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Deriving the Kelly Criterion to Maximise Profits

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Re: Deriving the Kelly Criterion to Maximise Profits

#11

Post by uoaei »

KK7NIL wrote: post #10
roenxi wrote: post #8

A word that is good to know here is ergodic [0]. Which I must admit to not really understanding although it is something like the average system behaviour being equivalent to a typical point's behaviour. If a process is non-ergodic then E[X] is usually not as helpful as it seems in formulating a strategy. [0] https://en.wikipedia.org/wiki/Ergodic_process

Some math/finance nerds made a whole YouTube channel about ergodicity, which I've been really enjoying: https://youtu.be/VCb2AMN87cg Nassim Taleb also talks about this quite a lot: https://youtu.be/91IOwS0gf3g TL;DR: while a single investment may be ergodic, portfolio management (the math behind weighting successive and concurrent investments/bets) is not, as it has a strong dependence on all prior states.

this comment may be confusing and I doubt this will help much but:

Ergodicity is less about memorylessness and more about the constraints on transitions into this or that state. A system is ergodic if "anything that can be an outcome, eventually will happen".

Re: Deriving the Kelly Criterion to Maximise Profits

#12

Post by uoaei »

wenc wrote: post #9
eftychis wrote: post #5

Earlier quoted context omitted.

I am not sure what you mean by "never used as is." The Kelly criterion is an optimization of capital growth (its logarithm) method/guide. Not using it doesn't change its correctness. But yes you need to know the advantage/the edge you have. Like with pricing methods eg for European options for Black Scholes you need to know the volatility and there is no way to know it, you estimate. This is where all the adjusting f…

But do you calibrate p (say through estimation) and then apply the Kelly criterion in your portfolio? I don’t think it is used in this way. It swings too much with a given p.

You calibrate for a reasonable distribution of p and use that to estimate (Monte Carlo, etc.) expected gain, optimizing your investment based on that. With this technique your estimate will probably end up somewhere around the common heuristics.

Re: Deriving the Kelly Criterion to Maximise Profits

#13

Post by bdjsiqoocwk »

gwbrooks wrote: post #4

A million years ago, when you could still find video poker games with 100%+ theoretical return or poorly thought-out promotions offering enough cash-back to get you over 100%, we'd calculate the Kelly number for a given opportunity -- the bankroll necessary to ride out hills and valleys in favorable situations. Spoiler: It's almost always 3-4x the value of a royal flush. So you needed $12-16k if you were playing a $1…

I would like to understand in detail what you just wrote.

"$1 per coin game" is this a game where you put in $1 to play and get paid either $2 or $0 with 50-50 probability (0 expected).

And the what does it mean %1 edge? Does it mean the probabilities are such that the expected payout is 1c per coin flip?

Re: Deriving the Kelly Criterion to Maximise Profits

#14

Post by diab0lic »

roenxi wrote: post #8

A word that is good to know here is ergodic [0]. Which I must admit to not really understanding although it is something like the average system behaviour being equivalent to a typical point's behaviour. If a process is non-ergodic then E[X] is usually not as helpful as it seems in formulating a strategy. [0] https://en.wikipedia.org/wiki/Ergodic_process

An example that may be useful to aid in understanding… Casinos are non ergodic.

A million players each placing a single bet will have an expectation of losing the house edge.

A single player placing a million bets has an expectation of $0.

The fact that the aggregate and the single entity Experience different expectations despite both placing a million bets is what makes this ergodic.

Re: Deriving the Kelly Criterion to Maximise Profits

#15

Post by intuitionist »

wenc wrote: post #2

The Kelly criterion is almost never used as-is because it is very sensitive to probability of success, which is hard to know accurately and in many cases, dynamically changing. This is easy to see in an Excel spreadsheet. Changing the probability by even 0.01 percent can vastly shift the results. The article calls this out in the last paragraph. The article mentions fractional Kelly is a hedge. But what fraction is o…

Yeah, but I think this misses the point a bit. The fact that your true edge isn’t knowable wouldn’t be so bad except that if you’re betting full-Kelly and overestimate your edge even a little bit, your probability of ruin in the long run goes to 1. Whereas if you underbet, you’ll compound wealth at a little lower rate but won’t risk ruin.

