[Dark Horse]

I would never go all-in before a game, but at halftime of a basketball game, if the numbers are ridiculously tilting one way, I might under very rare conditions. At halftime there are cards on the table, so the poker comparison is closer.

[donjuan]

The time you would need to consider going "all-in" would be when the bet is of a quality that makes it a scarce opportunity. i.e. Would you call a all-in bet with AA in holdem (preflop) if your entire liferoll were on the line?


The "opportunity" to wager your whole BR with such a great advantage (~85%*) may not come along again for quite a while.

* Assuming the other player doesn't have AA

The opportunity is not relevant here. You're a moron if you put your entire bankroll for your lifetime on an 85% chance at +100 odds.

[SBR Lou]

I would go all in at poker in different situations, but one that comes up every now and then is a rainbow flop, harmless turn, all in before the river. Most of the time it's calling an all in unless the pot odds are right and it makes more sense to push. I will take that chance every time holding the nuts on the turn with no straight or flushdraw possibilities, and if my opponent hits a case card on me so be it. One outer losses hurt, but that's the nature of the game. Two outers suck too.

[donjuan]

I think you're getting confused with what you're talking about. Going all-in for your buy in at a poker table is different than doing so with 100% of your bankroll, whether it's poker or sports.

[20Four7]

I think you're getting confused with what you're talking about. Going all-in for your buy in at a poker table is different than doing so with 100% of your bankroll, whether it's poker or sports.

That is correct, but there are a few casino's with big enough games that the buy in is unlimited. For whatever that means with your bank roll.

Now back to the question at hand. In the NBA section appparently GS going into Miami was enough to make Miami a "go all in" pick. As much as I did like golden state, I still have under 5% of my roll on them.

[SBR Lou]

I think you're getting confused with what you're talking about. Going all-in for your buy in at a poker table is different than doing so with 100% of your bankroll, whether it's poker or sports.

I was just injecting myself in the conversation on terms of when I'd go all in at cards, stakes aside but what I said would remain true even if one was a degenerate nincompoop and wanted to bet it all. You can sit with over 50k at higher limit tables at most sites.

[slacker00]

That is correct, but there are a few casino's with big enough games that the buy in is unlimited. For whatever that means with your bank roll.

What kind of person buys in with their entire bankroll, unless they are prepared to use it?

As for pro players taking the AA preflop for their entire bankroll prop, I think many pro players would take that wager. Sure, there's 10-20% chance they get busted, but most of them could instantly get staked by other pros and be back in the game instantly. Sure it would suck building their bankroll back up, but these guys are in the business of gambling. I'd imagine only the tightest nits would pass on this type of prop.

As for sports betting, I've never seen a widely available line that has even close to this much of an edge.

As for people investing their whole net worth into one company or opportunity, it happens every day and people lose their life savings every day. What about Enron? What about pyramid scheme scams? Real estate? Even starting your own business is a huge risk, but people do it every day. What it the stat on that one? 9 out of 10 startup businesses fail in the first 3 years? People betting and losing their whole bankroll is as old as money itself.

[operaman]

The opportunity is not relevant here. You're a moron if you put your entire bankroll for your lifetime on an 85% chance at +100 odds.

liferoll = 100% of ones current net worth

Sorry if that was unclear.


I wouldn't wager my lifetime eanings on one bet in my current situation, but to say a person would be a moron for taking such a wager is incorrect and inflamatory.

[donjuan]

I wouldn't wager my lifetime eanings on one bet in my current situation, but to say a person would be a moron for taking such a wager is incorrect and inflamatory.

It's not incorrect at all. Expected growth >>>>>>>> expected value

[operaman]

What kind of person buys in with their entire bankroll, unless they are prepared to use it?

As for pro players taking the AA preflop for their entire bankroll prop, I think many pro players would take that wager. Sure, there's 10-20% chance they get busted, but most of them could instantly get staked by other pros and be back in the game instantly. Sure it would suck building their bankroll back up, but these guys are in the business of gambling. I'd imagine only the tightest nits would pass on this type of prop.

As for sports betting, I've never seen a widely available line that has even close to this much of an edge.

As for people investing their whole net worth into one company or opportunity, it happens every day and people lose their life savings every day. What about Enron? What about pyramid scheme scams? Real estate? Even starting your own business is a huge risk, but people do it every day. What it the stat on that one? 9 out of 10 startup businesses fail in the first 3 years? People betting and losing their whole bankroll is as old as money itself.

There was a poll taken on 2 + 2 a long time ago and the posters who said they play poker for a living were about %50-%50 on calling allin liferoll AA(no opp AA).

[donjuan]

There was a poll taken on 2 + 2 a long time ago and the posters who said they play poker for a living were about %50-%50 on calling allin liferoll AA(no opp AA).

Half of them are retards who think purely in terms of EV and not EG. Well, actually more than half but this post confirms that half are.

[SBR Lou]

Half of them are retards who think purely in terms of EV and not EG. Well, actually more than half but this post confirms that half are.

What I hate most about 2+2 is an overwhelming majority speak in their own forum lingo. Here I don't think anybody really does that. Go Read 2+2 and it's like they communicate in their own language....

"newbzors, HU4ROLLZ, DONKAMENT, Andy Beal...Who is he?".

[donjuan]

What I hate most about 2+2 is an overwhelming majority speak in their own forum lingo. Here I don't think anybody really does that. Go Read 2+2 and it's like they communicate in their own language....

"newbzors, HU4ROLLZ, DONKAMENT, Andy Beal...Who is he?".

SBRaments, lol.

[operaman]

It's not incorrect at all. Expected growth >>>>>>>> expected value

So you can't think of any examples where it would be correct
for a person to take the wager you thought I was advocating?

[donjuan]

I'd love to hear one.

[operaman]

I'd love to hear one.

Player has terminal cancer and will die in about 1 month.
He is intending to give all his money to a large charity.

[donjuan]

Optimal play for player is to spend time with friends and family. Even so, it would depend on the marginal utility of each dollar donated, among other things.

[operaman]

Optimal play for player is to spend time with friends and family. Even so, it would depend on the marginal utility of each dollar donated, among other things.

What does spending time with family have to do with anything we are talking about?

The reason "LARGE" was used was to get rid of any marginal utility issues.

[donjuan]

It has to do with you selecting an example that is so far removed from the theme of this thread and your above post about the poll on 2p2 that it's laughable.

There are still going to be marginal utility issues no matter how large the charity is. Unless the charity is so large that the money donated is of no consequence and is thus a waste of money in the first place.

[operaman]

It has to do with you selecting an example that is so far removed from the theme of this thread and your above post about the poll on 2p2 that it's laughable.

There are still going to be marginal utility issues no matter how large the charity is. Unless the charity is so large that the money donated is of no consequence and is thus a waste of money in the first place.

So you are using strawman AND weakman arguments?
The example I gave you is just one of a myriad of situations
that would present a logical option in favor of taking the bet.


