Kelly with correlated outcomes

[Tomato]

Denmark-Netherlands

Pinnacle probabilities:
Netherlands -179 = 62.958%
Draw +309 = 23.992%
Denmark +652 = 13.049%

Netherlands -1 +102 = 49.6975%
Denmark +1 -109 = 51.3025%

BetOnline odds:
Draw +350 = 22.2222% = 7.967% EDGE
Denmark +1 +110 = 47.619% = 7.735% EDGE

Should you take into consideration the correlation between the draw and +1 (If draw wins, +1 +110 wins) and adjust the structure of your bets (and how would you do this), or treat them as separate markets and bet both full Kelly?

[Optional]

I think you should take the correlation into account if trying to follow Kelly, as it assumes independent bets. (but don't know how)

[Pancho sanza]

Denmark-Netherlands

Pinnacle probabilities:
Netherlands -179 = 62.958%
Draw +309 = 23.992%
Denmark +652 = 13.049%

Netherlands -1 +102 = 49.6975%
Denmark +1 -109 = 51.3025%

BetOnline odds:
Draw +350 = 22.2222% = 7.967% EDGE
Denmark +1 +110 = 47.619% = 7.735% EDGE

Should you take into consideration the correlation between the draw and +1 (If draw wins, +1 +110 wins) and adjust the structure of your bets (and how would you do this), or treat them as separate markets and bet both full Kelly?

Yes you should consider the correlation.

There are 4 unique outcomes, Denmark Win, Draw, Netherlands by exactly 1 , Netherlands By > 1, you need the probabilities (P) for each. You have all but Netherlands by exactly 1 which you can figure out from the other probs.

Next, calculate the log of your ending bankroll under each unique outcome, the formula is P*LOG(END BANKROLL), end bankroll is just Sum of opening bankroll + $$ W/L under each scenario.

Sum these amounts and maximize them using excel solver.

[Ganchrow]

Denmark-Netherlands

Pinnacle probabilities:
Netherlands -179 = 62.958%
Draw +309 = 23.992%
Denmark +652 = 13.049%

Netherlands -1 +102 = 49.6975%
Denmark +1 -109 = 51.3025%

BetOnline odds:
Draw +350 = 22.2222% = 7.967% EDGE
Denmark +1 +110 = 47.619% = 7.735% EDGE

Should you take into consideration the correlation between the draw and +1 (If draw wins, +1 +110 wins) and adjust the structure of your bets (and how would you do this), or treat them as separate markets and bet both full Kelly?

Of course you need to take correlation into account.

Based on Pinnacle these are the underlying probs for the spread bet:


And the conditional probabilities (i.e., conditioned on the Netherlands NOT winning by exactly 1) for the ML Bet:

(Note the slight error correction shown in red.)

So the unconditional probabilities and bet results for each of the four possible relevant outcomes are

So if we label the bet on the draw as X_Dr and the bet on Denmark as X_D+1 then out full Kelly expected utility function looks like this:

E[U(X_Dr, X_D+1)] =

13.0492% * log(1 - X_Dr + 1.1*X_D+1) +
23.9927% * log(1 + 3.5*X_Dr + 1.1*X_D+1) +
27.7970% * log(1 - X_Dr) +
35.1611% + log(1 - X_Dr - X_D+1)

Which when maximized wrt X_Dr and X_D+1, yields the following full-Kelly wagers:

Some years back, I posted the attached spreadsheet to optimize just these sorts of straightforward ML/Spread combinations. It's simplistic and reflects the work of a younger, gentler, sloppier man (), but it appears I wasn't totally inept and it does do the trick. NO warranties, either expressed or implied.

Note that on the spreadsheet the ML refers to the Draw bet which, perhaps somewhat confusingly is marked as a "Win" if the teams Draw and a loss otherwise.

[Tomato]

Thank you Pancho villa and Ganchrow.

Tomato is just starting to bet soccer and implement Kelly with his edge and this is very helpful.

[ojs69]

Good stuff you guys

[roon]

I am new to the mathematical approach to sports wagering. Am I missing something in these calculations? I get

Netherlands -179 = 64.2%
Draw +309 = 24.4%
Denmark +652 = 13.3%

when I add these percentages, it comes to 101.9, so I would assume this is where I am off and I need to perform a second calculation. Do I take the 64.2 and divide by the 101.9 .642/1.019? And the again with the draw and Denmark? When I do this I get the correct percentages from the first post. Is this to compensate for the vig?

I would really appreciate any help. I am pretty strong with math and am very intrigued with these concepts. I'm just beginning to put this info into practice. Gonna take awhile though...

[Poogs]

The extra 1.9 is the theoretical hold of the book.