vpFREE2 Forums

So you go to the Casino with a plan ?

#41

I think the difficulty is discussing this topic arises from the fact that there is not a common
on understanding of what "kelly" means. I also don't think you are claiming that Kelly
applies to eveyone or anyone.. I am just trying to make clear that Kelly isn't simply a
strategy that optimized geometric bankroll growth (or however you phrased it)-- because
it isn't.

To begin, I suggest taking a look at how Kelly is derived. There is supposedly easy version
of it here: http://www.tightpoker.com/poker/kelly_easy.html though it is not easy or
complete (lol)

You will notice that "kelly" requires a utility function of the form U(x) = log(x).

This particular choice "log" is trully an arbitrary choice. I define "kelly" to be the strategey
that comes from optimizing things given utility functions of either log(x) or similar, as in
log(x+c) or maybe log(ax+b) etc. So for me... all "kelly" strategies require some version of
this model of utility... that an complete loss is infinitely bad, and that everything follows a
log.

Inspired by kelly, many people have come up with different but similar utility functions
that aim to capture a more realistic valuation ("feelings") about a win/loss. I'd call these
"kelly-like" or "kelly-inspired" or maybe "modified-kelly". Kelly to me still requires the
"log" function (or one with very similar behaviors, like a polynomial expansion of a log).
Kelly doesn't mean just "optimized mean geometric bankroll growth", but rather an
optimized betting stratgey for bankroll growth (or loss) given a particular ulitily function of
U(x) = log(x). The log is important. Very important. Without it (or something very similar
to it), you don't have Kelly.

BTW, if you have any stock tips... I'de love to hear them.

And I'm not claiming Kelly applies to everyone.

···

> Discussions of Kelly are often difficult.

Speaking of which:

Title: Fortune's Formula : The Untold Story of the Scientific Betting
System That Beat the Casinos and Wall Street
Author: William Poundstone
Publisher: Hill and Wang (September 14, 2005)
ISBN: 0809046377

#42

I am constantly amazed people have time to think about this kind of
stuff + classify what kind of people play VP (another DOA thread).
Must have memorized all penalty cards & scouted all good VP options
already, I guess. Stock tip: own some.

I think the difficulty is discussing this topic arises from the

fact that there is not a common

on understanding of what "kelly" means. I also don't think you are

claiming that Kelly

applies to eveyone or anyone.. I am just trying to make clear that

Kelly isn't simply a

strategy that optimized geometric bankroll growth (or however you

phrased it)-- because

it isn't.

To begin, I suggest taking a look at how Kelly is derived. There

is supposedly easy version

of it here: http://www.tightpoker.com/poker/kelly_easy.html though

it is not easy or

complete (lol)

You will notice that "kelly" requires a utility function of the

form U(x) = log(x).

This particular choice "log" is trully an arbitrary choice. I

define "kelly" to be the strategey

that comes from optimizing things given utility functions of either

log(x) or similar, as in

log(x+c) or maybe log(ax+b) etc. So for me... all "kelly"

strategies require some version of

this model of utility... that an complete loss is infinitely bad,

and that everything follows a

log.

Inspired by kelly, many people have come up with different but

similar utility functions

that aim to capture a more realistic valuation ("feelings") about a

win/loss. I'd call these

"kelly-like" or "kelly-inspired" or maybe "modified-kelly". Kelly

to me still requires the

"log" function (or one with very similar behaviors, like a

polynomial expansion of a log).

Kelly doesn't mean just "optimized mean geometric bankroll growth",

but rather an

optimized betting stratgey for bankroll growth (or loss) given a

particular ulitily function of

U(x) = log(x). The log is important. Very important. Without it

(or something very similar

···

--- In vpFREE@yahoogroups.com, "cdfsrule" <groups.yahoo@v...> wrote:

to it), you don't have Kelly.

BTW, if you have any stock tips... I'de love to hear them.

#43

> The problem with Kelly is that you have to define "bankroll" to include
> one's entire net worth, and to exclude the possibility of income from
> another source. Then, if you lose everything, you are truly screwed.

I thought this might be interesting to flush out a bit with an
example. Kelly strategy accepts 0 ror.

This is often cited as one of Kelly's "wonderful" properties, but it
really isn't all that special. In fact, this is really just a side effect
of _any_ system that bets a fixed fraction of bankroll. If you always
bet 90% of bankroll, then you can still never go broke, just like Kelly.

