General Betting
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Where am I going wrong with this?
Assume someone is tossing a coin and offering £1.50 every time you call right but you only have to pay £1 when you are wrong.
If you have a starting bank of £100. According to the Kelly Criterion on your first bet you should wager 16.67% of your bank and continue to wager that on each spin.
Now, suppose we call heads each time and the sequence of ten spins is as follows:
Betting bank stake outcome win/loss
100 16.67 (h) won +25
Now, our next bet is now 16.68% of what is now in the bank (£125 = £28.85)
125 20.85 (h) won +31.27
156 26 (t)lost -26
130 21.67 (h)won +32.5
162.5 27.10 (t)lost -27.10
135.4 22.57 (t)lost -22.57
112.43 18.74 (h)won +28.1
140.5 23.42 (t)lost -23.42
117.80 19.64 (t)lost -19.64
98.16 16.36 (h)won +24.5
total in bank after 10 spins = £122.7
Had we just flat-betted £10 on each spin we would have made £25 so using Kelly, in such favourable circumstances, seems to have made things worse, or certainly no better.
Over to you. . .
Assume someone is tossing a coin and offering £1.50 every time you call right but you only have to pay £1 when you are wrong.
If you have a starting bank of £100. According to the Kelly Criterion on your first bet you should wager 16.67% of your bank and continue to wager that on each spin.
Now, suppose we call heads each time and the sequence of ten spins is as follows:
Betting bank stake outcome win/loss
100 16.67 (h) won +25
Now, our next bet is now 16.68% of what is now in the bank (£125 = £28.85)
125 20.85 (h) won +31.27
156 26 (t)lost -26
130 21.67 (h)won +32.5
162.5 27.10 (t)lost -27.10
135.4 22.57 (t)lost -22.57
112.43 18.74 (h)won +28.1
140.5 23.42 (t)lost -23.42
117.80 19.64 (t)lost -19.64
98.16 16.36 (h)won +24.5
total in bank after 10 spins = £122.7
Had we just flat-betted £10 on each spin we would have made £25 so using Kelly, in such favourable circumstances, seems to have made things worse, or certainly no better.
Over to you. . .
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Extend the same sequence to 20 bets...
100.00 16.67 win 25.01 125.01 20.84 win 31.26 156.26 26.05 lost -26.05 130.21 21.71 win 32.56 162.77 27.13 lost -27.13 135.64 22.61 lost -22.61 113.03 18.84 win 28.26 141.29 23.55 lost -23.55 117.74 19.63 lost -19.63 98.11 16.36 win 24.53 122.64 20.44 win 30.67 153.31 25.56 win 38.34 191.65 31.95 lost -31.95 159.70 26.62 win 39.93 199.63 33.28 lost -33.28 166.35 27.73 lost -27.73 138.62 23.11 win 34.66 173.28 28.89 lost -28.89 144.40 24.07 lost -24.07 120.33 20.06 win 30.09 150.41 vs 150.00 |
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30 bets....