If you feel the choice to refuse this bet is absolute then at what % would you say it might be good to take the bet?
95% 99% 99.999%?

Are you a ghost of curious or something?

[donjuan]

If you feel the choice to refuse this bet is absolute then at what % would you say it might be good to take the bet?
95% 99% 99.999%?

If you don't understand expected growth and the Kelly criterion, please read Ganchrow's 2 excellent introductory posts to the topic:

http://www.sportsbookreview.com/forum/handicappe...on-part-i.html
http://www.sportsbookreview.com/forum/players-ta...n-part-ii.html

Basically any time you bet 100% of your bankroll on something that has a non-100% chance of happening you will have expected growth of -100%. Now if you use the Kelly calculator on this site, you may find that if you put enough 9s on the end of 99.999 you will get the calculator to come up with a Kelly stake that is 100%. I am fairly certain that this is simply due to rounding as you can't bet .99999 cents.

[operaman]

If you don't understand expected growth and the Kelly criterion, please read Ganchrow's 2 excellent introductory posts to the topic:

http://www.sportsbookreview.com/forum/handicappe...on-part-i.html
http://www.sportsbookreview.com/forum/players-ta...n-part-ii.html

Basically any time you bet 100% of your bankroll on something that has a non-100% chance of happening you will have expected growth of -100%. Now if you use the Kelly calculator on this site, you may find that if you put enough 9s on the end of 99.999 you will get the calculator to come up with a Kelly stake that is 100%. I am fairly certain that this is simply due to rounding as you can't bet .99999 cents.

So if I understand you correctly, you say the following bet is illogical to take?

Person earns 100,000 / year
40 years to live
4 million for simplicity
death upon loss (wasn't sure how to deal with loss of life earnings)
99% chance of winning

4M/Death 99% to win?


p.s. Yes I know about kelly.

[operaman]

There are still going to be marginal utility issues no matter how large the charity is. Unless the charity is so large that the money donated is of no consequence and is thus a waste of money in the first place.

Marginal utility could be negative in some situations?
i.e. you are in desert for 3 days without food or water. 1 can of soda is of little value.

[MrX]

I'm not sure why we're discussing such contrived situations.

Anyone who is a full-time +EV gambler with a small enough bankroll that his bills are close to the same size as his expected earnings would most likely want to take this bet.

I disagree with DonJuan (maybe a first) that only a moron would take that bet. Although, anyone who thinks it is always (or even usually) correct to do so is certainly in the moron neighborhood.

Especially in the days before online poker, a lot of young +EV poker players would find themselves in a situation where it was nearly impossible to save enough of their earnings to build a bankroll and move up in limits. Such a player was left with the options of getting a real job until they saved a more appropriate bankroll or "taking a shot."

"Taking a shot" is just another name for temporarily over-betting Kelly in an attempt to increase bankroll. If losing your bankroll isn't the end of world (you can get a job), taking a shot can definitely be a non-moron's preference.

Their were times early in my career that I would have jumped at 85% +100 for my bankroll, and it wouldn't have been out of stupidity. I was just in that situation.

Now I wouldn't even consider 99.5% at +100.

[donjuan]

Mr. X,

I have always used and understood the word bankroll to mean a source of money that cannot easily be replenished. If it is easily replenished, what they are saying is their bankroll is not really their full bankroll. It may be semantics but how you define bankroll is important to the question. In the case you propose, that may be right. I guess it could also be right if your utility curve was so messed up that having $100k or $0 made little difference to you but having $200k was far better than $100k.

[Ganchrow]

donjuan really gets it right is his last post. (Although I do disagree with his effective labelling of certain preferences illogical in his earlier posts. While people with more standard utility functions might find a certain course of actions suboptimal, there's why such might not make perfect sense given some other atypical preference set).

As I've mentioned on this forum before the term "bankroll" as it relates to Kelly, doesn't precisely correspond to the word "bankroll" as used in common parlance.

A Kelly bankroll refers to the entirety of all one's assets (financial and otherwise), including all mortgageable and stealable assets, as well as the present value of the future stream of one's income. If losing everything implies infinitely negative utility then losing everything needs to be a fate as generally unfavorable as death.

For full-Kelly to correspond to the maximization of expected growth, losing one's entire bankroll needs to correspond to a state from which no recovery is ever possible. (The problem implied here is that Bankroll represents an expectation that also needs to also account for the utility of non-monetary considerations). For example, if I can steal $100,000 from the local bank and have a 90% probability of being caught and sent to jail for 25 years (a fate likely better death) ... what's the value of that "asset" within Kelly? Clearly such value will change drastically as a function of current bankroll. (It'd be worth more to me, for example, if I'd already sold my kidneys and leased my wife into slavery.) As long as I'm willing to attempt that robbery then my bankroll isn't really zero even after exhausting every other option. An academic point, to be sure.

As such, the all-in analogy with AA's for the entirety of one's "bankroll" is invalid. If one's able to build up a bankroll again (whether through begging, borrowing, blowing, or stealing) after losing then that wasn't really one's entire "Kelly bankroll" to begin with. It might have been a "bankroll" in the vernacular sense ... but not in the Kelly sense.

[Arilou]

I don't think it's academic at all; sensible poker bankrolls and the same person's true Kelly bankroll are very different. At one point I did have the vast majority of my money in my bankroll, but vast majority is very different from all. Still, I would have taken the wager for everything most likely, assuming I wasn't worried about cheating, for the simple reason that I consider my human capital to be a substantial portion of my value and that's not going to be on the line even if you theoretically cut out all my potential credit lines.

[operaman]

As I've mentioned on this forum before the term "bankroll" as it relates to Kelly, doesn't precisely correspond to the word "bankroll" as used in common parlance.

A Kelly bankroll refers to the entirety of all one's assets (financial and otherwise), including all mortgageable and stealable assets, as well as the present value of the future stream of one's income. If losing everything implies infinitely negative utility then losing everything needs to be a fate as generally unfavorable as death.

In the kelly literature I have read, the term "bank roll" is just a lump of money(weath, assets, cash). It stipulates nothing about the future earnings of the player as it doesn't address the player at all. Ganch, if I am wrong about this please illuminate me where you are getting the idea of future earnings being part of kelly.

Kelly doesn't tell YOU how to bet to max growth, but tells your BANK ROLL how to bet.

Kelly is dependant on being able to size ones bets.
The all-in choice is an all or nothing decision with sizing off the table.

It doesn't handle the situation where one has the possibility of an outside source of money.

It is not well suited for finite timelines.

[Ganchrow]

In the kelly literature I have read, the term "bank roll" is just a lump of money(weath, assets, cash). It stipulates nothing about the future earnings of the player as it doesn't address the player at all. Ganch, if I am wrong about this please illuminate me where you are getting the idea of future earnings being part of kelly.

Kelly doesn't tell YOU how to bet to max growth, but tells your BANK ROLL how to bet.

