what the heck is it? splain please

[sagefr0g]

try this again, lol.....
lol, i don't feel so bad now, others have struggled with this concept:
http://www.bjmath.com/bjmath/loggrow/1225.htm (Archive copy)

ok here's a question, errhh whatever i've never fully understood what the heck does "maximize the expected logarithm of your total bankroll" mean?
what the heck is an expected logarithm of a bankroll?
or is there even such a thing, what's it mean, how's it tick, lol......
or can anyone explain the concept from a layman's perspective?
errh, i sorta half a$$ know what logarithms are.......

from: http://www.bjmath.com/bjmath/kelly/kellyfaq.htm (Archive copy)

Q1: What is wrong with maximizing your expected winnings?

A1: Although at first glance it seems obvious that it is best to maximize the expected (i.e., predicted average) amount of your winnings, this is in fact not true for most people. If this were really your goal then whenever you had the slightest advantage you would mortgage your house, car, and boat and bet your entire fortune. Although this gives you the greatest net win, on average, this is entirely too risky for most people.

Q2: What is the "Kelly Criterion" and what are "Utility Functions"?

A2: The Kelly utility function and other utility functions give a economically justified and mathematically precise way to compute optimal bets that leads to large winnings but limits the total amount of risk. The Kelly criterion dictates that you should try to maximize the expected logarithm of your total bankroll rather than trying to maximize the expected bankroll itself. In other words, you should try to maximize the exponential rate of bankroll growth.

edit: errhh so i think maximizing a exponential rate of anything probably means making it grow at the most accelerated rate possible, ie. exponentially, sorta thing....
so would doing that to an expected bankroll growth mean optimally betting to the advantage you are able to find?, sorta thing?
the right amount to bet for a given advantage all the while taking into consideration the bankroll one has and the risk that entails?

 

[blackjack avenger]

Think Exponential

A Kelly resizing bettor experiences exponential growth. Think compound interest.

What is magical about Kelly theory is it maximizes this exponential growth while not going broke. One can bet more or less but they won't perform as well in the long run.

A critique of kelly resizing is variance; it's a roller coaster, one has a 50% chance of losing half bank. A way around this is to bet less then kelly. I don't know, 1/8th kelly resizing comes to mind for minimizing variance. One has a theoretical; and probable real world, 0% chance of losing half bank.

:joker::whip:
good cards

 

[London Colin]

I tried to get my head around this a while ago. The understanding I reached, which may well be flawed, is as follows.

The goal of the Kelly Criterion is to maximise the expected rate of growth of your bankroll, given that you can re-invest any winnings and increase your bet size as a result, and that conversely you will have to drop your bet size after a loss.

This is the exponential growth which you wish to maximise. You want to find the sweet spot between overbetting and underbetting.

The mathematics of how to maximise this thing throws up a fiendishly complicated formula involving the expected value of the logarithm of growth. But since it is known that if ln(x) > ln(y) then x > y, we can keep things simple by making it our new (equivalent) goal to maximise this new thing instead.

So, if I have understood correctly, logarithms only come into the picture as part of the mathematical journey towards an answer, and it's a bit misleading to mention them upfront as if they were central to the initial goal.

The above could be nonsense, but it's the impression I came away with after much searching of bjmath etc.

 

[Sucker]

I notice that the article was written by Richard Reid. I didn't even know he was a mathematician. I know he's not very good at blowing up tennis shoes.

 

[Automatic Monkey]

I notice that the article was written by Richard Reid. I didn't even know he was a mathematician. I know he's not very good at blowing up tennis shoes.

No that's Abdul Jalib.

 

[Sharky]

splain? don't know, but Sands is building in Spain, in today's WSJ

 

[sagefr0g]

splain? don't know, but Sands is building in Spain, in today's WSJ

yah, splain Sharky, cause if some one explained it they'd probably just go back over the math of A New Interpretation of Information Rate By J. L. Kelly, jr. and i'd never get a handle on that.
just it would seem if one had a real solid understanding of the reasoning behind the sense of trying to maximize the expected logarithm of your total bankroll rather than trying to maximize the expected bankroll itself that one would have a tactical footing for making decisions out in the field, sorta thing, even given that we should already have a game plan to follow from the get go. especially maybe if one was making decisions about more betting situations than just card counting.

