[sl8r]

One interpretation of Kelly is maximizing the e.v. of the log return. Another (equivalent) interpretation is maximizing the expected IRR -- which is where the "in the long run" comment comes in, since "in the long run" you care more about the internal rate of return than the expected value of any individual bet.

E.g., imagine a bet that costs $1 to play and pays out $2 with probability 60% and $0 with probability 1%. How much would you bet? The expected value of the bet is $1.2 per dollar you bet, so for a single bet, you might wager 100% of your bankroll. But "in the long run" you'll loose all your money doing this. Instead, Kelly would recommend that you bet only 20% of your bankroll. "In the long run", you'll make infinite money doing this. (Not only that, but there's no other strategy that will make you money faster.)

[OscarCunningham]

>(Not only that, but there's no other strategy that will make you money faster.)

That's not true. If you bet 99% of your money each time then there's still no probability that you go bankrupt (it's literally impossible to go bankrupt unless you bet all your money), and you make money much faster.

Perhaps we could add in a lower bound, like you have to stop betting if you have less than $1. But then it's possible to go bankrupt even if you use the Kelly criterion. Furthermore we've introduced a fixed quantity into the problem, which means there's no longer any justification for saying that your bet should be the same proportion of your wealth every turn.

I've never yet seen a convincing argument for Kelly betting aside from the when utility is logarithmic.

[ThrustVectoring]

>If you bet 99% of your money each time then there's still no probability that you go bankrupt (it's literally impossible to go bankrupt unless you bet all your money), and you make money much faster.

Go back and read through the Math for the Kelly Criterion - when you know your edge and odds, it's the optimal solution. It's basically the balance point between taking advantage of current betting opportunities and preserving capital to take advantage of future betting opportunities.

If you bet 99% of your money on a coin flip, you'll eventually lose a flip and have too little money to take advantage of future coin flips.

Let me try another explanation: your return from a series of coinflips comes from two sources. The first is the return from the next coin flip, which when you have an edge, makes you want to bet as much as possible on this flip. The second is the return from all future flips, which makes you want to bet less so that a poor result doesn't permanently diminish your ability to make bets. Mathematically, the Kelley Criterion is the point where adding or removing bet sizing moves these two values the same amount, resulting in a change in expected value per bet size of zero, which means it's a maximum.

[bluecalm]

>>If you bet 99% of your money on a coin flip, you'll eventually lose a flip and have too little money to take advantage of future coin flips.

You are comparing Kelly bets to being stupid so of course Kelly wins. Kelly maximizes just one thing - log of bankroll. If your utility is not logarithmic it's not optimal to use Kelly bet sizings and if your utility is logarithmic with some multiplier then you need to adjust Kelly as well (which btw gamblers using Kelly are doing as pure Kelly criterion is universally considered too risky).

>>The first is the return from the next coin flip, which when you have an edge, makes you want to bet as much as possible on this flip. The second is the return from all future flips, which makes you want to bet less so that a poor result doesn't permanently diminish your ability to make bets.

While pure result diminish (or kills) your ability to make money from further bet betting it all and winning increases it. If you want to maximize EV of total amount of money you bet all at every turn and this is the optimal solution to optimize that. If you want to optimize logarithm of total amount of money you bet Kelly. If you want to maximize more conservative utility then you bet something else. There is nothing magical about Kelly criterion other than that.

[OscarCunningham]

I guess one thing you can say about the Kelly criterion without mentioning utility is that if Alice uses the Kelly criterion and Bob uses some other strategy (which is still of the form "bet some fixed proportion of your money each turn") then the probability that Alice has more money than Bob tends to 1 as the number of turns increases.

Of course in the cases where Bob has more money he might have much more money, so this fact isn't very relevant to them unless they have appropriate utility functions.

Another thing that occurs to me is that your utility function is changed by the opportunities you expect to encounter. If your utility function for money would normally be U_0, and you are about to be allowed to make a bunch of bets, then your current utility function, U_-1, is equal to the expectation of U_0 under the probability distribution that results from you betting optimally starting with however much money you have.

Maybe there's a family of utility functions for which if U_0 is in that family then U_-1 is approximately logarithmic? Then that would be a good justification for using the Kelly criterion if you have a long string of bets ahead of you. On the other hand I just checked the HARA (https://en.wikipedia.org/wiki/Hyperbolic_absolute_risk_avers...) family of utility functions, and they're all stable under the process I described. So there are certainly a lot of functions that don't become logarithmic.

[bluecalm]

Thanks for the interesting comment and a link. I will be using your the first paragraph from your post in the future :)

[cardmagic]

> That's not true. If you bet 99% of your money each time then there's still no probability that you go bankrupt (it's literally impossible to go bankrupt unless you bet all your money), and you make money much faster.

The huge mistake you're making is in assuming that money is infinitely divisible. Let's say you start with $1 and you bet 99 cents and lose. Now try betting 99% of 1 cent and see if you don't go bankrupt.

It's very simple to verify Kelly's findings by building a naive simulator.

[OscarCunningham]

>The huge mistake you're making is in assuming that money is infinitely divisible. Let's say you start with $1 and you bet 99 cents and lose. Now try betting 99% of 1 cent and see if you don't go bankrupt.

But then it's possible to go bankrupt even if you use the Kelly criterion.

[cardmagic]

Kelly criterion never claimed to prevent bankruptcy. It only tells you how to bet to achieve the maximum theoretical return possible given a set of odds and payouts.

I say theoretical because unless you're in a casino, you don't know the actual odds and edge you are playing with (because they move) which makes your Kelly ideal bet size a guess rather than a fixed number.

Here's more information: https://en.wikipedia.org/wiki/Kelly_criterion