This isn't quite right.

Kelly was ostensibly the 1950s-era brainchild of one John L. Kelly of Bell Labs.

In reality, however, the theoretical underpinnings of Kelly had already been developed by Bernoulli in the 18th century (whose methodology of the maximization of geometric mean was ultimately quite similar to Kelly’s) and then later formalized within the context of the isoelastic utility and constant elasticity of substitution production within microeconomics (I’d have to really hunt around to figure out it its true originator – Keynes, perhaps?)

Although Kelly most likely developed his criterion without prior knowledge of the existing work on the subject, it's nevertheless only original in the sense that he explicitly applied it to gambling (and only gambling -- horse betting in particular).

Kelly's derivation (and you can download his original paper in PDF format here) while extremely clever, is clearly not the work of an economist. Kelly doesn't derive a solution by appealing to the preferences of a gambler but rather by assuming that maximizing expected bankroll growth is a self-evident logical goal in and of of itself for any bettor able to reinvest profits. That such a strategy might not be applicable for certain classes of risk averse bettor never enters in to Kelly’s discussion.

As such Kelly doesn't touch on “fractional Kelly” (which doesn't have quite so intuitive an explanation as simply "maximizing expected bankroll growth") and instead talks only about growing generic "capital" or "wealth" exponentially. The precise meanings of these two phrases are never explicitly defined and I suppose are just meant to be taken axiomatically. Nevertheless, by examining Kelly from an economic perspective we can get a clearer picture.

The n-Kelly functions are the sole members of a class of a set of utility functions known as isoelastic utility. These are functions depending on a single parameter α and are of the form:

U(x;α) = α/(α-1) * x ^(α-1)/α for α ≠ 1, > 0
U(x;α) = ln(x) for α = 1

so that the marginal utility (i.e., the first derivative of utility with respect to x, corresponding to the increase in utility of obtaining slightly more of x) will equal:

U_x = x^-1/α for α > 0

One feature of this class of function is that they exhibit what's known as "constant relative risk aversion". This means that regardless of any increase or decrease in bankroll a player's aversion to risk (his dislike for outcome uncertainty) will remain unchanged when evaluating a given percentage change in bankroll (constant absolute risk aversion, on the other hand would correspond to equivalent aversion to risk for a given dollar change).

Putting it another way, if two individuals both exhibit isoelastic utility, each with the same α parameter, then each will treat an x% change in bankroll identically irrespective of their relative bankroll levels. In other words, if I have $10,000 to my name and you have a $1,000,000 then an additional $100 would increase my happiness by the same amount as an additional $10,000 would increase yours.

Now whether you agree with the above depiction or not is largely irrelevant. These are the utility functions implied by Kelly (with α representing the so-called "Kelly fraction") and if this depiction seems inapplicable to your personal preferences, then while that may well be a trait that you share with most human beings, it in no way implies the inefficacy of n-Kelly at maximizing the utility of a hypothetical individual whose preferences are in line with isoelastic utility.

As I alluded to earlier (and have mentioned countless times on this forum), one by-product of full-Kelly maximization is that it will also maximize the expected growth rate of a (still axiomatic) bankroll. This means that over an arbitrarily long stretch of time the probability that bankroll being bet with Kelly will be larger than a bankroll bet using some other strategy will approach certainty (100%)..

While this may be a by-product of isoelastic utility, it was nevertheless the precise and only effect intended by Kelly. Kelly didn't consider any of the other implications of this mode of thought, nor did he even consider its applicability. He simply made the a priori declaration that such a strategy was effectively preference-independent (I mean who wouldn't want to maximize bankroll growth? Duh.) Now that in no way is meant to marginalize the contributions of Dr. Kelly. The guy wasn't an economist --he just derived his results from the perspective of maximizing expected growth, rather than from the perspective of maximizing a broader class of utility function.

So really there are two schools of thought from which to consider Kelly:

1. the John L. Kelly/Bernoulli manner -- Kelly is the strategy maximizing the expected growth of a bankroll. Fractional Kelly, while not explicitly defined, can be considered the full-Kelly stake multiplied by the Kelly fraction. The fractional-Kelly stake may not be well-defined for all classes of bets, nor may all questions of Klely-optimality be easily answered.
2. the utility function manner -- Kelly is the strategy maximizing the utility function for a bettor with Kelly risk aversion parameter α. The fractional Kelly stake will always be well-defined for any class of bets, and in general, for small stakes, will closely mirror the fractional Kelly stake defined in 1) above. This methodology is readily extensible to include all forms of wealth manipulation.

So what I’m going to argue is that neither of these interpretations is consistent (in most cases) with an easily replenished bankroll.

In the Kelly/Bernoulli school the Ultimate Goal of Existence is getting as rich as possible as deliberately and as quickly as possible. Losing one's entire is infinitely bad (log(0) ~ -∞) because once one's bankroll drops to 0 it's impossible for it to grow ever again. It's not just bad, or very bad, or really bad, really really very very bad, but infinitely, no possible to do any worse than this, Book of Revelations bad. Now that's bad.

Even were a bettor's wealth to drop to only $0.01, then as long as she could continue making bets (i.e., there are no betting minimums) then she still has a possibility of growing it again (and in fact will do so with probability approaching 1 over an increasing time horizon). So while this may be "bad" it's still infinitely better than losing that last penny from which no recovery is possible.

Within this interpretation it's not about being happy or sad but about the ability to grow one's bankroll. If however, another source of betting capital existed (for example, you just needed to transfer more money from your checking account) then losing that penny is necessarily stripped of its "God Almighty Has Struck You Down" bad-ness. While going broke in this limited sense might "suck" (or even "suck really, really hard") it does not represent a situation from which no recovery is possible and as such can not correspond to a negative Kelly-score.

In the microeconomics Kelly-as-isoelastic-utility School the Goal of Existence is to maximize a utility function for no existential purpose other than "that's what people do". Bankroll growth is a necessary and sufficient by-product of full-Kelly maximization (like love and marriage) but does not follow from fractional-Kelly maximization (for fractions other than 1).

Losing one's entire bankroll is again infinitely bad (for Kelly mulitiples ≤ 1) because marginal utility grows unboundedly larger as one's bankroll approaches zero. As such, a Kelly bettor will never risk any possibility of going broke (regardless of current wealth) and such an outcome would result in "sadness unyielding".

What's must illustrative, however, is the limiting case where win probability of a given bet (holding payout odds constant) approaches 100%. What we see is that as the probability of winning a bet increases (and we're not just talking from 90% to 95%, but more like from 99.9999999% to 99.99999999%) a bettor will become increasingly more willing to bet every dollar on which he can get his hands. Borrow from the in-laws, borrow from your in-laws’ in-laws, raid your daughter's college fund, second and third mortgage your house, second and third mortgage your granddaughter’s virginity, etc.)

