5 ms·

The Kelly criterion is to financial math as the Fibonacci sequence is to mathematics. Yes it's neat, no it's not special, please stop bringing it up all the tim

The Kelly criterion is to financial math as the Fibonacci sequence is to mathematics. Yes it's neat, no it's not special, please stop bringing it up all the time.

bobbylarrybobby 5y ago
The Fibonacci sequence is not special because it is just one element of the family of linear recurrences. Is there a larger family to which the Kelly criterion belongs? (Not being snide — I’m genuinely asking)
- murbard2 5y ago
  Yes, any utility function will give you a Kelly-like criterion. Kelly is log utility.
  - spekcular 5y ago
    Kelly is special though, it's not just log utility. In fact, viewing it as maximizing log utility is ahistorical; the original derivation was in terms of information theory. (The paper's title is "A new interpretation of information rate" [0].) Further, and most importantly, Kelly betting has certain favorable asymptotic properties that betting strategies motivated by other utility functions don't have. See Breiman's "Optimal gambling systems for favorable games" [1]. This point is well known in the literature, for instance see [2]: > Perhaps one reason is that maximizing E log S suggests that the investor has a logarithmic utility for money. However, the criticism of the choice of utility functions ignores the fact that maximizing E log S is a consequence of the goals represented by properties P1 and P2, and has nothing to do with utility theory. [0] https://www.princeton.edu/~wbialek/rome/refs/kelly_56.pdf https://www.princeton.edu/~wbialek/rome/refs/kelly_56.pdf [1] http://www-stat.wharton.upenn.edu/~steele/Resources/FTSResources/KellyBreiman/Breiman61.pdf http://www-stat.wharton.upenn.edu/~steele/Resources/FTSResou... [2] https://pubsonline.informs.org/doi/abs/10.1287/moor.5.2.161 https://pubsonline.informs.org/doi/abs/10.1287/moor.5.2.161
    - rssoconnor 5y ago
      Exactly. The Kelly strategy is the strategy that, with probability 1, eventually and permanently beats any other strategy. This isn't true for any other strategy and this criterion has nothing to do with utility.
  - jsw97 5y ago
    Paul Samuelson wrote an article in 1979 on this. It contains only one-syllable words, except for the last word, which is "syllable". Title: "Why We Should Not Make Mean Log of Wealth Big though Years to Act Are Long." His 1971 paper, "The 'Fallacy' of Maximizing the Geometric Mean in Long Sequences of Investing or Gambling" is more readable.
  - LudwigNagasena 5y ago
    That’s wrong. The assumptions behind the Kelly criterion lead to the log utility only in a particular case. What if each week you can bet $1 that in 50% of cases will triple your bet and in 50% of cases will give you $0? According to the log utility whether you should bet depends on your current wealth. According to the assumptions behind the Kelly criterion you should take the bet every week.
    - kqr 5y ago
      Technically the Kelly criterion suggests you should take this bet if your wealth is greater than $1.23, so it does depend on your current wealth. However, this might still look dumb, and it's because my calculation assumes you could pick a smaller stake for the same bet, which isn't necessarily the case. If you can't, the calculation gets a little bit more complicated, but sure, the Kelly criterion can deal with that too.
      - LudwigNagasena 5y ago
        According to the assumptions behind the Kelly criterion (maximizing long-term growth rate) you should always take the bet. Kelly criterion can’t be applied here because we have a different setup (you can’t choose your bet size), that’s my entire point!
- paulpauper 5y ago
  the entire field of martingale pricing
  - arthurcolle 5y ago
    I can hear your sigh in answering this all the way from here :D
ltbarcly3 5y ago
This so much. Like anything that is 'optimal' it is optimal with respect to some criterion. For the Kelly Criterion it is to maximize the logarithm of the weighted sum of the expected value across all outcomes. This is probably not what you actually want in almost any situation. The one time it actually makes sense to bring it in is if you are forced to make a certain number of wagers in some game AND you have good-enough knowledge of the odds of the game AND your payout is proportional to how much money you have left at the end of the game, AND the wager size does not effect the outcome of the game in any way. This scenario never happens. Even when you meet many of the necessary prerequisites to use Kelly, it still doesn't make sense at all. For example, Blackjack tournaments. You are given a set amount of starting money, you play for a set number of hands (or time), and you know the odds perfectly. However, the payout structure isn't proportional to your final amount of money, so using Kelly has more or less no chance of winning any money in the tournament. They usually pay for the top N results, which means you have to go with a very high variance strategy AND win consistently to place. Poker: Nope, not even close, bet size is a direct input into the dynamics of the game. Predictable betting of any kind is a maximally bad strategy. Stock Market: utility of money isn't logarithmic, so it is not worth maximizing, even if you knew probabilities and were forced to make wagers, which you don't and aren't. If you could even approximate probabilities you could use that power to basically print money on derivatives, so even if Kelly applied, the prerequisites are too strong and would make far far better strategies available. The only valid use case for bringing in the Kelly Criteria is for gamblers to feel better about burning money at the tables by improperly applying it.
