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Kelly Criterion and Middling
A poster PM'd me the following question:
First let's get the easy part out of the way -- how to calculate the EV on a middle.A total for an NCAAF game is 59.5. Based on my handicapping, I've come to the conclusion that the "fair" line is 59.5 +100.
Two books are offering o58.5 -107 and u60 -102. Using the half-point calculator with a “fair” line 59.5 +100, the edges for the two totals and prices are (approximately) 6.01% and 1.04% respectively.
Using the arbitrage calculator, I find that for each $100 bet on the -107 (over) side I should bet $97.69 on the -102 (under) side and that the "profit" for each $197.69 bet is -$4.23.
How would I go about calculating the edge of the combined position and could I use such a calculated edge for Kelly considerations?
If you have two bets with edges E% and F% that push with probabilities P% and Q%, respectively, and you’re risking X units on the first bet and Y units on the second bet, your total edge will be X/(X+Y) * E%*(1-P%) + Y/(X+Y) * F%*(1-Q%)
So with the above allocation your total edge would be $100/$197.60 * 6.01% + $97.60/$197.60 * 1.04%*(1-2%) ≈ 3.54%.
What you’re doing is adjusting each edge for its push probability (recall that we quote edges conditioned on not pushing), and then taking the average of the two weighted for the relative bets sizes. Note that if the two bets could simultaneously push, we’d then need to divide the result through by (1 – simultaneous push prob) in order to yield an edge conditioned on not pushing.
But the real issue we need to consider is to what extent it’s even appropriate to use the arbitrage calculator in this regard. What the calculator will output are the relative bet sizes which will serve to equate losses were the middle not to hit. While this is certainly a reasonable sounding objective, we have no reason to believe it would result in an optimal relative bet allocation from the perspective of Kelly.
(Realize that we can’t simply use the Kelly calculator tool because it only works for either uncorrelated binary bets or mutually exclusive binary bets where every outcome is covered. The former condition is clearly not satisfied. The latter condition is not satisfied either and because the middle is not mutually exclusive with either the over or the under.)
Referring back to the half-point calculator, we see that the over wins and the under loses with probability 48%, the under wins and the over loses with probability 45.2%, the over wins and the under pushes with probability 2%, and both bets win with probability 4.8%.
In this post I detailed how one might go about solving this type of problem.
(For those posters who after reading this post have decided they don’t feel like implementing the methodology on their own I’ve posted a Kelly spreadsheet for middles. The editable cells are those displayed in light gray. The spreadsheet can also be used to for reverse middles and for other overlapping bets, such as a money line and spread on the same team.)
As is frequently the case with arbs and middles the optimal bet allocation will be highly dependent on maximum bet sizes. In fact, with full-Kelly what we find is the optimal allocation would ideally utilize 100% of bankroll -- 51.8094% of bankroll on the over 58.5 and 48.1906% on the under 60. Putting this another way, this would imply that for every $100 bet on the over, $93.02 should be bet on the under. You’ll notice that this corresponds to a smaller allocation on the under than indicated by the arbitrage calculator. This is because Kelly prefers the higher EV/shorter odds over bet to the under bet and as such is willing to risk losing more in the case of the over losing.
Now as I’ve mentioned, it’s not reasonable to expect that a bettor will be able to place 100% of his bankroll on any given pair of bets. For one, the bettor’s bankroll is likely to be spread out over a number of different books and financial institutions, and for another the books offering these bets are likely to have fairly low maximum wager sizes relative to an advantage player's bankroll.
So if we reduce the maximum allowable wager sizes on each bet to, let’s say, 20% of bankroll, what we find is that for every $100 placed on the over we’d want to risk $96.09 on the under. In other words, with lower limits we’d risk relatively more on the under relative to the over. The reasoning behind this is that with lower limits we would have less money at stake and consequently we wouldn't worry as much about the risk of being “out-of-hedge”.
If we further reduce maximum bet sizes to 2% of bankroll, then the above effect becomes even more pronounced, and we find that the full-Kelly optimal allocation would be the max on each bet (a hedge ratio of 1:1).
So what this suggests is that there’s nothing fundamentally “special” about the allocation that equates losses (i.e., the allocation suggested by the arbitrage calculator), and depending upon the terms of the bets and max wager considerations, a player’s actual bet allocation may differ substantially.
Nevertheless, for some bettors there may be something intuitively appealing about equating losses, and the fact is that for middles consisting of bets with similar odds and where the payout region is fairly centrally located between to the two bet figures, such a method will typically yield results that are fairly close to Kelly optimal. The advantage to doing is that we then might be able to use the standard Kelly calculator.
If we decided that that was how we wanted to proceed we’d have to treat the loss-equated middle as a binary bet that wins if the middle hits and loses if it doesn’t. But that’s a problem for us because one of the bets has a non-zero probability of pushing. Hence the bet is not binary and neither is the middle (both are ternary).
So in order to demonstrate how to do this with binary bets, we’d need to eliminate the possibility of either bet pushing. To that end, let’s assume that instead of the u60 -102, the book were offering u60.5 -108 (corresponding to an edge of 0.15%, a bit less than in the original example). This implies that (as before) the over wins and the under loses with probability 48% and the under wins and the over loses with probability 45.2%. The only other outcome, both bets winning occurs with probability 6.8%.
Plugging the -108 over and -107 under into the arbitrage calculator we see that for every $100 risked on the -107 we’d risk $100.45 on the -108. This would correspond to a loss of $6.99 or 3.4879% were the middle not to hit. If the middle were to hit, profit would be ($100*100/107 + $100.45*100/108) ≈ $186.47 or $186.47/$200.45 ≈ 93.0243%. This is the equivalent of a line of about 100 * 93.042%/3.4879% ≈ +2,667.1.
So if we plug in a line of +2,667.1 and a win probability of 6.8% into the Kelly calculator we get a total bet size of 3.3056%, which corresponds to the percentage of bankroll we’d be willing to risk losing on this bet. Because for every unit risked on the middle we’ll lose 3.4879% of a unit, the total allocation will be 3.3056% / 3.4879% ≈ 94.7734% of bankroll. This corresponds to 94.7734% * 100/200.45 ≈ 47.28% of bankroll on the over and 94.7734% * 100.45/200.45 ≈ 47.49% of bankroll on the under, resulting in expected bankroll growth of 1.1725%. By contrast, the expected growth from the actual Kelly optimization (where each bet may be independently sized) is 1.2222%.
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Originally Posted by sofos
