Kelly criterion

[Hulu]

Probably the two books that best demonstrate basic quantitiave analysis technique in sportsbetting are Fixed Odds Sports Betting: Statistical Forecasting and Risk Management by Jospeh Buchdahl and Sharp Sports Betting by Stanford Wong.

Thank you sir. Those 2 books are winging their way to me as we speak (along with several CDs...damn you Amazon!)

[Ganchrow]

Does the fact that the stock market and the sports wagering market are- to certain degrees- fluid and self correcting have any impact on the predictability of such models?

To some extent ... as the market adjusts itself the predictive strength of any given forecast will, in general, decay over time. This will tend to manifest itself in one of two ways.

Firstly, there might be a decrease in the number and strength of forecasts that meet your hurdle rate.

Secondly, and potentially more harmful to a market participant, the forecasting power of the model might decrease, resulting in biased forecasts. The problem is that the player would be overestimating his edge on any given bet, meaning that not only would he make less money (as in the "firstly"), but also he'd be unable to optimally manage his risk. But ultimately, it would just be up to the player to create a robust enough model to properly account for this factor and a flexible enough modelling framework to rapidly adjust for changes in regime.

[Ganchrow]

I think we're back to sequences (aka as streaks).

I'm talking about projecting a sequence of events in advance (different from evaluating each single event). If a chance of failure is 1/75, that failure will occur with certainty. It's just a matter of time.

In simple math. The chance of failure is 1.
It just may not be this time.

I'm not sure entirely certain what you're getting at here, but I fear you might be treading dangerously close to the Gambler's Fallacy.

Just to be totally clear: assuming a 1/75 fail probability, the probability of the shuttle failing at least once at some point over the next 1,000 launches would be (1-1/75)^1000 ≈ 99.99985%.

However, given that the shuttle has not failed at any point in the last 999 launches, the probability of it failing on the thousandth launch would still be 1/75.

[Ganchrow]

So from that perspective, an astronaut stepping on a shuttle flight would first have to embrace his own death. Only then could he operate without fear. Only then, in real life, would the end result no longer matter. (as it doesn't in the abstract world of math).

You just lost me.

If I translate this (things may be lost in translation) to what Ganch said about what one's true Kelly bankroll really is (everything!), then a gambler using Kelly must either embrace bankruptcy upfront or live in a state of constant fear (controlled by math based assurances).

Well guess what ... gambling can be a bit of a gamble.

[Dark Horse]

Always good to meet someone who has truly embraced risk.

(Time still has a few mysteries. ).

[trustbutverify]

To some extent ... as the market adjusts itself the predictive strength of any given forecast will, in general, decay over time. This will tend to manifest itself in one of two ways.

Firstly, there might be a decrease in the number and strength of forecasts that meet your hurdle rate.

Secondly, and potentially more harmful to a market participant, the forecasting power of the model might decrease, resulting in biased forecasts. The problem is that the player would be overestimating his edge on any given bet, meaning that not only would he make less money (as in the "firstly"), but also he'd be unable to optimally manage his risk. But ultimately, it would just be up to the player to create a robust enough model to properly account for this factor and a flexible enough modelling framework to rapidly adjust for changes in regime.

My thoughts exactly. In my original post on modeling i didn't mean to imply that it was impossible to create historical models with very strong predictive results.

I meant to say that it would be a very formidable task- considering many issues- to aquire target pcts with enough stability to depend on for practical kelly implementation for sports. Like most capper/investors I've played with the idea of learning enough about modeling, probability, database design etc.. and aquiring a massive amount of reliable data to do this. It's too much for me.

If i was a card counter i would use an optimizing strategy. But most sports wagerers are doing well if they find a bunch of strategies that beat the line in the long run. IMO- the money mgmnt approach best suited for long term success for most cappers is to treat all situations/angles the same(even though they are, of course, not) and then bet flat with occasional readjustments. Along with finding the best price, this will create the largest gap between the breakeven point(avg) and the avg advantage. That gap will grind out the profit- and drive the bet level and the bankroll up.

