This is the standard said by many especially using the kelly staking, but I'm not sure I agree.


I'm talking about sports betting and other gambling codes where the probability is not fixed like in poker when cards are face up and displayed and probabilities can be calculated, using kelly staking its simply a opinion for example I think this team is even money but they are offering 2-1 so bet big. Yeah I agree with this, but not betting smaller when the edge is smaller.

A bet is either profitable or not, so why decrease bet size if the bet is profitable. Some bets are going to returns bigger ROI's say 10% while other types of bets smaller ROI's say 4%, so why bet smaller on the smaller returning bets?

Yes because its a smaller ROI and those bets are returning a smaller ROI, but then why are you betting that bet at in the first place?

I'd rather grow my money than live in a Sklansky-mansion inside a homeless shelter (which is where EV-maximization would lead). And I'd rather grow it at a faster rate than a slower rate. Kelly maximizes expected growth, which to me is better than maximizing EV. It leads to the fastest exponential growth rate (as calculus shows).

It's not about the edge, it's mostly about the likelihood of winning. If you have a 10% edge on a 1:1 payout, Kelly says to bet bigger on that than if you have a 20% edge with only a 1% chance of winning.

Not necessarily. Kelly says that the optimum bet is bigger. But if the house limit is smaller than the optimum bet size, you should bet that house limit not only with your biggest edges but with somewhat smaller ones as well.

Agreed

I didn't quite understand your reply.

Why bet smaller when the edge is smaller? say compared to betting 2% or 4% of bankroll.

If kelly says to bet less then 2% say for this example.

If the bet is profitable why not maximum bet it, either by using kelly staking or at least 2% of bankroll, if kelly is saying to bet less then 2% isn't this leaving money on the table because 2% of bankroll isn't over betting. And if the bet is profitable then why bet smaller then 2%?

Also how do you have a 20% edge but 1% chance of winning? to have edge means the probability is greater the the odds offered.

So if you bet 2/3 of the numbers on a Roulette wheel, then you have the edge over the house? No, edge is the % of a dollar you win per dollar wagered. In Roulette, you lose 5.26 cents per $1 wagered, so the edge is -5.26%. In my hypothetical game where you have a 20% edge but 1% chance of winning, that means the payout is big enough to make that true.

Huh? You just said the Kelly stake was less than 2%

If 2% is higher than Kelly, then it is over-betting. If kelly is 1% and you bet 2%, you're actually over-betting by so much that you'll go broke if you keep doing that. If Kelly is 1.5%, you won't go broke but you'll profit at a slower rate than had you only bet 1%, so it's more risk for less reward! That may sound weird, but it's proven by basic calculus.

Not only that, like you said in your OP, when applying this to sports wagers, you never know your chance of winning like you do in say Blackjack, what you calculate to be the Kelly stake is only an estimate. So really you should be betting less than that in case your estimate is too high. If Kelly says 2% then maybe bet 1.5%, or less depending on your margin of error. You wouldn't want to accidentally overbet by too much. And under-betting is always better than over-betting, because you can achieve the same rate of growth (compared to over-betting) but with less risk than full kelly. Over-betting, on the other hand, has no upside whatsoever.

A example

Probability of winning bet 40%
Odds offered $2.55 decimal


With a 40% probability of bet winning the "Max likely loosing run per 1000 bets = 14

So what I am saying is, if we bet 2% of bank which is more then kelly suggests (1.29%) it should not be a overbet, yes to kelly it is but because the risk of ruin calculator says with a 40% probability, -> "likely no more then 14 losing bets in a row", which is easily covered betting only 2% of bank..

which = leaving money on the table 0.71%, which is the difference between 1.29 kelly vs 2% for this bet example.

With a run of out no more then 14 >> betting bets with a 40% probability of winning. this bet could actually wager 14% of bank vs the 1.29% suggested by kelly. But lets not use 14% of bank, but instead 2% vs kelly's 1.29 for this example.

Also regarding your roulette example yes you're correct, but I always assumed edge meant profit. So for example if a bettor says they have a edge vs the bookie for this particular game I assumed edge equalled profit in dollars, not say the roulette example, sure you have a edge but not a profitable edge. Is the word edge used wrong?

A losing streak isn't the only way you can be ruined. "Likely no more than 14 in a row" is a useless stat. If it's a risk of ruin calculator, it should tell you the risk of ruin (the % likelihood of you going broke).

No, what I'm trying to impart to you is that you leave money on the table by betting more than kelly! You're hung up on what happens if you win one wager. I'm talking about what happens in the long run. In the long run, you'll have more money in your account by Kelly betting than by using any other sizing strategy.

Let's take your example and pretend you'll win exactly 400 out of 1000 wagers. You started with $1000.

If you bet 2% each bet, then assuming you don't get wiped out (or pretending there is no minimum wager), you'll end up with the following amount of money regardless of the order of wins and losses.

$1000 * (1 + .02*1.55)^400 * .98^600 = $1094.24

If you bet 1.29% each bet, your ending balance = 1000 * (1 + .0129*1.55)^400 * (1-.0129)^600 = $1137.39

Which one left money on the table?

