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Comment by cubefox | Hacker News Reader

The author of the video says that the Kelly criterion is used in investing, but that's not really the case. Kelly should only be used when the probabilities of winning/losing are known exactly. When they are only estimated, like in investing, the Kelly criterion is still far too risky. Here is a bit more about this problem:

https://youtube.com/watch?v=-9JM9suCIHs

Unfortunately there doesn't yet seem to exist an equivalent to the Kelly formula which would apply to subjective probabilities. I haven't even found an explanation of why uncertainty about probabilities requires betting more conservatively than Kelly would recommend.

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> I haven't even found an explanation of why uncertainty about probabilities requires betting more conservatively than Kelly would recommend.

It’s because the risk of ruin increases quickly if you overestimate optimal f, as calculated by Kelly criterion. So OVER-estimating f, dramatically increases your risk of bankruptcy, while UNDER-estimating f only results in not optimizing your returns.

If you’re not certain of your probabilities because (you’re estimating unknowns), you’d prefer having sub-optimal gains over going bankrupt.

I guess something like that is indeed the reason. However, as stated (in terms of "risk of ruin") the argument doesn't quite work. Because even if the probabilities are known with certainty, betting the Kelly optimum would increase the risk of ruin compared to betting less. But Kelly betting would nonetheless be better.

The thing about Kelly is that it assumes risk of ruin is zero because you never bet your entire bankroll. It assumes you can always bet a fraction of a fraction of what you have remaining.

Sure, but you can get arbitrarily close to zero over time.

If you don't make any assumptions about the underlying distribution you're associating with the risk, it can be anything, so I'm not surprised you haven't found an explanation. For example, the risk could be dependent on previously values, dependent on unobservable data, completely independent, etc. etc.

If you make some basic assumptions about independence, along with some others that I may not understand so well, you can start to grapple with the idea that you should be decreasing the amount you wager on each round as you're hedging against risk. If you make a mistake in your probability estimation in your favor, then great, you win more, but if you make a mistake against, it bites you and you lose more than you would have thought.

In the extreme, your "deflation factor" is 0, meaning you wager nothing so you're completely hedged against risk.

In terms of the video you provided, he talked about using a sample of 10 to estimate the probability but notice he didn't even mention the variance of such a limited sample. I don't claim to have any deep knowledge of this but Evan Miller's "how not to sort by average rating" post comes to mind in order to get some handle on this particular setup [0]. From the Miller post:

    What we want to ask is: Given the ratings I have,
    there is a 95% chance that the “real” fraction of
    positive ratings is at least what?

Miller advocates for using a lower bound Wilson score confidence interval for a Bernoulli parameter. Again, I'm pretty naive when it comes to this stuff but I suspect this is exactly what would be wanted when trying to apply the Kelly criterion in the example from the video you provided.

The reality is that risk management is a big subject precisely because the risk model can be complex. The Kelly criterion is meant to be a starting point for complex real world examples, not the destination.

I also found a post that has a nice graphic for deflating the Kelly criterion [1], so maybe that might be helpful to you.

[0] https://www.evanmiller.org/how-not-to-sort-by-average-rating...

[1] https://nickyoder.com/kelly-criterion/ (just above https://nickyoder.com/kelly-criterion/#theory-vs-reality)

> If you make a mistake in your probability estimation in your favor, then great, you win more, but if you make a mistake against, it bites you and you lose more than you would have thought.

There are two mistakes we can make due to inaccurate probability estimation that are not on our favor: Betting too little, and betting too much (relative to the actual Kelly optimum with the true probabilities). I believe betting too much is disproportionately worse than betting too little. But I haven't seen a direct argument for why this would be so.

The Nick Yoder Blog post is interesting, but he basically says: people just happen to be risk averse, and that's the reason we happen to want to bet less than the Kelly optimum.

> Most people assign a negative value to risk. That’s why we pay a premium to insurance companies to haul away excess risk.

But I don't think a psychological disposition is the right explanation here. The original video I linked to suggests that it is actually irrational to bet the estimated Kelly optimum when the probabilities are estimated. No matter whether you are psychologically risk averse or not.

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