This provides the clearest indication of what a "complete Kelly-bankroll" really is. If the consequences of losing sound disastrous well that's precisely the point. As I mentioned before, losing a bankroll can't just suck, it has to suck infinitely. It has to suck so bad that the only time that a bettor would ever be willing to risk all of it would be if he possessed 100% certainty that he'd win.

Now I fully grant that this interpretation might not be computationally convenient. But that’s beside the point. Convenience and intellectual probity, after all, don’t necessarily go hand-in-hand. From an operational perspective what should be taken from this is that a Kelly bankroll should be maximally construed (and remember that this doesn't take into account withdrawals from one's bankroll for the purposes of living expenses) and the real fudging should come in the form of adjusting one's Kelly multiplier.

If you're (truly) a professional bettor with a $1,000,000 betting bankroll and $3,000,000 in savings you're probably better off assuming a $4MM bankroll with around a 5% Kelly fraction than a $1MM bankroll with around a 20% Kelly fraction (and in this simplified analysis we're admittedly neglecting the costs of transferring money from savings to betting accounts, as well as the opportunity costs (in terms of interest, dividends, capital appreciation, etc.) of removing capital from savings). Most of the time the difference between the two will be minimal, however where the contrast may well become apparent would be in times of several exceptionally good bets or during a period where a huge number of fairly good bets come along (such as, perhaps, during the Super Bowl or March Madness).

As I've said before this is certainly a rather academic point, but that doesn't detract from its representing an unsullied view of the Kelly criterion. While any given individual’s personal preferences may well deviate substantially from those implied by Kelly, that in no way implies those preference more indicative of Kelly than those I’ve outline above.

I will grant, however, that there are some circumstances where within Kelly "wealth" may be narrowly construed. These situation will follow from scenarios such as those to which you allude where a bankroll is taken in a vacuum. While such a scenario could fit in with the Kelly-Bernoulli interpretation where one's trying to grow an isolated set of capital for no particular reason (if it were general wealth enhancement which one were after, then isolating the bankroll would be suboptimal) it's difficult to see how it could fit in with general concepts of utility maximization in all but the most of contrived of examples.

All I can think of would involve two individuals competing in a bankroll growth contest for a fixed prize given some arbitrary starting values. Obviously, in such a contest considering any other wealth not available to the contest itself would be meaningless.

The isoelastic utility view of Kelly is simply an extension of the very well-defined preference-theory of financial economics. And the only straightforward way for the conclusions of Kelly to match the implications stated isoelastic preferences is to construe bankroll/wealth/capital in a certain manner. This isn't necessarily most convenient (or least dangerous for a self-deluding "advantage" bettor) but it is nevertheless most accurate and in keeping with underlying theory.

Kelly is dependant on being able to size ones bets.
The all-in choice is an all or nothing decision with sizing off the table.

It doesn't handle the situation where one has the possibility of an outside source of money.

It is not well suited for finite timelines.

While all these criticisms of Kelly might be difficult to refute utilizing the narrow Kelly/Bernoulli interpretation, they simply become issues of increasing computational complexity when the Kelly criterion is considered as isoelastic utility maximization. I believe I’ve touched on all these concepts on this forum.

[RickySteve]

It's alright when you're this guy:

[Data]

Ganchrow, you bring up an interesting point. I personally cannot agree with it, though. While talking about bankroll, you mix business and personal life and, while some people may like this, in fact, that approach will not let you achieve many things that the money cannot buy.

Once you separate your personal life from your business all the drama that came up in your post disappears. I think that borrowing money from in-laws just does not belong on the business side. Say, you are a hedge fund manager. The fund does not have in-laws to borrow from but does have a "bankroll". So, why not to think about your personal bankroll along these lines?

[square1]

This isn't quite right.

Kelly was ostensibly the 1950s-era brainchild of one John L. Kelly of Bell Labs.

In reality, however, the theoretical underpinnings of Kelly had already been developed by Bernoulli in the 18th century (whose methodology of the maximization of geometric mean was ultimately quite similar to Kelly’s) and then later formalized within the context of the isoelastic utility and constant elasticity of substitution production within microeconomics (I’d have to really hunt around to figure out it its true originator – Keynes, perhaps?)

Although Kelly most likely developed his criterion without prior knowledge of the existing work on the subject, it's nevertheless only original in the sense that he explicitly applied it to gambling (and only gambling -- horse betting in particular).

Kelly's derivation (and you can download his original paper in PDF format here) while extremely clever, is clearly not the work of an economist. Kelly doesn't derive a solution by appealing to the preferences of a gambler but rather by assuming that maximizing expected bankroll growth is a self-evident logical goal in and of of itself for any bettor able to reinvest profits. That such a strategy might not be applicable for certain classes of risk averse bettor never enters in to Kelly’s discussion.

As such Kelly doesn't touch on “fractional Kelly” (which doesn't have quite so intuitive an explanation as simply "maximizing expected bankroll growth") and instead talks only about growing generic "capital" or "wealth" exponentially. The precise meanings of these two phrases are never explicitly defined and I suppose are just meant to be taken axiomatically. Nevertheless, by examining Kelly from an economic perspective we can get a clearer picture.

The n-Kelly functions are the sole members of a class of a set of utility functions known as isoelastic utility. These are functions depending on a single parameter α and are of the form:

U(x;α) = α/(α-1) * x ^(α-1)/α for α ≠ 1, > 0
U(x;α) = ln(x) for α = 1

so that the marginal utility (i.e., the first derivative of utility with respect to x, corresponding to the increase in utility of obtaining slightly more of x) will equal:

U_x = x^-1/α for α > 0

One feature of this class of function is that they exhibit what's known as "constant relative risk aversion". This means that regardless of any increase or decrease in bankroll a player's aversion to risk (his dislike for outcome uncertainty) will remain unchanged when evaluating a given percentage change in bankroll (constant absolute risk aversion, on the other hand would correspond to equivalent aversion to risk for a given dollar change).

Putting it another way, if two individuals both exhibit isoelastic utility, each with the same α parameter, then each will treat an x% change in bankroll identically irrespective of their relative bankroll levels. In other words, if I have $10,000 to my name and you have a $1,000,000 then an additional $100 would increase my happiness by the same amount as an additional $10,000 would increase yours.

Now whether you agree with the above depiction or not is largely irrelevant. These are the utility functions implied by Kelly (with α representing the so-called "Kelly fraction") and if this depiction seems inapplicable to your personal preferences, then while that may well be a trait that you share with most human beings, it in no way implies the inefficacy of n-Kelly at maximizing the utility of a hypothetical individual whose preferences are in line with isoelastic utility.

As I alluded to earlier (and have mentioned countless times on this forum), one by-product of full-Kelly maximization is that it will also maximize the expected growth rate of a (still axiomatic) bankroll. This means that over an arbitrarily long stretch of time the probability that bankroll being bet with Kelly will be larger than a bankroll bet using some other strategy will approach certainty (100%)..

