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In the real world not many wagers go off at even money. To handle this situation we rewrite the formula for G from Kelly - Part 2 as
G = (( 1 + B * F ) ^ P) * (( 1 - F ) ^ Q)
where B is the payoff per dollar wagered.
If, for example a team were going off at +120 then the value of B would be 1.2. A favourite at -200 would have a B value of 0.50. In the case of an even money wager the value of B is 1 and the formula reduces to what we had in the previous post.
If you do the calculus thing again you find that the value of F that maximizes G is now
F = ( B * P - Q ) / B
If B is set equal to 1, F simplifies to our previous result, P - Q.
Note that in the general case where B is not equal to 1 the optimum fraction to bet is NOT the same as your potential profit per wager. If you bet 1 dollar, win B dollars with probability P and lose 1 dollar with probability Q then the average profit per dollar wagered is equal to
B * P - Q * 1 = B * P - Q
which is not the same as the value of F.
Since F is the profit divided by the value of B, you bet a greater fraction than your potential profit if you're betting favourites, a smaller fraction if you're betting dogs. You bet a much smaller fraction if you're making long shot plays like parlays.
Two examples: 100 game samples with a starting capital of 1000 dollars
1) Betting on +200 dogs so B is 2. Assume we have a P of 0.36 and a Q of 0.64.
F = ( 2 * 0.36 - 0.64 ) / 2 = 0.04
profit = 2 * 0.36 - 0.64 = 0.08
F After 100 games
---------------------
0.01 1072
0.02 1126
0.03 1159
0.04 1171
0.05 1159
0.06 1127
0.07 1075
0.08 1006
0.09 925
0.10 835
2) Betting on -200 favourites so B is 0.50. Assume we have a P of 0.70 and a Q of 0.30.
F = ( 0.50 * 0.70 - 0.30 ) / 0.50 = 0.10
profit = 0.50 * 0.70 - 0.30 = 0.05
F After 100 games
---------------------
0.05 1208
0.06 1237
0.07 1259
0.08 1276
0.09 1286
0.10 1289
0.11 1286
0.12 1276
0.13 1259
0.14 1235
0.15 1205
Note that betting a fraction equal to your potential profit can be serious indeed if you're betting dogs. In the first example betting 0.08 of capital leads to a finishing capital of 1006 dollars, a net win of 6 dollars after 100 games despite the fact that we have an 8 percent ROI (return on investment).
In general it is not serious to bet a smaller amount than is called for by Kelly. It can be very bad news if you bet a larger amount.
More to come tomorrow...
G = (( 1 + B * F ) ^ P) * (( 1 - F ) ^ Q)
where B is the payoff per dollar wagered.
If, for example a team were going off at +120 then the value of B would be 1.2. A favourite at -200 would have a B value of 0.50. In the case of an even money wager the value of B is 1 and the formula reduces to what we had in the previous post.
If you do the calculus thing again you find that the value of F that maximizes G is now
F = ( B * P - Q ) / B
If B is set equal to 1, F simplifies to our previous result, P - Q.
Note that in the general case where B is not equal to 1 the optimum fraction to bet is NOT the same as your potential profit per wager. If you bet 1 dollar, win B dollars with probability P and lose 1 dollar with probability Q then the average profit per dollar wagered is equal to
B * P - Q * 1 = B * P - Q
which is not the same as the value of F.
Since F is the profit divided by the value of B, you bet a greater fraction than your potential profit if you're betting favourites, a smaller fraction if you're betting dogs. You bet a much smaller fraction if you're making long shot plays like parlays.
Two examples: 100 game samples with a starting capital of 1000 dollars
1) Betting on +200 dogs so B is 2. Assume we have a P of 0.36 and a Q of 0.64.
F = ( 2 * 0.36 - 0.64 ) / 2 = 0.04
profit = 2 * 0.36 - 0.64 = 0.08
F After 100 games
---------------------
0.01 1072
0.02 1126
0.03 1159
0.04 1171
0.05 1159
0.06 1127
0.07 1075
0.08 1006
0.09 925
0.10 835
2) Betting on -200 favourites so B is 0.50. Assume we have a P of 0.70 and a Q of 0.30.
F = ( 0.50 * 0.70 - 0.30 ) / 0.50 = 0.10
profit = 0.50 * 0.70 - 0.30 = 0.05
F After 100 games
---------------------
0.05 1208
0.06 1237
0.07 1259
0.08 1276
0.09 1286
0.10 1289
0.11 1286
0.12 1276
0.13 1259
0.14 1235
0.15 1205
Note that betting a fraction equal to your potential profit can be serious indeed if you're betting dogs. In the first example betting 0.08 of capital leads to a finishing capital of 1006 dollars, a net win of 6 dollars after 100 games despite the fact that we have an 8 percent ROI (return on investment).
In general it is not serious to bet a smaller amount than is called for by Kelly. It can be very bad news if you bet a larger amount.
More to come tomorrow...
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