Bet size during testing phase of new system?

[Dark Horse]

Imagine following scenario. You discovered a system that looks promising. You start playing it, but your sample size isn't big enough to bet it at a full unit. You estimate that it will take about a month to have a large enough sample size. What should be your bet size each of these weeks (with each week being a marker to see if the system still produces)?

For a 1K bettor, I came up with:
week 1 - 200 (initial results 38-14 or 73%)
week 2 - 300
week 3 - 450
week 4 - 675
week 5 - 1000

So I'm adding 50% to the bet size after each successful week during the testing phase. But I'm not sure if this is the best way to approach it from a mathematical point of view. Would appreciate your thoughts.

[raiders72002]

I wouldn't use short term success to quantify any type of betting system.

[EBone]

I wouldn't use short term success to quantify any type of betting system.

Agreed. I'd go 1/2 unit, maybe even a 1/4 unit flat for a substantial period until you feel you have the data to go full bore for the future.

Even if it looks good, collect the hard data first. That would be my opinion. If you must bet it, I'd go very low and make it flat.




E

[jjgold]

Why use any money? Keep testing it Son and see if it holds up and then maybe start playing it small.

[Ganchrow]

So I'm adding 50% to the bet size after each successful week during the testing phase. But I'm not sure if this is the best way to approach it from a mathematical point of view. Would appreciate your thoughts.

You've previously initimated that maximizing bankroll growth is not your goal as a bettor, so before I attempt to answer this, you'd first need to give some indication as to what your goal actually is.

[Korchnoi]

You've previously initimated that maximizing bankroll growth is not your goal as a bettor, so before I attempt to answer this, you'd first need to give some indication as to what your goal actually is.

cheese stackin'

[Dark Horse]

Ganch, if I would have to state a goal it would be to get the bet size up to normal, in a gradual way that finds a good balance between testing stage(s) and not leaving money on the table.

I do use something called the Z-factor to determine the relationship between winning percentage and sample size, but beyond that I'm more or less left to my intuition.

[Ganchrow]

Ganch, if I would have to state a goal it would be to get the bet size up to normal, in a gradual way that finds a good balance between testing stage(s) and not leaving money on the table.

If you're saying all you want to do is maximize expectation, then the recommendation would be quite simple: bet every dollar you have on this strategy.

But I don't think that that's really what you're looking to do.

The truth is that mathematically speaking this first and foremost isn't so much a question of bet sizing (if it were then we'd get into another discussion of Kelly staking), but rather of how to estimate a system's return characteristics based on historical results.

That's actually not a simple problem issue at all, and to properly address it one issue you'd have to consider would be to what extent you're looking at a purely in-sample phenomenon. So I'll ask you this -- have you tested this out-of-sample yet? If so, what were your results? Is this the only strategy you've considered, or did you look at numerous strategies and this strategy is simply the result of picking and choosing the one that yielded the best return?

These are important considerations that you need to address (to your own satisfaction) before you can accurately determine appropriate bet sizing.

[Dark Horse]

Just a system based on filters. Two main filters, and a secondary filter to tilt the balance in some cases (if one of the primary filters is too evenly balanced). I started looking at it and it immediately went 10-1, so I played most of the qualifying plays for a 38-14 early record. So 28-13 since the first step. I don't consider it an established system (which requires a larger sample size), nor expect it to keep up at this rate, yet I feel pretty good about playing at a lowered bet size instead of building a larger database without betting on it.

[Ganchrow]

Just a system based on filters. Two main filters, and a secondary filter to tilt the balance in some cases (if one of the primary filters is too evenly balanced). I started looking at it and it immediately went 10-1, so I played most of the qualifying plays for a 38-14 early record. So 28-13 since the first step. I don't consider it an established system (which requires a larger sample size), nor expect it to keep up at this rate, yet I feel pretty good about playing at a lowered bet size instead of building a larger database without betting on it.

If this is the only system you've been considering, and if you haven't tweaked the system at all since going 10-1, then assuming 50/50 bets (meaning we're not talking money lines) the probability of these results occurring by chance alone would be 1.38%.

Of course, insofar as you have tweaked the strategy, or that this was one simply of many possible strategies, then that 1.38% figure would go up substantially.

EDIT: Assuming your experimental results were properly determined out-of-sample, the 95% confidence interval for true pick probability would be 54.46%-79.92% (meaning a 2.5% chance the system's pick probability is < 54.64% and a 2.5% chance it's > 79.92%).

