When your wife is making you rip out the basement it doesn't leave a lot of time for writing.

The story to date - if you know your P's and Q's, the Kelly aproach tells you to bet a fraction F of your bankroll equal to

F = ( B * P - Q ) / B

to maximize your return. This assumes you are betting one game at a time. What if you want to bet 2, 3, 4... games simultaneously.

Dr. Thorp makes a stab at this in his paper posted at

//www.bjmath.com/bjmath/thorp/paper.htm

He looks at the special case where you are betting two games at once and the games are going off at even money, i.e. B = 1. He shows that if you have two games to wager on, both with identical values of P and Q then the optimal value of F is

F1 / ( 1 + F1 ^ 2 )

where F1 is the optimal value if you are betting on one game only.

For example, if P = 0.55, Q = 0.45 and B = 1 then the optimum fraction to wager for one game is

F1 = ( 0.55 - 0.45 ) = 0.10

If you have two games you would wager the following fraction on each of the two games

F = 0.10 / ( 1 + 0.10 ^ 2 ) = 0.099

The difference is not very large.

Dr. Thorp does not give a solution for the 2 game case where B is not equal to 1. Neither does he look at the general case of more than 2 games. I suspect that these betting situations are algebraically intractable. They can be solved numerically however by any interested party like me who has access to a computer.

Following are the results for betting 1, 2, 3, 4... 10 games simultaneously when the values of P, Q and B are 0.50, 0.50 and 1.2 respectively.

#games F

---------

1 0.0833

2 0.0827

3 0.0820

4 0.0813

5 0.0806

6 0.0799

7 0.0792

8 0.0784

9 0.0776

10 0.0768

Note that F drops slowly as more and more games are bet. When you approach the number of games where you would be betting your whole bankroll the F value comes closer and closer to BANKROLL / #games but never quite gets there.

The program I wrote to build the above table was written in Visual FoxPro and runs into problems with exponent underflow as #games increases in value. Also roundoff problems are worrisome. You'll perhaps understand why when you see the formula that you have to optimize. It'll be in the next installment. Before then I'll also rewrite the program using Mathematica. Mathematica gives you the correct answer to any desired degree of accuracy.

The story to date - if you know your P's and Q's, the Kelly aproach tells you to bet a fraction F of your bankroll equal to

F = ( B * P - Q ) / B

to maximize your return. This assumes you are betting one game at a time. What if you want to bet 2, 3, 4... games simultaneously.

Dr. Thorp makes a stab at this in his paper posted at

//www.bjmath.com/bjmath/thorp/paper.htm

He looks at the special case where you are betting two games at once and the games are going off at even money, i.e. B = 1. He shows that if you have two games to wager on, both with identical values of P and Q then the optimal value of F is

F1 / ( 1 + F1 ^ 2 )

where F1 is the optimal value if you are betting on one game only.

For example, if P = 0.55, Q = 0.45 and B = 1 then the optimum fraction to wager for one game is

F1 = ( 0.55 - 0.45 ) = 0.10

If you have two games you would wager the following fraction on each of the two games

F = 0.10 / ( 1 + 0.10 ^ 2 ) = 0.099

The difference is not very large.

Dr. Thorp does not give a solution for the 2 game case where B is not equal to 1. Neither does he look at the general case of more than 2 games. I suspect that these betting situations are algebraically intractable. They can be solved numerically however by any interested party like me who has access to a computer.

Following are the results for betting 1, 2, 3, 4... 10 games simultaneously when the values of P, Q and B are 0.50, 0.50 and 1.2 respectively.

#games F

---------

1 0.0833

2 0.0827

3 0.0820

4 0.0813

5 0.0806

6 0.0799

7 0.0792

8 0.0784

9 0.0776

10 0.0768

Note that F drops slowly as more and more games are bet. When you approach the number of games where you would be betting your whole bankroll the F value comes closer and closer to BANKROLL / #games but never quite gets there.

The program I wrote to build the above table was written in Visual FoxPro and runs into problems with exponent underflow as #games increases in value. Also roundoff problems are worrisome. You'll perhaps understand why when you see the formula that you have to optimize. It'll be in the next installment. Before then I'll also rewrite the program using Mathematica. Mathematica gives you the correct answer to any desired degree of accuracy.

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