Twitter user @optibrebs recently made me aware of the generalised Kelly Criterion (a.k.a. The Real Kelly) discussed in this @Pinnacle article The Real Kelly
To be honest I have been successfully using the simple/naïve Kelly Criterion for years and haven’t bothered looking anywhere else.
This is how the simplified Kelly looks
s = b * (o / f – 1) / (o – 1)
where
s…Kelly Stake
b…Bankroll
o…Odds offered
f…Odds fair
The simplified Kelly Criterion has some potential flaws and can give some head-scratching results which are explained in the article above.
I therefore recommend using the more sophisticated generalised Kelly Criterion wherever possible as it is superior to the simplified Kelly Criterion. The Real KC is very useful for outright betting as shown below.
In this article I will explain how to use the real KC in order to find optimal bet sizes for a set of futures bets on several different teams to win the same division or championship. However please note that the algorithm can be applied to ANY market with mutually exclusive outcomes (e.g. it could also be applied to a 3-way betting market as explained in example #1 in the @Pinnacle article).
Let’s dive in.
The excel template can be downloaded here real_kelly-mutually_exclusive_outcomes-
Please note
(1) Outcomes must be mutually exclusive (= EXACTLY one outcome will happen). The algorithm will work for markets like ‘Liverpool To Win The EPL’, but will not work for markets like ‘Liverpool Top-4’.
(2) Probabilities need to add up to 1 (or very close to it).
(3) You can adapt the number of outcomes to your needs, but be aware to also update the variable cells within the solver.
(4) You can use excel’s built-in solver using method ‘GRG Nonlinear’, but be advised that it might not give you the best set of results. I recommend using the GlobalMinimize macro instead which can be downloaded here GlobalMinimize Excel Macro.
The algorithm will very likely find a global maximum (instead of just a local maximum) for the objective and will yield the optimal solution.
The drawback is that GlobalMinimize is limited to 30 variable cells. Excel’s solver has a limit of 250 variable cells.
INSTRUCTIONS
- Make sure to open both GlobalMinimize and the Real Kelly worksheet.
- Fill out the fields in the yellow columns (‘BET’, ‘BOOK_ODDS’, ‘MY_ODDS’) as shown below. ‘MY_ODDS’ is what you consider to be the fair odds.
- Set all stakes to 0.
- Enter your bankroll in the yellow cell ‘START BANKROLL’
- If you have any existing bets on this tournament enter those in columns (O, P, Q). Note that any strings in column O must exactly match those in column A.
- Press CTRL+M to open GlobalMinimize settings.
- Adjust the stake range/constraints according to your needs.
- Select Differential-Evolution (DE) followed by Nelder-Mean (NM) and hit ‘Minimize’.
Differential-Evolution (DE) followed by Nelder-Mean (NM) should be your default option, as it increases the likehood of finding the global maximum, and NM ‘polishes off’ the optimum with a high degree of accuracy.
Depending on the field size the algorithm can now take several minutes until excel populates the optimal set in the stakes column.
What I like about this application is that you could potentially use it from week to week as the season progresses. The algorithm will find optimal bets and/or (partial) hedges whenever the sheet is updated with new prices.
Have fun!