[Doug]

Any way to back test it over a bigger sample ?

[Justin7]

Be very wary of trend-based handicapping.

As Ganchrow mentioned, you have a 1.38% chance of your initial result. If you tested 75 systems, you'll find one that did this.

Take your filter, and test it against an earlier year. I'd never bet money on filter-systems though - the market gets more efficient every year. Look at NFL - 7.5+ dogs used to hit 56% long term. Last 5 years, it doesn't work anymore.

[Dark Horse]

If this is the only system you've been considering, and if you haven't tweaked the system at all since going 10-1, then assuming 50/50 bets (meaning we're not talking money lines) the probability of these results occurring by chance alone would be 1.38%.

Of course, insofar as you have tweaked the strategy, or that this was one simply of many possible strategies, then that 1.38% figure would go up substantially.

EDIT: Assuming your experimental results were properly determined out-of-sample, the 95% confidence interval for true pick probability would be 54.46%-79.92% (meaning a 2.5% chance the system's pick probability is < 54.64% and a 2.5% chance it's > 79.92%).

I bow to your understanding of math. Does this mean I'm being to conservative with bet size at this point? (BTW, this is a 50/50 system, not ML).

Doug and Justin, this system is almost impossible to back test, unfortunately. Also, I didn't look for the best possible combination of filters to come up with the highest winning percentage. I started with a couple of basic filters just to see how they would combine. So, in that sense, the early results are not distorted by 'desire'.

[cobra_king]

Dark

I had the exact same scenario as you. A system based on filters that was working just fine on paper. I tested it for 508 games, and it was running at an unbelieveable +60 units. So even tho my plan was to go to a minumum 1500 games before actually investing money into it if it was productive, (fully being aware that a 1500 game sample still isn't enough data) i decided that i'd invest a 1/4 unit of a normal bet into it because it was going along so well. Now i fully expected it not to continue clicking at this rate but i felt that it could still have a positive expectation no matter how small, and if it did i'd make some money off of it. I decided to go only a 1/4 unit since i also was fully aware that as the sample size grew, it was possible that things would begin to even out (for lack of a better term) and if that was the case i wasn't going to get clobbered chasing something that wasn't truly time tested. So for this system i used a 1/4 unit of what i would normally have bet and was willing to lose up to 15 of those units before pulling the plug. Well i'm sure you know what ended up happenning. I lost those 15 units faster than you could imagine. I will continue to track this tho as in the end i may have just jumped on the bandwagon on one of the inevitable downward spirals that anyone who bets has. As the saying goes, sometimes you have to pay for your education and i was willing to pay a certain price, which i did. Just wanted to share my similiar experience with you. Good luck to you if you do decided to dive ahead with this.

[tacomax]

I had the exact same scenario as you. A system based on filters that was working just fine on paper. I tested it for 508 games, and it was running at an unbelieveable +60 units. So even tho my plan was to go to a minumum 1500 games before actually investing money into it if it was productive, (fully being aware that a 1500 game sample still isn't enough data)

It depends on what results you get. As Ganch pointed out, DH's "theory" appears to have a 98% chance of being down to the system rather than pure chance based upon only 41 games. Justin has also pointed out that the longer you follow your system, the more chance that the lines will adjust to that system.

[cobra_king]

Justin has also pointed out that the longer you follow your system, the more chance that the lines will adjust to that system.

Agreed. Spotting this and being willing and able to adapt is imperative.

[Ganchrow]

I actually kind of dread having to respond to threads like this. The problem is that it's hard to express my point accurately while still being respectful. Nevertheless, DH, you're a big boy, and I know you can give it as well as you can take it.

That said, and with the proviso that you acknowledge that I say this with all due respect, and with no intention whatsoever of condescension:

Are your results, if obtained using proper methods of data sampling significant? Most definitely. And if your results were properly obtained, are you most likely being too conservative in your staking. I would say so.

But here's the rub. Here's the problem that I've found with 99%+ of the strategies people have presented to me over the past 10 or so years (both in quantitative finance and in sports betting). I suspect that your results were not properly obtained. Now that's not a dig at you personally and that's not a statement that should be taken as anything other than a well-intentioned note of caution.

Time and time again, the problem with the vast majority of otherwise well thought out quantitative strategies stems from improper data sampling procedure. And I've seen this coming from all strata of mathematical ability -- from the savvy, happy-go-lucky, street-smart AMEX options trader, to the bookish and diligent MIT physics PhD. Flaws in data sampling are by far the biggest problems most people encounter when doing quantitative analysis.