While this may be a by-product of isoelastic utility, it was nevertheless the precise and only effect intended by Kelly. Kelly didn't consider any of the other implications of this mode of thought, nor did he even consider its applicability. He simply made the a priori declaration that such a strategy was effectively preference-independent (I mean who wouldn't want to maximize bankroll growth? Duh.) Now that in no way is meant to marginalize the contributions of Dr. Kelly. The guy wasn't an economist --he just derived his results from the perspective of maximizing expected growth, rather than from the perspective of maximizing a broader class of utility function.

So really there are two schools of thought from which to consider Kelly:

1. the John L. Kelly/Bernoulli manner -- Kelly is the strategy maximizing the expected growth of a bankroll. Fractional Kelly, while not explicitly defined, can be considered the full-Kelly stake multiplied by the Kelly fraction. The fractional-Kelly stake may not be well-defined for all classes of bets, nor may all questions of Klely-optimality be easily answered.
2. the utility function manner -- Kelly is the strategy maximizing the utility function for a bettor with Kelly risk aversion parameter α. The fractional Kelly stake will always be well-defined for any class of bets, and in general, for small stakes, will closely mirror the fractional Kelly stake defined in 1) above. This methodology is readily extensible to include all forms of wealth manipulation.

So what I’m going to argue is that neither of these interpretations is consistent (in most cases) with an easily replenished bankroll.

In the Kelly/Bernoulli school the Ultimate Goal of Existence is getting as rich as possible as deliberately and as quickly as possible. Losing one's entire is infinitely bad (log(0) ~ -∞) because once one's bankroll drops to 0 it's impossible for it to grow ever again. It's not just bad, or very bad, or really bad, really really very very bad, but infinitely, no possible to do any worse than this, Book of Revelations bad. Now that's bad.

Even were a bettor's wealth to drop to only $0.01, then as long as she could continue making bets (i.e., there are no betting minimums) then she still has a possibility of growing it again (and in fact will do so with probability approaching 1 over an increasing time horizon). So while this may be "bad" it's still infinitely better than losing that last penny from which no recovery is possible.

Within this interpretation it's not about being happy or sad but about the ability to grow one's bankroll. If however, another source of betting capital existed (for example, you just needed to transfer more money from your checking account) then losing that penny is necessarily stripped of its "God Almighty Has Struck You Down" bad-ness. While going broke in this limited sense might "suck" (or even "suck really, really hard") it does not represent a situation from which no recovery is possible and as such can not correspond to a negative Kelly-score.

In the microeconomics Kelly-as-isoelastic-utility School the Goal of Existence is to maximize a utility function for no existential purpose other than "that's what people do". Bankroll growth is a necessary and sufficient by-product of full-Kelly maximization (like love and marriage) but does not follow from fractional-Kelly maximization (for fractions other than 1).

Losing one's entire bankroll is again infinitely bad (for Kelly mulitiples ≤ 1) because marginal utility grows unboundedly larger as one's bankroll approaches zero. As such, a Kelly bettor will never risk any possibility of going broke (regardless of current wealth) and such an outcome would result in "sadness unyielding".

What's must illustrative, however, is the limiting case where win probability of a given bet (holding payout odds constant) approaches 100%. What we see is that as the probability of winning a bet increases (and we're not just talking from 90% to 95%, but more like from 99.9999999% to 99.99999999%) a bettor will become increasingly more willing to bet every dollar on which he can get his hands. Borrow from the in-laws, borrow from your in-laws’ in-laws, raid your daughter's college fund, second and third mortgage your house, second and third mortgage your granddaughter’s virginity, etc.)

This provides the clearest indication of what a "complete Kelly-bankroll" really is. If the consequences of losing sound disastrous well that's precisely the point. As I mentioned before, losing a bankroll can't just suck, it has to suck infinitely. It has to suck so bad that the only time that a bettor would ever be willing to risk all of it would be if he possessed 100% certainty that he'd win.

Now I fully grant that this interpretation might not be computationally convenient. But that’s beside the point. Convenience and intellectual probity, after all, don’t necessarily go hand-in-hand. From an operational perspective what should be taken from this is that a Kelly bankroll should be maximally construed (and remember that this doesn't take into account withdrawals from one's bankroll for the purposes of living expenses) and the real fudging should come in the form of adjusting one's Kelly multiplier.

If you're (truly) a professional bettor with a $1,000,000 betting bankroll and $3,000,000 in savings you're probably better off assuming a $4MM bankroll with around a 5% Kelly fraction than a $1MM bankroll with around a 20% Kelly fraction (and in this simplified analysis we're admittedly neglecting the costs of transferring money from savings to betting accounts, as well as the opportunity costs (in terms of interest, dividends, capital appreciation, etc.) of removing capital from savings). Most of the time the difference between the two will be minimal, however where the contrast may well become apparent would be in times of several exceptionally good bets or during a period where a huge number of fairly good bets come along (such as, perhaps, during the Super Bowl or March Madness).

As I've said before this is certainly a rather academic point, but that doesn't detract from its representing an unsullied view of the Kelly criterion. While any given individual’s personal preferences may well deviate substantially from those implied by Kelly, that in no way implies those preference more indicative of Kelly than those I’ve outline above.

I will grant, however, that there are some circumstances where within Kelly "wealth" may be narrowly construed. These situation will follow from scenarios such as those to which you allude where a bankroll is taken in a vacuum. While such a scenario could fit in with the Kelly-Bernoulli interpretation where one's trying to grow an isolated set of capital for no particular reason (if it were general wealth enhancement which one were after, then isolating the bankroll would be suboptimal) it's difficult to see how it could fit in with general concepts of utility maximization in all but the most of contrived of examples.

All I can think of would involve two individuals competing in a bankroll growth contest for a fixed prize given some arbitrary starting values. Obviously, in such a contest considering any other wealth not available to the contest itself would be meaningless.

The isoelastic utility view of Kelly is simply an extension of the very well-defined preference-theory of financial economics. And the only straightforward way for the conclusions of Kelly to match the implications stated isoelastic preferences is to construe bankroll/wealth/capital in a certain manner. This isn't necessarily most convenient (or least dangerous for a self-deluding "advantage" bettor) but it is nevertheless most accurate and in keeping with underlying theory.

While all these criticisms of Kelly might be difficult to refute utilizing the narrow Kelly/Bernoulli interpretation, they simply become issues of increasing computational complexity when the Kelly criterion is considered as isoelastic utility maximization. I believe I’ve touched on all these concepts on this forum.

Honestly, I'm a little taken aback to see ANYONE defend this position, much less someone who's obviously better-versed in both economic and gambling literature than I am.

Nevertheless, I will try my best to refute it.

Utility theory is built on consumption, not wealth. The only purpose of wealth per se is to transfer consumption across time periods. Of course one wishes to grow wealth when possible, but only because that enables one to consume more later.

