What is the best way to bet in a +EV game ?

Say I play a game where I have a 10% edge. - What happens if:

a) after a lose I double my initial stake, and after that second game (win or lose) I drop back to the initial stake for the next game

b) after a win I double my initial stake, and after that second game (win or lose) I drop back to the initial stake for the next game

c) I bet same stake all the way.. - I guess all are equal in the end, but a) and b) will only make the variance to be really doubled ?

It's not all equal in the end. You are betting more with a or b than you are with c.

The exact numbers depend on your hit rate and variance. There is more than one way to have a 10% edge. Are you flipping fair coins and getting paid 6:5 on wins? Or are you getting paid 1:1 but have a 55% chance of winning? Or are you getting paid 10:1 on a bet that you win 10% of the time?

These all have 10% edges, but the numbers work out differently. This is why you need to divide edge by variance when applying the Kelly Criterion. (the lower the variance, the better for you, obviously)

Is about an online skill game, where I am paid 1:1 but I have a 55% win chance. Weird that seems with a) works better than with b)

Quote: PlayHunter

Weird that seems with a) works better than with b)

Is your probability of winning an event related to either the outcome on the previous event or the amount of the bet or a relationship between bet amounts? If any of those were the case, it could explain the difference. (I have not attempted an analysis.)

(b) will get the most money on the table and therefore result in the largest long-term win out of the 3 (assuming that the higher variance does not cause you to go broke).

But, these are all terrible choices. Figure out what your bankroll is (if it's replenishable, and you don't mind topping up, then figure out a good "effective bankroll" -- ie, how much you are willing to invest in this venture over the long term) and use Kelly betting. Your edge is 10%, as you pointed out. The variance is (0.55)(0.9^2)+(0.45)(1.1^2) = 0.99. The Kelly amount to bet is 10.1% of your bankroll on each bet (recompute your bankroll after each bet). It will be a wild ride (random walks are like that) but it will maximize your long term bankroll growth.

If you can make this bet very often, and continue to have the same edge as you go up in stakes, you will be very rich relatively quickly. I hope the maximums are high!

Although if you're playing against a live opponent, you may want to take into account that that opponent may not want to play lower or higher stakes. In that case, and b/c I'm generally risk averse as compared to many APs, I would not bet 10.1%, but closer to 2-5% depending on the specifics. (Of the gamble.)

Well, if you are playing against a live person, sure, you need to bet whatever the sucker wants to bet, not whatever you want. It's not clear what your ceiling should be, but I would suggest not going above 2x Kelly.

I'm not sure why you wouldn't go all the way up to full Kelly if you get to choose the bet size. I don't really understand the "risk averse" argument here. I'd argue that the risk is higher (money not won is the same as money lost). This all assumes that you resize your bets after each resolution, of course.

I meant that any significant downswing may prevent you from playing if you use full kelly. (Assuming that your opponent only wants to play at X stakes and up) There's really not enough info given, though.

I tend to be risk averse. It's most certainly not more risky to bet less, if you're always playing at an advantage. That's not necessarily true of BJ, where you don't always play at an advantage. Obviously, if you don't spread your bets large enough, for example, your RoR is 100%, but if you bet to big your RoR is also 100%. Kelly is optimal, but I tend to go for slightly less risk even if it costs me long term growth. IMO, with my current bankroll that I don't want to lose, it makes more sense to bet in a risk averse manner. But half kelly has a much smaller risk than fully kelly. (With a big enough bankroll, you can realistically deal with full kelly since you'll basically never drop down so far that you "feel ruined".)

Smaller bankrolls, IMO, are much better off using something like 2/3 kelly or 1/2 kelly. Of course it depends completely on the gamble. For BJ I advocate around 2/3 kelly, or less if you have a small bankroll you don't want to risk losing.

For a low variance game with a 10% advantage, I'd bet half kelly if I could pick my own bet sizes constantly. If I relied on playing a certain stake, I'd like to start with at least double the necessary kelly bankroll. I guess it's not necessary, but I'd like to be able to keep betting even if I lost half my bankroll.

