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Comment by zucker42 | Hacker News Reader

I wonder how professional gamblers approach bet sizing. It seems to me that for most applications the Kelly Criterion is not the right choice. The utility of money is asymmetric; gaining $25000 is worse than not losing $25000. Relatedly, most actual gamblers want to ensure good returns while not going broke, so minimizing risk of ruin is often more important than maximizing return rate. Further complicating the matter is that in real life you don't know your actual probability of success, but you may have an estimation. And finally, though this is less commonly significant, your rate of return in a given game might depend on your the amount you bet.

From what I've seen in the poker community, no one has really approached this type of bet sizing from a rigorous perspective beyond the relatively simple Kelly Criterion.

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The Kelly criterion takes into account the fact that "gaining $25000 is worse than not losing $25000". A game where you have a 50/50 chance of gaining $25k or losing $25k has a negative expectation in the log domain, so per the Kelly Criterion one would not bet on this game.

You are right that "your rate of return in a given game might depend on your the amount you bet", this is actually very common. Consider a stock market: buying 1000 shares and selling them a year later will generate less than 1000x the return of buying 1 share and selling it a year later (assuming the stock goes up), because you pay more per share to buy 1000 shares and make less per share when you sell 1000 (because the share price moves as you buy/sell).

Related, I really enjoyed this treatment of the Kelly Criterion by Thorp and highly recommend it http://www.eecs.harvard.edu/cs286r/courses/fall12/papers/Tho...

Kelly Criterion maximizes the wealth in the long-run. It doesn’t take asymmetric utility into account. It just happens to coincide with log-utility. If there is a fixed amount of bets the Kelly criterion will be suboptimal, but as the number of bets grows the optimal strategy will asymptotically reach the Kelly criterion.

It does not. It maximizes log-wealth. Maximizing expected wealth gives a strategy which results in $0 a lot of the time but much much more than Kelly occasionally, achieving a higher average wealth. Kelly gives that up, getting far less expectation of actual wealth, but far more expectation of log-wealth (which is -inf at $0 so avoids the $0 results). If you don't believe this, pick any one scenario and actually do the math, take the limits, etc.

> It does not. It maximizes log-wealth. Maximizing expected wealth gives a strategy which results in $0 a lot of the time but much much more than Kelly occasionally, achieving a higher average wealth.

Log utility function is u(w) = log(w), that’s what it is called in economics. Surely maximizing log of wealth means you are maximizing log utility function.

> Kelly gives that up, getting far less expectation of actual wealth, but far more expectation of log-wealth (which is -inf at $0 so avoids the $0 results).

That’s a crucial misunderstanding of the Kelly’s result. He doesn’t give anything up. He showed than in a nonterminating game the strategy of betting as if you have log utility will give you superior results to any other strategy in terms of long-term wealth growth. What if the game is terminating? Well, in that case you need to know the utility function of the gambler to determine a superior strategy. But in a non-terminating game it is a bit irrelevant because almost all strategies will lead to infinite utility.

(If memory serves right), for bets that either win or double, the Kelly criterion is equivalent to maximize the median wins. Which coincidentally also maximizes expected log wins.

That's an interesting connection! I've heard risk managers talk about how people put too much faith in the mean outcome and don't focus enough on the median outcome. I didn't know there's a correspondence to the Kelly criterion.

Professional gamblers don't play pure random games (like craps or roulette) for a living. They play games where they feel that knowledge/skill have some influence (poker, horse racing).

To take poker, bet sizing is an important factor in play, but it has as much to do with the impression the action makes on other players as the actual underlying odds.

With regards to poker, "bet sizing" in the original article is equivalent to "bankroll management" in poker parlance. Bet sizing within individual hands is indeed a matter of game strategy, but the stakes you should play (i.e. blinds or tournament buyin amounts) is closely related to the original article.

Professional sports betters absolutely think in probabilistic terms -- "I think there's a 25% chance we'll win, the market thinks there's a 20% chance we'll win" etc. Kelly bet sizing absolutely makes sense, though market depth and Bayesian uncertainty also act to reduce bet sizes.

Most of the groups I’m aware of use some variation on Kelly (e.g. half Kelly), but it’s worth pointing out that in some sports it’s actually quite hard to deploy your full bankroll at any one time anyway.

Exactly I'm a pro gambler and data scientist and the optimal staking strategy is mostly just stake sizes that are inversely proportional to odds, plus a little increase/decrease relative to perceived value (what the kelly criterion does).

Check my golf betting algorithm out if you're interested: https://wwww.golfforecast.co.uk

Link doesn't work for me, https://www.golfforecast.co.uk seems better

> no one has really approached this type of bet sizing from a rigorous perspective beyond the relatively simple Kelly Criterion.

Kelly Criterion takes into account your bankroll at the point of betting, so it does take that into account

I haven't understood your points further than that - minimizing risk of ruin (taken literally) is typically equivalent to not ever gambling at all. No gambler would agree with that, by definition of them being a gambler.

This comment sounds so confused that I have to break it down into parts. Maybe I have just misunderstood you.

> I wonder how professional gamblers approach bet sizing.

Some variation of the Kelly criterion, whether they admit it or not. Any time you make a decision with the goal of maximising your growth of wealth relative to a level of risk, you're using the Kelly criterion.

> The utility of money is asymmetric; gaining $25000 is worse than not losing $25000.

Right. Not only is it asymmetric -- it is concave. Just like Bernoulli realised when he invented the proto-Kelly criterion. The Kelly criterion can be interpreted as incorporating an asymmetric, concave utility of wealth. (It's not assuming such a utility -- it's just that log utility happens to maximise long-run growth and be asymmetric and concave.)

But critically, it depends on the size of your bankroll. When you are talking about small stakes compared to your wealth, maximising expected value (i.e. running the risk of losing your entire wager) is the growth optimal choice, because in a local enough region, any continuous utility is linear.

> Relatedly, most actual gamblers want to ensure good returns while not going broke, so minimizing risk of ruin is often more important than maximizing return rate.

This is exactly what the Kelly criterion does. It sacrifices expected value in every gamble for larger long-run growth (which requires also that you don't go bust, or lose too much too quickly.)

> Further complicating the matter is that in real life you don't know your actual probability of success, but you may have an estimation.

But decomplicating the matter is that the Kelly criterion is very forgiving of estimation error, as long as you make your errors intelligently. The optimal bet sizing under the Kelly criterion tends to look like a quadratic function. This means that there is almost a plateau around the optimal bet size where small amounts of misestimation does not actually make your growth much different from the optimal growth. (And if you're worried about large amounts of misestimation, the linear combination of keeping money in your wallet and the full Kelly bet form a sort of "efficient frontier" (to borrow MPT terminology) of risk--growth tradeoffs. Meaning you can always bet less than the full Kelly bet, and get an optimal growth for the risk level that corresponds to. Hence the popularity of half-Kelly and similar variants.)

In the end, estimating the joint distribution of outcomes for the Kelly criterion is a far more forgiving requirement than e.g. estimating the parameters needed for modern portfolio theory type analysis.

> And finally, though this is less commonly significant, your rate of return in a given game might depend on your the amount you bet.

A fact also easily plugged into the Kelly criterion. It might make the search for optimality harder than setting a derivative to zero, but we have computers for numerical optimisation!

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