RoR is somewhat meaningless if the player is permitted to divide the
bankroll into ever small fractions, as required by Kelly. If you put
a realistic minimum wager in place, then RoR is no longer zero.

What if you are willing to
take, say, a 10% risk of losing it all? All being defined as the point
at which you will quit this particular project, i.e. your ultimate
stop loss limit, not all your worldly possessions. (Again the
gambler's anonymous issue of having a stop loss limit that you
actually stick to.) And, you have a choice of two games, either FPDW
or a monster DDB progressive (rf=1600) that no one has hit yet but
it's ready to go any moment now, so they say. Using:
http://wizardofodds.com/videopoker/analyzer/CindyProg.html , the FPDW
has an er=1.0076 and 10% ror bankroll of 3525 bets, the DDB
progressive has an er=1.0127 and 10% ror bankroll of 8503 bets. Which
is the better play? Based on er, the progressive is better. Based on
average bankroll growth
(FPDW:1.0076/3524=0.03%,DDB=1.0127/8503=0.01%), FPDW is better.

Based
on Kelly criterion, if your current bankroll is 2924 bets, then the
FPDW is the optimum bet, if your current bankroll is 8310 bets, then
the DDB progressive is the optimum bet.

I don't think you're interpreting these numbers correctly. If you have a
bankroll of 8310 bets, then that doesn't necessarily make DDB the
better game. You'll be underbetting on the FPDW game, but the growth
rate could still be higher than what you'd get from DDB.

In other words, if you plot growth rate vs. bet size, the Kelly bankroll
indicates which bet size will line up with the peak of the growth curve.
The bet size at that point doesn't tell you the magnitude of that peak.

···

On Saturday 24 December 2005 07:32 pm, nightoftheiguana2000 wrote:

--- In vpFREE@yahoogroups.com, Steve Jacobs <jacobs@x...> wrote:

#44

Like "iguana", I don't understand why you think you have to define
bankroll that way. Why can't bankroll be defined as a portion of
your net worth that you are willing to devote to AP gambling? A
typical example of partitioning would be keeping one's home separate
from the bankroll devoted to gambling.

--Dunbar

The problem with Kelly is that you have to define "bankroll" to

include

one's entire net worth, and to exclude the possibility of income

from

another source. Then, if you lose everything, you are truly

screwed.

Even if you do that, I believe there are people who would risk it

all

···

--- In vpFREE@yahoogroups.com, Steve Jacobs <jacobs@x...> wrote:

for a good shot at a huge reward. For such people, the Kelly model
doesn't apply (at least, not perfectly).

#45

> > The problem with Kelly is that you have to define "bankroll" to

include

> > one's entire net worth, and to exclude the possibility of income

from

> > another source. Then, if you lose everything, you are truly

screwed.

>
> I thought this might be interesting to flush out a bit with an
> example. Kelly strategy accepts 0 ror.

This is often cited as one of Kelly's "wonderful" properties, but it
really isn't all that special. In fact, this is really just a side

effect

of _any_ system that bets a fixed fraction of bankroll. If you always
bet 90% of bankroll, then you can still never go broke, just like Kelly.

RoR is somewhat meaningless if the player is permitted to divide the
bankroll into ever small fractions, as required by Kelly. If you put
a realistic minimum wager in place, then RoR is no longer zero.

Isn't risk of ruin always zero? You reach a point where the minimum
wager is 2x Kelly and at that point you stop, because growth is zero?
There is a risk of being forced to stop, and a risk of bankroll
halving ... If the Kelly bankroll for FPDW is 2924, and you stop if
you hit 2924/2, the risk of that happening is 38%. That's the risk of
having your bankroll halved, not complete ruin.

···

--- In vpFREE@yahoogroups.com, Steve Jacobs <jacobs@x...> wrote:

On Saturday 24 December 2005 07:32 pm, nightoftheiguana2000 wrote:
> --- In vpFREE@yahoogroups.com, Steve Jacobs <jacobs@x> wrote:

#46

This is one of the better explanations:
http://www.bjmath.com/bjmath/thorp/paper.htm
Kelly's original paper is here:
http://www.arbtrading.com/reports/kelly.pdf

To begin, I suggest taking a look at how Kelly is derived. There is

supposedly easy version

of it here: http://www.tightpoker.com/poker/kelly_easy.html though

it is not easy or

···

--- In vpFREE@yahoogroups.com, "cdfsrule" <groups.yahoo@v...> wrote:

complete (lol)