Applying isoelastic utility, or any univariate utility function, to financial economics is of course standard practice. But I think it incorporates as an assumption that the agent has already made a decision to consume some portion of his wealth today, and save (invest) the rest to enable consumption tomorrow. Thus the agent's portfolio (read: bankroll) is taken as given not because that's everything he owns, plus whatever he can "beg borrow or steal"; rather, it is everything he decided not to consume. And his utility function represents his preferences regarding risk - that is, how much consumption will he give up in a future world where he has a lot to gain a bit more in a future world where he has only a little.

And by "consume" we mean not only direct purchases like Mercedes-Benz's and trips to Tahiti, but also choosing to forgo opportunities to acquire wealth, such as mortgaging your granddaughter's virginity (!).

While you acknowledge withdrawals for living expenses, you seem to think of them as an afterthought; when in reality, "living expenses" are the only reason anyone is interested in maximizing anything bankroll-related at all.

What does this have to with Ganchrow's post? Just this: Trying to justify Kelly via isoelastic utility is flawed, not only because utility may not be isoelastic, but much more importantly, wealth itself has no utility at all. Therefore, we have no reason at all to believe the goal of maximizing the growth of expected wealth will coincide with the true goal of maximizing expected utility.

Thus, we are left with only No. 2, the "narrower" Kelly/Bernoulli interpretation. Here, I again have to disagree with Ganchrow. He argues losing one's entire bankroll is the worst thing that could possible happen, as shown by the fact that the log function is unbounded in the negative direction as one's bankroll tends to zero. But remember Kelly himself did not use utility functions. He did not, to my knowledge, address just how bad it would suck to lose one's bankroll. The only significance the lack of a lower bound has is that it implies a user of the Kelly Criterion will NEVER assign a positive probability to losing the entire bankroll.

What, then, is your "Kelly bankroll"? I would argue it is the amount of money you would be willing to lose betting on sports before you stopped. So losing your roll - theoretically impossible though it is - would not represent the end of the world; simply the end of your sports gambling. This is consistent with Ganchrow's argument that even if your roll dwindles to $0.01, you can still rebuild it - but if it goes to zero, you are absolutely done.

I want to emphasize that the ONLY point on which I am disagreeing is the size of the bankroll. All of the mathematical properties of the Kelly Criterion that Ganchrow has so eloquently and clearly illustrated in his many useful and informative posts are absolutely correct. In my limited experience, most skilled advantage bettors use Kelly, or some variant of it. If I were convinced that I could consistently and accurately estimate probabilities, I would use some variant of fractional-Kelly myself (but I'm not).

[Ganchrow]

Utility theory is built on consumption, not wealth. The only purpose of wealth per se is to transfer consumption across time periods. Of course one wishes to grow wealth when possible, but only because that enables one to consume more later.

Applying isoelastic utility, or any univariate utility function, to financial economics is of course standard practice. But I think it incorporates as an assumption that the agent has already made a decision to consume some portion of his wealth today, and save (invest) the rest to enable consumption tomorrow. Thus the agent's portfolio (read: bankroll) is taken as given not because that's everything he owns, plus whatever he can "beg borrow or steal"; rather, it is everything he decided not to consume. And his utility function represents his preferences regarding risk - that is, how much consumption will he give up in a future world where he has a lot to gain a bit more in a future world where he has only a little.

And by "consume" we mean not only direct purchases like Mercedes-Benz's and trips to Tahiti, but also choosing to forgo opportunities to acquire wealth, such as mortgaging your granddaughter's virginity (!).

While you acknowledge withdrawals for living expenses, you seem to think of them as an afterthought; when in reality, "living expenses" are the only reason anyone is interested in maximizing anything bankroll-related at all.

Youll get no real argument from here. (Although I'd note that utility theory doesn't need to be built on consumption ... it could just as easily be built on wealth, health, the happiness of others, or the average number of women that say "hi" to you in a given day. There exist misers, for example, for whom consumption comprises but a minor portion of utility.)

Just realize that while the flavor of utility function you've outlined would doubtlessly be better in practice for most (if not all) humans than "real" Kelly, this just highlights the fact that Kelly is less than ideal for real-world applications. So we can certainly try to improve on Kelly but doing so as you've suggested (by limiting bankroll to a constant fraction of wealth does not necessarily make sense microeconomically. The question I'd pose to you would be this: What well-behaved utility function would imply your conclusion? Answer: None that I can think of.

Now I'm not saying that we couldn't design a utility function that took consumption into account ... just that for it to be well-behaved it would be considerably more complicated than limited-bankroll Kelly.

Trying to justify Kelly via isoelastic utility is flawed, not only because utility may not be isoelastic, but much more importantly, wealth itself has no utility at all. Therefore, we have no reason at all to believe the goal of maximizing the growth of expected wealth will coincide with the true goal of maximizing expected utility.

That Kelly utility is isoelastic utility of wealth is fundamentally true by the definition. Argue that isoelastic of wealth is an imperfect gauge of real world utility all you like (and you'd be right) .. but that doesn't change what Kelly is.

Thus, we are left with only No. 2, the "narrower" Kelly/Bernoulli interpretation. Here, I again have to disagree with Ganchrow. He argues losing one's entire bankroll is the worst thing that could possible happen, as shown by the fact that the log function is unbounded in the negative direction as one's bankroll tends to zero. But remember Kelly himself did not use utility functions. He did not, to my knowledge, address just how bad it would suck to lose one's bankroll. The only significance the lack of a lower bound has is that it implies a user of the Kelly Criterion will NEVER assign a positive probability to losing the entire bankroll.

The aim of the methodology outlined in John Kelly paper's was to maximize the expected growth rate of a bankroll over time. This is 100% functionally equivalent (i.e., implies and is implied by) maximizing expected log utility.

What, then, is your "Kelly bankroll"? I would argue it is the amount of money you would be willing to lose betting on sports before you stopped. So losing your roll - theoretically impossible though it is - would not represent the end of the world; simply the end of your sports gambling. This is consistent with Ganchrow's argument that even if your roll dwindles to $0.01, you can still rebuild it - but if it goes to zero, you are absolutely done.

The points you're making aren't invalid and represent well-articulated and legitimate criticisms of the Kelly Criterion. But Kelly is itself just a theory and if our goal is to better understand it, then a big part of that is realizing its limitations. In the real world I'd never suggest anyone use his entire discounted net worth as his Kelly bankroll (and not just because that would imply a linearity that didn't exist), but nevertheless we should still be cognizant of the fact that Kelly necessarily describes a set of preferences. If those preferences don't apply to me or you or even to anyone we can't just go back and say, "Oh no Kelly doesn't really imply this." No we just need to accept that Kelly is imperfect.

