different EV games, max bet and Kelly criterion

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shmely492
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shmely492

different EV games, max bet and Kelly criterion
« on: December 05, 2019, 10:35:13 PM »

Hey arbusers,

I have 2 different games with same standard deviation, but different EV and max bet.

For example,
1st game: 200 max bet, 1% EV. I need placing 900 bet based on Kelly criterion
2nd game: 500 max, 0.5% EV. 600 bet based on Kelly criterion

I can choose one game only. What's the game I need to play for the maximum bankroll growth?
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pythonic
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pythonic

Re: different EV games, max bet and Kelly criterion
« Reply #1 on: December 06, 2019, 10:13:51 AM »

I guess 1st as that has similar expected value but much lower variance because you are essentially using a low fractional kelly strategy with 200 max.
But an optimal strategy would probably be a mixture of both.
If you have the odds then you can evaluate the expected log-value.
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shmely492
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shmely492

Re: different EV games, max bet and Kelly criterion
« Reply #2 on: December 06, 2019, 10:47:02 AM »

EV is not similar. Variance is almost similar.

If you have the odds then you can evaluate the expected log-value.
How?

Ok, real data below:
1st: EV is 1.07%, 200 max bet, std dev 1.13 (variance is 1.69)
2nd: EV is 0.75%, 500 max bet, std dev 1.14 (variance 1.96)

I know that betting half the Kelly amount reduces bankroll volatility by 50%, but growth by only 25%.

I betting 1/4 the Kelly amount if I choose 1st game. How growth reduces in this case?

I think I need simulations here. Just need compare how much bets need to double my BR for both games.
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Wolfie
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Wolfie

Re: different EV games, max bet and Kelly criterion
« Reply #3 on: December 06, 2019, 11:10:50 AM »

Second one is to risky. I would go with the first one.
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shmely492
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shmely492

Re: different EV games, max bet and Kelly criterion
« Reply #4 on: December 06, 2019, 11:52:45 AM »

Who care about risk, if this strategy increase your BR faster?
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pythonic
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pythonic

Re: different EV games, max bet and Kelly criterion
« Reply #5 on: December 06, 2019, 02:24:21 PM »

EV is not similar. Variance is almost similar.
In the first post you wrote
200 max bet, 1% EV
500 max, 0.5% EV
So the expected value in absolute terms would be 100*1% = 2 in the first and 2.5 in the second case.
Now with the numbers from the second post that is somewhat different.
You get 2.14 for the first and 3.75, almost twice as much for the second strategy, but also more risk which reduces the expected growth rate.
Btw std dev should be the square root of variance, so it seems you have an error there somewhere.
Quote
If you have the odds then you can evaluate the expected log-value.
How?
You have 2 possible outcomes:
- You win your bet with probability p and end up with a bankroll of b1
- You lose your bet with probability 1 - p and end up with a bankroll of b2
Now B = b1*p + b2*(1-p) is your expected bankroll and log(B) is your expected log bankroll. Kelly says that you should maximise log(B). But remember that Kelly is essentially risk-indifferent, so there will be large swings with full Kelly.

Quote
I betting 1/4 the Kelly amount if I choose 1st game. How growth reduces in this case?

I think I need simulations here. Just need compare how much bets need to double my BR for both games.
If you have the odds and the ev or the odds and the real probability, then it is a straightforward calculation to see which one gives higher expected growth rate.
But of course you can do a Monte Carlo simulation if you prefer that.
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dnf
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dnf

Re: different EV games, max bet and Kelly criterion
« Reply #6 on: January 10, 2020, 01:23:03 PM »

You have 2 possible outcomes:
- You win your bet with probability p and end up with a bankroll of b1
- You lose your bet with probability 1 - p and end up with a bankroll of b2
Now B = b1*p + b2*(1-p) is your expected bankroll and log(B) is your expected log bankroll. Kelly says that you should maximise log(B). But remember that Kelly is essentially risk-indifferent, so there will be large swings with full Kelly.

Minor nitpick, but worth pointing out—in the standard derivation of the Kelly criterion you're not maximizing log(B) (this would be identical to maximizing B as the logarithm is a monotonic function) but rather the quantity B*=log(b1)*p + log(b2)*(1-p).
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pythonic
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pythonic

Re: different EV games, max bet and Kelly criterion
« Reply #7 on: January 11, 2020, 12:10:31 PM »

Oops, indeed. Thanks!
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