Can someone answer this probability problem - it may involve the Kelly Criterion

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jammymike

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Sep 14, 2023
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I've tried getting and answer from chat GPT, but it kept giving me answers that I'm sure are wrong. Here is the text I put in chat GPT:

I would like to understand the maths involved in a bet. I start with £10. I can bet a minimum of 1p up to a maximum of all my money I have (£10+winnings-losses) per bet. If I run out of money I can no longer place any more bets. The bet involves a 6 sided die. I place a bet and roll the die. If I roll a 1 to 5 I lose my bet, but if I roll a 6 I win 10 times my bet. I can role the die up to 100 times before I cannot roll it any more. My question is, what is the best strategy to make the most money? Also, what are the maths equations involved?

Chat GPT said it used the Kelly Criterion - but the conclusion it gave was to not make a bet at all as the number it came out with was negative. I then ask it the same question, but changed the winning amount from 10 times the bet to one million times the bet. It still said I shouldn't make the bet! I believe the error was something to do with the fact that each individual roll was unlikely to be a win (1 in 6) regardless of how much you would win if you did roll the 6. My point is if you bet all your money you are likely to not win anything as the first roll will likely be a loss, so you need enough rolls to start getting an average of 1 in 6 rolls being a win. At the same time you only have 100 rolls so you want to bet enough to maximise your winnings. I would assume the bet should be a percentage of your maximum bet. I would like to understand the equation involved in finding the optimum wager amount.

Thanks
 
I don't have much experience with GPT, but I saw one GPT-generated answer someone posted on this forum, and that answer looked like complete garbage. Seemed that GPT is good at making answer look reasonable, but making no sense at the second look. Actually, in the past I've met some people working in a similar MO :)
 
Having voiced my opinion of GPT I have to admit that the reference to Kelly Criterion looks useful. But I see two thing to keep in mind about the criterion:
  1. It optimizes the mean value of the logarithm of the balance, which will not necessarily optimize the mean balance itself.
  2. It assumes that you always bet the same fraction of your balance on every toss of the dice, but I haven's seen any proof that it is an optimal way to play. I.e., is it possible that smartly varied bet might yield better expected balance?
 
I think that this is a problem for someone in the field of operations research.
You need to first decide on the risk you are willing to take. A low risk will have a lower the expected profit while a higher risk will yield a higher expected profit. A low risk betting scheme means that you have a low chance of ruin (losing all your money). A higher risk betting scheme means that you have a higher chance of ruin.
 
Hi. I've looked into this some more and I think I've found the answer. I won't be using chat GPT for equations in the future as it doesn't full understand the formulas. For anyone interested I came up with betting 8.3% of the maximum you can bet on any bet.
 
jammymike said:
Hi. I've looked into this some more and I think I've found the answer. I won't be using chat GPT for equations in the future as it doesn't full understand the formulas. For anyone interested I came up with betting 8.3% of the maximum you can bet on any bet.
Click to expand...
Can you share the details of your solution?
Thank you.

P.S. Unless I've misunderstood it, the Kelly criterion recommends 2/27 [imath]\approx[/imath] 0.074.
 
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