Question for YOU on Kelly and Bet Sizing

[GameTheory]

Originally Posted by Robert Fischer

Seems to be a consesnsus that the Law of Total Probability says that we can in fact basically "average" the win% to come up with an "overall probability".

So, using Kelly for:
Win Percentage: 45% & Odds at 3/2 odds for ALL bets

should give the same results as If we had:

Win Percentage: 50% & Odds at 3/2 odds for 9/10 of bets, and then Win Percentage = 0% at 3/2 Odds for 1/10 of bets ?

Is there a way to simulate these two conditions separately, for a hard-head such as myself?

You make 100 bets. 90 of them with the normal coin, and win 50% of those = 90 * .5 = 45. Those are the only winners you get since the other 10 have no chance, so 45 out of 100 is your probability of winning

.9*.5 + .1*0 = .45

[Dave Schwartz]

Why would you simulate separately if you cannot tell them apart before placing your wager?

That is kind of like saying that some horse races are lost. I would like to divide the races into those I lose and those I win and bet them separately.

I would venture a guess that in the races/coin tosses that are lost the optimum bet would be zero.

[Robert Fischer]

Originally Posted by Dave Schwartz

Why would you simulate separately if you cannot tell them apart before placing your wager?

That is kind of like saying that some horse races are lost. I would like to divide the races into those I lose and those I win and bet them separately.

I would venture a guess that in the races/coin tosses that are lost the optimum bet would be zero.

Hey, I wish!

The mental-block for me is understanding that a simulation with a "bad-apple", will grow my bankroll just the same as a "smooth", homogeneous, 45%@3-2 simulation.

So i wouldn't want to simulate the coins separately.
What my question is about is whether we can just "average" our hit% and edge using Kelly, and obtain the same growth and risk.

You are right, I can't tell them apart before placing the wager.

So maybe my perspective is simply the issue?

After all, a 50% winner, loses half the time anyway, "overall".

[Robert Fischer]

Originally Posted by GameTheory

I'm assuming you must make your bet before the coin is drawn (otherwise this exercise makes no sense). I'm also assuming the the coins are replaced after you drawn them and can be drawn again, so the percentages stay fixed. Then you just have to use the information you have at that time of your bet, which is that you have a 45% chance of winning your bet at odds of 3-2. The chance of getting the tails coin is built-in to the win percentage, and so it is fairly simple and you bet accordingly.

That makes sense.

Thanks to everyone who contributed here or via PM.


Now to apply this principle to my game...
It is pretty much straight forward in concept. If you believe that this is a real phenomenon (occasionally getting a "dud" horse) then you can simply lower you estimate for hit%.

The fact that I can now see that it all comes together in the 'Law of Total Probability' to be 'built in' to the win percentage, simplifies things quite a bit.

One aspect is that occasionally (perhaps less frequently?) you will have horses that have more ability or value than you estimated. Now that I understand that the occasional 'dud' horses don't greatly increase your chance of ruin (provided you adjust your hit% estimate down) it looks a lot better.

In general I try to be conservative.
I make a low estimate for final odds, and a low estimate for hit percentage. I already touched on the concept of of reaching the pool-size ceiling, and how I believe that provides some incentive to be cautious.

[GameTheory]

And at this point I will repeat that you DO NOT want to use average win percentage and average odds to plug-in to a Kelly formula for use with horse racing. In your example the odds were FIXED at 3-2. Horse racing payoffs are not fixed. Neither are winning chances (they are heavily correlated with the odds). Kelly bet percentages should be computed PER RACE (per bet, really) based on the information you have for THAT RACE (for that particular bet). If you average, you will get the wrong answer.

[Robert Fischer]

Originally Posted by GameTheory

And at this point I will repeat that you DO NOT want to use average win percentage and average odds to plug-in to a Kelly formula for use with horse racing. In your example the odds were FIXED at 3-2. Horse racing payoffs are not fixed. Neither are winning chances (they are heavily correlated with the odds). Kelly bet percentages should be computed PER RACE (per bet, really) based on the information you have for THAT RACE (for that particular bet). If you average, you will get the wrong answer.