Now without knowing more I couldn't say what mistake you may have made, but just based on my experience it's my strong suspicion that you've done something wrong with your data sampling. Why? Because everyone errs at some point when it comes to data sampling.

If you want me look at what you've done I'd be happy to do so. You can send me a PM or an e-mail and we can discuss this further offline. If not (and I completely understand why you might not want to share anything about the specifics of your strategy), then my best and most heartfelt advice would be to proceed extremely slo-o-o-o-owly.

Continue tracking your results over another maybe 100 or so more trials, betting some relatively small quantity per game (in order to keep score), and making absolutely sure not to change anything whatsoever. Keep your bet size constant, don't modify the strategy at all, and don't allow anything to be subjective in your decision making process. Your results need to be scientifically verifiable and reproducible.

Make sure to count every game on which you bet, and don't count a game if you don't bet on it. Be strict about this. What I've often found useful in this regard is to make a small deposit at a book that I don't typically use and then "keep score" with that balance. Obviously you won't get the best line by doing that but at least you'll have a very good idea of where you stand after 100 or so bets.

So anyway, that's my advice. Probably not what you wanted to hear, I know. Unless you can say with an extremely degree of certainty that you've properly this strategy, then bet small for a stretch in order to keep score, and only once you can be sure there's been no mistakes in your sampling should you seriously start ramping up.

Anyway, just my 2&cent;.

[Dark Horse]

I appreciate the feedback. I've ridden enough short term fluctuations to realize that what goes up usually comes down (some systems are seasonal so if you don't ride them while they're hot you could miss out). Let's see what happens. I'm not attached. I'll be back here with results for the second week.

[Ganchrow]

I appreciate the feedback. I've ridden enough short term fluctuations to realize that what goes up usually comes down (some systems are seasonal so if you don't ride them while they're hot you could miss out). Let's see what happens. I'm not attached. I'll be back here with results for the second week.

Cool. Best of luck.

[Dark Horse]

Also appreciate the advice to go slow. I sort of like the 1/5 unit to start out with and adding 1/2 of the amount each consecutive profitable week. After one month that should result in betting a full unit, if the system passes the test.

[Arilou]

I will state the obvious, which is that no simple system of filters could possibly expect this win rate. Holes like that wouldn't last long. So when you go 34-14 for 73%, you know that the majority of that MUST be random. That doesn't mean those are not promising results, but there is an upper limit to how you can do playing this many spots.

Ganchrow is right that you are in grave danger of having made a data error, so be very careful with that. But that's not your question.

Your question was, how to size your bets? There is no easy answer, but one thing I like as a practical solution is: Create a bankroll dedicated to your strategy. Say "I'm prepared to risk 20 units on this strategy," or whatever number you decide. Then you have a second bankroll, and each one bets as if it were your only bankroll. So when you bet with the secondary bankroll, assume you have an edge that is small but solid and proceed from there; Ganchrow has provided excellent advice on how to manage said bankroll. Then, as the strategy succeeds, the bankroll expands and your unit size for it increases. Eventually its bankroll would exceed your main one, and at some point you'd have enough data and confirm your idea. I also like this for attacking a new sport - it's the same idea as having a bankroll overall. Of course it's probably mathematically not correct, but that's not the right goal. You want to not spit in math's face, but estimating your edge here is impossible at this stage.

[Dark Horse]

Twenty units? That's a lot to risk on an unproven system. If an experimental system doesn't produce in the first ten bets, I'll toss it aside. No point forcing things. Of course, ten bets at my suggested initial 1/5th unit risks only two units total. If the system continues to produce it will pay for itself.

You are right. My question was not about whether or not I made an error, but about how to best ease into an unproven, yet promising system. I haven't read anything about this in the betting literature, but it could be something worth figuring out for those who regularly experiment with new systems and angles.

[Ganchrow]

You are right. My question was not about whether or not I made an error, but about how to best ease into an unproven, yet promising system. I haven't read anything about this in the betting literature, but it could be something worth figuring out for those who regularly experiment with new systems and angles.

What you're asking for is really not well-defined mathematically. The notion of easing in to an unproven system is one of personal preference rather than mathematical rigor.

The reason I make such a big deal out of flawed sampling procedure is because this concept is crucial to determining proper bet size. Based on the concept of expected growth maximization, bet sizes are only functions of the first moment (i.e., the expectation) of our edge estimates.

