http://en.wikipedia.org/wiki/Kelly_criterion

In a universe of +EV bets, the Kelly Criterion is a mathematical formula used to size bets for optimal bankroll growth. Given that you know the current size of your bankroll, the odds on your bet, and an estimate on the probability on your bet winning, it tells you exactly what % of your bankroll you need to bet in order to grow your bankroll the fastest. It is used widely in the investing world, and sharp gamblers use it for sports betting as well.

One of the reasons that we don’t hear about the tool in poker is that usually we are more preoccupied with other concerns – most notably our opponent’s reaction to our decisions, but also the playability of hands, metagame, planning for later streets, etc. Especially damning is the standardized size of bets in Limit Holdem. However, there are a few situations where you do have some control over your bet-sizing, the most notoriously debated on this forum is that of whether or not to raise with a “marginal” appearing holding when it is folded to you in one of the two blind spots. You can choose to check and wager nothing or raise and wager 1 SB.

Example: You are playing in a normal 20/40 game and have QT in the BB.

There are 3 limpers to you and the SB completes. Your move.

The assumed ranges are appended below in the Poker Stove simulation, but the summary is that the limper #1 is assumed to have the top middle 30.3% if his range, limper #2 -37.3%, limper #3 - 44.5%, and the SB – 87.6%. You can argue with these assumptions quite a bit without changing the primary result – our equity is estimated at 25.8%

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25,456,273 games 53.375 secs 476,932 games/sec

Board:
Dead:

equity win tie pots won pots tied
Hand 0: 25.850% 24.12% 01.73% 6139937 440547.77 { QTs }
Hand 1: 19.894% 18.41% 01.48% 4686334 377824.93 { AhAs, KhKs, 88-55, A9s-A2s, K6s+, Q8s+, J8s+, T7s+, 96s+, 86s+, 75s+, 65s, 54s, A9o-A7o, K8o+, Q9o+, J8o+, T8o+, 97o+, 87o, 76o }
Hand 2: 18.732% 17.35% 01.39% 4415755 352710.10 { 88-22, A9s-A2s, K5s+, Q7s+, J7s+, T6s+, 96s+, 85s+, 75s+, 64s+, 54s, 43s, A9o-A6o, KJo-K8o, Q8o+, J8o+, T7o+, 97o+, 86o+, 76o, 65o }
Hand 3: 18.709% 17.39% 01.32% 4426374 336145.02 { 88-22, A9s-A2s, K2s+, Q5s+, J6s+, T6s+, 95s+, 85s+, 74s+, 64s+, 53s+, 43s, A9o-A4o, KJo-K7o, Q8o+, J8o+, T7o+, 97o+, 86o+, 75o+, 65o, 54o }
Hand 4: 16.816% 15.73% 01.09% 4004091 276554.18 { TT-22, AJs-A2s, K2s+, Q2s+, J2s+, T2s+, 92s+, 82s+, 72s+, 62s+, 52s+, 42s+, 32s, AQo-A2o, K2o+, Q2o+, J2o+, T2o+, 95o+, 84o+, 73o+, 63o+, 53o+, 43o }
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Plugging these variables into a Kelly Calculator(5-1 odds, 25.8% chance of winning) recommends that you should wager 10.96% of your bankroll if allotted.

Of consideration is that the Kelly Criterion concerns itself primarily with maximal bankroll growth. Reducing risk of ruin is an implicit consideration but a secondary goal. Thus, it is a common procedure to reduce the amount Kelly advises, the most common recommendation being one-half, also known as half-kelly betting. In this situation Half-kelly would still call for 5.48% of your bankroll. For a 300 BB(600 SB) bankroll that would be equivalent to ~32 SBs. (This figure may seem big to you, but it takes into account the factor below, and also is a comment on how antediluvian our standard bankroll requirements are.) To not put in 1 more SB would be a ridiculous undersizing of a bet and to want to reduce variance is at the expense of a significant amount of expected growth(EG).

