All the examples up 'til now have assumed that all bets have the same values of P, Q and B. This of course will never happen. It has been pointed out by people in other threads that it is next to impossible to predict a P and a Q for all your bets individually. Even if it were the amount of work involved in doing this and calculating the correct values of F would be excessive. How much would we lose if instead of trying to calculate F values for all bets individually, we simply calculate an average value for all our wagers?

Let's assume the following scenario. We will make 100 wagers broken down as follows.

25 wagers with a P of 0.50, a Q of 0.50 and a B of 1.10

25 wagers with a P of 0.50, a Q of 0.50 and a B of 1.15

25 wagers with a P of 0.50, a Q of 0.50 and a B of 1.20

25 wagers with a P of 0.50, a Q of 0.50 and a B of 1.25

The correct F values for each of these groups of 25 wagers is 0.04545, 0.06522, 0.08333 and 0.10000 respectively. From these we can calculate the "expected" return over 100 wagers as

( ( ( 1 + 1.10 * 0.04545 ) ^ 0.50 ) * ( ( 1 - 0.04545 ) ^ 0.50 ) ) ^ 25 *

( ( ( 1 + 1.15 * 0.06522 ) ^ 0.50 ) * ( ( 1 - 0.06522 ) ^ 0.50 ) ) ^ 25 *

( ( ( 1 + 1.20 * 0.08333 ) ^ 0.50 ) * ( ( 1 - 0.08333 ) ^ 0.50 ) ) ^ 25 *

( ( ( 1 + 1.25 * 0.10000 ) ^ 0.50 ) * ( ( 1 - 0.10000 ) ^ 0.50 ) ) ^ 25

which is equal to 2.00726. i.e. our capital approximately doubles.

The average P and Q for all 100 wagers is 0.5 but the average B is ( 1.1 + 1.15 + 1.2 + 1.25 ) / 4 or 1.175. So the average F will be 0.07447. After 100 wagers with this F value the "expected" return will be

( ( ( 1 + 1.10 * 0.07447 ) ^ 0.50 ) * ( ( 1 - 0.07447 ) ^ 0.50 ) ) ^ 25 *

( ( ( 1 + 1.15 * 0.07447 ) ^ 0.50 ) * ( ( 1 - 0.07447 ) ^ 0.50 ) ) ^ 25 *

( ( ( 1 + 1.20 * 0.07447 ) ^ 0.50 ) * ( ( 1 - 0.07447 ) ^ 0.50 ) ) ^ 25 *

( ( ( 1 + 1.25 * 0.07447 ) ^ 0.50 ) * ( ( 1 - 0.07447 ) ^ 0.50 ) ) ^ 25

which is equal to 1.91315. So we don't do as well. This is to be expected since almost all the time we will be wagering less than or more than the optimum value. The greater the deviation from the optimum value the greater will be the penalty.

To see how good or bad Kelly is we have to build a simulation model that is relatively realistic and test various betting strategies against it. This I will do in the next installment.

Let's assume the following scenario. We will make 100 wagers broken down as follows.

25 wagers with a P of 0.50, a Q of 0.50 and a B of 1.10

25 wagers with a P of 0.50, a Q of 0.50 and a B of 1.15

25 wagers with a P of 0.50, a Q of 0.50 and a B of 1.20

25 wagers with a P of 0.50, a Q of 0.50 and a B of 1.25

The correct F values for each of these groups of 25 wagers is 0.04545, 0.06522, 0.08333 and 0.10000 respectively. From these we can calculate the "expected" return over 100 wagers as

( ( ( 1 + 1.10 * 0.04545 ) ^ 0.50 ) * ( ( 1 - 0.04545 ) ^ 0.50 ) ) ^ 25 *

( ( ( 1 + 1.15 * 0.06522 ) ^ 0.50 ) * ( ( 1 - 0.06522 ) ^ 0.50 ) ) ^ 25 *

( ( ( 1 + 1.20 * 0.08333 ) ^ 0.50 ) * ( ( 1 - 0.08333 ) ^ 0.50 ) ) ^ 25 *

( ( ( 1 + 1.25 * 0.10000 ) ^ 0.50 ) * ( ( 1 - 0.10000 ) ^ 0.50 ) ) ^ 25

which is equal to 2.00726. i.e. our capital approximately doubles.

The average P and Q for all 100 wagers is 0.5 but the average B is ( 1.1 + 1.15 + 1.2 + 1.25 ) / 4 or 1.175. So the average F will be 0.07447. After 100 wagers with this F value the "expected" return will be

( ( ( 1 + 1.10 * 0.07447 ) ^ 0.50 ) * ( ( 1 - 0.07447 ) ^ 0.50 ) ) ^ 25 *

( ( ( 1 + 1.15 * 0.07447 ) ^ 0.50 ) * ( ( 1 - 0.07447 ) ^ 0.50 ) ) ^ 25 *

( ( ( 1 + 1.20 * 0.07447 ) ^ 0.50 ) * ( ( 1 - 0.07447 ) ^ 0.50 ) ) ^ 25 *

( ( ( 1 + 1.25 * 0.07447 ) ^ 0.50 ) * ( ( 1 - 0.07447 ) ^ 0.50 ) ) ^ 25

which is equal to 1.91315. So we don't do as well. This is to be expected since almost all the time we will be wagering less than or more than the optimum value. The greater the deviation from the optimum value the greater will be the penalty.

To see how good or bad Kelly is we have to build a simulation model that is relatively realistic and test various betting strategies against it. This I will do in the next installment.

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