The "Just One More" Paradox – Kelly Criterion [video]

> 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.

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)

Agreed.

Most people understand quickly that after a 50% loss a 50% gain is not enough to break even.

That's not what it's about. You're only talking about why the median goes down if you bet 100%.

This explains why the median (and mode) goes up if you bet 37.5% instead of 100% each time.

The St Petersburg lottery is a good counterexample against basing decisions on (arithmetic) mean outcome; AKA the expected value.

In particular, the arithmetic mean is useful in situations where the source of value is irrelevant, we only care about the total. For example, if I can earn $X/hour doing job 1 and $Y/hour doing job 2, then (assuming they're equally difficult, fulfilling, etc.) my total value should be the arithmetic mean of spending the fraction t of my time doing job 1 versus job 2:

  t * $X/hour + (1 - t) * $Y/hour

This makes sense, since the money I get is the sum of the money from each job. In particular if I "waste" some time on a low-paid job, I can "make it up" by spending some time on a well-paid job, resulting in a total value somewhere in-between. To maximise my value, I go "all in" on whichever job pays most.

Games of chance are not like this, since the source of value is highly relevant. We'll only receive one of the possible payoffs: if it's a low payout, we can't "make it up" using some of a high payoff that didn't happen. Hence we don't directly care about the "total" of all the payouts, since we can't aggregate winnings from alternate timelines. Instead, we should maximise the geometric mean of the payouts, which makes each individual outcome the least-bad. The result is the Kelly Criterion.

The St Petersburg lottery makes this clear: I don't care how exponentially-huge one of the payoffs might have grown, if the payoff I actually receive is $1!

The same logic works across other sources of value, e.g. GDP-per-capita is an arithmetic mean; a government can maximise it by going all-in on the sector with the highest return (e.g. the US could divert its military budget, food subsidies, etc. into the tech sector); or the region with the highest return (e.g. the UK could invest everything in London); or the age demographic that's most productive; etc. However, as an individual human I only have one age; I work in one sector; I live in one place; etc. so I'd prefer a government that spreads its investments more like Kelly betting.