Quote: Boney526

I meant that any significant downswing may prevent you from playing if you use full kelly. (Assuming that your opponent only wants to play at X stakes and up) There's really not enough info given, though.

This is a really interesting point. Ok, so, I have a question. Say you have an edge in a game with a high minimum relative to your bankroll (perhaps because your opponent will only play for certain stakes, or because the flashing dealer only works in the uber high limit room of the uber high limit casino which has $5000 minimums, or something). If you drop below the minimum bet, you can't play any more. How should you size your bets?

I think that the answer is, you treat your bankroll as (bankroll - min bet) and apply Kelly to that number (call the result k). Then:

If k < min/2, don't play
If min/2 < k < min, bet the minimum.
If min < k, bet k.

But that is just off the top of my head. I haven't tried to prove it yet.

Quote:

I tend to be risk averse. It's most certainly not more risky to bet less, if you're always playing at an advantage.

The problem here is that we haven't defined "risky". Risk of what, exactly?

Quote:

That's not necessarily true of BJ, where you don't always play at an advantage. Obviously, if you don't spread your bets large enough, for example, your RoR is 100%, but if you bet to big your RoR is also 100%.

RoR is usually defined as your risk of going broke before you double up. Betting too big will never send it to 100%. You can bet your whole BR on one hand and it will never go higher than the probability of losing that one hand (around 60% in blackjack?)

While I think that measure is useful if you are playing blackjack as a business where you have backers and a defined beginning and ending (they give you money; you play until you either double up or lose it) I don't think it's particularly useful if you have an actual bankroll (your own money that you're going to play with forever). I think that the standard Kelly betting for the first case gives a RoR of 13% or something (from memory; possibly inaccurate; I'd have to check one of my books to be sure) This assumes no resizing of bets, though. If you are playing with a real bankroll you definitely want to be resizing bets.

Quote:

Kelly is optimal, but I tend to go for slightly less risk even if it costs me long term growth. IMO, with my current bankroll that I don't want to lose, it makes more sense to bet in a risk averse manner. But half kelly has a much smaller risk than fully kelly. (With a big enough bankroll, you can realistically deal with full kelly since you'll basically never drop down so far that you "feel ruined".)

Again, I don't understand what you are saying. Risk of what? See above. With bet resizing, you are extremely unlikely to ever go broke.

Quote:

Smaller bankrolls, IMO, are much better off using something like 2/3 kelly or 1/2 kelly. Of course it depends completely on the gamble. For BJ I advocate around 2/3 kelly, or less if you have a small bankroll you don't want to risk losing.

For a low variance game with a 10% advantage, I'd bet half kelly if I could pick my own bet sizes constantly. If I relied on playing a certain stake, I'd like to start with at least double the necessary kelly bankroll. I guess it's not necessary, but I'd like to be able to keep betting even if I lost half my bankroll.

I think we are using the terms "bankroll" to mean different things. I use it to mean, you are a professional gambler (it's your only source of income). You have a certain amount of money with which to play. If you lose it all, you are bankrupt. Even if you are not a professional gambler, you can count all your liquid assets, and also use Kelly to determine things like how much you should be willing to invest in things like the stock or bond market (remember that investing is no different from gambling!)

I suspect (and, please, correct me if I'm wrong!) that you mean, you start with a certain amount of money that you are willing to devote to that game, and you are going to stop when you lose it all or when you hit some set win amount. Then, you will start over with a new "bankroll" (possibly of a different size) and do it all over again. I don't think that this is a great way to approach it, mostly because you lose a lot by not resizing your bets correctly.

Quote: AxiomOfChoice

you start with a certain amount of money that you are willing to devote to that game, and you are going to stop when you lose it all ... Then, you will start over with a new "bankroll" (possibly of a different size) and do it all over again.

Such a person actually has a larger bankroll than he will admit to himself.

Seems to me one risk here is very large: the risk that you will get no more action! Such a sweet deal is almost guaranteed to vanish and only the lucky indeed see continued action for any length of time. Does that mean you should bet larger than Kelly? I would certainly push the limit for that kind of action if I could get it.