Its not optimal- but it might be the best approach for most.

[Milkin' it Slowly]

Thanks for the info .... From what I can gather, It's more based on money management... But the inherent problems in regards to handicapping, is you never no your true chance of winning. If applied correctly, you may be withered down to micromorsels, which in gambling terms is going BROKE. On the other hand if things are goin well, you can increase your stake based on the chance you think you have ... So in other words it can keep you in the game for a while but it still can't take into consideration the actual factors of a contest (weather, injury, pshycological high and low streaks...etc) Pretty complex ....

Anyway, Thanx again...

[Ganchrow]

Even if you didn't feel comfortable making the frequent edge approximations required by Kelly and have resigned yourself to assuming an equivalent edge on each bet you placed, that doesn't necessarily mean that traditional flat-betting would be your best option. In fact, it probably won't be.

If lowering the variance of your bankroll's growth were of concern to you (and it certainly should be) and you frequently find yourself betting across a wide variety of money lines, then you might want to consider moving away from fixed unit staking towards a "fixed-profit" staking plan. Fixed-profits staking refers to betting to win a constant amount on all bets. So in other words, if a fixed-profits staker were to bet 1 unit at a line of +100, he would be betting 1.1 units on a money line of -110, and ½ of a unit on a a money line of +200.

Joseph Buchdahl in <a href=http://www.amazon.com/Fixed-Odds-Sports-Betting-Statistical/dp/1843440199/sr=8-1/qid=1164261995/ref=pd_bbs_sr_1/103-6849374-8616645?ie=UTF8&s=books TARGET=_blank>Fixed Odds Sports Betting: Statistical Forecasting and Risk Management</a> demonstrates how a bettor engaging in fixed-profits staking can reduce both his standard deviation and his risk-of-ruin versus a flat bettor with the same average bet size.

(Fixed-profits staking, btw, is actually implicit in Kelly betting.)

[trustbutverify]

Even if you didn't feel comfortable making the frequent edge approximations required by Kelly and resigned yourself to assuming an equivalent edge on each bet you placed, that doesn't necessarily mean that traditional flat-betting would be your best option. In fact, it probably won't be.

If lowering the variance of your bankroll's growth were of concern to you (and it certainly should be) and you frequently find yourself betting across a wide variety of money lines, then you might want to consider moving away from fixed unit staking towards a "fixed-profit" staking plan. Fixed-profits staking refers to betting to win a constant amount on all bets. So in other words, if a fixed-profits staker were to bet 1 unit at a line of +100, he would be betting 1.1 units on a money line of -110, and ½ of a unit on a a money line of +200.

Joseph Buchdahl in Fixed Odds Sports Betting: Statistical Forecasting and Risk Management demonstrates how a bettor engaging in fixed-profits staking can reduce both his standard deviation and his risk-of-ruin versus a flat bettor with the same average bet size.

(Fixed-profits staking, btw, is actually implicit in Kelly betting.)

I see alot of people who bet fixed profit on neg payouts- not too many on pos. Greed.

Another method of getting more money in on higher expectation positions and vice-versa. I'm curious- have you ever run say, 3000 of your own bets through a test of fixed stake vs fixed profit? If so- how did it come out.

[Ganchrow]

I see alot of people who bet fixed profit on neg payouts- not too many on pos. Greed.

I'd say it's often as much naïveté as it is greed. The decision to bet fixed profits on payout odds less 1:1 but fixed stake on payout odds greater than 1:1 might be little more than the product of US-style lines display and an unimaginative mind.

Another method of getting more money in on higher expectation positions and vice-versa. I'm curious- have you ever run say, 3000 of your own bets through a test of fixed stake vs fixed profit? If so- how did it come out.

It generally comes out as Buchdahl's simulation predicts: Roughly equal return and lower standard deviation of finishing bankroll than with fixed profits.

