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This so much. Like anything that is 'optimal' it is optimal with respect to some criterion. For the Kelly Criterion it is to maximize the logarithm of the wei

This so much. Like anything that is 'optimal' it is optimal with respect to some criterion. For the Kelly Criterion it is to maximize the logarithm of the weighted sum of the expected value across all outcomes. This is probably not what you actually want in almost any situation. The one time it actually makes sense to bring it in is if you are forced to make a certain number of wagers in some game AND you have good-enough knowledge of the odds of the game AND your payout is proportional to how much money you have left at the end of the game, AND the wager size does not effect the outcome of the game in any way. This scenario never happens. Even when you meet many of the necessary prerequisites to use Kelly, it still doesn't make sense at all. For example, Blackjack tournaments. You are given a set amount of starting money, you play for a set number of hands (or time), and you know the odds perfectly. However, the payout structure isn't proportional to your final amount of money, so using Kelly has more or less no chance of winning any money in the tournament. They usually pay for the top N results, which means you have to go with a very high variance strategy AND win consistently to place. Poker: Nope, not even close, bet size is a direct input into the dynamics of the game. Predictable betting of any kind is a maximally bad strategy. Stock Market: utility of money isn't logarithmic, so it is not worth maximizing, even if you knew probabilities and were forced to make wagers, which you don't and aren't. If you could even approximate probabilities you could use that power to basically print money on derivatives, so even if Kelly applied, the prerequisites are too strong and would make far far better strategies available. The only valid use case for bringing in the Kelly Criteria is for gamblers to feel better about burning money at the tables by improperly applying it.