Putting this another way: if an estimate of edge (appropriately determined) is truly an unbiased indicator of actual edge, then the growth maximizing bet quantity is determined completely irrespective of estimate volatility.

Now that claim might come as a bit of a surprise to most but that's probably for two reasons: 1) because we implicitly realize that our data sampling techniques usually are generally somewhat flawed and as such tend to produce very biased estimates, and 2) in the real world, raw sample probabilities are themselves probably biased estimators of actual probabilities.

So 1) is why I bring up sampling methodology. If you're convinced that your sampling methodology is completely correct and hence producing unbiased results (at least after adjusting for issue #2), and if your ultimate goal were to maximize bankroll growth, then the corresponding mathematically correct course of action would be to proceed just as if you considered your strategy were already "tested". In other words, you shouldn't treat your strategy any differently just because the volatility of your edge estimates happens to still be rather large.

<input type=button style="font-size: 9px;" value="Show Additional Blather" onclick='if (this.value == "Show Additional Blather") { document.getElementById("divBlather").st yle.display = ""; this.value="Hide Additional Blather"; } else { document.getElementById("divBlather").st yle.display = "none"; this.value="Show Additional Blather"; } '> <div id=divBlather style="display: none;"><hr><BLOCKQUOTE>2) is a lot more complicated but certainly extremely important when coming up with unbiased probability/edge estimations. Accounting for this factor involves making use of our a priori knowledge of the relative likelihood of underlying system probabilities or edges. We can probably say that due to volatility inherent to sports betting no system (given some odds level), can produce better than X% winners. Even better we might be able to impute an actual distribution of conceivable prediction probabilities or edges.

So for example, even if we saw a system producing 75% winners after 100 trials, based on some system probability distribution determined a priori, we might still be able say that we're 95% sure that the true system pick probability is lower than 70% (because it's a lot more likely that we simply got very, very lucky than it is that we managed to find a system producing 75% winners).

As a first order approximation of this phenomenon I'd introduce a manual correction factor such that the farther away from about 0% (perhaps a little higher) our no-vig edge as determined by sample success rate lies, the more we adjust towards about 0%. So in other words, if our sample were at were producing 73% winners over 48 trials, we might adjust that estimate downwards to let's say 58% and then treat that figure as our unbiased estimate of system pick probability for purposes of bet sizing. If our system were producing 54.5% winners over 100 trials, however, we might only adjust that downwards to 54.25%. Note that this is NOT a confidence interval but rather reflects our implicit understanding of the increasing difficulty of identifying progressively higher edge systems.</BLOCKQUOTE> <hr> </div>

[jjgold]

What really is a system?? How many profitable years makes it a system? Most systems go bad just like most stock market systems. A system is all about timing and catching it before it goes cold.

[wrongturn]

Dark Horse, I don't understand why you would increase bet size in order not to "leave money on the table". If you find a profitable system, you will make it up later. More than likely you are going to lose money in testing a system. So I would use whatever the bet size that keeps me interested in tracking the system and make tracking easily while not risking too much.

[Dark Horse]

Ganch, you mentioned something about 58% and 54% but I think you edited it out.

This is interesting, because based on that you would have to NOT increase bet size for a system that has started out at a 73% rate, because it is almost certain to self-correct.

Would be interesting to see the initial curves for systems that proved to have long term value. Especially the point where they start to correct towards a winning percentage that reflects the boundaries of 'reality'. By overlaying such curves for a large number of systems, perhaps it would be possible to come up with a standard approach for bet sizes for systems in experimental stages.

[EBone]

Dark Horse, I don't understand why you would increase bet size in order not to "leave money on the table". If you find a profitable system, you will make it up later. More than likely you are going to lose money in testing a system. So I would use whatever the bet size that keeps me interested in tracking the system and make tracking easily while not risking too much.

I'd agree 100% with this post. That "leaving money on the table" comment bothered me as well. We all want to get the maximum dollar for our time but, in a test phase, I believe that anything you make on it during testing is an added bonus. Just an opinion.


E

[Dark Horse]

So we're all suddenly using systems with sample sizes in the hundreds?

In that case, I'm impressed. I must be the only one who knows plenty of bettors who jump on a 10-1 trend.

[EBone]

So we're all suddenly using systems with sample sizes in the hundreds?

In that case, I'm impressed. I must be the only one who knows plenty of bettors who jump on a 10-1 trend.