What can't easily be determined from looking at a single sequence, however, is the fact fixed-profit staking also boasts a higher probability of being profitable over any given stretch, along with a lower risk-of-ruin probability.

[trustbutverify]

Less risk and more profit probability( probability of making A profit) is certainly appealing. And the avg break even pct should- I think (correct me if I'm wrong) be the same as fixed stake-even with varying bet sizes. Of course I'm refering to to break even translated and averaged out over the whole set of outcomes.

Maybe I should read some more and take another look at fixed odds.

[Ganchrow]

And the avg break even pct should I think (correct me if I'm wrong) be the same as fixed stake-even with varying bet sizes. Of course I'm refering to to break even translated and averaged out over the whole set of outcomes.

No. If you're betting more on short odds relative to long odds, then your average break-even win percentage would have to increase.

But all this really proves is that break-even win percentage is not a very meaningful statistic when considering bets of varying odds.

[ugard]

This is actually indicative of probably the single biggest "smart-person misconception" about Kelly -- specifically that one's bankroll only includes that which the bettor can afford to lose. This is in fact untrue. One's Kelly bankroll is actually one's entire marked-to-market cash balance (properly discounted of course). That means your bankroll would consist of the value of not just your offshore betting account, but also the value of your checking account, the value of your savings account, the equity in your house, the maximum cash advance level on each of your credit cards, the maximum amount you could borrow from your family and friends, the maximum amount you could borrow from your loan shark, the $3,000 cash your elderly neighbor keeps under her mattress, etc., etc. Of course each of these sums would need to be properly discounted to reflect the cost of obtaining them (a cost which could potentially be so great as to make the sources of cash essentially valueless, but that's beside the point), but as far as Kelly is concerned your bankroll should represent the dollar figure such that if you lost it your life would be as good as over. Another way to look at it is like this, let's say you had an even odds bet that you knew a priori would win with 100% likelihood -- how much would you bet? The logical answer would of course be, "every dollar you could safely get your hands on."

Now people might very well object when they read this, saying that this bankroll valuation just doesn't make any sense, and that no one would want to bet in this matter, etc. etc. And you know what? You'd probably be right. Kelly assumes logarithmic preferences and as I've mentioned many times before most human just don't have log prefs. So to get around this issue, people often claim (in fact I don't know anyone who doesn't) that a Kelly bankroll is only what the bettor would feel comfortable losing. That's all well and good -- but to be perfectly clear that's a compromise position and doesn't represent "true" Kelly.

In conclusion, the Kelly stake represents the optimal bet size as percentage of total bankroll that should be bet if the bettor's goal were to maximize the expected growth rate of that bankroll. (In fact, this is equivalent to saying that the bettor has log preferences.) Were that bit your goal, and it probably isn't, then strictly, strictly speaking Kelly's not for you. (The ambitious might consider implementing Kelly using a amore appropriate utility function. This actually isn't too difficult to figure computationally for well-behaved, convex preferences.) But that doesn't mean that you couldn't use a version of it with which you're sufficiently comfortable.

Although this is my first post to this forum, I have been reading for going on two years. Firstly, I, like many others I am sure, would to like to thank the many posters who spend valuable time and effort explaining matters calmly, precisely and in detail. In particular, Ganchrow.

As a economist (in training!) it is especially interesting to see mathematical rigour being applied to gambling (which, of course, as a profit maximiser, is what it's all about ).

On to my points: I have been reading Kelly's 1956 paper, and there are a few things about maximising expected utility of money and expected value of money that the above paragraphs are a little confusing about. Of course, I may be misunderstanding your terminology, or may have misunderstood Kelly altogether.

