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OK, finally, the big day. Who wins, Kelly or flat betting? The results may surprise you.
What I did was set up a simulation using a random number generator in Visual FoxPro.
Taking baseball as an example, I assumed a 180 day season, betting 4 games per day, 720 games in all. I assumed that all the bets were on dogs with odds varying uniformly from +100 to +160. Each of these wagers had a ROI between 2 and 8 percent. Again the distribution was uniform.
Since we can calculate B from the odds and since we know that B, P, Q and ROI are related by the formula
B * P - Q = ROI / 100
we can calculate
P = ( 1 + ROI / 100 ) / ( 1 + B )
remembering that Q = 1 - P.
For each game a random value was compared with the calculated P to see if we won or lost the game.
I first ran the simulation 1000 times calculating the optimal Kelly factor for 1 game from the formula
F = ( B * P - Q ) / B
I didn't adjust the Kelly factor for multiple games since the adjustment is pretty small. The starting capital was 10000 dollars.
The results are as follows. The value "i" in column 1 is the exponent of the number 2. The value in column 2 is the number of times our capital at the end of the season is in the range 2 ^ i to 2 ^ ( i + 1 ).
-6 0
-5 2
-4 8
-3 29
-2 88
-1 203
0 236
1 206
2 145
3 52
4 27
5 3
6 1
7 0
The average end of season capital is 32158 dollars.
The second simulation was run with F values of ( 0.05 / B ). i.e. I used the average expectation of 5 percent and adjusted for the B value - longer odds, less money bet. The results are
-6 0
-5 4
-4 10
-3 35
-2 116
-1 200
0 231
1 180
2 131
3 65
4 23
5 2
6 3
7 0
The average end of season capital is 33952 dollars.
The third simulation was run with F values of 0.80 * 0.05. i.e. I used just the average expectation of 5 percent - no adjustment for B. The factor 0.80 is used to lower the average bet size to make the total amount wagered at least initially comparable with the first 2 simulations.
-6 1
-5 4
-4 11
-3 44
-2 114
-1 203
0 215
1 188
2 122
3 64
4 28
5 3
6 2
7 1
The average end of season capital is 34921 dollars.
The fourth simulation was run with flat bets of 400 dollars. The bet size was never adjusted during the course of a simulation.
-6 0
-5 0
-4 0
-3 5
-2 11
-1 58
0 258
1 452
2 79
3 0
4 0
5 0
6 0
7 0
The average end of season capital is 20896 dollars. Note that 137 results are missing from the table. Those simulations ended in bankruptcy.
The fifth simulation was run with flat bets starting at 400 dollars ( 4 percent of capital ). The bet size was adjusted if capital increased by fifty percent or decreased by fifty percent.
-6 2
-5 6
-4 18
-3 57
-2 128
-1 168
0 210
1 178
2 139
3 67
4 20
5 2
6 2
7 0
The average end of season capital is 32090 dollars.
So what does all this mean - it means that staying at flat bets especially if your bets are too high (4 percent is definitely too high) is BAD. It also means that plateau betting is NO DIFFERENT from Kelly betting. In fact, as I mentioned in a long-ago post that started all this furor about Kelly - plateau betting IS Kelly betting.
Pure Kelly betting - i.e. varying your bet size according to win probability and expected payoff will give better results than plateau betting for higher F values than are used in the simulation. But I felt it was better to give an example using realistic expectations.
If you reduce the bet size to about 2 percent of capital you find that flat betting doesn't go bankrupt as often as at 4 percent but that Kelly betting and plateau betting still give better and equivalent results.
'mute
What I did was set up a simulation using a random number generator in Visual FoxPro.
Taking baseball as an example, I assumed a 180 day season, betting 4 games per day, 720 games in all. I assumed that all the bets were on dogs with odds varying uniformly from +100 to +160. Each of these wagers had a ROI between 2 and 8 percent. Again the distribution was uniform.
Since we can calculate B from the odds and since we know that B, P, Q and ROI are related by the formula
B * P - Q = ROI / 100
we can calculate
P = ( 1 + ROI / 100 ) / ( 1 + B )
remembering that Q = 1 - P.
For each game a random value was compared with the calculated P to see if we won or lost the game.
I first ran the simulation 1000 times calculating the optimal Kelly factor for 1 game from the formula
F = ( B * P - Q ) / B
I didn't adjust the Kelly factor for multiple games since the adjustment is pretty small. The starting capital was 10000 dollars.
The results are as follows. The value "i" in column 1 is the exponent of the number 2. The value in column 2 is the number of times our capital at the end of the season is in the range 2 ^ i to 2 ^ ( i + 1 ).
-6 0
-5 2
-4 8
-3 29
-2 88
-1 203
0 236
1 206
2 145
3 52
4 27
5 3
6 1
7 0
The average end of season capital is 32158 dollars.
The second simulation was run with F values of ( 0.05 / B ). i.e. I used the average expectation of 5 percent and adjusted for the B value - longer odds, less money bet. The results are
-6 0
-5 4
-4 10
-3 35
-2 116
-1 200
0 231
1 180
2 131
3 65
4 23
5 2
6 3
7 0
The average end of season capital is 33952 dollars.
The third simulation was run with F values of 0.80 * 0.05. i.e. I used just the average expectation of 5 percent - no adjustment for B. The factor 0.80 is used to lower the average bet size to make the total amount wagered at least initially comparable with the first 2 simulations.
-6 1
-5 4
-4 11
-3 44
-2 114
-1 203
0 215
1 188
2 122
3 64
4 28
5 3
6 2
7 1
The average end of season capital is 34921 dollars.
The fourth simulation was run with flat bets of 400 dollars. The bet size was never adjusted during the course of a simulation.
-6 0
-5 0
-4 0
-3 5
-2 11
-1 58
0 258
1 452
2 79
3 0
4 0
5 0
6 0
7 0
The average end of season capital is 20896 dollars. Note that 137 results are missing from the table. Those simulations ended in bankruptcy.
The fifth simulation was run with flat bets starting at 400 dollars ( 4 percent of capital ). The bet size was adjusted if capital increased by fifty percent or decreased by fifty percent.
-6 2
-5 6
-4 18
-3 57
-2 128
-1 168
0 210
1 178
2 139
3 67
4 20
5 2
6 2
7 0
The average end of season capital is 32090 dollars.
So what does all this mean - it means that staying at flat bets especially if your bets are too high (4 percent is definitely too high) is BAD. It also means that plateau betting is NO DIFFERENT from Kelly betting. In fact, as I mentioned in a long-ago post that started all this furor about Kelly - plateau betting IS Kelly betting.
Pure Kelly betting - i.e. varying your bet size according to win probability and expected payoff will give better results than plateau betting for higher F values than are used in the simulation. But I felt it was better to give an example using realistic expectations.
If you reduce the bet size to about 2 percent of capital you find that flat betting doesn't go bankrupt as often as at 4 percent but that Kelly betting and plateau betting still give better and equivalent results.
'mute
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