I'm not saying that. I guess my point is: proceed with caution. I have a "glass half empty" type of view point.

Having said that, I just think that "leaving money on the table" with a small sample size is a foreign concept to me. I would want to make money as well on a trend that is 10-1 but also know that there isn't a lot of history. So, I'd proceed with caution until you have more data then fire away.



E

[Dark Horse]

I understand.

I used to bet full units on new discoveries like this, but now am more cautious. So it was with some surprise that I read, or I think I read, that Ganch is, or could be, in favor of betting a full unit. Unless a mistake was made in the record, but there was no mistake.

I have quite regularly run into systems that take off (last season a 57-19 run to start out a system) only to fall under gravity's spell and come back down to earth. So I'm still not sure what is the best approach. Full units or caution. The point being that when you discover a system, you do so precisely because it is hot at that time... If it started out 10-10 you'd toss it aside.

It would help if we could tie in systems to certain parts of the season (and switch from one to another at the right time), but so far I've only succeeded doing that for one; or so I've convinced myself.

[EBone]

It would help if we could tie in systems to certain parts of the season (and switch from one to another at the right time), but so far I've only succeeded doing that for one; or so I've convinced myself.

Now, this statement is something that we all need to be thinking about. I don't think there is a "one size fits all" system to sports throughout a year. Sure, you can probably grind out a profit staying "fixed" but catching the turns in a particular sports season is probably the most elusive thing of all. If I could model the turns on a consistent basis, I'd quit my full time job.

If you think you've succeeded at catching the turns in a sport as suggested above, then major league hats off to ya! That is a fantastic accomplishment and I know that takes a whole helluva lot of work!!!!!



E

[Arilou]

One other thing to keep in mind is that while jj's statement is MOSTLY nonsensical, there is a related point: Different phases of a season have different gambling dynamics, the market gets its fads and its opinions, and even the sport itself can change. Even if your edge is real when you took the sample, rather than being lucky, that does not mean that you have anything now. Of course, I have no idea what you're doing so I have no idea if this applies, but estimating an edge based on past results has even more issues than Ganchrow is raising.

[Ganchrow]

Ganch, you mentioned something about 58% and 54% but I think you edited it out.

No. It's still there. Click the "Show" button in my post.

This is interesting, because based on that you would have to NOT increase bet size for a system that has started out at a 73% rate, because it is almost certain to self-correct.

There seems to be some blurring together of disparate issues here. Let me try to enumerate the considerations I've outline:

1. What is the bettor's objective? This is the most single most fundamental issue at play whenever trying to determine optimal behavior, and is just what I addressed in my very first post of this thread. One can't answer the question, "What's my best course of action?" without first defining the surprisingly nebulous concept of "best".
   
   For example:
   For a full-bettor, "best" would imply the course of action that maximizes the expected growth of one's bankroll.
   
   For a perfectly risk-neutral bettor (aka a "0-Kelly bettor"), "best" would imply the course of action that maximizes the expected value of one's bankroll without any regard to risk.
   
   For a perfectly risk-averse bettor (aka an "&infin;-Kelly bettor"), "best" would imply the course of action that guarantees a constant bankroll will be equivalent regardless of any bet's outcome.
   
   So the idea is that while in plain English terms, the concept of "best" might appear axiomatic, in economics, it needs to be explicitly spelled out, as varying definitions of prefeernces may result in widely varying strategy recommendations.
   
   Now within the context of this thread you appear to have defined your version of "best" as something to the effect of, the course of action which "leaves the least money on the table while easing in to a still untested strategy". But this definition is extremely problematic, because while it might sense intuitively, in reality it says nothing that can be used to illuminate your objective mathematically.
   
   What do the terms "leaves the least money on the table" and "easing in" to a strategy actually mean scientifically? You're going to need to explicitly define these terms mathematically before you can even attempt to, as you put it, "approach [bet sizing] from a mathematical point of view".<br>
   
2. Biased results stemming from improper data sampling procedure. This is the issue to which I've given the most attention, because as I've said it's 99+ times out of 100 it's the biggest sticking point in quantitative modeling. If you're consistently finding strategies that yield the equivalent of 60%+ winners (adjusting for issue #3 below as necessary) during the induction and initial testing out-of-sample phases with p-values < 2-3%, then this is is indicative of either flawed sampling methodology and concomitant selection bias, or of a facility in quantitative forecasting that's simply leagues beyond anything I've ever seen. I do freely admit that the latter remains a possibility.
   
