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Kelly Criterion

#1

As I understand the Kelly Criterion, it basically says that, to maximize your win rate while minimizing your risk, you should scale your bet to be a percentage of your bankroll equal to your advantage on the game. Thus, if you have a $1,000 bankroll, and you're playing Deuces Wild at 100.76% ER, your bet should be 0.76% of $1,000, or $7.60. Great! All we have to do is find a 5-coin $1.52 FPDW game. (OK, let's settle for a $1.00 game, and we won't bother to be concerned that the RoR on $1 DW with that small bankroll is over 87%.) If your bankroll drops below $658.00, your bet should be less than $5.00, so you need to move to a lower denomination machine. Oops, where do we find such a machine?

Of course, most players are more conservative, and play half-Kelly or perhaps smaller bets, but even if we go to a quarter machine with that bankroll, the RoR is over 70%. Of course, we could be much more conservative and play a much smaller fraction of Kelly bets, but the main reason that Kelly can't be applied to video poker is because, as I've shown in previous posts, the RoR is not directly related to either the player advantage or the game's variance. The Kelly Criterion is effective only for low variance games, such as blackjack.

BTW, I replaced my old Attractiveness Quotient with a new Attractiveness Index well over a year ago. The AI is described in VP Times issues 12.3 and in the second edition of VP-Optimum Play. It is still a function of player advantage and standard deviation, but I suggest it only for use as a quick comparison of situations, not as a gauge for bet size or RoR. While there are several approximation formulae available, including Tomski's formula, the Sorokin formula is the only way to get a good handle on RoR.

Dan

···

nightoftheiguana2000" <nightoftheiguana2000@yahoo.com> wrote:

If it's some function of advantage/variance, or the inverse, it's the
Kelly Criterion.

If it's some function of advantage/sqrt(variance), or the inverse,
it's the Sharpe ratio.

Dan's AQ formula, I think, is 1000 x %advantage/VI where VI=
volatility index = standard deviation = sqrt(variance), thus AQ is the
Sharpe ratio scaled by 100,000. For example, Dan in Video Poker
Optimum Play, lists the AQ of full pay 2 pair or better joker as 376,
which is calculated from 1000 x 2/5.34

--
Dan Paymar
Author of best selling book, "Video Poker - Optimum Play"
Editor/Publisher of VP newsletter "Video Poker Times"
Developer of VP analysis/trainer software "Optimum Video Poker"
Visit my web site at www.OptimumPlay.com

"Chance favors the prepared mind." -- Louis Pasteur

#2

From: Dan Paymar <Dan@OptimumPlay.com>

>If it's some function of advantage/variance, or the inverse, it's the
>Kelly Criterion.
>
>If it's some function of advantage/sqrt(variance), or the inverse,
>it's the Sharpe ratio.
>
>Dan's AQ formula, I think, is 1000 x %advantage/VI where VI=
>volatility index = standard deviation = sqrt(variance), thus AQ is the
>Sharpe ratio scaled by 100,000. For example, Dan in Video Poker
>Optimum Play, lists the AQ of full pay 2 pair or better joker as 376,
>which is calculated from 1000 x 2/5.34

As I understand the Kelly Criterion, it basically says that, to
maximize your win rate while minimizing your risk, you should scale
your bet to be a percentage of your bankroll equal to your advantage
on the game. Thus, if you have a $1,000 bankroll, and you're playing
Deuces Wild at 100.76% ER, your bet should be 0.76% of $1,000, or
$7.60.

This is a misconception.

Kelly actually said nothing about betting, his paper was on signal noise, and people discovered that it could be applied to wagering systems.

The described simplified "Kelly Criterion" only applies to wagers where the variance is close to 1, such as blackjack.

For "Kelly" or "log-optimal" growth, the function is more complex if the game has a series of possible outcomes other than +1/-1 bet. In fact it's more complex for blackjack, too, but people use the approximation of only +1/-1 outcomes to simplify.

Assume you are always betting a fixed fraction of your bankroll. All possible outcomes of a wager can be described as having a multiplier effect on your bankroll. Say you bet 1%, if you lose your bankroll is now .99 what it used to be, if you break even, it's 1 of what it used to be, if you win one bet, it's 1.01 of what it used to be, and so on. If you take the log of all the possible outcomes, and take the weighted average of those logs of outcomes, you have the "log" effect of that wager on your bankroll. If the log effect of the wager on your bankroll is negative, then even though you may be making a positive expectation bet, you are overbetting your bankroll and will go broke (well, experience exponential shrinkage of your bankroll) eventually if you continue betting that same fraction of your bankroll indefinitely. If the log effect is positive, then your bankroll should experience exponential growth in the long term if you keep making the same percentage wager.

"Kelly" or "log optimal" betting is picking the fraction of your bankroll that maximizes the log effect. That determines the optimal wager that gives you the maximum exponential growth of your bankroll.

True Kelly betting requires the ability to continuously change the size of your wager for each bet as your bankroll size changes.

For situations where you cannot conveniently change your bet size, some people have taken the approach of computing the optimal Kelly percentage for their current bankroll, then using 1/2 or 1/3, or 1/4 or some other fraction of that number for their wagers, and rescaling their bets when their bankroll undergoes signficant change.