 

[sagefr0g]

I tried to get my head around this a while ago. The understanding I reached, which may well be flawed, is as follows.

The goal of the Kelly Criterion is to maximise the expected rate of growth of your bankroll, given that you can re-invest any winnings and increase your bet size as a result, and that conversely you will have to drop your bet size after a loss.

This is the exponential growth which you wish to maximise. You want to find the sweet spot between overbetting and underbetting.

The mathematics of how to maximise this thing throws up a fiendishly complicated formula involving the expected value of the logarithm of growth. But since it is known that if ln(x) > ln(y) then x > y, we can keep things simple by making it our new (equivalent) goal to maximise this new thing instead.

So, if I have understood correctly, logarithms only come into the picture as part of the mathematical journey towards an answer, and it's a bit misleading to mention them upfront as if they were central to the initial goal.

The above could be nonsense, but it's the impression I came away with after much searching of bjmath etc.

so, another way of saying it might be, it's better to try and exponentially maximize the expected growth of your bankroll than to try and maximize the expected bankroll?
or perhaps less accurately one could say it's better to try and maximize the rate of expected growth of your bankroll than to try and maximize the expected bankroll?
i guess that way of looking at it falls in with the idea where most people wouldn't want to 'bet the farm' every time they have an advantage, knowing that if they win, they win big, but if they lose they go broke, sorta thing.
so but like you say if you find that sweet spot sorta bet between over betting and under betting, errrh well you have a chance of keeping down the possible damage while still having a chance of improving, adding to the roll and then being able to either re-size down or up.

 

[k_c]

so, another way of saying it might be, it's better to try and exponentially maximize the expected growth of your bankroll than to try and maximize the expected bankroll?
or perhaps less accurately one could say it's better to try and maximize the rate of expected growth of your bankroll than to try and maximize the expected bankroll?
i guess that way of looking at it falls in with the idea where most people wouldn't want to 'bet the farm' every time they have an advantage, knowing that if they win, they win big, but if they lose they go broke, sorta thing.
so but like you say if you find that sweet spot sorta bet between over betting and under betting, errrh well you have a chance of keeping down the possible damage while still having a chance of improving, adding to the roll and then being able to either re-size down or up.

The best article I've seen on the consequences of betting various fractions of bankroll on positive EV, imho, is at this link:

http://www.marvinfrench.com/p1/blackjack/optimal.pdf

Loosely stated it seems the optimal fraction of bankroll to bet on +EV is defined as the fraction that yields the greatest growth in bankroll for average luck. Betting a greater or lesser fraction will result in a lesser bankroll growth or even a bankroll shrinkage, assuming average luck. However, luck can be better or worse than average with varying consequences for differing fractional betting strategies. Adopting a specific fractional betting strategy on +EV is a personal choice. The general choices are high risk for a high reward if lucky, low risk for a small reward even if somewhat unlucky, or something in between.

 

[MangoJ]

But since it is known that if ln(x) > ln(y) then x > y, we can keep things simple by making it our new (equivalent) goal to maximise this new thing instead.

This is true for ordinary numbers, but not for random numbers regarding their expectation value.

i.e. if EV(x) > EV(y), then it doesn't necessarily mean that EV(ln(x)) > EV(ln(y)).
The relation holds, when we are underbetting, but failes if we are overbetting. The reason is variance, this difference is the essence of a Kelly bet.

First see what we mean with x (and y). It is the total bankroll after our bet has settled. Hence x = bankroll + betsize*pay.

Lets first take a look maximizing EV(x) (it is the wrong way to bet):
max(EV(x)) = max(EV(bankroll + betsize*pay))
= bankroll + max(betsize*EV(pay)).
So whenever EV(pay) is negative, we shouldn't bet (betsize=0) or even offering this bet (betsize<0). However if EV(pay) is slightly positive, it tells us to max-bet on every small advantage. This will almost surely lead to the gamblers ruin, although he had the edge.