The Kelly Criterion also assumes that all of your bets are not correlated with any other bets(covariance = 0). While it is true that your decision to raise or check has a small to no change on whether you are more likely to win the hand, you will be making more bets on the same hand for 3 more streets. Therefore, it would be prudent to make sure the maximum amount you are wagering during the hand is still within the confines of the recommended size. However, there are few to no hands you play with QTs that you should expect to lose 16BBs.

At this point, it’s worth examining the other factors that you will be subjected to if you decide to raise – you will be playing the hand out of position and are increasing the size of the pot with initiative. This obviously makes hands with good implied odds and can allow your opponents to draw thin or incorrectly(big paint suited connectors) much better than to raise with than hands that are the opposite of that(unsuited big aces). There are other considerations at play, but you can find them in a number of other old threads in this forum debating preflop. There are of secondary importance here. (I will add that most people argue irrationally against raising in these spots because they are uncomfortable with checking certain flops, check-raising certain flops, etc.)

If you are anywhere near the recommended bankroll for the game you are playing in, it is an absolutely gross underbet to check a lot of these hands we routinely debate in this forum from a growth perspective. It’s not even close. You may feel relieved when the board comes blank and you get to “save a bet”, and just as happy when you turn your flushes, but your bankroll will suffer immensely in the long run. Though your play will still be +EV, it will often be –EG.

The Bryce was right when he proclaimed that “guessing is out and math is king”. Once you’ve attained a level of competence in LHE, it’s important to examine the rule of thumbs you use every time you play and wonder if they are compounding errors that are compromising the growth of your bankroll. What you feel is something you want to apply to reads, not when you are doing a math problem. Being concerned only with +EV is so last season anyways. The future is all about +EG.

Very nice post. Can you think of any other situations besides preflop edges where the concept of Kelly EG applies to limit holdem?

Great post Professor and 100% agreed but I think the above factor is a little under empahized. This criterion assumes that we are going to play perfect post-flop whereas sometimes raising pre-flop with small edges will cause players to make mistakes post flop which end up making raising pre-flop overall -EV. For example, since I have no check button, if I raised QTs in your example above, I would likely be betting entirely too many flops/turns that I miss/don't hit hard enough whereas had I just checked pre-flop, I could check fold the fold at no additional cost.

Doesn't this then assume that your will opponents play perfectly post-flop and not make similar off-setting mistakes? (Obviously something that 3 limpers probably won't do to often.)

BTW: Thanks for a great post Ben.

No, I'm assuming that my opponents will play just as ******edly whether I raise or check. That said, in my last few sessions if seen a ridiculous number of flop folds for 1 SB in bloated pots.

i believe that the kelly criterion is only useful preflop since your equity changes drastically from preflop to the flop the flop is like an unforeseen event its like betting a large amount in blackjack when the count is high your bet is equivalent to preflop in holdem then when you actually get dealt your cards in blackjack thats like seeing the flop in holdem if that analogy makes any sense

Well, postflop in limit you can only do stuff like jam your draws, which everyone knows how to do already with pot equity. Since the cap is generally going to be less than Kelly's optimal bet size (assuming you have a decent bankroll), I don't see how you can apply it.

Still, nice post Ben. Now I see why you like to play under-rolled, so your bet size can approach the proper size in relation to your roll for maximal growth.

taking shots/bankroll management, but that's an area i don't have as much empirical data for, but enough to say that we are generally too conservative.

ill comment more on this later during MNF, college basketball openers coming out right now.

i really dont know too much about sports but i had fun last year betting on them

my system was id look on pinacle sports online since they have millions of bettors and id get a good gauge of what the line should be

id then go to the stratosphere and fitzgeralds since they offered some of the best lines in las vegas

if the lines differed from pinnacle sports id consider betting on them to whichever way they favored like if pinnacle was -3 on a line and stratosphere was -2 id consider the bet at -2

my dad watches a lot of sports and is up to date on whos injured and whatnot so id then confirm with him do you like team X vs team Y with this line?

if he said yes id then place the bet

wow. nice post.

Good, this makes me feel that my aggressive approach to bankroll is better than most peoples

good post, read it all[even the wiki].

I have been playing a little more conservative lately, a few adjustments are in order. Although my shot taking gets me in trouble, but its kinda what got me here in the first place.