   
3. Structural "bias" in properly conducted sampling. (This relates to issue #2 in this post after clicking the "Show" button"). This is clearly the most difficult issue mathematically, which is exactly why I added it above with as spoiler-type text. The idea is that we actually have some a priori knowledge of the likelihood of uncovering systems of a given degree of predictability.
   
   For example, I give you a bag filled with 100 marbles and tell you that between 0-100 marbles are white and between 0-100 marbles are black (all possibilities equally likely). You pull out a marble, record its color, replace it, and repeat this process 9 more times, finding in total 9 black marbles and 1 white.
   
   From your sample mean (90% black and 10% white) alone you might hypothesize that the bag contains 90 black marbles and 10 white (this is known as the “maximum likelihood estimator”).
   
   If however, we assume that every initial configuration of marbles were equally likely, then our sample mean will not be the best estimator of the number of black marbles in the bag (although asymptotically it will be so, meaning that as the number of marbles we sample approaches &infin; the sample mean approaches the expectation of the number black marbles in the bag).
   
   So in other words, even though the single most likely possibility is that there are indeed 90 black marbles and 10 white marbles in the bag, the expectation (that is the expectation of the posterior distribution) of the number of blacks in the bag is in fact lower -- it happens to be about 83.32. (If instead of 9 black and 1 white, we had instead found 90 black and 10 white, our unbiased estimator would go all the way up to 89.22.)
   
   So let’s relate this to sports betting. Given proper modeling methodology, if after 10 trials you’ve experienced 6 winners and 4 losers, the single most likely success rate of your system is 60%. But that, however, represents an imperfect estimate of your expected rate of success conditioned on the knowledge available -- that is your win/loss record up to that point. (So if you came up with 1,000,000 different systems based on completely different datasets, all of which happened to go 6 – 4 over the first 10 trials, and you ran each system 100 more times, you would expect to win fewer than 60 games per system on average).
   
   Now remember that we’re assuming that the success rates of potential systems are uniformly distributed. In reality, that’s not a realistic assumption. More likely, the distribution of true success rates will be wide peaked with flat tails (implying a negligible probability of locating a strategy with a very high success rate). This will only serve to intensify this phenomenon I’ve outlined above.
   
   So the point here is that given an experimental success rate of a binomial event, due to the inefficiency of the maximum likelihood estimator in predicting the expected "true" success rate, your expected success rate will always need to be shifted towards 50% (or whatever the expected success rate of the typical strategy might be.
   
   Does this make sense?

[Ganchrow]

One other thing to keep in mind is that while jj's statement is MOSTLY nonsensical, there is a related point: Different phases of a season have different gambling dynamics, the market gets its fads and its opinions, and even the sport itself can change. Even if your edge is real when you took the sample, rather than being lucky, that does not mean that you have anything now. Of course, I have no idea what you're doing so I have no idea if this applies, but estimating an edge based on past results has even more issues than Ganchrow is raising.

There's no question that past performance doesn't guarantee future results, but ostensibly this consideration (along with possible cyclicality) should be factored in to the formulation (and continuous refinement) of the model itself.

[Dark Horse]

Now within the context of this thread you appear to have defined your version of "best" as something to the effect of, the course of action which "leaves the least money on the table while easing in to a still untested strategy". But this definition is extremely problematic, because while it might sense intuitively, in reality it says nothing that can be used to illuminate your objective mathematically.

What do the terms "leaves the least money on the table" and "easing in" to a strategy actually mean scientifically? You're going to need to explicitly define these terms mathematically before you can even attempt to, as you put it, "approach [bet sizing] from a mathematical point of view".<br>

Not realizing it would have such impact, I wrote down the first vague 'definition' that came to mind. I would have to think more about a mathematically correct definition (if I should require one). Personal changes in money management and investing make this a little unclear at the moment. In sports gambling my objective is not to maximize profit. This is true. One reason is that profit cannot be maximized in this field because there simply isn't enough money in the market. In that sense the stock market holds greater promise.

So I don't see sports betting as a way to riches (unless one sold advice to a group of investors, or would go to such extremes as Walters). But sports betting can definitely be a way to make a living. Anyone with a bankroll of 100K would have to be up two or three (2%) units a month to live comfortably in most countries. That is not particularly hard for those who take the subject seriously.

To me ideal money management would find just the right balance between being profitable and low in stress. Kelly is much too stressful to my liking. Of course, this is a personal preference, and not close to a definition of any use in a mathematical sense.