[concreteblock]

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]

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".

[concreteblock]

> 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 you're just going to throw away the criterion because you think the results are unintuitive? That's the argument you're making here.

To take your reasoning seriously, the reason why you might not want to bet 1 cent at a time is because the Kelly bet is guaranteed to eventually overtake your 1-cent-bet-strategy. Furthermore, it is completely incorrect to say that the Kelly bet has a 1/4 chance of losing all your money in the given situation. If you lose your first bet, the Kelly criterion tells you not to bet the whole house on the next bet.

Nothing you have written so far suggests that you actually understand the sense in which the Kelly criterion is optimal, which I attempted to explain in my other reply to you. You keep writing as though it only maximizes the expectation of log-utility. In fact it's not clear that you even understand what the Kelly criterion is telling you to do.

[ltbarcly3]

Kelly is not optimal if you can't estimate the probabilities with great precision, which you can't outside of contrived examples or casinos. In casinos you have negative expected value on every bet, and Kelly tells you to not play at all. Contrived examples don't matter.

You aren't addressing what I said, you are cherry picking things you find easy to rebut. You are right that I messed up and that you can't lose all your money with two 50% bets. However you ignore the stronger argument, which I opened with and repeatedly pointed out, which is that you can't estimate probabilities well enough to use it, and that if you can estimate probabilities well enough to benefit from Kelly, that ability to estimate probabilities itself almost always unlocks strategies that strictly dominate any benefit Kelly gives you. It's useless in practice.

[concreteblock]

I don't really care about winning any global arguments. I see bad logic and I try to point it out. If you didn't like that some of your points were easily rebutted then you shouldn't have written them. Leaving them unchallenged makes your position seem artificially strong.

I couldn't resist bringing up the poker bankroll example because I think your in-game-poker example was poorly chosen. To me, it looked like you came up with a situation where the criterion obviously had no hope of being applicable and then used it to argue that the criterion is useless. E.g. I could find a whole list of things for which calculus is not applicable, but that would not be a good argument for 'calculus is useless'. The example I gave is at least closer to the assumptions of the Kelly Criterion.

I think the main thing I wanted to do was to correct the misconception that Kelly is only maximizing expected log utility, because it is a shame if someone (including other readers) thinks that the Kelly Criterion is just a fancy name we gave for the argmax of E f(S) where f happens to be the logarithm.

After all this, you (and other readers) might still conclude that the criterion is useless. But the set of justifications, and maybe the certainty, in that position, should change.

[anonymoushn]

You don't have a 1/4 chance of losing all your money if you repeatedly bet half of it on the trillion-to-one coin flip.