Multiple exclusive outcome Kelly stake

[CptPicard]

Hello gents,

I notice there has been interesting advanced discussion of betting math on this forum, so let's try asking this question here..

I haven't quite been able to find a satisfactory explanation of the mathematics involved for the Kelly formula in the case where a certain bet can produce a win in multiple different, exclusive ways, each of which have their own individual probability and outcome. Just adding up the subcases' kelly stakes doesn't seem like the way to go

Most of the sources I find on the net are "almost there" but not quite, and I'm not quite sure if the multiple-parlay thread on this forum is applicable... is this computable, and how would I go about that?

Another complication of the situation is that the possible distinct bets' possible subcases overlap (meaning that betting X and Y can both win if one of their mutual possibilities is the realized outcome) so the overall optimization is more complicated, but it would be handy to at first at least gain an understanding of how one evaluates the single bet that is divided into subcases...

[Justin7]

The idea of Kelly betting is to maximize the derivative of the natural log of your bankroll growth. The formula you often see for risk staking is the simplified solution in one case. If you give me a (fairly simple) problem that you described, I can walk you through the risk staking.

[LT Profits]

CptPicard, have you played with Ganchrow's tool?

(And I mean that in a clean way...LOL)

http://www.sbrforum.com/Betting+Tools/Kelly+Calculator.aspx

[Data]

Check out this thread:

http://forum.sbrforum.com/handicapper-think-tank/62456-another-kelly-conundrum.html

There you will learn how to maximize the expected utility (Kelly) and maybe more.

[Ganchrow]

The idea of Kelly betting is to maximize the derivative of the expected natural log of your bankroll

FYP

[CptPicard]

Hi guys, sorry for having been away for rest week.

If you give me a (fairly simple) problem that you described, I can walk you through the risk staking.

The case is really simple on the surface... suppose you have a bet that you either stake part x of your bankroll or don't. If you do, the bet can come out with either probability p1 and odds o1, or alternatively p2 and o2. Finally, with probability (1 - p1 - p2) you lose. This can be extended to arbitrary number of cases... the stuff I'm really considering has some 3500 possible outcomes with probability/odds pairs per bet

So the question is.. how much should one bet on that kind of a proposition? It's important to note that the bet is placed on the whole distribution of outcomes, not the elements of the distribution individually.

The expected value part is easy of course, but I'm not really sure how to move to the bankroll growth side of the argument

[The HG]

Hello gents,

I notice there has been interesting advanced discussion of betting math on this forum, so let's try asking this question here..

I haven't quite been able to find a satisfactory explanation of the mathematics involved for the Kelly formula in the case where a certain bet can produce a win in multiple different, exclusive ways, each of which have their own individual probability and outcome. Just adding up the subcases' kelly stakes doesn't seem like the way to go

Most of the sources I find on the net are "almost there" but not quite, and I'm not quite sure if the multiple-parlay thread on this forum is applicable... is this computable, and how would I go about that?

Another complication of the situation is that the possible distinct bets' possible subcases overlap (meaning that betting X and Y can both win if one of their mutual possibilities is the realized outcome) so the overall optimization is more complicated, but it would be handy to at first at least gain an understanding of how one evaluates the single bet that is divided into subcases...

Can you clarify what you mean by "a certain bet can produce a win in multiple different, exclusive ways, each of which have their own individual probability and outcome"?

Do you mean for instance, a "bet" on a starting pitcher doing well can "produce" a win with a side, an under, a team total under, a 5-inning under, etc etc? So that all those bets are distinct but related? It seems like you are getting at something like that.

[CptPicard]

Well, you can think of betting on a die roll, with the die being weighted somehow, and there being different payoffs for each side, and possibly one side making you lose.

[Data]

Can you clarify what you mean by "a certain bet can produce a win in multiple different, exclusive ways, each of which have their own individual probability and outcome"?

Sounds like the slots.

[CptPicard]

Sounds like the slots.

Yes, exactly. The question is how much you're willing to bet on the outcome on the whole, knowing the discrete distribution of payoffs, and after you've placed your bet, the outcome is selected according to that distribution...

[Ganchrow]

So your full-Kelly expected utility function would look like:

E[U(x)] = p₁*ln(1+x*w₁) + p₂*ln(1+x*w₂) + ... + (1-Σp_i)*ln(1-x)

Where p_i and w_i correspond to the probability of and the payout for (net of initial bet ... so decimal odds -1) respectively for the i^th outcome, and x corresponds to the percentage of bankroll.

Taking the first derivative wrt x and setting to 0 gives us:

Σ [p_iw_i/(1+xw_i)] = (1-Σp_i)/(1-x)

Which for n discrete outcomes yields an (n-1)^th degree polynomial.

Anyway, attached is a simple spreadsheet that demonstrates a solution using Excel Solver. Fields intended for editing are the probability and outcome columns and the Kelly multiplier cell.

[RickySteve]

Your optimal Kelly stake is 0.

[CptPicard]

Big thanks, very informative, will have to dig into this

[milligan]

Hello aal,
My first post on this topic because this is what lead me here. I have also been searching for an answer to a similar question to CptPicard (Data, Picard, maybe I should have called myself ?Worf?). I am looking for a solution to making several mutually exclusive bets in the one event and trying to apply Kelly to that.

For example (I will use horse racing as that what I know) say you have a field of 10 runners and you assess 4 of these as main chances. Your work out your odds line and now only concentrate on the "overs." Lets say 2 of these represent good value. No. 1 is paying $4 and you rate it at $2.50. No 2. is paying $12 and you rate it at $7.

Only one of these can win (barring a dead heat, ofcourse). My problem is, other than being very confused, is how to crunch the numbers. From what I have seen thus far it requires excel solver which I have. The bit I dont have is all the fancy numbers and letters that fill the spreadsheet.

Can someone enlighten me.

BTW, sorry for making my first post a request. I know thats rude but I have been up all night searching for an answer. Please forgive

[Ganchrow]

My first post on this topic because this is what lead me here. I have also been searching for an answer to a similar question to CptPicard (Data, Picard, maybe I should have called myself ?Worf?). I am looking for a solution to making several mutually exclusive bets in the one event and trying to apply Kelly to that.

For example (I will use horse racing as that what I know) say you have a field of 10 runners and you assess 4 of these as main chances. Your work out your odds line and now only concentrate on the "overs." Lets say 2 of these represent good value. No. 1 is paying $4 and you rate it at $2.50. No 2. is paying $12 and you rate it at $7.

http://www.sbrforum.com/Betting+Tool...alculator.aspx

Select "Exclusive Outcomes".

[milligan]

Thanks Ganchrow. I was getting confused with the events and series columns. Just selected the number of runners in the events drop down and everything is sweet. Sorry about that.

A couple of things. Is this sort of thing difficult to program into a spreadsheet. I already have one with the required data in it. This could be some additioonal columns to this sheet therefore keeping everything together and minimizing double handling.

Also, if I wanted to use a proportion of the full kelly can I just multiple the result % by that fraction or are the results not linear???

Thanks again for the speedy response.

[Ganchrow]

A couple of things. Is this sort of thing difficult to program into a spreadsheet. I already have one with the required data in it. This could be some additioonal columns to this sheet therefore keeping everything together and minimizing double handling.

It should be a straightforward task in Visual Basic.

Also, if I wanted to use a proportion of the full kelly can I just multiple the result % by that fraction or are the results not linear???

The results are definitely nonlinear. However, for fairly small edges (relative to odds) it should be a workable approximation.

I haven't figured out the closed-form solution to mutually exclusive outcome partial-Kelly, but it's probably not that difficult.