If you were to only bet 1%, you'd win less than with 1.29% but you'd still win more than you do betting 2% (and with less risk of ruin). You'd win $1130.02, leaving $7.37 on the table.

And if you were to bet 2.59%, you'd be overbetting by so much that you're losing money. You'd wind up with $999.26 after 1000 bets starting with $1000.

Wow!






Thanks for explaining man, much appreciated!

Although your're are right, I'm still confused how one can bet a OVERLAY and not be profitable?

The probability of this bet winning is 40% (2.50) and securing odds of $2.55 is a overlay.

I mean if you get $2.01 about a coin flip which is Even money $2 its profitable, so why is this scenario not profitable since you're getting 2.55 for a 2.50 probability of winning the bet if betting 2% of bank? Level staking would show a profit but if I understand this correctly 2% of bank doesn't

If you're betting a fixed percentage of your bankroll, then the profitability of a series of bets depends on the change in the size of your bankroll. Consider that if you start with a bankroll of 100, betting 10% each bet, then if you lose and then win, you lose 10 and then win 9 (ending down 1). This is called, I think, "volatility drag". If you win and then lose, you win 10 and then lose 11 (also ending down 1)

I don't know if this is right, but my interpretation is that betting "too much" means that bad runs will affect your future profit a lot more due to lowering the amount of future bets you can safely make.

2% would profit, just not the max. Anything higher than 2.58% would lose. Why, because of the volatility drag Rusty mentioned. Betting too big makes that the bigger force. I don't have a good explanation why. It's obvious why you'd lose wagering 100%, but why you would when wagering 2.6% is something different.

Maybe it would be instructive to find an example with easier numbers. Like, let's say you were getting 2:1 on a coin flip.

f = (bp - q) / b = (2*.5 - .5) / 2 = .5/2 = .25

So the kelly fraction says to bet 25%. What happens if you bet 50%?

Starting with 100% units, let's look 3 bets into the future.

WWW: 100 -> 150 -> 225 -> 337.5
WWL: 100 -> 150 -> 225 -> 112.5
WLW: 100 -> 150 -> 75 -> 112.5
WLL: 100 -> 150 -> 75 -> 37.5
LWW: 100 -> 50 -> 75 -> 112.5
LWL: 100 -> 50 -> 75 -> 37.5
LLW: 100 -> 50 -> 25 -> 37.5
LLL: 100 -> 50 -> 25 -> 12.5

Only one of these outcomes is a substantial winner. 3/8 win a little. The rest lose a lot.

Hopefully I haven't messed up the numbers. Getting such a good bet is rare and you'd think you could bet the farm on it, but you'd lose money betting 50% of your roll.

(Interestingly, it seems like the average of these 3-bet runs is breaking even, I wonder if that's a function of the (arbitrary) bet size I chose given the odds/payout?)

If you bet over the kelly max there is too high a chance you go bankrupt, also too high a chance of a large draw down, which impacts future bet sizes/profitability. I always thought betting under kelly max would be ideal, generally because there can be errors in edge calcs

Yeah interesting and explains the bank difference as you say, although up 10% then down 10% you would think square but obviously not due to volatility drag, thanks for explaining.



Yeah I worded it wrong when i said wouldn't profit I understood your previous posts which you explained well, just meant weird that betting a overlay would show a minus but just shows that getting over the odds doesn't always = profit in fact as you say betting 2.59% - $999.26 after a 1000 bets using a original 1k bank.

Thanks again.

yep, thanks.

Nope.

Alright, to be precise, anything higher than 4/155 (2.58064516%). That's double-kelly. If you bet exactly 4/155, you'll break even, except eventually you'll get unlucky and go broke.

No matter how much or how long you bet your EV is positive.

Which has nothing to do with my statement, "You'll lose money." You'll win Sklansky Bucks, but unfortunately nobody accepts those as payment.

If you bet more than 2x kelly in OP's scenario, your expected log growth is negative, which means your bankroll shrinks. I showed how betting 2.59% would lead to a shrunken bankroll after 1000 wagers, and I doubt you disagree with the math I used. After many more wagers, your bankroll would approach zero. To take things to the extreme, try betting 100% of bankroll every bet and see if you don't quickly go broke.

The key is that we're talking about a constant fraction, not a decreasing one. A 3% wager would be $30. If you were to keep betting $30 forever, then you'd make money (edit: probably, but with a nonzero risk of ruin), because eventually $30 would be less than 2f of bankroll.

You should not use the term "lose money". If a googolplex different people all bet triple Kelly, their combined results will certainly be positive until the end of the universe. It is important to make this clear. Otherwise you leave the impression that somehow a set of good bets somehow has the worst of it.

No they won't! When a googolplex different people each suffer bankroll decay, the combined result of that is not a positive bankroll growth. Just like you can't add -EV's to get a +EV, you can't add -EG's to get a +EG.

A -EG bet is not a good bet!

Nope.