Prompted in part by heehaww's post above, I thought I would try to solve for the simplest case. Two independent bets each with exactly two outcomes (win and lose) and each with positive EV. (Based upon heehaww's post, I think that he and probably others have previously derived these formulas so I claim no primacy or originality.)
Without loss of generality, consider your original bankroll to be 1. We are interested in determining the optimal fraction of your bankroll to bet on each of two simultaneous and independent bets.
Let there be the following two bets offered to you.
Bet 1: prob win = P; if you win you receive Ax in profit if you bet x units
Bet 2: prob win = Q, if you win you receive By in profit if you bet y units
Based upon P, Q, A, and B, we seek the optimal x and y.
Mathematically, the Kelly criterion prescribes that a bettor bet an amount that maximizes the expected logarithm of the bettor's ending bankroll. In the single bet environment, this leads to a fairly simple (and well-known) formula. We will come back to this below.
Setting partial derivatives to zero and solving yields two very messy equations that do not appear to present a closed-formed solution. However, since you can plug one equation into the other equation, you can essentially wind up with a complicated non-linear equation in one variable. So numerical techniques can be used to solve for the optimal pair in a very short amount of time.
We are going to have a bunch of really long terms. I attempted to "simplify" all of the expressions, but looking back on it maybe I should have kept the terms in their original forms. For simplicity in what follows I have omitted all of the multiplication symbols. So, for example, ABPQ stands for A*B*P*Q and AAQY stands for A*A*Q*Y.
X3TERM = AA
X2TERM = AAQY-AAPQY-AP-AAP+AAPY-ABPQY-APQY-AABPQY+APY+AABQY+2A-AY+ABY-AA-AABY
X1TERM = 2AQY-AQYY+2ABQY+AAP+AABPY-AAPY-AABPYY-P-BPY+PY+BPYY+ABBQYY+1+BY-Y-BYY-2A-3ABY+AY+ABYY-ABBYY
X0TERM = (1+BY)*(1-Y)*(-APQY+QY-PQY+AP+ABPY-ABPQY-1+P-BY+BPY+BQY-BPQY)
Let E = (3*X3TERM*X1TERM-X2TERM*X2TERM)/(3*X3TERM*X3TERM)
Let F = (2*X2TERM*X2TERM*X2TERM-9*X3TERM*X2TERM*X1TERM+27*X3TERM*X3TERM*X0TERM)/(27*X3TERM*X3TERM*X3TERM)
Then the optimal X can be expressed in terms of Y and the other variables:
X = 2*SQRT(-E/3)*cos((1/3)*arccos((3*F/2E)*SQRT(-3/E))-2*PI/3)-(X2TERM/(3*X3TERM))
Similarly, by partial derivatives we can solve for Y in terms of X.
Y3TERM = BB
Y2TERM = -BBQ+BBQX+BBPX-ABPQX-BBPQX+2B+ABX-BX-BB-ABBX+ABBPX-BQ+BQX-BPQX-ABBPQX
Y1TERM = BBQ+ABBQX-BBQX-ABBQXX+2BPX-BPXX+1+AX-X-AXX-2B-3ABX-AABXX+BX+ABXX-Q-AQX+QX+AQXX+2ABPX+AABPXX
Y0TERM = (1+AX)*(1-X)*(BQ-BPQX+ABQX-ABPQX-PQX+PX+Q-1+APX+AQX-APQX-AX)
Let G = (3*Y3TERM*Y1TERM-Y2TERM*Y2TERM)/(3*Y3TERM*Y3TERM)
Let H = (2*Y2TERM*Y2TERM*Y2TERM-9*Y3TERM*Y2TERM*Y1TERM+27*Y3TERM*Y3TERM*Y0TERM)/(27*Y3TERM*Y3TERM*Y3TERM)
Then the optimal Y can be expressed in terms of X and the other variables:
Y = 2*SQRT(-G/3)*cos((1/3)*arccos((3*H/2G)*SQRT(-3/G))-2*PI/3)-(Y2TERM/(3*Y3TERM))
As I said above, we need to solve these two equations for X and Y. One approach is to plug the X equation (X as a function of Y) into the Y equation. This yields a single highly non-linear complicated equation in one unknown (Y). Numerical techniques are available to solve this equation in a matter of seconds. Of course, once we have the optimal Y, the optimal X is immediate.
Here are three cases for which I have calculated the optimal simultaneous bets and compared them to the original optimal single-bet Kelly condition (as if each were the only bet available).
|Variable|| Case 1|| Case 2|| Case 3|
where SingleKelly refers to the optimal bets under the Single-Bet Kelly Criterion and SimulKelly refers to the optimal bets under the Simultaneous-Bet Kelly Criterion.
Anyway, not sure that this is overly interesting, especially since, even after the partial derivatives are solved, numerical techniques are still required to derive the optimal bet sizes. That is, the partial derivatives are nice and everything (lol), but numerical techniques could have simply been applied to the original expectation equation without knowing anything about the partial derivatives.
Any and all comments are of course welcome.