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100 bets in one day - kelly adjustment?
Assume that on one morning I make a total 100 bets with my $100,000 bankroll, each independent of one another, all even money, and I have a 3% edge on all of them. Each bet will be won/lost by the end of the day, and I won't be making any more bets that day. How much should I bet on each assuming I want to maximize growth? What would the bet amount be if I had a 5% edge on all of them?
Assume I can bet on credit, and I will not hit any house limits.
See http://www.sportsbookreview.com/forum/handicappe...tml#post215383, Kelly Calculator, and Advanced Kelly Criterion Excel Spreadsheet for some ideas on how to proceed.
Thank you for these references. I looked at the implementation of your algorithm (the kelly calculator) and looked at the results of 2 even money wagers with 3% edge, and, assuming I am using it right, the calculator produced 2.91% as the optimum kelly stake for each bet.Originally Posted by Ganchrow
I thought this might be a bit low, so I went ahead and did a fairly simple calculation in Excel by expanding on all of the 4 win/loss possibilities (no ties assumed), to get the following:
Seq Net Result Prob. Result Log(Growth)*Prob
--- ---------- ------------ -----------------
LL -6 0.235225 -0.014554642
LW 0 0.249775 0
WL 0 0.249775 0
WW +6 0.265225 0.015454371
Assuming a 3% kelly stake for each bet, this yields an expected growth of exp(-0.014554642 + 0.015454371) = 1.0009001342. Using solver Excel computes the optimal kelly stake at 2.9973%, which yields an expected growth of 1.0009001349.
So I wonder if I am using your calculator in error, or if there is some explanation for the differences in your calculations of optimal kelly stake and that I described above.
Click the "2" tab "Stakes for parlays of size:" label. You'll see that the optimal solutions also includes a bet of 0.09% of bankroll on the parlay of the two bets.
Set initial bankroll to $100.0000000000 (to force a precision of 10 decimal places) and you'll see that the calculator gives an implied expected growth of $100.0900540292/$100 - 1 = 0.0900540292%, a tad higher than the single bet only results you obtained of 0.090013493%.
To constrain the max values of any given parlay (or to assign an initial position, or tweak any individual parlay payouts) see my http://www.sportsbookreview.com/forum/handicappe...ml#post1215635.
The general assumption should be that parlays would not be available, i.e. my example noted above, or at least your calculator would provide that option to exclude parlays. (I understand that your spreadsheet has that option, just not as simple to use) This explains the 2.91% - 2.9973% disparity. But please don't infer my ingratitude for the calculator, just a suggestion to add that option would be simple to implement and make a lot of sense for real world situations.
So with that in mind (and notwithstanding your suggestions to solve the problem) my question above remains. I would be interested to see what you or other posters think the kelly stake for 100 independent bets should be.
Last edited by BigCap; 06-13-09 at 07:48 PM. Reason: typos
The problem is that like it or not parlays ARE a part of the complete unconstrained solution, which is the solution that provides maximal expected growth irrespective of the simulatenity of the given bets. This what the JavaScript calculator provides and corresponds to the methodology I outlined in http://www.sportsbookreview.com/forum/handicappe...tml#post215383.
Excluding parlays makes the problem far less tractable and in order to determine exact solutions would in general require solving (in this case) a 100-degree polynomial.
Due to the sheer size of the problem even an approximate (optimized) solution wold be hard to come by. That said, given the relative simplicity of what you've laid out (insofar as all bets are at even odds), a solution is nevertheless in theory easily found in Excel. However, due to the limited precision of Excel, all we can say is that the optimal solution would be somewhere ≥ 0.9999999999999% and < 1% of bankroll per bet.
Anyway, the exact problem to be solved would be:
Maximize wrt. X, s.t. 0 ≤ X < 1:Good luck and have fun.
This may be correct, but it does not change the fact that parlays are not always available, due to the book offering the best odds do not allow parlays (e.g. matchbook) or finding the best bet at book A and another bet at book B cannot be parlayed.