You mean, you think it will be your last +EV opportunity ever? How horrible!!! That would give me nightmares! :)

Assuming that you will get more +EV opportunities in the future, I think you are always better betting Kelly. If you bet too much, then you are risking losing "too much" which will cause you to not be able to bet enough at the next +EV opportunity. The point is, you need to balance this risk with the opportunity (betting more = more EV, but also betting more = not being able to bet as much next time if you lose). And that balance is hit right at the Kelly number.

I had this long post written out answering all your questions but it got lost. I'm gonna re do most of it soon, but I've got some stuff to do first.

Doc, no the probability of winning is per each game in part and should not be dependent of the previous stake. Say the game in cause is a Backgammon game of 1 point. Whoever takes all checkers off the table first, wins. I have a group of few people which I know (after many game analyses against each one in part) that I have a 55% win chances when I play a game with them. All play at the same skill level, so I have the same edge over each one in part. Some of them are willing to apply for a) and some others are willing to apply for b) and of course there are some others which would like continue playing on the same stake. Seems A wins more than B.

Quote: AxiomOfChoice

This is a really interesting point. Ok, so, I have a question. Say you have an edge in a game with a high minimum relative to your bankroll (perhaps because your opponent will only play for certain stakes, or because the flashing dealer only works in the uber high limit room of the uber high limit casino which has $5000 minimums, or something). If you drop below the minimum bet, you can't play any more. How should you size your bets?

I think that the answer is, you treat your bankroll as (bankroll - min bet) and apply Kelly to that number (call the result k). Then:

If k < min/2, don't play
If min/2 < k < min, bet the minimum.
If min < k, bet k.

But that is just off the top of my head. I haven't tried to prove it yet.

Seems like a reasonable answer. I'm not really sure, myself. I usually won't play near the level that's above my bankroll, but I have a couple of times. So I guess I'm not always risk averse, it just depends on a lot of different variables.

Quote:

The problem here is that we haven't defined "risky". Risk of what, exactly?

Risk tolerance is dependent on the person's tolerance for variance and their ability to keep playing after experiencing negative variance. EG: A card counter that barely meets the bankroll requirments to beat the game with a $5-60 spread may not be able to find a beatable $3 game if he sees a downswing. So that's a risk. Some people can't deal with losing a good chunk of their bankroll over two games. I understand that this player won't be capitalizing on all of the potential for growth. Sometimes, that's OK.

Quote:

RoR is usually defined as your risk of going broke before you double up. Betting too big will never send it to 100%. You can bet your whole BR on one hand and it will never go higher than the probability of losing that one hand (around 60% in blackjack?)

I didn't know that. I thought RoR was defined as the probability of losing your stake before bringing RoR (to some predetermined point) very close to 0%. In that case, betting far more than called for at a +EV would send your "RoR" to nearly 100%, but I got the definition wrong.

Quote:

While I think that measure is useful if you are playing blackjack as a business where you have backers and a defined beginning and ending (they give you money; you play until you either double up or lose it) I don't think it's particularly useful if you have an actual bankroll (your own money that you're going to play with forever). I think that the standard Kelly betting for the first case gives a RoR of 13% or something (from memory; possibly inaccurate; I'd have to check one of my books to be sure) This assumes no resizing of bets, though. If you are playing with a real bankroll you definitely want to be resizing bets.


Again, I don't understand what you are saying. Risk of what? See above. With bet resizing, you are extremely unlikely to ever go broke.

The primary factor that I think could influence me to be risk averse is that a small bankroll may not be able to handle 2 or 3 large downswings. I realize that's not big enough for professional play.

Quote:

I think we are using the terms "bankroll" to mean different things. I use it to mean, you are a professional gambler (it's your only source of income). You have a certain amount of money with which to play. If you lose it all, you are bankrupt. Even if you are not a professional gambler, you can count all your liquid assets, and also use Kelly to determine things like how much you should be willing to invest in things like the stock or bond market (remember that investing is no different from gambling!)