Firstly, your example of betting everything one owned in a single (certain to pay off) bet does not require Kelly. The logic to act in such a way can be derived from expected value maximisation, of the form ER=p(w+x) - (1-p)(w-x), where p is one and x>0. Admittedly, Kelly does also suggest betting everything (and confirms that you should do so repeatedly), but the point is that one does not require "log preferences"* to rationalise this behaviour, as you imply ("Kelly assumes logarithmic preferences and as I've mentioned many times before most human just don't have log preferences"). In fact (if risk were reintroduced by making p<1), log preferences would mean the individuals minimum required ER to take the gamble would have to be higher than that required without log preferences. In other words, log preferences make this behaviour harder to explain, by requiring bigger expected returns.

Secondly, a more fundamental problem, is whether Kelly actually implies "log preferences". Kelly uses logs, but these have "nothing to do with the value function he attached to his money" (Kelly (1956) p925). Logs are used solely as a mathematical device to maximise the function that determines growth rate. Does wanting to maximise growth rate, rather than expected value, in the first place assume some sort of diminishing marginal utility of wealth? But, surly this is an entirely different problem (repeated choices) to that in which the term "preferences" are normally used.

As mentioned above, I am not sure whether what you have written is misleading (or at best incomplete), or whether I have been previously mislead, and your points as guiding me back towards the light of reason, even though I don't realise it.


* By "log preferences", I take you to mean those in the standard one time period, do-I-bet-all-or-nothing EV maximisation problem (of the form above). That is, wrapping a log function around peoples wealth, to give them diminishing marginal utility of wealth.

If log preferences are used above (replacing (w+x) with log(w+x), and (w-x) with log(w-x)), the result is that the excepted return (or "edge") must be slightly greater than one to reach the point of indifference between betting and not betting.

[Ganchrow]

Although this is my first post to this forum, I have been reading for going on two years. Firstly, I, like many others I am sure, would to like to thank the many posters who spend valuable time and effort explaining matters calmly, precisely and in detail. In particular, Ganchrow.

That's very nice of you to say. Thank you.

Firstly, your example of betting everything one owned in a single (certain to pay off) bet does not require Kelly. The logic to act in such a way can be derived from expected value maximisation, of the form ER=p(w+x) - (1-p)(w-x), where p is one and x>0.

To an economist, this is obviously quite clear. I never meant to imply this was solely applicable to Kelly qua Kelly. If you read through much of the forum literature on Kelly then what you'll find are that many to most fairly quantitative posters mistakenly claim that the Kelly bankroll is a subset of total net worth. Obviously these good people aren't economists.

Admittedly, Kelly does also suggest betting everything (and confirms that you should do so repeatedly), but the point is that one does not require "log preferences" to rationalise this behaviour, as you imply ("Kelly assumes logarithmic preferences and as I've mentioned many times before most human just don't have log preferences").

Again, I hadn't mean to imply imply that this behavior was solely applicable to log prefs. Clearly its applicable to a set of preferences of which logarithmic are but one. The important point isn't that given the certainty of an event one would bet all under log utility, but rather that given a sufficiently near certainty, log utility would imply one would bet so much as to risk any specified fate not quite so bad as death (assuming death to be infinitely bad).

Secondly, a more fundamental problem, is whether Kelly actually implies "log preferences"*. Kelly uses logs, but these have "nothing to do with the value function he attached to his money" (Kelly (1956) p925). Logs are used solely as a mathematical device to maximise the function that determines growth rate. Does wanting to maximise growth rate, rather than expected value, in the first place assume some sort of diminishing marginal utility of wealth?

The answer is yes it does. Maximizing log utility is functionally equivalent to maximizing expected bankroll growth (or some constant fraction thereof). If a market participant's goal is the latter then that implies his utility function is logarithmic. If a market participant's utility is logarithmic, then that implies he will act to maximize expected bankroll growth. Realize that we can talk theoretically about maximizing expected bankroll growth even if we're only dealing with a single time period.

But, surly this is an entirely different problem (repeated choices) to that in which the term "preferences" are normally used.