This is basically my method shown above, but with 100 events instead of 2.
I did not say the answer was easy to derive, I was just curious what you or others might think the answer should be. Again, I appreciate your theoretical approach to this, I was just looking for a number (an educated guess would be fine).
If you are claiming the answer is near 1%, then either you did not read my initial post closely enough (betting on credit is accepted), or reinforces my suspicion that bettors typically reduce their kelly stake more than they should in these type of situations (although 100 events is exaggerated I picked a high number to illustrate this).
You should be including the credit as part of your bankroll already.
First, I don't use credit but if I did, I would still use half-Kelly. 1.5% of bankroll on each event for 150% of bankroll assuming credit. I use half-Kelly because I am comfortable with that and I don't get overly concerned about maximum growth rate.
Joe.
You are confusing "the credit" that I can obtain from a book that is due the following day/week, and traditional credit that is not due for some time, which may be available to me because of my earnings capacity, home equity, etc.
In the case I described, you may assume that my "bankroll" noted above is the maximum available to me which I am able to pay when I settle with the book (i.e. the following day). So in that regard I have already "maxxed out" my credit.
Any guess on what the correct kelly stake would be for the 3% example?
Your guess of 1.5% is closer than 1%, but still a ways off.
I really don't understand the point of this thread. If you're including the credit that you can pay back as part of your bankroll, how exactly would you exceed 1% wagered on 100 simultaneous events? I'm not even sure you know where you're going with this. Obviously, there is some constraint here which would lead to 100% of your bankroll. If you're going to speak of "real world" situations with some notion of infinite bankroll, well that's certainly ridiculous.
I'd certainly have to agree with Monkey that the notion of "credit" to which you appeal in your above posts is vastly underspecfied. If you have no constraints on borrowing then what would your utility function look like?
See this post for some discussion on how best to construe Kelly bankroll.
I guess both of you are missing the general point. Money I am required to pay back tomorrow is not part of my bankroll; it only allows me to use those funds today to make wagers knowing I will have the funds to pay it back tomorrow. For example, you could use these funds in a hedge in which you are not exposed. Books that allow a player to play on credit do not increase that player's bankroll, assuming that they balance out the next day.
If you are trying to compare exposure of losing 100 consecutive, independent, even money bets to some theoretical point that this would ever happen in real life, then that point can be dismissed easily.
Last edited by BigCap; 06-15-09 at 10:56 AM.
Right, but in the example you've given you would be exposed to a loss of greater than your current bankroll. We're not talking about a riskless arbitrage here. What would happen, for example, if you were to lose more than the value of your current bankroll?
That full-Kelly suggests an optimal bet size of a tad less than 1% of bankroll per wager is unquestionably correct. The only bone of contention is the proper manner in which bankroll ought be construed.
As I recall this topic was already discussed in some depth in the above referenced post. Any comments in response to that post would of course be welcome.
Again, if you are considering a possible loss of 100 consecutive, independent, 51.5% win probability bets as "possible" in a real sense, (which would be equivalent of winning 4 powerballs in a row), then that just does not make any sense at all.
If this is true, then the theoretical application is not appropriate (and does not produce the correct result) for this example. As I noted, the $100,000 bankroll is already "maxxed out" in the sense that no additional money can be obtained by the player. And I already distinguished a book's offering of credit than obtaining credit for bankroll.
I'm not sure if it was or not, but the question from my original post remains.
Irrespective of the likelihood of such an outcome, I have indeed given you the correct answer a la Kelly (i.e., the bankroll growth maximizing solution). Whether or not this represents the optimal staking for you personally is but a function of your own particular preferences.
As I've previously mentioned your issue with the Kelly-optimal solution likely stems from a misunderstanding regarding the operative definition of the term "bankroll" as it applies to Kelly.
Well please do give it a look as it deals in some depth with just this issue. Also, I'm kind of hoping that the number of posters who actually read it through in its entirety will soon reach double digits.