Re: Deriving the Kelly Criterion to Maximise Profits

#16

Post by [deleted] »

diab0lic wrote: post #14
roenxi wrote: post #8

A word that is good to know here is ergodic [0]. Which I must admit to not really understanding although it is something like the average system behaviour being equivalent to a typical point's behaviour. If a process is non-ergodic then E[X] is usually not as helpful as it seems in formulating a strategy. [0] https://en.wikipedia.org/wiki/Ergodic_process

An example that may be useful to aid in understanding… Casinos are non ergodic. A million players each placing a single bet will have an expectation of losing the house edge. A single player placing a million bets has an expectation of $0. The fact that the aggregate and the single entity Experience different expectations despite both placing a million bets is what makes this ergodic.

[deleted]

Re: Deriving the Kelly Criterion to Maximise Profits

#17

Post by sobriquet9 »

roenxi wrote: post #8

A word that is good to know here is ergodic [0]. Which I must admit to not really understanding although it is something like the average system behaviour being equivalent to a typical point's behaviour. If a process is non-ergodic then E[X] is usually not as helpful as it seems in formulating a strategy. [0] https://en.wikipedia.org/wiki/Ergodic_process

An illustrative example to explain ergodicity. Consider the following game. Players start with $100. At every turn, a fair coin is flipped. If tails, the amount of player's money is increased by 50%. If heads, the amount of player's money is decreased by 40%. To play or not to play, that is the question.

Re: Deriving the Kelly Criterion to Maximise Profits

#18

Post by CrazyStat »

I had never paid much attention to Effective Altruism or SBF before FTX blew up, but when that happened I spent some time reading old EA forum posts and SBF tweets and interviews. One of the things that absolutely shocked me was the dismissal of the Kelly criterion by SBF and other EAs. The argument was that the Kelly criterion was only rationalized by a logistic utility function, and if you were going to use your money for altruistic purposes a linear utility function is more appropriate (at least up into the trillions of dollars) because you can help twice as many people with $200 billion as you can with $100 billion.

This argument was used--by SBF and others--to justify truly absurd risk taking. I don't think it's an exaggeration to suggest that this misunderstanding may have been one of the primary drivers of Alameda's (and hence FTX's) downfall. For a group with as many smart people as EA and as many people obsessed with existential risks as EA not to have started screaming en masse when SBF suggested he would take a 51-49 bet on doubling utility or deleting all known life out of existence[1] is insane.

The mathematical misunderstanding is one part of it. Kelly betting dominates any other betting strategy in the sense that as the number of bets increases the probability that the Kelly better will have more money than someone following any other strategy approaches 1. You don't need a logarithmic utility function. If I bet Kelly and you follow some other strategy, eventually I will almost surely end up with more money and more utility than you.

I suspect another part of it is a misunderstanding by SBF (and perhaps others) of Jane Street's trading strategy. Jane Street encouraged their traders to be "risk neutral", which can be expressed as maximizing expected utility with a linear utility function. They wanted their traders to be willing to take big risks. But any individual trader is only working with a tiny fraction of Jane Street's capital, so even if they're risking all the money they've been given to work with on a bet that's still a small bet relative to the entire company. SBF seems to have taken that same risk neutral idea and applied it to the entirety of Alameda/FTX's available capital (and indeed expressed a willingness to apply it to the combined utility of the entire world), with predictably disastrous results.

[1] https://elmwealth.com/a-missing-piece-of-the-sbf-puzzle/

Re: Deriving the Kelly Criterion to Maximise Profits

#20

Post by evrydayhustling »

CrazyStat wrote: post #18

I had never paid much attention to Effective Altruism or SBF before FTX blew up, but when that happened I spent some time reading old EA forum posts and SBF tweets and interviews. One of the things that absolutely shocked me was the dismissal of the Kelly criterion by SBF and other EAs. The argument was that the Kelly criterion was only rationalized by a logistic utility function, and if you were going to use your mo…

What a great example of Dunning-Kruger as applied to elites. I remember the spike of interest in DK bias during the pandemic [1], largely as a way of explaining how uneducated folks could be so confidently incorrect about vaccination strategies. In reality it can strike in any social strata -- like a bunch of professional traders wielding billions of dollars, smugly misunderstanding Kelly.

[1] https://trends.google.com/trends/explore?date=today%205-y&ge...

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