An analogy might be Newtonian mechanics, which tells us that the kinetic energy of a body is equal to ½mv². Special relativity, however, tells us that while this is an excellent approximation for slow-moving bodies it is in reality only the first term of a Taylor expansion of a more involved energy equation. So what do we do? Do we say, "Oh no, Newtonian mechanics doesn't really imply this"? No. We simply accept the fact that while Newtonian mechanics is exceedingly useful, it is what it is -- an approximation of real world phenomena.

Anyway, an excellent post.

[square1]

Just realize that while the flavor of utility function you've outlined would doubtlessly be better in practice for most (if not all) humans than "real" Kelly, this just highlights the fact that Kelly is less than ideal for real-world applications. So we can certainly try to improve on Kelly but doing so as you've suggested (by limiting bankroll to a constant fraction of wealth does not necessarily make sense microeconomically. The question I'd pose to you would be this: What well-behaved utility function would imply your conclusion? Answer: None that I can think of.

Now I'm not saying that we couldn't design a utility function that took consumption into account ... just that for it to be well-behaved it would be considerably more complicated than limited-bankroll Kelly.

I'm not sure we're on the same page yet. The classic consumer's problem is to maximize the expected value of the summation over all t of

d^t * u[c(t)]

given w(0) and subject to the constraints:
c(t) + i(t) <= w(t) t=0,1,2...
w(t+1) = i(t) * X(t) t=0,1,2...

where c is consumption, 0 < d < 1 is a discount factor, i is investment, w is wealth, and X is a stochastic variable representing return on investment.

A great deal depends on how we structure X. But if we assume X(t) is iid for all t, then the consumer's problem is identical in every period, with only the initial wealth changing from period to period. Hence, both present consumption c(0) and present investment i(0) = w(0) - c(0) will be functions exclusively of initial wealth (for a given pdf of X, that is). Obviously future consumption/investment depends on the realization of X.

(For details on solving these sorts of problems, I'll have to refer the reader to Sargent's Dynamic Macroeconomic Theory or Stokey and Lucas's Recursive Methods in Economic Dynamics. I was never that good to begin with, and what skill I did have has rusted away after years of disuse. Be prepared for Bellman Equations and Envelope Criteria).

But the intuition is not too difficult. The consumer will evaluate the how lucrative the investment is (look at the distribution of X). If X is composed of many different events (such as a sequence of n sporting events the consumer may wager on), the consumer will have to choose an investment strategy, as well as an investment amount. Under some circumstances (if n is sufficiently large to overcome the consumer's risk aversion), the consumer may well choose full-Kelly as an investment strategy. This absolutely does not imply that the consumer will choose his entire wealth as the investment amount. Indeed, if the consumer has logarithmic utility, he will NEVER invest all of his wealth. In short, limiting bankroll to a fraction of one's wealth CAN make sense microeconomically, and it has very little dependence on the particular functional form of u. Instead, it depends heavily on modelling the problem as maximizing the utility of consumption versus maximizing the utility of wealth. And I think we have agreed the former is far more realistic.

Of course there are lots of reasonable utility functions and distributions of X for which the consumer will NOT choose full-Kelly. But theoretical justification of partial-wealth Kelly simply isn't an exercise is esotericism - it's fairly straightforward. The key is to think about how we want to set up the problem, and to realize the only purpose of wealth is to enable future consumption.

That Kelly utility is isoelastic utility of wealth is fundamentally true by the definition. Argue that isoelastic of wealth is an imperfect gauge of real world utility all you like (and you'd be right) .. but that doesn't change what Kelly is.

The aim of the methodology outlined in John Kelly paper's was to maximize the expected growth rate of a bankroll over time. This is 100% functionally equivalent (i.e., implies and is implied by) maximizing expected log utility.

Maybe what's happening here is that we have different definitions of "Kelly". To me, Kelly is a decision rule used to determine an optimal investment/betting strategy. "Kelly utility" doesn't mean anything to me. Who defined it? Unless I grossly misunderstood your post, we have both agreed Kelly never bothered with utility at all. I agree 100% that one who follows the Kelly criterion will be maximizing the expected value of the log of one's bankroll, and vice versa. This is terrifically convenient from a computational standpoint. But it has no theoretical significance. "Utility" is a convenient analytic representation of preferences. One's preferences for one's bankroll may well be radically different than one's preferences for one's wealth. There is no contradiction here. To go from isoelastic utility for one's bankroll (for the sole purpose of calculating which bets will maximzing expected growth) to isoelastic utility for one's wealth seems like an enormous leap of faith to me.

In summary, I would say this: Kelly-as-utility and/or Kelly-as-wealth-management we have both agreed to be pretty lousy ideas for anyone to follow in practice. But whereas you seem to interpret that as a "flaw" in theoretical Kelly, I think it is an "overextension" of theoretical Kelly-as-investment/bankroll management, which by itself is pretty nifty, though not ideal for all people or situations.

P.S. http://www-stat.wharton.upenn.edu/~s...uelson1979.pdf
is an interesting refutation of Kelly-as-wealth by famous economist Paul Samuelson. It was written using only one-syllable words. (He was sarcastically implying anyone who needs an explanation of what's wrong with Kelly is an idiot, who needs things excessively simplified). No one less respected than him could ever have gotten that published. FWIW, I think he makes some valid points, but I'm not sure why he had such an attitude problem. It's a shame, but as Lisa pointed on the Simpsons where Homer had the crayon removed from his brain: "As intelligence goes up, happiness often goes down. I made a graph. I make a lot of graphs."

[Ganchrow]

the consumer will have to choose an investment strategy, as well as an investment amount. Under some circumstances (if n is sufficiently large to overcome the consumer's risk aversion), the consumer may well choose full-Kelly as an investment strategy. This absolutely does not imply that the consumer will choose his entire wealth as the investment amount. Indeed, if the consumer has logarithmic utility, he will NEVER invest all of his wealth. In short, limiting bankroll to a fraction of one's wealth CAN make sense microeconomically, and it has very little dependence on the particular functional form of u. Instead, it depends heavily on modelling the problem as maximizing the utility of consumption versus maximizing the utility of wealth. And I think we have agreed the former is far more realistic.

Of course there are lots of reasonable utility functions and distributions of X for which the consumer will NOT choose full-Kelly. But theoretical justification of partial-wealth Kelly simply isn't an exercise is esotericism - it's fairly straightforward. The key is to think about how we want to set up the problem, and to realize the only purpose of wealth is to enable future consumption.

OK. Let's try to keep this tractable. Let's work out an example where utility is defined a function of consumption rather than wealth.

Let's imagine a 2 period model where at the start of each period the player decides how much to invest and at the end of the period the investment return is realized and he then determines how much to consume. We'll further assume that each investment is a binary outcome event.

So this gives us the following variables:

Let B = Initial bankroll (we can set this to 1 unit without loss of generality)
Let I1 = Investment in period 1
Let Cw = Consumption in period 1 given period 1 investment was win
Let Cl = Consumption in period 1 given period 1 investment was loss
Let Iw2 = Investment in period 2 given period 1 investment was win
Let Il2 = Investment in period 2 given period 1 investment was loss

Note that period 2 consumption isn't a variable insofar as the player will simply consume his entire bankroll. Putting it another way: "You can't take it with you."