#47

another search that might be useful:
http://www.amazon.com/gp/product/0809046377

--- In vpFREE@yahoogroups.com, "nightoftheiguana2000"
<nightoftheiguana2000@y...> wrote:

···

This is one of the better explanations:
http://www.bjmath.com/bjmath/thorp/paper.htm
Kelly's original paper is here:
http://www.arbtrading.com/reports/kelly.pdf

--- In vpFREE@yahoogroups.com, "cdfsrule" <groups.yahoo@v...> wrote:
> To begin, I suggest taking a look at how Kelly is derived. There is
supposedly easy version
> of it here: http://www.tightpoker.com/poker/kelly_easy.html though
it is not easy or
> complete (lol)

#48

I thought I might take another wack at explaning the usefulness of
Kelly for video poker: Let's say you had access to a FPDW machine that
was full pay for any bet size. What should you bet? At first glance,
seems like you should bet your whole bankroll, afterall that optimizes
$ev/hand, right? But by betting your whole bankroll on one hand, your
risk of ruin is ROR(1)=0.999347^1=99.93%! If you happen to be one of
the people who is disturbed by that particular statistic, you might
take the other extreme of preserving bankroll at all costs and bet the
minimum, let's say 1/millionth of your whole bankroll per hand, now
your risk of ruin is ROR(million)=0.999347^million=0%! So, you've
solved the risk of ruin problem, however, now your $ev/hand is squat!
The question is: is there a happy medium inbetween? Is there an
optimum bet fraction that maximizes $ev/hand while minimizing ror? The
Kelly Criterion says yes, and that value for FPDW is 1/2924 of your
bankroll, and further, that if you bet 2x Kelly you no longer are
achieving any bankroll growth in the long term. How is this applied in
the real world of video poker? Where generally your choices are
something like bet, bet/2 or 2xbet? Start at bet=bankroll/2924, if
your bankroll halves, drop the bet to bet/2, if you double up your
bankroll, bump the bet up to 2xbet. If you want to get fancy, you can
also modify your hand drawing strategy, at bet size = kelly/2 optimal
strategy is max-ER, at bet size = 2xkelly optimal strategy is min-ROR,
the rest are inbetween. See http://members.cox.net/vpfree/FAQ_S.htm
for explanation of strategies.

#49

Happy Chriskwanukah
  to all
   
Les

···

---------------------------------
Yahoo! DSL Something to write home about. Just $16.99/mo. or less

[Non-text portions of this message have been removed]

#50

The underlying goal of maximizing the expected value of log(bankroll) is
mathematically equivalent to maximizing the expected value of the geometric
mean of the final bankroll. You can certainly claim that is isn't _simply_
that, but someone who sets out to do _simply_ that will end up with the
same result.

Of course, Kelly is also a major religion amongst some gamblers, but I don't
think that is what your getting at. Suggesting that people take a look at how
Kelly is derived doesn't help -- we've already done that. So, just what is
your point?

···

On Saturday 24 December 2005 11:18 pm, cdfsrule wrote:

I think the difficulty is discussing this topic arises from the fact that
there is not a common on understanding of what "kelly" means. I also don't
think you are claiming that Kelly applies to eveyone or anyone.. I am just
trying to make clear that Kelly isn't simply a strategy that optimized
geometric bankroll growth (or however you phrased it)-- because it isn't.

#51

I thought I might take another wack at explaning the usefulness of
Kelly for video poker: Let's say you had access to a FPDW machine that
was full pay for any bet size. What should you bet? At first glance,
seems like you should bet your whole bankroll, afterall that optimizes
$ev/hand, right? But by betting your whole bankroll on one hand, your
risk of ruin is ROR(1)=0.999347^1=99.93%!

Umm, no. ROR(1) doesn't mean RoR from playing one hand of VP. Rather,
it means the risk of ultimately going broke if you start with a single unit
and attempt to play indefinitely. The probability of going broke from a
single hand is closer to 55%.