But this does not produce the correct answer. You should be able to understand this better by putting yourself in my example: I know for an absolute fact that you would never seriously consider the possibility of losing 100 consecutive 51.5% independent events as possible in any real world situation. So in that case you would not wager a "tad less than 1%", if you had an opportunity to wager on a book's credit due the next day.Irrespective of the likelihood of such an outcome, I have indeed given you the correct answer a la Kelly (i.e., the bankroll growth maximizing solution). Whether or not this represents the optimal staking for you personally is but a function of your own particular preferences.
As I've previously mentioned your issue with the Kelly-optimal solution likely stems from a misunderstanding regarding the operative definition of the term "bankroll" as it applies to Kelly.
I gave it a look, and I see some of the positions posted. They are not quite the same as my example, but I can see how pure theory could be misapplied in extreme examples. The positions there were discussing merit of evaluating bankroll. My example is different in that regard, i.e. bankroll is already maxxed out.Well please do give it a look as it deals in some depth with just this issue. Also, I'm kind of hoping that the number of posters who actually read it through in its entirety will soon reach double digits.
And upon what criteria are you basing this decision?
And what would happen were you to lose your entire (pre-credit) bankroll?
(1)I think you can see that if you do not consider losing 100 consecutive said bets as possible in a real sense, then you would wager more than 1%. I am absolutely sure that you do not consider this scenario possible in the real world.(1)And upon what criteria are you basing this decision?
(2)And what would happen were you to lose your entire (pre-credit) bankroll?
(2)How would this be possible in a real sense? I think we are past this point, no?
This is not Kelly.
Who said it was? I think I already indicated above that this does not produce the correct result. That is why I titled this thread "Kelly Adjustment". So I think you agree now that Kelly does not maximize growth in this case.
Let me digress a bit here. Maximizing one's bankroll only requires an array of outcomes each described by potential probabilities, and the returns on those outcomes.
If one feels compelled to consider out-of-this-world possibilities in those array of outcomes, then that model may not produce the maximized growth model one could obtain in the real world. For example, one would never bet 100% (or more, if possible) on a pure arbitrage because of the minute risk that the counter-party to the winning side would not pay off. But no reasonable player would consider this risk, if that player believed there was no real world risk of this default occurring.
So any guess on the correct bet for the 3% example?
Last edited by BigCap; 06-15-09 at 01:53 PM.
I.e., this is not the strategy that maximizes expected bankroll growth).
If your goal were to maximize expected bankroll growth without using parlays then your strategy would be to bet a tad less than 1% of bankroll (properly defined) per wager. Period.
If you'd like to attempt to prove otherwise then by all means give it a shot. Who knows ... it could even mean a Nobel Prize in Economics.
I do not agree and it does.
It's not that I feel "compelled" to consider these unlikely cases, it's just that they're an explicit part of the Kelly utility function. If you want to design a NEW utility functions that excludes events that occur with sufficiently low likelihood, then by all means go for it, enjoy, and have fun.
But alas, it won't maximize expected bankroll growth.
As I said if you can prove that the Kelly solution does not maximize expected bankroll growth then please do give it a try. (But guess what ... you'd be wasting your time, because it most certainly does).
Listen, Kelly is far from the be all and end all of utility functions. But that it and it alone maximizes expected bankroll growth is simply irrefutable. (But go ahead and try if you like.)
Correct, i.e. Kelly does not produce the correct result in this case.
This is the Kelly theoretical result, which I already indicated above considers out-of-this-world possibilities which I know you would never consider. So, no this is not the correct answer.
I am obviously not out to prove Kelly Criterion incorrect, just that you are not applying it in its correct context here, i.e. considering out-of-this-world events in a real world scenario.
If you do not agree that Kelly produces the incorrect result, then you are considering the "4 powerball in a row" a real world possibility, which I am absolutely sure you cannot consider possible in the real world. So you are basically contradicting yourself.