For simplicity we'll assume that both investments are identical, paying out at fractional odds f, and winning with probability p. We'll further assume that period 2 consumption is discounted at a rate of k (so 1 unit of utility in period 2 is worth k times as much as 1 unit of utility now -- for most players k < 1 ).

Period 2 starting bankroll given a win would be:

Bw = B + I1 * f - Cw

And given a loss:

Bl = B - I1 - Cl

So period 2 consumption (in other words ending bankroll after the 2nd investment was realized) given a win/win would be:

Cww = Bw + Iw2 * f

Period 2 consumption given a win followed by a loss would be:

Cwl = Bw - Iw2

Period 2 consumption given a loss followed by a win would be:

Clw = Bl + Il2 * f

Period 2 consumption given a loss/loss would be:

Cll = Bl - Il2

So expected utility looks like this:

E(U) = p * ( U(Cw) + k * ( p * U(Cww) + (1-p) * U(Cwl) ) )
+ (1-p) * ( U(Cl) + k * ( p * U(Clw) + (1-p) * U(Cll) ) )

If we assume logarithmic utility then we know that period 2 investment will necessarily be the player's Kelly stake (you can't take it with you, remember?)

Iw2 = Bw/f * (p*f - (1-p))
Il2 = Bl/f * (p*f - (1-p))

So substituting in:

Cww = (B + I1 * f - Cw) * p * (2f - 1)
Cwl = (B + I1 * f - Cw) * (1-p) * (f+1)/f
Clw = (B - I1 - Cl) * p * (2f - 1)
Cll = (B - I1 - Cl) * (1-p) * (f+1)/f

So this gives us (after setting starting bankroll, B, to 1 unit) the following expected utility as a function of the decision variables Cw, Cl, and I1:

E(U) = p*( log(Cw) + k * ( p * log((1 + I1 * f - Cw) * p * (2f - 1)) + (1-p)*log((1 + I1 * f - Cw) * (1-p) * (f+1)/f) ) )
+ (1-p)*( log(Cl) + k * (p*log((1 - I1 - Cl) * p * (2f - 1)) + (1-p)*log((1 - I1 - Cl) * (1-p) * (f+1)/f) ) )

Differentiating wrt to Cw, Cl, and I1 and setting to zero gives us

0 = p*(k*(-((1 - p)/(1 + f*V - W)) - p/(1 + f*V - W)) + W^(-1))
0 = (1 - p)*(L^(-1) + k*(-((1 - p)/(1 - L - V)) - p/(1 - L - V)))
0 = k*(1 - p)*(-((1 - p)/(1 - L - V)) - p/(1 - L - V)) + k*p*((f*(1 - p))/(1 + f*V - W) + (f*p)/(1 + f*V - W))

Solving then yields:

I1 = ( f*p - (1-p) ) / f
Cw = ( f*I1 + 1 ) / (1+k)
Cl = ( 1 - I1 ) / (1+k)

(I'll leave it as an exercise for the motivated reader to verify that is indeed a global maximum for f*p - (1-p) > 0, in other words for positive edge. I'll also note that this result is contingent on isoelastic utility, so partial Kelly would yield the same results, but another utility function would not.)

Of particular interest is the variable I1 (the amount invested at the start of period 1), which you'll note is simply the Kelly stake based solely on wealth. So in other words targeting consumption in Kelly leaves the solution completely unchanged! The investment amount is even independent of the discount rate. Now of course the more you discount future consumption (i.e., the lower the value of k) the more you'd choose to consume now but the discounting won't effect how much you choose to invest.

Now granted this is a rather simplified general example (although you'd find the same results even if you went out an infinite number of periods for k < 1 -- in other words even without the "You can't take it with you assumption) but the point is clear. Kelly staking of full bankroll is in this model completely consistent with maximizing utility of consumption and inconsistent with partial wealth Kelly maximization.

Maybe what's happening here is that we have different definitions of "Kelly". To me, Kelly is a decision rule used to determine an optimal investment/betting strategy. "Kelly utility" doesn't mean anything to me. Who defined it? Unless I grossly misunderstood your post, we have both agreed Kelly never bothered with utility at all. I agree 100% that one who follows the Kelly criterion will be maximizing the expected value of the log of one's bankroll, and vice versa. This is terrifically convenient from a computational standpoint. But it has no theoretical significance.

Kelly utility is the utility function that both implies and is implied by the conclusions of John Kelly's paper. While it's true that Kelly himself never appealed to the notion of utility per se, he did however make the a priori assumption that an agent acts to maximize the expected rate of growth of bankroll. This both implies and is implied by log utility. You've fully agreed with this.

But the point (and I believe I addressed this in my earlier post) is that losing one's entire bankroll needs to be infinitely bad for the Kelly conclusions to have force of logic. Why else would a player with a readily replenishable bankroll never choose to invest 100% of said bankroll (minimally defined) unless said bet won with certainty? Why else would a player choose inaction over a 99.9999% probability of multiplying his bankroll 50-fold with a 0.0001% probability of losing everything? Indeed why else would a player actually be willing to forfeit half his bankroll to avoid making such a monstrously +EV bet? Answer? Because a lost bankroll is infinitely bad.

A player can define his goals anyway he sees fit. But we should still be able analyze these goals using traditional utility theory and determine what preferences a player would need to have for them to be rational. To this end, I'm still waiting to hear of a nonpathological example of how a player's preferences would need to look for partial wealth Kelly to be rational.

If a player's goal were solely to maximize some subset of his bankroll for no reason other than "that's what he wanted to do" then sure -- partial wealth Kelly would work. But without making that rather contrived assumption -- I just don't see how one reaches partial wealth any other way.

I understand your point. I really do. You're saying that we can get around the problems with Kelly simply by minimally construing the concept of bankroll. The problem I have with that is that while that may be convenient I can see no theoretical justification for making that assumption.

Either losing one's entire bankroll is infinitely bad or it isn't. If it isn't then Kelly isn't strictly appropriate. If it is then how can one's bankroll merely be some minimally construed subset of wealth?

Now I'm not saying that partial-wealth Kelly is useless. But let's just be clear that the further away we move from full-wealth Kelly the less economically meaningful its conclusions become.

http://www-stat.wharton.upenn.edu/~s...uelson1979.pdf
is an interesting refutation of Kelly-as-wealth by famous economist Paul Samuelson.

Yeah I read that paper when I first started looking at Kelly for sports betting (although now that I recall I might have actually first encountered that paper many years earlier in a class of the late, great Herschel Grossman's -- he was always in to fun stuff like that). It was certainly "cute" but definitely annoying. As I recall his point was essentially that most real people don't have log utility. I agree.

I also think it apparent that most people don't have log utility when it comes to some constant subset of bankroll either.