If you happen to be one of
the people who is disturbed by that particular statistic, you might
take the other extreme of preserving bankroll at all costs and bet the
minimum, let's say 1/millionth of your whole bankroll per hand, now
your risk of ruin is ROR(million)=0.999347^million=0%! So, you've
solved the risk of ruin problem, however, now your $ev/hand is squat!
The question is: is there a happy medium inbetween? Is there an
optimum bet fraction that maximizes $ev/hand while minimizing ror? The
Kelly Criterion says yes, and that value for FPDW is 1/2924 of your
bankroll, and further, that if you bet 2x Kelly you no longer are
achieving any bankroll growth in the long term. How is this applied in
the real world of video poker? Where generally your choices are
something like bet, bet/2 or 2xbet? Start at bet=bankroll/2924, if
your bankroll halves, drop the bet to bet/2, if you double up your
bankroll, bump the bet up to 2xbet. If you want to get fancy, you can
also modify your hand drawing strategy, at bet size = kelly/2 optimal
strategy is max-ER, at bet size = 2xkelly optimal strategy is min-ROR,
the rest are inbetween. See http://members.cox.net/vpfree/FAQ_S.htm
for explanation of strategies.

I don't believe that a bet size of kelly/2 is small enough to make the
log-optimal strategy become identical to max-ER strategy. It is true that
a bet size _near_ twice Kelly gives zero geometric growth and causes
the log-optimal strategy to become identical to min-risk strategy. The
factor of bet size isn't exactly two, partly because of strategy changes.

···

On Sunday 25 December 2005 01:16 pm, nightoftheiguana2000 wrote:

vpFREE Links: http://members.cox.net/vpfree/Links.htm

Yahoo! Groups Links

#52

It seems to me that if you define bankroll as some small fraction of
your net worth, then losing that entire bankroll can't reasonably be
thought of as "infinitely bad". Losing a trip bankroll wouldn't disuade
anyone from giving up gambling forever, but the Kelly model treats
such a loss as total devastation. It effectively weights that last unit
of your bankroll as infinitely more important than the second to last
unit. To me, that doesn't seem reasonable in the context of a trip
bankroll.

···

On Sunday 25 December 2005 01:04 am, dunbar_dra wrote:

Like "iguana", I don't understand why you think you have to define
bankroll that way. Why can't bankroll be defined as a portion of
your net worth that you are willing to devote to AP gambling? A
typical example of partitioning would be keeping one's home separate
from the bankroll devoted to gambling.

#53

> > > The problem with Kelly is that you have to define "bankroll" to

include

> > > one's entire net worth, and to exclude the possibility of income

from

> > > another source. Then, if you lose everything, you are truly

screwed.

> > I thought this might be interesting to flush out a bit with an
> > example. Kelly strategy accepts 0 ror.
>
> This is often cited as one of Kelly's "wonderful" properties, but it
> really isn't all that special. In fact, this is really just a side

effect

> of _any_ system that bets a fixed fraction of bankroll. If you always
> bet 90% of bankroll, then you can still never go broke, just like Kelly.
>
> RoR is somewhat meaningless if the player is permitted to divide the
> bankroll into ever small fractions, as required by Kelly. If you put
> a realistic minimum wager in place, then RoR is no longer zero.

Isn't risk of ruin always zero? You reach a point where the minimum
wager is 2x Kelly and at that point you stop, because growth is zero?

I don't make a distinction between being forced to stop playing, and
going broke. Either event precludes the player from any further
bankroll growth.

By the same standard, you could simply make a pact never the wager
the final unit of your playing bankroll. Then your RoR is zero whether
you use proportional betting or not. But, I think doing that is just kidding
yourself -- in effect your RoR is ROR(bank-1) rather than ROR(bank).

There is a risk of being forced to stop, and a risk of bankroll
halving ... If the Kelly bankroll for FPDW is 2924, and you stop if
you hit 2924/2, the risk of that happening is 38%. That's the risk of
having your bankroll halved, not complete ruin.

My understanding of Kelly is that the probability of your bankroll dropping
to X% is X/100, so that the risk of dropping to 50% is 1/2 and the risk of
dropping to 1% is 1/100. This is based on continuously adjusting bet size
to be the proper Kelly bet.

···

On Sunday 25 December 2005 02:33 am, nightoftheiguana2000 wrote:

--- In vpFREE@yahoogroups.com, Steve Jacobs <jacobs@x...> wrote:
> On Saturday 24 December 2005 07:32 pm, nightoftheiguana2000 wrote:
> > --- In vpFREE@yahoogroups.com, Steve Jacobs <jacobs@x> wrote:

#54

Well said Michael.