100.00 16.67 win 25.01 125.01 20.84 win 31.26 156.26 26.05 lost -26.05 130.21 21.71 win 32.56 162.77 27.13 lost -27.13 135.64 22.61 lost -22.61 113.03 18.84 win 28.26 141.29 23.55 lost -23.55 117.74 19.63 lost -19.63 98.11 16.36 win 24.53 122.64 20.44 win 30.67 153.31 25.56 win 38.34 191.65 31.95 lost -31.95 159.70 26.62 win 39.93 199.63 33.28 lost -33.28 166.35 27.73 lost -27.73 138.62 23.11 win 34.66 173.28 28.89 lost -28.89 144.40 24.07 lost -24.07 120.33 20.06 win 30.09 150.41 25.07 win 37.61 188.02 31.34 win 47.02 235.04 39.18 lost -39.18 195.86 32.65 win 48.97 244.83 40.81 lost -40.81 204.02 34.01 lost -34.01 170.01 28.34 win 42.51 212.52 35.43 lost -35.43 177.09 29.52 lost -29.52 147.57 24.60 win 36.90 184.47 vs 175.00 |
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Having put thought into why this would be, it's because if you know it's going to be 50-50 (which is a logical place to start if testing vs level stakes) you always bet more on the first loser than the first winner:
100 LOSER 16.67 gives a bankroll of 83.33, so you only bet 13.89 on the next one, even if it wins 100 WINNER gives 125.01, so you're now betting 20.84 and so on until you hit a loser. Either way your loser is staked bigger than your winner. That sounds crap of course, but you make up for it when your bankroll starts to build up (as seen in the examples above , hopefully) as it starts to take more than one loser to wipe out your profits you've made over time. That is also true of level stakes, but level stakes begins to fall behind due to not pressing home it's advantage. It's also true if you bet more, but it can be shown that betting more actually lowers your expected growth (but it would not be immediately obvious unless you charted some bad runs) despite raising your expected value per bet. |
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Actually, the loser/winner thing I just wrote is utter tosh, I shouldn't do maths in the morning
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Here's an easy one for you Lori. What's 1 plus 1?
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So, trying that again, now I've confused people - likely including myself.
100.00 16.67 lost -16.67 83.33 13.89 win 20.84 104.17 vs 105.00 (Note the order doesn't matter with Kelly) 100.00 16.67 lose -16.67 83.33 13.89 lose -13.89 69.44 vs 80.00 100.00 16.67 win 25.01 125.01 20.84 win 31.26 156.26 vs 130.00 So with Kelly, we have a 50% chance of 104.17 bankroll, a 25% chance of 69.44 and 25% chance of 156.26 that means on average we have .52.09 + 17.36 + 39.07 = 108.51 using Kelly and 52.5 + 20 + 32.5 = 105 Using level stakes. That looks a better example, sorry about that. Hopefully this one is right :) |
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I got a sheet for that one Jimmy, but I can't format it.
Original Value number to be added Sum 1 1 2 |
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In a similar vein, if you extend the maths to 10 throws:
Winners Prob (K) (L) 0 0.001 16.151 0 1 0.010 24.226 25 2 0.044 36.339 50 3 0.117 54.508 75 4 0.205 81.762 100 5 0.246 122.643 125 6 0.205 183.965 150 7 0.117 275.947 175 8 0.044 413.921 200 9 0.010 620.882 225 10 0.001 931.323 250 Kelly gives an expected roll of 150.41, Level Stakes an expected roll of 125. |
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Now...who's going to win the 2 o'clock at Fontwell...
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Many thanks for that Lori. I have been pressing buttons on my calculator for some time trying to get the 10-bet sequence to 'work' better than level stakes. I have been using half-Kelly betting for some time but have relied on using a calculator to work out how much to wager so did not really understand the maths. Then I started going through a book on Probability and ended up confusing myself.
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Where's the £10 level stake come from?
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Just a comparison bushy (Comparing Kelly to levels)
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Yes, but in the first example the choice of £10 appears to be arbitrary (I think it comes from dividing £100 by 10 tries, thus ensuring you can't run out of money before the 10 tries are up - something Kelly, by definition, doesn't have an issue with). For the sequence ProPlayer has chosen you could use a level stake of £100 !
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No wonder the poor man topped himself. RIP.
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the optimal kelly fraction simply falls out of the derivative of the profit curve from first principles.
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what you might find instructive is to look at your results if you build in a known error in your edge estimate.
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Lori wrote:
So, trying that again, now I've confused people - likely including myself. 100.00 16.67 lost -16.67 . . . . . 108.51 using Kelly and 52.5 + 20 + 32.5 = 105 Using level stakes. That looks a better example, sorry about that. Hopefully this one is right :) Blooming brilliant post - best maths post of the year imho, kelly explained for people who forgot or could never do calculus. Cheers Lori, and happy christmas. |