I agree with the point you are making.

I actually break down my bread-and-butter plays into "wager types", and calculate a fractional Kelly based on a low-estimate for the hit% and payoffs for several common sub-types, rather than actually calculating on the fly.

So overall average isn't a big concern to me, but it was important in being able to illustrate and grasp the concept in this thread, - about the occasional under-performing horse that can occur within a long term profitable sequence, and how exactly it actually impacts the entire system.

[Dave Schwartz]

Now to apply this principle to my game...
It is pretty much straight forward in concept. If you believe that this is a real phenomenon (occasionally getting a "dud" horse) then you can simply lower you estimate for hit%

You already did that when you reduced the win probability to 45%.


Everything GT has said is right on the money. The guy knows his stuff.

[GameTheory]

Originally Posted by Robert Fischer

I agree with the point you are making.

I actually break down my bread-and-butter plays into "wager types", and calculate a fractional Kelly based on a low-estimate for the hit% and payoffs for several common sub-types, rather than actually calculating on the fly.

So overall average isn't a big concern to me, but it was important in being able to illustrate and grasp the concept in this thread, - about the occasional under-performing horse that can occur within a long term profitable sequence, and how exactly it actually impacts the entire system.

Yes, well overall average or average of subtype, it will still be wrong if you have varying payoffs which we always do. (Not every winning bet pays the exact same amount.) So you do have to "estimate low" as the average odds will always be too high.

There actually is no easy formula for this (I don't think, it has been a while since I worked on this) -- you have to find the optimum Kelly percentage in the face of varying payoffs using iterative search/optimization. What you want to do is maximize the geometric mean return -- that is generalized point behind what Kelly is -- finding the percentage of bankroll to bet that will maximize the geometric mean of your returns will grow your bankroll the most in the long run. And here's another interesting tidbit about this: the "long run" means an infinite time series, but of course you won't have an infinite number of bets. If you choose some finite number of bets you are going to make (even something as low as 20 or 50 bets if you are doing "session play" like Dave likes), then that changes the results a bit and you'll actually get a different optimum bet percentage (again, if memory serves, I have to dig up my code for this) -- I think it becomes more aggressive, i.e. somewhat bigger bets.

And then there is "multiple-Kelly" where you are betting 2 or 3 things in a single race and want to find the optimum percentage of each. Again not easy because you can't just compute the single optimum percentage of each favorable wager you are going to make -- you must compute the whole batch simultaneously as one big bet but with multiple parts. (Although there is a simplified method that computes them one at a time which is less computationally demanding but is slightly less accurate than computing them all at once as a multi-dimensional optimization problem. In years past that was needed as the computers weren't powerful enough but modern machines can do it the better way without much trouble.)

[Robert Fischer]

Originally Posted by GameTheory

...
-- you have to find the optimum Kelly percentage in the face of varying payoffs using iterative search/optimization. What you want to do is maximize the geometric mean return -- that is generalized point behind what Kelly is -- finding the percentage of bankroll to bet that will maximize the geometric mean of your returns will grow your bankroll the most in the long run. And here's another interesting tidbit about this: the "long run" means an infinite time series, but of course you won't have an infinite number of bets. If you choose some finite number of bets you are going to make (even something as low as 20 or 50 bets if you are doing "session play" like Dave likes), then that changes the results a bit and you'll actually get a different optimum bet percentage (again, if memory serves, I have to dig up my code for this) -- I think it becomes more aggressive, i.e. somewhat bigger bets.

