Kelly Can't Fail

Quoting the same page: One easy way to dismiss the paradox is to note that Kelly assumes that probabilities do not change.

That's good to know. Kelly is good if you know the probabilities AND they don't change.

If you don't know or if they can change, I expect the right approach has to be more complex than the Kelly one.

In particular, then the right approach has to be more risk averse than Kelly would recommend. In reality, most probabilities can only be estimated, while the objective probabilities (e.g. the actual long run success rate) may well be different and lead to ruin. That's also what makes the title "Kelly can't fail" more wrong than right in my opinion.

Unfortunately, in the real world, playing the game changes the game.

For instance, casinos have different payout schedules for Blackjack based on minimum bet size and number of decks in the shoe. Payouts for single deck Blackjack are very small in comparison to multi-deck games, as well as requiring larger minimums (and they shuffle the deck after only a few hands).

For the issue in Proebsting's paradox, one simple approach I've found successful is to gradually accumulate your full bet as the betting lines progress to their ultimate positions. This works even in illiquid markets where your bets affect the lines, since it gives the other participants less room to suddenly react to what you're doing. (Though you always have the slight worry of a huge last-second bet that you can't react to, eBay-auction style.)

As for the actual probability being different from the expected probability, that's not too difficult to account for. Just set up a distribution (more or less generous depending on your risk tolerance) for where you believe the actual probability may lie, and work out the integrals as necessary, recalling that you want to maximize expected log-value. It's not the trivial Kelly formula, but it's exactly the same principle in the end.

The title is gentle clickbait applies to one specific game with 0 variance, not to all uses of Kelly.

I think the problem with estimating a distribution is the same, it might simply not match reality (actual unknown success rates, actual unknown variance of your estimation of success rates being correct). In particular, if you are too optimistic relative to reality, Kelly betting will lead to ruin with high objective probability.

There are people who can verifiably estimate unknown probabilities with accuracy finer than 5 percentage points. These are called superforecasters.

Almost all people can achieve at least 20 point accuracy with a few hours of training. Unknown probabilities are not so problematic as people make them out to be.

Probabilities are literally measures of uncertainty. It's okay for them to be uncertain. They always are.

Yeah, that's why I mentioned creating a very generous distribution for the probability if you don't want the risk of being wrong. You can set up the maximization step just the same, and the wider the distribution, the more conservative it will tell you to be, until ultimately it says not to take the bet at all, since you have no edge. If you're really risk-adverse, you can go with a full min-max approach, where you bet as if the actual probability is unfavorable as possible. You just end up making suboptimal choices compared to someone who (accurately) puts a narrower distribution on the probability.

Also, your estimate of the true probability doesn't have to be that exact, if your edge is big enough to begin with. Once I made great profits (betting with fake internet points) just by naively taking the sample proportion from a small sample. In fact, the event turned out to be more 'streaky' than a weighted coin flip, but it didn't matter, since everyone else there was betting on vibes.

In any case, it's not like there's just the trivial Kelly formula, and woe to you if its toy model doesn't apply precisely to your situation. It's a general principle that can be adapted to all sorts of scenarios.

Kelly betting is a rare case where the difference between subjective and objective betting is actually a big deal, because any difference in those probabilities can make a large difference in the amount of money the Kelly formula says you should bet. I don't know the exact source, but there was a YouTube video on Kelly showing the impact of miscalibrated probabilities using simulations.