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#3

http://www.bjmath.com/bjmath/kelly/kellyfaq.htm

>From: Dan Paymar <Dan@O...>
>
>nightoftheiguana2000" <nightoftheiguana2000@y...> wrote:
> >If it's some function of advantage/variance, or the inverse, it's the
> >Kelly Criterion.
> >
> >If it's some function of advantage/sqrt(variance), or the inverse,
> >it's the Sharpe ratio.
> >
> >Dan's AQ formula, I think, is 1000 x %advantage/VI where VI=
> >volatility index = standard deviation = sqrt(variance), thus AQ

is the

> >Sharpe ratio scaled by 100,000. For example, Dan in Video Poker
> >Optimum Play, lists the AQ of full pay 2 pair or better joker as 376,
> >which is calculated from 1000 x 2/5.34
>
>As I understand the Kelly Criterion, it basically says that, to
>maximize your win rate while minimizing your risk, you should scale
>your bet to be a percentage of your bankroll equal to your advantage
>on the game. Thus, if you have a $1,000 bankroll, and you're playing
>Deuces Wild at 100.76% ER, your bet should be 0.76% of $1,000, or
>$7.60.

This is a misconception.

Kelly actually said nothing about betting, his paper was on signal

noise,

and people discovered that it could be applied to wagering systems.

The described simplified "Kelly Criterion" only applies to wagers

where the

variance is close to 1, such as blackjack.

For "Kelly" or "log-optimal" growth, the function is more complex if

the

game has a series of possible outcomes other than +1/-1 bet. In

fact it's

more complex for blackjack, too, but people use the approximation of

only

+1/-1 outcomes to simplify.

Assume you are always betting a fixed fraction of your bankroll. All
possible outcomes of a wager can be described as having a multiplier

effect

on your bankroll. Say you bet 1%, if you lose your bankroll is now

.99 what

it used to be, if you break even, it's 1 of what it used to be, if

you win

one bet, it's 1.01 of what it used to be, and so on. If you take

the log of

all the possible outcomes, and take the weighted average of those

logs of

outcomes, you have the "log" effect of that wager on your bankroll.

If the

log effect of the wager on your bankroll is negative, then even

though you

may be making a positive expectation bet, you are overbetting your

bankroll

and will go broke (well, experience exponential shrinkage of your

bankroll)

eventually if you continue betting that same fraction of your bankroll
indefinitely. If the log effect is positive, then your bankroll should
experience exponential growth in the long term if you keep making

the same

percentage wager.

"Kelly" or "log optimal" betting is picking the fraction of your

bankroll

that maximizes the log effect. That determines the optimal wager

that gives

you the maximum exponential growth of your bankroll.

True Kelly betting requires the ability to continuously change the

size of

your wager for each bet as your bankroll size changes.

For situations where you cannot conveniently change your bet size, some
people have taken the approach of computing the optimal Kelly

percentage for

their current bankroll, then using 1/2 or 1/3, or 1/4 or some other

fraction

of that number for their wagers, and rescaling their bets when their
bankroll undergoes signficant change.

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#4

As I understand the Kelly Criterion, it basically says that, to
maximize your win rate while minimizing your risk, you should scale
your bet to be a percentage of your bankroll equal to your advantage
on the game.

advantage/variance
In Blackjack, a low variance game, most people ignore the variance
part, but they shouldn't. See Wong, he doesn't ignore variance.

Oops, where do we find such a machine?

Today, there are many "oddball" machines, pennies from 1 to 250 bets,
etc., some with good paytables. Typically once over 5 bets, the royal
is full pay, i.e. 800.

While there are several approximation
formulae available, including Tomski's formula, the Sorokin formula
is the only way to get a good handle on RoR.

Agreed. Sorokin is exact, others are approximate. Sometimes an
approximation will do.

···

--- In vpFREE@yahoogroups.com, Dan Paymar <Dan@O...> wrote:

#5

Just to make clear:

Thus, if you have a $1,000 bankroll, and you're playing
Deuces Wild at 100.76% ER, your bet should be 0.76% of $1,000, or
$7.60.

Kelly bet is bankroll x advantage / variance

For FPDW: advantage=.76%, variance=26

$1,000 x .76% / 26 = $0.29

Let's say you want to play quarter 5 coin deuces and want to know how
much of a bankroll you need?

Kelly bankroll = (bet per hand) x variance / advantage

For FPDW, 5 coin quarters:

$1.25 x 26 / .76% = $4,276

You can use "Sorokin" to get the risk of ruin:

0.999346833^($4276/$1.25)=10.6983019%

but the
main reason that Kelly can't be applied to video poker is because,

as

I've shown in previous posts, the RoR is not directly related to
either the player advantage or the game's variance.

That is incorrect.

The Kelly
Criterion is effective only for low variance games, such as

blackjack.

That is incorrect.

···

--- In vpFREE@yahoogroups.com, Dan Paymar <Dan@O...> wrote:

#6

No it isn't. If you read the FAQ from the link that you posted, that's just an approximation, useful for blackjack and some other games.

Actual Kelly bet formulation is much more complicated than that and is not a function of advantage and variance.

···

From: "nightoftheiguana2000" <nightoftheiguana2000@yahoo.com>
Kelly bet is bankroll x advantage / variance

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#7

Agreed. It is an approximation. Nevertheless, approximations can
sometimes be useful.

>Kelly bet is bankroll x advantage / variance

No it isn't. If you read the FAQ from the link that you posted,

that's just

an approximation, useful for blackjack and some other games.

Actual Kelly bet formulation is much more complicated than that and

is not a

function of advantage and variance.

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how to

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>From: "nightoftheiguana2000" <nightoftheiguana2000@y...>
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