Kelly only deals with "snapshot" utility. If you wanted to include a notion of maximization over some time horizon you'd have to come up with some estimate of the distribution future betting opportunity. We would expect this to be superior to Kelly.

By "log preferences", I take you to mean those in the standard one time period, do-I-bet-all-or-nothing EV maximisation problem (of the form above). That is, wrapping a log function around peoples wealth, to give them diminishing marginal utility of wealth.

We're wrapping the log around wealth to cause players to choose to maximize expected bankroll growth. Diminishing marginal utility naturally follows. Again, if we wanted to throw either an integral or a summation around the utility function and add subscripts and some time-value-of-money measure we certainly could do so. Is it worth the effort? Unless you're typically operating near full capacity with trades expiring over some wide swath of time periods, I'd say that from a nonacademic point of view it's probably not.

If log preferences are used above (replacing (w+x) with log(w+x), and (w-x) with log(w-x)), the result is that the excepted return (or "edge") must be slightly greater than one to reach the point of indifference between betting and not betting.

At an edge of exactly 0% (defining edge as expected return -- equivalent to an edge of 1 as you define it) a Kelly player will be indifferent between betting and not betting (the Kelly stake). For any positive edge (assuming no minimum bet size) the player will strictly prefer to bet. For a bet paying out at 1:1, a Kelly player will choose to bet his edge.

As a economist (in training!) it is especially interesting to see mathematical rigour being applied to gambling (which, of course, as a profit maximiser, is what it's all about).

Cool. Out of curiosity what program are you in?

[ugard]

That's very nice of you to say. Thank you.

Your welcome. It's not sycophantic praise, but wholeheartedly meant (if that doesn't disprove my point ).

To an economist, this is obviously quite clear. I never meant to imply this was solely applicable to Kelly qua Kelly. If you read through much of the forum literature on Kelly then what you'll find are that many to most fairly quantitative posters mistakenly claim that the Kelly bankroll is a subset of total net worth. Obviously these good people aren't economists.

Again, I hadn't mean to imply imply that this behavior was solely applicable to log prefs. Clearly its applicable to a set of preferences of which logarithmic are but one. The important point isn't that given the certainty of an event one would bet all under log utility, but rather that given a sufficiently near certainty, log utility would imply one would bet so much as to risk any specified fate not quite so bad as death (assuming death to be infinitely bad).

Agreed. I was wasn't disputing the point about using one's total worth as a bankroll, just that you seemed to imply that gambling everything was strictly a result of log preferences.

The answer is yes it does. Maximizing log utility is functionally equivalent to maximizing expected bankroll growth (or some constant fraction thereof). If a market participant's goal is the latter then that implies his utility function is logarithmic. If a market participant's utility is logarithmic, then that implies he will act to maximize expected bankroll growth. Realize that we can talk theoretically about maximizing expected bankroll growth even if we're only dealing with a single time period.

Understood. I was mistakenly bringing in multiple time periods, when Kelly does not consider this.

Kelly only deals with "snapshot" utility. If you wanted to include a notion of maximization over some time horizon you'd have to come up with some estimate of the distribution future betting opportunity. We would expect this to be superior to Kelly.

Would this distribution would give an indication of whether the expected "expected return of future opportunities" was higher or lower than current expected returns? If so, how would this affect current choices? Are you referring to some sort of bizzare attempt at consumption smoothing, whereby people see returns to future bets will be higher and so become less risk averse and chase expected returns rather than percentage growth in the present? Of course, as you state, the future returns would also have to be discounted.

At an edge of exactly 0% (defining edge as expected return -- equivalent to an edge of 1 as you define it)

Yes, I was careless in my definition of edge, it should be ER - 1.

a Kelly player will be indifferent between betting and not betting. For any positive edge (assuming no minimum bet size) the player will strictly prefer to bet. For a bet paying out at 1:1, a Kelly player will choose to bet his edge.