And then there is "multiple-Kelly" where you are betting 2 or 3 things in a single race and want to find the optimum percentage of each. Again not easy because you can't just compute the single optimum percentage of each favorable wager you are going to make -- you must compute the whole batch simultaneously as one big bet but with multiple parts. (Although there is a simplified method that computes them one at a time which is less computationally demanding but is slightly less accurate than computing them all at once as a multi-dimensional optimization problem. In years past that was needed as the computers weren't powerful enough but modern machines can do it the better way without much trouble.)

That is interesting.

I wonder if it could provide a bit of an extra edge in certain horse racing markets?


I think with horse racing, you basically want a bet-size that is:
1. A percent of your bankroll, so that you grow exponentially.
2. Low enough to minimize risk of ruin.
3. High enough that you aren't needlessly leaving a lot of money on the table.

Kelly, and your basic understanding of the game, should be combined to find a percent of bankroll that is a safe, relatively efficient 'ballpark' estimate for a given wager or subset of wagers.

[jfdinneen]

Robert,

As you already know, many mathematical problems yield to both an analytical and a numerical solution.

In this instance, the numerical solution (using Solver in Excel) to minimizing the difference between expected value and volatility drag over a sequence of similar bets equals the analytical solution (using Kelly) for the same sequence!

John

[Robert Fischer]

Originally Posted by jfdinneen

Robert,

As you already know, many mathematical problems yield to both an analytical and a numerical solution.

In this instance, the numerical solution (using Solver in Excel) to minimizing the difference between expected value and volatility drag over a sequence of similar bets equals the analytical solution (using Kelly) for the same sequence!

John

Very Cool.

[Aaron Brown]

MJC922 and others are correct that in theory you can regard this as a bet where 45% of the time you win $1.50, and 55% of the time you lose $1. Kelly tells you to bet 1/12th of your bankroll on this.

On the other hand, you are right to be suspicious of that conclusion. It relies heavily on the Kelly assumptions being exactly true, and they are never exactly true.

Consider the following application. You develop a system that can identify horses that go off at 3-2 on average, and win 45% of the time. However, what really is going on is your horses should win 50% of the time, but 10% of the time their trainers hold them back.

You might say you don't care, Kelly only looks at the unconditional probability of winning, which is 45%. But in real life although you put your bet on when the odds are 3-2, you actually get 2-1 when the fix is in, and an average of an average of 9-4 the rest of the time. Although your average payout is 3-2, both when you put your bet on and at post time, you wins pay off only at an average of 9-4.

If anyone else knows which coin is drawn, they are likely to exploit the knowledge in a way that hurts you, so using the unconditional probability of winning can be dangerous.

Another practical example of where this matters is when probability of the bad coin only averages 10%, maybe you go through periods where it's 20% and periods where it's 0%. That would change your optimal betting, and make it more complicated.

In general, unconditional probabilities are dangerous and you should prefer to condition as much as possible in bankroll and optimal betting situations.

[lansdale]

Originally Posted by Aaron Brown

MJC922 and others are correct that in theory you can regard this as a bet where 45% of the time you win $1.50, and 55% of the time you lose $1. Kelly tells you to bet 1/12th of your bankroll on this.

On the other hand, you are right to be suspicious of that conclusion. It relies heavily on the Kelly assumptions being exactly true, and they are never exactly true.

Consider the following application. You develop a system that can identify horses that go off at 3-2 on average, and win 45% of the time. However, what really is going on is your horses should win 50% of the time, but 10% of the time their trainers hold them back.

You might say you don't care, Kelly only looks at the unconditional probability of winning, which is 45%. But in real life although you put your bet on when the odds are 3-2, you actually get 2-1 when the fix is in, and an average of an average of 9-4 the rest of the time. Although your average payout is 3-2, both when you put your bet on and at post time, you wins pay off only at an average of 9-4.

If anyone else knows which coin is drawn, they are likely to exploit the knowledge in a way that hurts you, so using the unconditional probability of winning can be dangerous.

Another practical example of where this matters is when probability of the bad coin only averages 10%, maybe you go through periods where it's 20% and periods where it's 0%. That would change your optimal betting, and make it more complicated.

