Gambles with the same Expected return and Kelly bet size

In our previous post we grouped gambles by their optimal bet sizes. Here we consider the subset of these groups in which the gambles share not only an optimal bet size, but also the same expected return.

1. Finding the set of gambles with same expected return and Kelly bet size

We consider gamble (p, a, b)  (p is the winning probability, a is the winning outcome and b is the losing outcome) and find the set of all gambles of the form (P, A, B) which have the same expected return and optimal bet size. Matching the expected value means that

$ap+b(1-p)=AP+B(1-P) = E$            eq. (1)

if we also match the optimal bet size, which we showed in our previous post, then

${\LARGE-\frac{ap+b(1-p)}{ab} = -\frac{AP+B(1-P)}{AB}}$

which is equivalent to

$-\frac{E}{ab}=-\frac{E}{AB}$

or

$ab=AB$                                                       eq. (2)

Substituting the equality in equation (2) into equation (1) we get

$PA^2-EA+(1-P)ab=0$                                         eq. (3)

which is a quadratic function of A with E, a and b constants defining the initial gamble.

Since a >0 is the positive outcome and b < 0 is the negative outcome, the determinant

$E^2-4abP(1-P)>0$

is positive, so the function has two real roots.

We next show that one of the roots is positive and the other is negative. We first rewrite equation (3) as

$A^2-\frac{EA}{P}+\frac{ab(1-P)}{P}=0$                                      eq. (4)

If we label the two roots as $A_1$ and $A_2$, based on the basic property of quadratic function, the product of the roots satisfies

$A_1A_2 =\frac{ab(1-P)}{P}$

Since a and P are positive and b is negative,

$\frac{ab(1-P)}{P} < 0$

so one root is positive and the other is negative. We take the positive root as A since it is the winning outcome of the new gamble (P, A, B). The negative root is

$\frac{B(1-P)}{P}$

in which B is the losing outcome of the new gamble times the losing odds ratio.

To see how a gamble maps the set of all gambles with the same expected value and optimal bet size, we consider a gamble (p = 0.5, a = 2, b = -1) as the initial gamble. This gamble has an expected value and optimal bet size of

E = 0.5*2-0.5*1 = 0.5

S = -E/(-1*2) = 0.25

We next pick a level of P and determine the values of the winning and losing outcomes. For instance, if we set P = 0.6, and solve for A and B in the gamble (P, A, B),  the quadratic function of A is

$0.6A^2-0.5A-0.8=0$

and the two roots are -0.8109 and 1.6442. The new gamble’s winning outcome A is the value of the positive root, 1.6442. Since

$ab = AB$                                eq.(2)

B = ab/A =  -1.2164.

The resulting gamble (P = 0.6, A = 1.6442, B = -1.2164) has the same expected value and optimal bet size as the initial gamble

E = 0.6 * 1.6442-0.4 * 1.2164 = 0.5

S = -E / (-1.2164 * 1.6442) = 0.25

The value of the negative root is

$\frac{B(1-P)}{P} = \frac{-1.2164(1-0.6)}{0.6} = -0.8109$

We can extend this analysis for other values of P from 0 to 1 and generate the graph below.

Next we consider how the range of positive outcomes, A, changes as the winning probability P approaches 1 and as it approaches 0.

When P approaches 1:

Even though P must be less than 1 to avoid creating a gamble with an infinite optimal bet size, we can determine the values of A and B in the open set as P approaches 1.

As P approaches one, the constant term of equation (3),  $ab(1-P)$, approaches zero, so equation  (3) converges to

$A^2-EA=0$

The positive root is E (the other root is zero), so even though A must be greater than E the value of A can be infinitely close to E.

On the other hand, as $P \to 1$ and $A\to 1$

$B = \frac{ab}{A} \to \frac{ab}{E} = -1/S$

This means that as the probability of winning approaches one, the winning outcomes approaches the expected value and the losing outcome approaches the maximum amount that one can lose without putting one’s entire bankroll at risk.

The pink point at the bottom-left of the graph represents when P = 1, A = E = 0.5 and B = -1/S = -4.

When P approaches 0:

When P approaches 0, there is no finite upper bound for the positive outcome, A. Instead, we consider the limit of the negative outcome, B, as A goes to infinity. As $A\to$ infinity,

$B=\frac{1}{A}ab\to0$

so B is bounded above by zero.

From our analysis of the range of P values, A ranges from E to infinity and B ranges from -1/S to 0 where S is the optimal bet size.

In the previous post we fixed the Kelly bet size, for a given quantity of P and drew a curve relating the positive and negative outcomes, A and B. However, if we also fix the expected return, we get a unique point in the space of A and B for each value of P. Next we consider how the curves for gambles with the same expected value and optimal bet size fit together to define the space of all gambles with the same optimal bet size.

2. Grouping gambles with same optimal bet size but different expected return.

From our earlier discussion about the expected value and optimal bet size we saw that:

$S = -\frac{E}{ab}$

which can be rewritten as

$ab = -\frac{E}{S}$

and outlines hyperbolas when plotted in the plane of winning and losing outcomes. Furthermore, when we replace the product of  a and b by -E/S in equation (4):

$A^2-\frac{E}{P}+\frac{ab(1-P)}{P}=0$

we get such equivalent equation

$A^2-\frac{E}{P}+\frac{(-E/S)(1-P)}{P}=0$

By setting values for E and S, we find the relationship between the values of A, B and P to create the graph below. The first curve of circles shows all the gambles with Kelly bet size 0.25 and expected return 0.5. Similarly, the second curve of triangles shows all the gambles with Kelly bet size 0.25 and expected return 1.

From our previous discussion, we know that when P approaches 1, A approaches E and B approaches -1/S. In this plot S = 0.25, so the lower bound of B is always -4. The lower bound of A differs but equals the expected return of each line. When P approaches 0, A increases without bound and B approaches 0.

From this picture we see that the space of gambles with a fixed optimal bet size is made up of hyperbolas for the gambles sharing the same expected value.

In our next post we will consider how risk measures divide the space of gambles.