This I think is the point I am misunderstanding. The relationship between Kelly (maximising expected growth) and standard EU theory (maximising expected value). According to Kelly, one is indifferent between a gamble at 0% edge and no gamble, but according to standard EU, the same person (who has log preferences) would refuse a bet with 0% edge (a fair bet), and hence the observed behaviour of payment of insurance premiums.

Cool. Out of curiosity what program are you in?

I'm in the UK, and assume by "program" you mean "course", in which case I'm reading largely economics (a bit of politics too).

[TLD]

Welcome ugard. Hope to hear more from you.

(“Program” I took to mean in which university’s economics department are you studying.)

[Ganchrow]

Would this distribution give an indication of whether the expected "expected return of future opportunities" was higher or lower than current expected returns? If so, how would this affect current choices? Are you referring to some sort of bizzare attempt at consumption smoothing, whereby people see returns to future bets will be higher and so become less risk averse and chase expected returns rather than percentage growth in the present? Of course, as you state, the future returns would also have to be discounted.

Maybe a simple example will illustrate:

Assume zero time-value-of-money and full-Kelly. Let's say that today we have a bet at 2.000 with and edge of 20%. This implies a win probability of 60%. Single-period Kelly stake is then 50%. The bet is settles tomorrow night. Tomorrow afternoon, after the other underlying event has already begun, there's a 10% chance there will be a betting opportunity at odds of 2.000 and edge of 90%. This implies a win probability of 100% the opportunity exists. Single-period Kelly stake on that would be 100%.

Call the quantity bet on the first event x₁ and the quantity bet on the second event x₂.

For multi-period optimization we'd have this:

maximize U =
90% * [ 60% * ln(1+x<sub>1</sub>) +
           40% * ln(1-x<sub>1</sub>) ] +
10% * [ 60% * ln(1+x<sub>1</sub>+x<sub>2</sub>) +
           40% * ln(1-x<sub>1</sub>+x<sub>2</sub>) ]
wrt x<sub>1</sub>, x<sub>2</sub>
subject to a budget constraint of x<sub>1</sub>+x<sub>2</sub> ≤ 1
Solving, we see that utility is maximized at:
(x<sub>1</sub>, x<sub>2</sub>) ≈ (14.89%, 85.11%).

Obviously, that's a pretty contrived example, and could still be solved with standard single-period contemporaneous Kelly (using a 90% push probability for bet # 2), but you should get the idea.

Yes, I was careless in my definition of edge, it should be ER - 1.

I think you mean to say that you are defining edge as ER + 1, right?

a Kelly player will be indifferent between betting and not betting. For any positive edge (assuming no minimum bet size) the player will strictly prefer to bet. For a bet paying out at 1:1, a Kelly player will choose to bet his edge.

This I think is the point I am misunderstanding. The relationship between Kelly (maximising expected growth) and standard EU theory (maximising expected value). According to Kelly, one is indifferent between a gamble at 0% edge and no gamble, but according to standard EU, the same person (who has log preferences) would refuse a bet with 0% edge (a fair bet), and hence the observed behaviour of payment of insurance premiums.

That's actually untrue. According to Kelly, one would strictly prefer no gamble to a gamble at 0% edge (0% edge => p = 1/o)

U(no gamble) = ln(1) = 0

maximize U(gamble of x at no edge) = [ln(1+(o-1)*x) + (o-1)*ln(1-x)]/o
wrt x
subject to 0 ≤ x < 1

U' = [ (o-1) / (1+(o-1)*x) - (o-1)/(1-x) ] / o = 0
implies x* = 0, U = 0 and U'' < 0 for o > 1.

Hence U(no gamble) = U(gamble of x at no edge) iff x = 0.
and U(no gamble) > U(gamble of x at no edge) iff 0 < x < 1.

Is Kelly an extension of standard EU that allows one to choose the optimal amounts to stake, rather than simply deciding whether the fixed gamble presented should be taken or not?