Well, I don't think it's necessary for it to be your only source of income for you to be a professional, I would argue that you just need consistent profits realized. Anyway, I think that Kelly is useful, and is especially useful as a benchmark of potential growth and risk aversion. You shouldn't overbet. You especially shouldn't bet double kelly. But if you'r willing to bet 150% of Kelly, you should prefer 50%, because it has the same long term growth but the latter has less risk. (I read that somewhere, so correct me if that's not true.) But in some forms of gambling, using Kelly is more dangerous than it seems. Poker is a good example, because a winning player may start keeping records and hit a long winning streak for above his what his average BB/100 should be, and if he used that number to estimate what his buy ins should be, his actual RoR would be pretty high (even with re sizing.) A player beating the game for 4BB/100 hands thinking he's winning at 10BB/100 could be disastrous, and is in the realm of possibility. In games of skill that don't have their edges a quantifiable as a player edge in BJ, I usually advocate risk aversion until you figure out a reason to take on more risk. Maybe there's really bad players at another table, which you have 18 buy ins for rather than your usual 25.

I started a small bankroll that I use for 25/50 cent NLHE home ring games. I strongly suspect I'm playing at an advantage, although I'm not entirely certain b/c my sample size isn't statistically significant. The games are pretty competitive, but have occasional badly losing players buy in for 20 once or twice. In this case, it's really hard to determine your advantage and variance, so it's better to have a relatively large amount of buy ins. Many suggest a (10, 20, 25 or 40) buy ins for NLHE. I argue that it's so dependent on the table and how you plan to play that you can't think of it as that simple. For example, I had a period with absolutely no games going for a few months at 25/50 and decided to take a shot at 1-2, knowing that even if I lost, if the 25/50 game ran at any point I'd still have more than ample buy ins for that level. I'd prefer to play 25/50 3 nights a week so I could grind my way up to a higher level with a very, very, very minuscule RoR assuming I'm winning, but I have to adapt to the environment around me.

Quote:

I suspect (and, please, correct me if I'm wrong!) that you mean, you start with a certain amount of money that you are willing to devote to that game, and you are going to stop when you lose it all or when you hit some set win amount. Then, you will start over with a new "bankroll" (possibly of a different size) and do it all over again. I don't think that this is a great way to approach it, mostly because you lose a lot by not resizing your bets correctly.

No, I only keep one gambling bankroll, which I've never lost and don't intend to, but it's significantly smaller than most people's bankrolls. That's why I don't play much in casinos. My bankroll won't support higher stakes. ATM it won't even support 1/2 if I have a downswing. But I also want to start small so I can build the experience cheaply, if I am losing. The BJ references I make are purely theoretical, because I don't count cards in casinos, I learned to do so as a hobby at home. I actually haven't practiced in about 1 month and a half. If I ever really wanted to, I could devote the time to getting back up to speed - but I really don't think I will. I enjoy poker way more.

Quote: PlayHunter

Doc, no the probability of winning is per each game in part and should not be dependent of the previous stake. Say the game in cause is a Backgammon game of 1 point. Whoever takes all checkers off the table first, wins. I have a group of few people which I know (after many game analyses against each one in part) that I have a 55% win chances when I play a game with them. All play at the same skill level, so I have the same edge over each one in part. Some of them are willing to apply for a) and some others are willing to apply for b) and of course there are some others which would like continue playing on the same stake. Seems A wins more than B.

With this info, I agree with a previous post that the best way to play is
B)
as you will ultimately get more action. You will win more frequently due to your advantage, which will cause you to play for doubled stakes more often. More action at an advantage equals highest profit.

This assumes that you have a negligible RoR, which if you are playing online backgammon I'm sure is fine, as I hope you are not betting more than 10% of your stake at once.

Thank you guys, fascinated by the posts ! Dwheatley I usually play 5% of my bankroll and 10% would be when the stake is doubled.

I see the point of why to play B. I started with this. But then I tried A. And seems that with A, I win more. (more constantly at least)