What's wrong with: I can make .002 of the gross $$ I run through
machine. This includes CB and coupons. I play at $2500 an hour and my
bankroll will grow at $5 per hour; if I play perfect. I probably don't.
Where do I go to calculate my risk of screwing up the Kelly system
because I'm a crummy player. So we use Kelly and apply ROR formula. Of
course, we have to consult with Dancer. (I'll probably call the wrong
Dancer and end up signing for tango lessons) Then we calculate crummy
player theory and add 5 beers and one iced tea from the bar and we know
there is a 42.00834% chance of going broke if we have 3.723 royals
bankroll. Gee, maybe I can write a book or something?

It does seem folks are making a big deal over something that is pretty
much simple.

Happy new year....Jeep

.

--- In vpFREE@yahoogroups.com, "Michael Boutot" <vegas_iwish@y...>
wrote:

···

I am constantly amazed people have time to think about this kind of
stuff + classify what kind of people play VP (another DOA thread).
Must have memorized all penalty cards & scouted all good VP options
already, I guess. Stock tip: own some.

--- In vpFREE@yahoogroups.com, "cdfsrule" <groups.yahoo@v...> wrote:

>
> I think the difficulty is discussing this topic arises from the
fact that there is not a common
> on understanding of what "kelly" means.

#55

> I thought I might take another wack at explaning the usefulness of
> Kelly for video poker: Let's say you had access to a FPDW machine that
> was full pay for any bet size. What should you bet? At first glance,
> seems like you should bet your whole bankroll, afterall that optimizes
> $ev/hand, right? But by betting your whole bankroll on one hand, your
> risk of ruin is ROR(1)=0.999347^1=99.93%!

Umm, no. ROR(1) doesn't mean RoR from playing one hand of VP. Rather,
it means the risk of ultimately going broke if you start with a

single unit

and attempt to play indefinitely. The probability of going broke from a
single hand is closer to 55%.

I made the assumption you would keep on playing as long as you had money.

···

--- In vpFREE@yahoogroups.com, Steve Jacobs <jacobs@x...> wrote:

On Sunday 25 December 2005 01:16 pm, nightoftheiguana2000 wrote:

#56

My understanding of Kelly is that the probability of your bankroll

dropping

to X% is X/100, so that the risk of dropping to 50% is 1/2 and the

risk of

dropping to 1% is 1/100. This is based on continuously adjusting

bet size

to be the proper Kelly bet.

The risk of getting your bankroll halved is 1/3, and the chance of
doubling it is 2/3, under Kelly strategy.

···

--- In vpFREE@yahoogroups.com, Steve Jacobs <jacobs@x...> wrote:

#57

"trip bankroll" is a 3rd entity all-together. It can easily be
treated as something different from "total gambling bankroll"
and "net worth". "Trip bankroll" considerations are the main
reason I am interested in short-term RoR calculations.

Quick definitions:
"Trip bankroll" = the amount of money you are prepared to lose
during a specified period of time, generally a single trip to a
gambling destination.

"Total gambling bankroll" = the amount of money you are prepared to
lose before you quit gambling.

"Net worth" = the sum of all your assets minus the sum of all your
liabilities.

I see these as 3 very different entities.

--Dunbar

> Like "iguana", I don't understand why you think you have to

define

> bankroll that way. Why can't bankroll be defined as a portion

of

> your net worth that you are willing to devote to AP gambling? A
> typical example of partitioning would be keeping one's home

separate

> from the bankroll devoted to gambling.

It seems to me that if you define bankroll as some small fraction

of

your net worth, then losing that entire bankroll can't reasonably

be

thought of as "infinitely bad". Losing a trip bankroll wouldn't

disuade

anyone from giving up gambling forever, but the Kelly model treats
such a loss as total devastation. It effectively weights that

last unit

of your bankroll as infinitely more important than the second to

last

···

--- In vpFREE@yahoogroups.com, Steve Jacobs <jacobs@x...> wrote:

On Sunday 25 December 2005 01:04 am, dunbar_dra wrote:
unit. To me, that doesn't seem reasonable in the context of a trip
bankroll.