It's not an extension, it's just expected utility given log prefs. The stake that maximizes E(U) given log prefs is the Kelly stake.

I'm in the UK, and assume by "program" you mean "course", in which case I'm reading largely economics (a bit of politics too).

I kind of guessed the UK part based on the combination of your proper diction and usage of "behaviour" and "maximisation". I meant in what school's economics department are you studying? Although perhaps you're currently an undergrad?

[ugard]

Before I reply: Can I stop vbullitin stripping out 'quotes within quotes', when I reply to a message that already has quotations? Are you re-adding the inner quotations manually, Ganchrow?

If the "Multi-Quote" button is something to do with this, its anchor seems to have been disabled with javascript.

[Ganchrow]

Before I reply: Can I stop vbullitin stripping out 'quotes within quotes', when I reply to a message that already has quotations? Are you re-adding the inner quotations manually, Ganchrow?

Yes. I am. I think this would be considered a "feature", btw.

If the "Multi-Quote" button is something to do with this, its anchor seems to have been disabled with javascript.

VB's Multi-Quote provides a facility for quoting multiple messages in the same response. Unfortunately neither "Quick reply" nor "Multi-Quote" are currently enabled on the forum.

[ugard]

I think you mean to say that you are defining edge as ER + 1, right?

I did mean ER - 1. What I meant was that the definition of "edge" that I implied was incorrect, and the correct definition is ER - 1. x2

This I think is the point I am misunderstanding. The relationship between Kelly (maximising expected growth) and standard EU theory (maximising expected value). According to Kelly, one is indifferent between a gamble at 0% edge and no gamble, but according to standard EU, the same person (who has log preferences) would refuse a bet with 0% edge (a fair bet), and hence the observed behaviour of payment of insurance premiums.

That's actually untrue. According to Kelly, one would strictly prefer no gamble to a gamble at 0% edge (0% edge => p = 1/o)

I rather thought it was untrue, hence the sentence about misunderstanding. I was just going by what you said in an earlier post, which I believe is a direct of contradiction the above:

At an edge of exactly 0% (defining edge as expected return -- equivalent to an edge of 1 as you define it) a Kelly player will be indifferent between betting and not betting.

U(no gamble) = ln(1) = 0

maximize U(gamble of x at no edge) = [ln(1+(o-1)*x) + (o-1)*ln(1-x)]/o
wrt x
subject to 0 ? x < 1

U' = [ (o-1) / (1+(o-1)*x) - (o-1)/(1-x) ] / o = 0
implies x* = 0, U = 0 and U'' < 0 for o > 1.

Hence U(no gamble) = U(gamble of x at no edge) iff x = 0.
and U(no gamble) > U(gamble of x at no edge) iff 0 < x < 1.

This is just what I was looking for, the Kelly optimisation problem (maths do have the virtue of being precise). I see now it is simply a normal E(U) maximisation problem. But, rather than the choice being a binary choice of rejecting or accepting a gamble (You see, the way this stuff is taught in basic economics courses is in the context of the agent being presented with a given profit/loss for a, say, p=0.5 gamble. So, the optimisation problem is a simple choice between rejecting or accepting the gamble, or calculating how much they would pay with certainty to avoid the gamble (an insurance premium)), it allows the agent to choose from a full range of gamble sizes, from all of their wealth to none of it.

It's not an extension, it's just expected utility given log prefs. The stake that maximizes E(U) given log prefs is the Kelly stake.

Your comment about being indifferent between a bet and no bet at 0% edge, combined with Kelly's "[the use of logs being] nothing to do with the value function he attached to his money", had thrown me into thinking it was some kind of departure from what I was calling "standard E(U)". Hence, the ad-hoc use of the term "extension".

I don't know how familiar you are with Kelly's 1956 paper, but I assume you'll agree that he's kidding himself that there is no "value function... attached to his money" (p926), he just sneaked it in at the start of the paper "[by assuming] that the gambler will always bet so as to maximise G [bankroll growth]" (p920).