Ben--

Awesome post. There will be haters--if not explicitly in this thread, implicitly in later threads--and people will extract arguments from thin air to try to argue against you (again, possibly only implicitly so in later threads), but amazing post.

Joker--

Where else does this apply? Bluffing the river comes to mind. Defending blinds more loosely against tough players does too.

--Nate

Isn't the example in the OP basically just a pokerstove exercise? I mean, usually the reason people don't raise preflop there is because either a) they don't know they have an equity edge, or b) they think being OOP postflop chews up that edge... not because they are playing under-rolled....

Ben,

Nice post, but I have to take issue with a couple of the points.

Although "antediluvian" is one of the most kick-ass words I have ever seen on 2p2, the Kelly criterion here doesn't invalidate the consensus bankroll requirements. Unfortunately, we can't say "I'm gonna play hold-em where I get dealt QTs in the bb and have 4 limping opponents...what limit should I choose?"--rather, we have to base our limit choice on the OVERALL edge we have in the game, not the one we have in one particular situation with one particular hand.

For example, an often cited edge around here is 2 BB/100 hands. To have the privilege of playing those hands, we have to bet about 9.375 BB--our blinds, assuming the table averages 8 players/hand. (Our edge comes from judiciously putting in bets when it is +EV to do so, and not putting in bets otherwise.) In other words, we are getting 11.375 to 9.375 on each hundred-hand bet.

With that assumption, I played around on the Kelly calculator and found that if you have a 45.4% chance of winning the hundred-hand bet, then the BR would need to be about 300 times the Kelly criterion...that sounds about right to me. Perhaps there is someone out there who has a dataset they could break into hundred hand chunks and see how often those are won.

Also,

I don't quite agree with this...any bet that is +EV will, in the long run, lead to bankroll growth. It IS true, however, that a non-maximal EV bet will not be maximally EG, which is what I think you are intending to point out here.

(and of course, you are correct in pointing out that this had is a raise, metagame considerations aside.)

Again, great post. Maybe a candidate for a magazine article?

--Dan

TYVM for all that work. If it wasn't for this forum I'd never be exposed to these concepts.

I was thinking the same thing.

Imagine a new poster who comes to us tomorrow. He plays perfectly, except for the fact that he always smooth calls his aces preflop. When we tell him to raise preflop and all the reasons for it, he agrees, but then responds that if he does this, he will feel like he needs to disguise his hand and thus check behind the flop and turn. What would we tell such a poster?

Put another way, avoiding a +EV play because it necessitates a -EV one can be solved by learning to play both situations correctly.

i would say that about a third of my total income comes from this exact process.

There's alot more I want to comment on in terms of playing higher limits faster, but after becoming alot more data-oriented in the last year, I hesitate to jump to any conclusions. However, knowing how aggressive optimal bankroll management is, what you guys are implying is dead on. GAMBOL!

Yeah, it pretty much confirms what all good value nazis/interwebs players already know(which doesn't surprise me that you two are impressed) - it's like when scientists report on something that's already common sense. We've debated these threads before though, and the conclusion is always drawn that because of the postflop considerations and other arguments I think hold little water, that checking, if a mistake is a small one. I think it's clear the evidence in OP shows how grand of a mistake it is. I do think it's worth quantifying our plays instead of just evaluating them as good or bad though.

I think your missing a few very important concepts in limit holdem.

We are out of position with a mediocre holding. Our advantage in this position is to control the pot size and have a check-raise at our disposal. If we increase the size of the pot we are giving more correct odds to draw against us when we flop a mediocre hand(top pair tens for example).

When we make a mediocre holding with this hand a preflop raise would diminsih our chances of wining in a loose limit holdem game.

Not to self. Stop being a bet monkey.

Wow, never thought of this.

gee, and here I'd raise QTs just because its fun! ... and it tilts everyone when I showdown what I raised out of the blinds ... and of course it makes me money.

Now I'm going to parrot other posters. I think bringing Kelly into the equation is not needed. We have a straight value raise here end of story. Unlike sports betting we must decide bet sizes before sitting down; not every hand is going to give us the same edge. Moving down is also not always an option.