Again, you are not understanding the point of this post. It is not to draw out a Kelly answer to a non-Kelly example. No person would ever consider the "4 powerball in a row" scenario as possible in the real world. This is why applying Kelly Criterion as you are in this case is not correct.It's not that I feel "compelled" to consider these unlikely cases, it's just that they're an explicit part of the Kelly utility function. If you want to design a NEW utility functions that excludes events that occur with sufficiently low likelihood, then by all means go for it, enjoy, and have fun.
You did not answer my question above regarding what you would do if you were the bettor in my example. I guess if you answered it in sincerity you would reveal that you would bet more than 1% of your bankroll. There are a multitude of examples in which, if you answered sincerely, would yield a result that does not coincide with the Kelly answer. This is the approach required to answer the question posed.
Again, my approach will maximize growth. Kelly does not in this specific case.
Once again, I am not out to disprove the Kelly Criterion. It does not hold in this case for the reasons I have described, which I am sure you recognize (i.e. viability of ruling out out-of-this-world scenarios).
Again, not the point of my post. I am only asking for an answer to the adjustments necessary to the 3% wager. 1% is clearly not the correct answer as I have shown above.
I don't see any reason to bring up a theory that is not applicable in this case. If you think it does apply, then I believe you are just being disingenuous, no disrespect of course.
I would appreciate your response to how you would approach my original question if you were in the shoes of the bettor. I know for an absolute fact that you would NOT bet "a tad less than 1%" of your bankroll because you would never consider losing 100 consecutive 51.5% independent events as anything close to possible in the real world.
Last edited by BigCap; 06-15-09 at 02:48 PM.
Have you read this post in its entirety?
With what part of it do you disagree?
Yes.
Again, I do not disagree with the Kelly Criterion. Only the application of it here in this example is incorrect. I do not think we need to go in circles with this. I am not going "all in".
Would you please provide me an answer for how you would approach this question if you were the bettor? I know it would be more than 1%, I am just wondering how much. Thank you.
Data venia, but 1% of what exactly?
The "maxxed out" bankroll I noted in my original post.
I'm honestly not trying to be difficult, but I really don't know to what precisely you're referring by "maxxed out bankroll". Is that just the $100K you had mentioned? Or is that the $100K + some unspecified additional amount of credit? Or is it something else entirely?
Exactly what I posted. You have $100,000 that you have "maxxed out" using all means of obtaining credit, e.g. home equity loans, borrowing against future income, etc. I believe I described this sufficiently above.
The book(s) where you are betting offer you the opportunity to make bets that are cleared the next day. So you will have no further means of increasing your bankroll in one day to pay off any losses. Again, I believe this has been stated clearly and sufficiently above.
You may wager as high a percentage of the $100,000 you want. I only ask that you maximize growth of this $100,000. Obviously betting extraordinary amounts via books' credit will not do this. Also, just as obvious, the percentage will NOT be 1% because you will never lose all of these bets. I thank you in advance for your answer.
OK. Then the solution that maximizes growth of this $100,000 is to bet slightly less than $1,000 per bet.
If you think this to be incorrect then I thank you in advance for providing a mathematical rationale supporting your contention.
I believe, however, that the reasons as to why the slightly less than 1% of bankroll per bet solution maximizes expected bankroll growth have already been stated stated clearly and sufficiently by John L. Kelly, Edward O. Thorp, and yours truly (to name but a few) ad nauseam.
But I know that this is NOT how much you would bet. This is the question I am asking. Thanks again in advance for your providing this answer.
Kelly, Thorp, et al, did not solve this specific problem. Or at least they would not use Kelly Criterion to solve it.
How is this a "real world" scenario precisely, BigCap? You're proposing a situation that is as likely as hitting the four Powerball numbers that you are railing against. It is certainly POSSIBLE to lose every bet no matter how slim the liklihood may be. AGAIN, there MUST be some bounds to the credit availability. And if there weren't, although I'm not quite certain how, you'd need a utility function that would be independent for each person's level of comfort with risk of ruin.
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