Now let's do better than busting and maximize EV(ln(x)): We treat the logarithm by noting that the bankroll is usually larger than the betsize.
We thus expand the logarithm in a Taylor-series, and get:
ln(bankroll + betsize*pay)
=~ ln(bankroll) + betsize/bankroll * pay - 1/2 (betsize/bankroll * pay)^2 + ...

Taking the EV yields:
EV(ln(x)) = ln(bankroll) + betsize/bankroll * EV(pay) - 1/2 (betsize/bankroll)^2 * EV(pay^2)
= ln(bankroll) - 1/2 + 1/2 bankroll^2 * ((EV(betsize * pay) + bankroll)^2 - VAR(betsize * pay) )

When we want to maximize the expected bankroll growth, we need to maximize this marked part (all other parts are constant).
We see, there is a fight between two terms.
The interpretation is: the first term depends on the EV of our bet EV(betsize*pay) and has a positive sign - thus it is favorable to play +EV games (who thought about that ^^). But the second term, depending only on the variance VAR(betsize*pay) of the bet, has a negative sign. So it is dangerous to play high-variance bets.
However, we are allowed to place any fixed betsize amount if our bankroll is large enough (contribution heavily to our first term).

So, regarding the optimal betsize (our original question) - there must be a balance between this betsize, your bankroll, EV and variance of the game.

 

[blackjack avenger]

Kelly Betting Resizing & the Looooooooooooooong Run

The long run for kelly resizing is staggering.

Considering NO (hands till expectation = standard deviation):
Let's take a 20,000 hand NO game

For kelly flat betting (13.35% ror)
NO 1 SD 20,000 hands
4 NO 2 SD 80,000 hands
9 NO 3 SD 180,000 hands

compared to:

for kelly resizing betting (0% ror)
4 NO 1 SD 80,000 hands
16 NO 2 SD 320,000 hands
36 NO 3 SD 720,000 hands

The NO for kelly resizing is 4 times that of kelly fixed betting.

But it gets worse:
For blackjack the number of hands needed to assure kelly resizing passes kelly flat betting is more hands then one will play in their lifetime!

:joker::whip:
good cards, you will need them

 

[blackjack avenger]

Maximize the Exponential Rate of Bankroll Growth, F### That!

or
Do we play to earn money or risk money?

With full kelly resizing one has a 50% chance of losing 50% or bank
With full kelly resizing one has a 20% chance of losing 80% of bank

How about considering:
A certainty of winning?
With 1/8th kelly resizing one has a .003% chance of losing 50% of bank
With 1/8th kelly resizing one has a 20% chance of losing 10% of bank

Can everyone see that with 1/8th kelly one does not have to play many hands to be ahead because your chances of being down a large amount are small? It does not maximize bank growth like kelly, but it does greatly increase one's chances of actually winning!

What are the probable reasons for AP failure?
Bank loss
Large bank drawdown
Failure of winning

Do we play to make money or risk money?

:joker::whip:
good cards

 

[MangoJ]

I think you ask the right question: Why are we playing.

The assumption of Kelly is that you always reinvest your full winnings into the bankroll, and that you never touch your bankroll for "-EV" things (like paying rent). With that you don't mind loosing half your bankroll, because you cannot do anything else with it anyway than playing.

Situation for professionals is different, because you have fixed running costs. A drop in bankroll is much severe because costs eat a much higher percentage if your bankroll - which further decreases betting size.
Your bankroll will collapse when - because of reduced betting size - your total EV does not even cover your costs anymore. Increasing betting size for more EV (to cover costs) will force you to overbetting. So you either die from overbetting, or the costs will eat you up.


The key should be to also account your fixed costs into your bankroll growth, which will modify your optimal bet size.

 

[blackjack avenger]

1/8th Kelly Resizing & Payouts

For every bank ATH half can be reinvested and half drawn off with a 0% ror.

or

For fixed costs
If EV>2 * expenses one still has 0% ror
If one has some losses, they would have to play more to keep the ratio of EV to expenses.

A near infinite bank offers serenity.

:joker::whip:
good cards