#58

Excellent. Glad you asked. See, Kelly very nicely folded in the
concept of utility function into his derivation (when he started),
then at the end, it seemed (too many casual readers) that he had
declared that his results are independant of the utility function.
What he was saying actually was a bit more tricky: perhaps better put
as if you want optimal bankroll growth (with no other constraints)
follow my strategy-- sinceit is optimal-- but it you have other
contratains (including different utility functions), you can approach
the problem in the same way I did, but you will get different
results. [Also, and this is for NOTI, the kelly criteria I beelive
optimizes MEDIAN bankroll growth. not average growth. I recall that
this was proven sometime in the 80's]

In the years since Kelly, it has established that while Kelly's
results are mathematically correct, if one uses a different utility
function, you don't necessarily get geometic bankrollg growth. (Using
this forma approach) To find the optimal stratgey for a
partiucalr "person", one must match a persons utility function (and
other constraints) with the appropriate strategey (in other words,
derive the optimal stragtegy by using the correct utility function).
The pure log-utility function of Kelly is just one choice. (This
gives a "kelly number of 1"). Most gamblers who do follow kelly,
don't share such a utility function exactly and so use a modified
stratgey (a kelly number other than 1). Likewise, there are all
kinds of other utility functions. While these will not produce
bankroll growth rates better than a kelly strategy, they nonetheless,
are optimal for certain players, since these players have a
different "feeling" about risk ( which is captured mathematically in
the the utility function) and do not like the idea of making "kelly"
bets. Many, many, classes of utility function shave been studied.
NOTI stated a few times that Kelly was optimal for a certain
(hypothesized) gambler-- though he never stated the gamblers utility
function. If indeed, kelly was optimal for said gambler, then this
gambler MUST have a logarithmic (or equivelent) utility function.
But even if said logarithmic gambler exists, he/she is certainlyt not
the only type of gambler, and I would guess, even the most common
type of gambler (as grouped by utility function/risk aversion,
etc). For example, in post 53678, NOTI, presents a discussion of
said gambler. He hints at the issue of utility function when he
mentions that the gambler my be "disturbed by that particular
statistic", but then drops it. Fact is, that same gambler who
doesn't want to bet it all, might not likewise feel logarithmically
about his/her bankroll. And hence for him/her, Kelly is not the
optimal strategy, given his/her personal tolerance/preference for
risk. Invoking personal preference ("utility function") in one
instance (in order to dismiss that betting it all is not a good idea)
and then dismissing it in the next (when claiming that Kelly is
optimal for that same person) just doesn't seem right to me.

···

--- In vpFREE@yahoogroups.com, Steve Jacobs <jacobs@x...> wrote:

So, just what is your point?

#59

Excellent. Glad you asked. See, Kelly very nicely folded in the
concept of utility function into his derivation (when he started),
then at the end, it seemed (too many casual readers) that he had
declared that his results are independant of the utility function.
What he was saying actually was a bit more tricky: perhaps better put
as if you want optimal bankroll growth (with no other constraints)
follow my strategy-- sinceit is optimal-- but it you have other
contratains (including different utility functions), you can approach
the problem in the same way I did, but you will get different
results. [Also, and this is for NOTI, the kelly criteria I beelive
optimizes MEDIAN bankroll growth. not average growth. I recall that
this was proven sometime in the 80's]

In the years since Kelly, it has established that while Kelly's
results are mathematically correct, if one uses a different utility
function, you don't necessarily get geometic bankrollg growth. (Using
this forma approach) To find the optimal stratgey for a
partiucalr "person", one must match a persons utility function (and
other constraints) with the appropriate strategey (in other words,
derive the optimal stragtegy by using the correct utility function).
The pure log-utility function of Kelly is just one choice. (This
gives a "kelly number of 1"). Most gamblers who do follow kelly,
don't share such a utility function exactly and so use a modified
stratgey (a kelly number other than 1). Likewise, there are all
kinds of other utility functions. While these will not produce
bankroll growth rates better than a kelly strategy, they nonetheless,
are optimal for certain players, since these players have a
different "feeling" about risk ( which is captured mathematically in
the the utility function) and do not like the idea of making "kelly"
bets. Many, many, classes of utility function shave been studied.
NOTI stated a few times that Kelly was optimal for a certain
(hypothesized) gambler-- though he never stated the gamblers utility
function. If indeed, kelly was optimal for said gambler, then this
gambler MUST have a logarithmic (or equivelent) utility function.
But even if said logarithmic gambler exists, he/she is certainlyt not
the only type of gambler, and I would guess, even the most common
type of gambler (as grouped by utility function/risk aversion,
etc). For example, in post 53678, NOTI, presents a discussion of
said gambler. He hints at the issue of utility function when he
mentions that the gambler my be "disturbed by that particular
statistic", but then drops it. Fact is, that same gambler who
doesn't want to bet it all, might not likewise feel logarithmically
about his/her bankroll. And hence for him/her, Kelly is not the
optimal strategy, given his/her personal tolerance/preference for
risk. Invoking personal preference ("utility function") in one
instance (in order to dismiss that betting it all is not a good idea)
and then dismissing it in the next (when claiming that Kelly is
optimal for that same person) just doesn't seem right to me.