arthurcolle 5y ago
Kelly must be the name of the wife in the WSB "wife's boyfriend" memes. More seriously, great analysis. Couldn't have said it better myself.
thenaturalist 5y ago
Hi there, you seem to know a good deal about the intricacies of where the Kelly Criterion lies within the range of possible options to calculate outcomes. Could you point me to any good resource you know of to learn more about the range itself and other options?
wizzwizz4 5y ago
> For the Kelly Criterion it is to maximize the logarithm of the weighted sum of the expected value across all outcomes. It's actually to maximise the expected logarithm of the monetary amount, and it's a pretty good heuristic (in most circumstances) given that most opportunities are exponential. (It usually takes a given amount of effort to double your money.)
concreteblock 5y ago
The first sentence of your post, while technically true, misses the point. This misunderstanding undermines many of your other points. The kelly criterion happens to be optimal with respect to log wealth but that's not the main reason why it's interesting. Many explanations, including the original post, make this mistake. Maybe because 'maximizing expected utility' is a more common idea. The first sentence of the wikipedia article: "In probability theory and intertemporal portfolio choice, the Kelly criterion (or Kelly strategy or Kelly bet), also known as the scientific gambling method, is a formula for bet sizing that leads almost surely (under the assumption of known expected returns) to higher wealth compared to any other strategy in the long run". In other words, pick a strategy. I'll pick the kelly strategy. There will be some point in time, after which I will have more money than you, and you will never overtake me. No logarithms involved. This is something you can easily check by simulation, but requires some heavier math to formulate precisely and prove. See also posts by spekcular and rssoconnor elsewhere in this thread.
- murbard2 5y ago
  If it's a mistake, it's a charitable one. Picking log utility is at least somewhat principled, trying to lead almost surely to higher wealth is way more arbitrary.
  - concreteblock 5y ago
    What exactly do you find arbitrary about this? When gambling or investing, more wealth is better than less wealth. 'Almost surely' is a technical term which means 'with probability 1'.
    - murbard2 5y ago
      At no point in time does Kelly give you more wealth with probability 1, and there is no reason to care about "more wealth with high probability", that's not even a transitive comparison function.
      - 5y ago
concreteblock 5y ago
You obviously wouldn't use Kelly criterion during a poker game because the assumptions don't fit. But on a larger scale it can be used for 'bankroll management' - what proportion of your wealth should you use on a tournament entry fee. Of course you don't have the exact parameter p but you can use an estimate to make sure you are not making a grossly over/under-sized bet.
- ltbarcly3 5y ago
  No, you can't use it for bankroll management, because you can't estimate the probabilities necessary. A rule like "never bet more than 15% of your bankroll on one thing" would work just as well and it doesn't require you to do a bunch of math to get to the same answer. Here, lets do some examples to see how dumb it is in practice: Lets say I want to enter a tournament where I guess I have a 5% chance of winning, and I have $1000, here is how much the KC says I should be willing to pay to enter based on the payout odds: 10:1 -> Don't enter (duh) 15:1 -> Don't enter (duh again) 19:1 (breakeven) -> $0.00 (duh) 20:1 -> $2.50 30:1 -> $18.33 100:1 -> $40.50have forced 1000:1 -> $49.05 10000:1 -> $49.95 10000000000000000000000000000000000:1 -> $50.00 oh, so this fantastic system tells me to never bet more than 5% of my money if I have a 5% chance of winning. So insightful! Ok, lets say we have a 95% chance of winning the tournament: 1:1 -> $900 2:1 -> $925 50:1 -> $949 1000000000000000:1 -> $950 So if I'm a sure thing, I should bet a bunch of money. Again, there's no way someone would do this without maximizing log expected value and doing a bunch of math. Maybe it gets more interesting if it's around 50/50: 1:1 -> $0 (ok, makes sense) 2:1 -> $250 3:1 -> $333 4:1 -> $375 100000000000:1 -> $500 Again, Kelly gives us terrible advice. If you have a trillion to one payout on a coin flip, you want to bet less, not more! Why would you risk half your money, and have a 1/4 chance of losing all your money, when you can bet 1 cent at a time and just wait to win one time so you can buy half of the stock market with your winnings? So I still contend that whatever it is that Kelly maximizes, it's a dumb thing to maximize outside of contrived situations where you are forced to bet and know exact odds, and where the expected value is positive (if you have negative expected value you should never play, and Kelly tells you that). Finally, it is very sensitive to your probability estimates. Going back to my first example, where you have a 5% chance of winning a tournament, lets fix the payout at 30:1 and look at what kelly tells us if the probability of winning isn't exactly what we thought it was: 5% (same as first example): $18.33 4%: $8.00 3%: Don't enter 6%: $28.67 7%: $39.00 So if I have to guess my probability of winning the tournament within 1% of the actual probability or Kelly is going to tell me drastically wrong amounts. Nobody can set odds on something like a tournament precisely enough for this to be useful. Just like with stocks, if you have the ability to estimate probabilities so well that Kelly stops telling you to do the wrong thing, you can make far more money just directly using your magical probability estimating powers and betting on derivatives. If I could estimate my odds of winning the tournament to within 1%, I can just go to the sports book and bet on who is going to win on the tournament and make far more money than I would in the tournament itself. It's like a system to sell a cake for 15% more profits, and it starts with "first, use your laser vision to preheat the cake pan".
tunesmith 5y ago
Can you or anyone expand on "utility of money isn't logarithmic"? It seems a whole bunch of economic theory is based on the wish ("because it has nice properties") that losing 10% of your money generally hurts the same amount no matter how much money you have. Even accounting for a non-elastic living wage on the bottom. I know there's a lot of political opinion here, but has there ever been an effort to empirically determine what the utility of money actually is?

arthurcolle 5y ago

Kelly must be the name of the wife in the WSB "wife's boyfriend" memes. More seriously, great analysis. Couldn't have said it better myself.

thenaturalist 5y ago

Hi there, you seem to know a good deal about the intricacies of where the Kelly Criterion lies within the range of possible options to calculate outcomes. Could you point me to any good resource you know of to learn more about the range itself and other options?

wizzwizz4 5y ago

> For the Kelly Criterion it is to maximize the logarithm of the weighted sum of the expected value across all outcomes. It's actually to maximise the expected logarithm of the monetary amount, and it's a pretty good heuristic (in most circumstances) given that most opportunities are exponential. (It usually takes a given amount of effort to double your money.)

concreteblock 5y ago

The first sentence of your post, while technically true, misses the point. This misunderstanding undermines many of your other points. The kelly criterion happens to be optimal with respect to log wealth but that's not the main reason why it's interesting. Many explanations, including the original post, make this mistake. Maybe because 'maximizing expected utility' is a more common idea. The first sentence of the wikipedia article: "In probability theory and intertemporal portfolio choice, the Kelly criterion (or Kelly strategy or Kelly bet), also known as the scientific gambling method, is a formula for bet sizing that leads almost surely (under the assumption of known expected returns) to higher wealth compared to any other strategy in the long run". In other words, pick a strategy. I'll pick the kelly strategy. There will be some point in time, after which I will have more money than you, and you will never overtake me. No logarithms involved. This is something you can easily check by simulation, but requires some heavier math to formulate precisely and prove. See also posts by spekcular and rssoconnor elsewhere in this thread.

