all in anyways. it’s much better to maximize your edge while you have it, then if the blinds raise and you are forced to move with a bad hand later.
If there are only a few players left in the tournament, it’s okay to target a number closer to, or even above the kelly. You shouldn’t AIM at over kelly, however, because kelly is based on the “long run” and the tournament won’t last that long, you may be willing to take a chance.
If you accept a number beyond kelly it WILL most likely reduce your kelly in the future should you win, so there are arguments for betting beyond kelly particularly in fast tournaments.
The slower the tournament and the greater the skill, the lower the kelly and more willing to fold bets that exceed the kelly you should be. If you do not have much skill, you will be forced all in anyways, so ignoring kelly and maximizing ev is not a bad thing for bad players.
The greater the skill factor, the longer you will last before you reach a point where acheiving 1/2 kelly becomes possible. So the greater the “Adjusted M” (true hands remaining based on skill) the less neccesary it is to take all in risks and play large pots.
Note that in most standard blind structures, by analyzing structure you can see it is consistent with the IRC method. The fact that skill factors drop dramatically during certain periods are reason to take on numbers greater than kelly prior to these points, and greater than kelly during these points as not doing so will result in your chipstack quickly deminishing to a point where avoiding all ins will become impossible anyways.
Now to take the IRC one step farther, you have to recognize that deendent upon the payout structure and your skill and tournament, you may need to accept an all in with lesser hands if you play to win. You will needed to be all in more than once most of the time, so lasting as long as you can before going all in isn’t neccesarily always the best strategy. Doubling up earlier may result in being able to reduce your bet sizes below Kelly for much longer, and reduce your overall risk, so there lways will be some reward associated with maximizing EV as opposed to maximizing long term growth by using the correct “kelly”. So if you have “100 true hands left” it may be wise to assume you only have 35 true hands left, and now the top 2.8% of hands become hands you’re willing to call an all in with, as opposed to the top 1%. The viewpoint of the IRC method is that you are playing to “survive” and this isn’t neccesarily the best way to maximize the cash. Now if ther are 100 entrants, winning WONT win you 100 buy ins, and this makes it not worth it as much to go for the win as it would be to play in a cash game. However, you must at some point recognize that a very high cash rate with a very low winrate is generally not as good as a medium win rate and a horrible cash rate. It may be worth your tournament life and higher variance to risk going bust at times. Obviously you want to minimize going bust as often as you can, but simply prolonging the inevitable all in for too much longer after the antes does not make as much sense. So I will use the IRC formula, and true hands left (adjusted M) up until the antes, then I will make a move, and sometimes keep an eye on how many more hands I have left in case I want to choose to play a style of maintaining big blinds, but generally after the antes the “true hands left” becomes less important, as at that point I will need to focuse on maximizing both my skill factor and EV, Ideally without violating kelly.
For me personally, this isn’t as neccesary, because when the antes approach, I get more aggressive thus increasing my skill factor, while also increasing my bets beyond kelly, and a more hyper aggressive and hyper active loose aggressive style at these points of a tournament will result in me generally having a greater number of “true hands remaining” then my spreadsheet will indicate, plus I generally aassume a conservative “skill factor” and most tournaments I play in, I am ahead of what my “skill factor” will indicate I should be. I may start stealing as much as Phil Gordon’s little green book indicates I must to keep up with the blinds, plus adjust my reraises and rereraises accordingly as well. This will certainly result in my moving in too often and violating kelly, however, If I am able to steal this often as the blinds go up, and players don’t play back at me, or when they do I reraise them enough, I can find myself accumulating tons of chips, maintaining a really high number of big blinds (like 40-60) when everyone else is low as the blinds rise.
Being able to make just a couple extra steals at these later levels with antes involved, is worth giving up all the value up until this point, and is worth the extra risk of elimination that you might take from trying to “eliminate any threats” preventing you from being able to control the table before the antes get involved, and is worth the potential risk of elimination.