···

--- In vpFREE@yahoogroups.com, Steve Jacobs <jacobs@x...> wrote:

So, just what is your point?

#60

My first response was too wordy & not to the point (lol). Try this
one.

From Kelly's paper:

"Suppose the situation were different; for example,suppose the
gambler's wife allowed him to bet one dollar each week but not to
reinvest his winnings. He should then maximize his expectation
(expected value of capital) on each bet. He would bet all his
available capital (one dollar) on the event yielding the highest
expectation. With probability one he would get ahead of anyone
dividing his money differently."

So...Different people, constraints, and/or utility functions,
different optimal strategies. But keep in mind that Kelly has been
shown to be correct when he said...

"The gambler introduced here follows an essentially different
criterion from the classical gambler. At every bet he maximizes the
expected value of the logarithm of his capital. The reason has
nothing to do with the value function which he attached to his money,
but merely with the fact that it is the logarithm which is additive
in repeated bets and to which the law of large numbers applies."

You see, Kelly provides optimal bankroll growth (independant of
utility function) but it is only applicable to the person who has a
logartihmic utility function. So, Kelly is not the optimal strategy
for anyone, unless that person happens to have a logarithmic utility
function. Do you?

> So, just what is your point?
>

Excellent. Glad you asked. See, Kelly very nicely folded in the
concept of utility function into his derivation (when he started),
then at the end, it seemed (too many casual readers) that he had
declared that his results are independant of the utility function.
What he was saying actually was a bit more tricky: perhaps better

put

as if you want optimal bankroll growth (with no other constraints)
follow my strategy-- sinceit is optimal-- but it you have other
contratains (including different utility functions), you can

approach

the problem in the same way I did, but you will get different
results. [Also, and this is for NOTI, the kelly criteria I beelive
optimizes MEDIAN bankroll growth. not average growth. I recall

that

this was proven sometime in the 80's]

In the years since Kelly, it has established that while Kelly's
results are mathematically correct, if one uses a different utility
function, you don't necessarily get geometic bankrollg growth.

(Using

this forma approach) To find the optimal stratgey for a
partiucalr "person", one must match a persons utility function (and
other constraints) with the appropriate strategey (in other words,
derive the optimal stragtegy by using the correct utility

function).

The pure log-utility function of Kelly is just one choice. (This
gives a "kelly number of 1"). Most gamblers who do follow kelly,
don't share such a utility function exactly and so use a modified
stratgey (a kelly number other than 1). Likewise, there are all
kinds of other utility functions. While these will not produce
bankroll growth rates better than a kelly strategy, they

nonetheless,

are optimal for certain players, since these players have a
different "feeling" about risk ( which is captured mathematically

in

the the utility function) and do not like the idea of

making "kelly"

bets. Many, many, classes of utility function shave been

studied.

NOTI stated a few times that Kelly was optimal for a certain
(hypothesized) gambler-- though he never stated the gamblers

utility

function. If indeed, kelly was optimal for said gambler, then this
gambler MUST have a logarithmic (or equivelent) utility function.
But even if said logarithmic gambler exists, he/she is certainlyt

not

the only type of gambler, and I would guess, even the most common
type of gambler (as grouped by utility function/risk aversion,
etc). For example, in post 53678, NOTI, presents a discussion of
said gambler. He hints at the issue of utility function when he
mentions that the gambler my be "disturbed by that particular
statistic", but then drops it. Fact is, that same gambler who
doesn't want to bet it all, might not likewise feel logarithmically
about his/her bankroll. And hence for him/her, Kelly is not the
optimal strategy, given his/her personal tolerance/preference for
risk. Invoking personal preference ("utility function") in one
instance (in order to dismiss that betting it all is not a good

idea)

···

--- In vpFREE@yahoogroups.com, "cdfsrule" <groups.yahoo@v...> wrote:

--- In vpFREE@yahoogroups.com, Steve Jacobs <jacobs@x> wrote:
and then dismissing it in the next (when claiming that Kelly is
optimal for that same person) just doesn't seem right to me.