I kind of guessed the UK part based on the combination of your good diction and usage of "behaviour" and "maximisation".

It's hopeless, but I do try to make my small contribution to the survival of British English. The popular culture that you Yanks export is like, sooo, popular, dude. .

I meant in what school's economics department are you studying? Although perhaps you're currently an undergrad?

Anonymity on the internet is an illusion (unless you really know what you are doing), but I like to stop people joining the dots too easily I have PM'ed you.

[ugard]

Welcome ugard. Hope to hear more from you.

(“Program” I took to mean in which university’s economics department are you studying.)

At the beginning of my first post, I didn't want to pompously give everyone the benefit of my views on who I think is worthy of recognition. But, since you popped up, TLD, you would have been on the list

[Ganchrow]

I did mean ER - 1. What I meant was that the definition of "edge" that I implied was incorrect, and the correct definition is ER - 1. x2

I'll admit it. I'm lost with this one. I guess I'll just have to assume it has something to do with that vaunted Brit humor I've heard so much about.

This is just what I was looking for, the Kelly optimisation problem (maths do have the virtue of being precise). I see now it is simply a normal E(U) maximisation problem. But, rather than the choice being a binary choice of rejecting or accepting a gamble (You see, the way this stuff is taught in basic economics courses is in the context of the agent being presented with a given profit/loss for a, say, p=0.5 gamble. So, the optimisation problem is a simple choice between rejecting or accepting the gamble, or calculating how much they would pay with certainty to avoid the gamble (an insurance premium)), it allows the agent to choose from a full range of gamble sizes, from all of their wealth to none of it.

Indeed as you've already discovered, the binary choice is but the tip of the proverbial iceberg.

Your comment about being indifferent between a bet and no bet at 0% edge,

I read back that initial sentence and realized I could have been much more clear. Here's how that sentence now reads (emphasis added), "At an edge of exactly 0% ... a Kelly player will be indifferent between betting and not betting the Kelly stake". Now the Kelly stake at an edge of zero is simply zero so this might appear just some silly restatement of the reflexive property couched in vague economics terms. However, the point of this when considered from the standpoint of continuous preferences is to illustrate that indifference between betting and not betting occurs not at some positive edge, but rather at zero edge. At any arbitrarily small positive edge (ε) the player will always prefer to bet, and at any nonnegative edge, epsilon the player will never prefer not to bet (or "weakly prefer to bet", just to load up on the micro jargon).

I don't know how familiar you are with Kelly's 1956 paper, but I assume you'll agree that he's kidding himself that there is no "value function... attached to his money" (p926), he just sneaked it in at the start of the paper "[by assuming] that the gambler will always bet so as to maximise G [bankroll growth]" (p920).

Kelly, you'll recall, was not an economist. If an economist had said that I would have thought him at best disingenuous. But from a physicist, especially this physicist, I'm willing to cut the guy some slack.

It's hopeless, but I do try to make my small contribution to the survival of British English. The popular culture that you Yanks export is like, sooo, popular, dude.

Popular culture? I just wanted to be like Sid Vicious in high school. He was American, right?

[ugard]

Popular culture? I just wanted to be like Sid Vicious in high school. He was American, right?

That's right. He played on that rotten song, "God save the President".

Many thanks for the clarifications.

[dwaechte]

I was just perusing through past threads and I found this give-and-take very interesting and enlightening. Thanks to both Ganch and ugard for all of these explanations.

[donjuan]

So from that perspective, an astronaut stepping on a shuttle flight would first have to embrace his own death. Only then could he operate without fear.

Do you realize that every time you get behind the wheel of a car there is some non-zero chance you will get in an accident and die? And as the number of times you get behind the wheel approaches infinity, your chance of dying in a car accident approaches 1 (100%)? Same goes for crossing the street, flying in a plane or for sitting in a movie theater. At some point you have to accept risks as part of life.