murbard2 5y ago
If it's a mistake, it's a charitable one. Picking log utility is at least somewhat principled, trying to lead almost surely to higher wealth is way more arbitrary.
- concreteblock 5y ago
  What exactly do you find arbitrary about this? When gambling or investing, more wealth is better than less wealth. 'Almost surely' is a technical term which means 'with probability 1'.
  - murbard2 5y ago
    At no point in time does Kelly give you more wealth with probability 1, and there is no reason to care about "more wealth with high probability", that's not even a transitive comparison function.
    - 5y ago

concreteblock 5y ago

You obviously wouldn't use Kelly criterion during a poker game because the assumptions don't fit. But on a larger scale it can be used for 'bankroll management' - what proportion of your wealth should you use on a tournament entry fee. Of course you don't have the exact parameter p but you can use an estimate to make sure you are not making a grossly over/under-sized bet.

ltbarcly3 5y ago
No, you can't use it for bankroll management, because you can't estimate the probabilities necessary. A rule like "never bet more than 15% of your bankroll on one thing" would work just as well and it doesn't require you to do a bunch of math to get to the same answer. Here, lets do some examples to see how dumb it is in practice: Lets say I want to enter a tournament where I guess I have a 5% chance of winning, and I have $1000, here is how much the KC says I should be willing to pay to enter based on the payout odds: 10:1 -> Don't enter (duh) 15:1 -> Don't enter (duh again) 19:1 (breakeven) -> $0.00 (duh) 20:1 -> $2.50 30:1 -> $18.33 100:1 -> $40.50have forced 1000:1 -> $49.05 10000:1 -> $49.95 10000000000000000000000000000000000:1 -> $50.00 oh, so this fantastic system tells me to never bet more than 5% of my money if I have a 5% chance of winning. So insightful! Ok, lets say we have a 95% chance of winning the tournament: 1:1 -> $900 2:1 -> $925 50:1 -> $949 1000000000000000:1 -> $950 So if I'm a sure thing, I should bet a bunch of money. Again, there's no way someone would do this without maximizing log expected value and doing a bunch of math. Maybe it gets more interesting if it's around 50/50: 1:1 -> $0 (ok, makes sense) 2:1 -> $250 3:1 -> $333 4:1 -> $375 100000000000:1 -> $500 Again, Kelly gives us terrible advice. If you have a trillion to one payout on a coin flip, you want to bet less, not more! Why would you risk half your money, and have a 1/4 chance of losing all your money, when you can bet 1 cent at a time and just wait to win one time so you can buy half of the stock market with your winnings? So I still contend that whatever it is that Kelly maximizes, it's a dumb thing to maximize outside of contrived situations where you are forced to bet and know exact odds, and where the expected value is positive (if you have negative expected value you should never play, and Kelly tells you that). Finally, it is very sensitive to your probability estimates. Going back to my first example, where you have a 5% chance of winning a tournament, lets fix the payout at 30:1 and look at what kelly tells us if the probability of winning isn't exactly what we thought it was: 5% (same as first example): $18.33 4%: $8.00 3%: Don't enter 6%: $28.67 7%: $39.00 So if I have to guess my probability of winning the tournament within 1% of the actual probability or Kelly is going to tell me drastically wrong amounts. Nobody can set odds on something like a tournament precisely enough for this to be useful. Just like with stocks, if you have the ability to estimate probabilities so well that Kelly stops telling you to do the wrong thing, you can make far more money just directly using your magical probability estimating powers and betting on derivatives. If I could estimate my odds of winning the tournament to within 1%, I can just go to the sports book and bet on who is going to win on the tournament and make far more money than I would in the tournament itself. It's like a system to sell a cake for 15% more profits, and it starts with "first, use your laser vision to preheat the cake pan".

tunesmith 5y ago

Can you or anyone expand on "utility of money isn't logarithmic"? It seems a whole bunch of economic theory is based on the wish ("because it has nice properties") that losing 10% of your money generally hurts the same amount no matter how much money you have. Even accounting for a non-elastic living wage on the bottom. I know there's a lot of political opinion here, but has there ever been an effort to empirically determine what the utility of money actually is?