In general, unconditional probabilities are dangerous and you should prefer to condition as much as possible in bankroll and optimal betting situations.

You seem to confuse a bet which has a strictly determined mathematical outcome, where the bet-size is easy to determine, like the one described by RF, and a horserace bet, where most bettors have, at best a subjective and usually overly optimistic estimate of their advantage. The former, like blackjack or vide-poker, is suitable for Kelly betting, the latter, for most bettors, is not. Kelly is a very self-destructive method for anyone playing either without an edge or without a very accurate per-race estimate of their edge, assuming they have one.

[Robert Fischer]

Originally Posted by Aaron Brown

MJC922 and others are correct that in theory you can regard this as a bet where 45% of the time you win $1.50, and 55% of the time you lose $1. Kelly tells you to bet 1/12th of your bankroll on this.

On the other hand, you are right to be suspicious of that conclusion. It relies heavily on the Kelly assumptions being exactly true, and they are never exactly true.

Consider the following application. You develop a system that can identify horses that go off at 3-2 on average, and win 45% of the time. However, what really is going on is your horses should win 50% of the time, but 10% of the time their trainers hold them back.

You might say you don't care, Kelly only looks at the unconditional probability of winning, which is 45%. But in real life although you put your bet on when the odds are 3-2, you actually get 2-1 when the fix is in, and an average of an average of 9-4 the rest of the time. Although your average payout is 3-2, both when you put your bet on and at post time, you wins pay off only at an average of 9-4.

If anyone else knows which coin is drawn, they are likely to exploit the knowledge in a way that hurts you, so using the unconditional probability of winning can be dangerous.

Another practical example of where this matters is when probability of the bad coin only averages 10%, maybe you go through periods where it's 20% and periods where it's 0%. That would change your optimal betting, and make it more complicated.

In general, unconditional probabilities are dangerous and you should prefer to condition as much as possible in bankroll and optimal betting situations.

Thanks for the response, and welcome to PA!

That is a problem/challenge regarding horseracing betting markets - we have to estimate the final odds. And what you imply is also correct - 'inside info' can often be reflected in lower-priced winners, and higher-priced losers within a given set of wager-types.

The player has to make an estimate for how he believes the crowd (market) will behave in general, and for the horses he is interested in. This is separate from who he the player, believes actually has the best chances.

When the crowd behaves differently than expected that can be a sign of randomness, or it can be a sign that you don't have insight into all of the important information.

Here is a post that applies some of that in an example.

[thaskalos]

Originally Posted by Aaron Brown

MJC922 and others are correct that in theory you can regard this as a bet where 45% of the time you win $1.50, and 55% of the time you lose $1. Kelly tells you to bet 1/12th of your bankroll on this.

On the other hand, you are right to be suspicious of that conclusion. It relies heavily on the Kelly assumptions being exactly true, and they are never exactly true.

Consider the following application. You develop a system that can identify horses that go off at 3-2 on average, and win 45% of the time. However, what really is going on is your horses should win 50% of the time, but 10% of the time their trainers hold them back.

You might say you don't care, Kelly only looks at the unconditional probability of winning, which is 45%. But in real life although you put your bet on when the odds are 3-2, you actually get 2-1 when the fix is in, and an average of an average of 9-4 the rest of the time. Although your average payout is 3-2, both when you put your bet on and at post time, you wins pay off only at an average of 9-4.

If anyone else knows which coin is drawn, they are likely to exploit the knowledge in a way that hurts you, so using the unconditional probability of winning can be dangerous.

Another practical example of where this matters is when probability of the bad coin only averages 10%, maybe you go through periods where it's 20% and periods where it's 0%. That would change your optimal betting, and make it more complicated.

In general, unconditional probabilities are dangerous and you should prefer to condition as much as possible in bankroll and optimal betting situations.

Don't tell me you are "THE" Aaron Brown. If you are, then I'm a big fan; of your books AND of your book reviews.