- arthurcolle 5y ago
  Kelly must be the name of the wife in the WSB "wife's boyfriend" memes. More seriously, great analysis. Couldn't have said it better myself.
- thenaturalist 5y ago
  Hi there, you seem to know a good deal about the intricacies of where the Kelly Criterion lies within the range of possible options to calculate outcomes. Could you point me to any good resource you know of to learn more about the range itself and other options?
ArtWomb 5y ago
I believe it's come into fashion again with the rise of legal sportsbook in the US (DraftKings, etc). Particulaly, with the fascination over long shot "parlays" or chain bets that can achieve very high payouts with very little outlays. Better odds than buying a lottery ticket by far, or so it is perceived. I think you have to have a few jokers short of a full deck to get into sports gaming and expect any outcome other than ruin. But the parlays are interesting. And the simplicity one could devise a winning strategy is enticing. Consider a typical NBA season: 120 game nights, about 8 games per night. Let's say you constructed a parlay strategy in which you pick the under to hit on every game played that night. If the payout is large, say $1000 for a $3 bet. That's $360 for the season risk. And a high probability that eventually it'll cash ;)
- confidantlake 5y ago
  If the odds of an under happening is 50% then that is indeed a great bet! The odds of winning would be (0.5)^8 or 1 in 256. So the EV of the bet would be E(x) = (1/256 * 1000) - (255/256 * 3) E(x) = .91796875 So you are expected to make nearly 92 cents on every 3 dollars bet. Over a 30% return! Any bookie offering these odds would quickly go broke.
  - nly 5y ago
    In practice they'd simply flag you as a sharp and ban your account before you made them broke
lordnacho 5y ago
When I was a young guy who happened to be partner in a hedge fund, I used to ask candidates about this. I kinda regret it now. I'm not sure what I was trying to find out from people. I guess in some way it's a cultural question masquerading as a technical question, because it reveals whether you've heard of it and whether you have heard of the standard stuff that's said about it: - Don't do full Kelly because if you're on the wrong side (too big bets) that's definitely bad. - Depends on the probabilities being static. - You can find a continuous form of it. You can also find the implied leverage from the Sharpe ratio. I wonder if my memory is even correct on these. But the point remains that it's not terribly useful as a thing to evaluate people with. I guess the question "How do you size your positions?" needs to start somewhere.
- sohei 5y ago
  > I guess the question "How do you size your positions?" needs to start somewhere. Maybe the question itself is a good place to start. If someone asked me about Kelly, my mind would immediately drift from "statistical models and dynamic programming" to something else. The real engineering problem is threefold: (1) how do I model my return generating process, (2) what is my utility function, and (3) do the first two steps yield a tractable Bellman equation? If the problem is framed right, most people will have something to say about 1 and 2. The third part is tricky. A real world solution will involve finite element methods. But in an interview, people may be hesitant to bring up approaches that don't yield closed form solutions. That's what's special about Kelly. It's assumptions for steps 1 and 2 that produce a closed form solution in step 3.
pradn 5y ago
Bloom filters are the equivalent in computer science. Just beyond the basics, but ubiquitous enough to be annoying when it’s on the front page every other week.

The Kelly criterion is to financial math as the Fibonacci sequence is to mathematics. Yes it's neat, no it's not special, please stop bringing it up all the time.

bobbylarrybobby 5y ago

The Fibonacci sequence is not special because it is just one element of the family of linear recurrences. Is there a larger family to which the Kelly criterion belongs? (Not being snide — I’m genuinely asking)

murbard2 5y ago
Yes, any utility function will give you a Kelly-like criterion. Kelly is log utility.
- spekcular 5y ago
  Kelly is special though, it's not just log utility. In fact, viewing it as maximizing log utility is ahistorical; the original derivation was in terms of information theory. (The paper's title is "A new interpretation of information rate" [0].) Further, and most importantly, Kelly betting has certain favorable asymptotic properties that betting strategies motivated by other utility functions don't have. See Breiman's "Optimal gambling systems for favorable games" [1]. This point is well known in the literature, for instance see [2]: > Perhaps one reason is that maximizing E log S suggests that the investor has a logarithmic utility for money. However, the criticism of the choice of utility functions ignores the fact that maximizing E log S is a consequence of the goals represented by properties P1 and P2, and has nothing to do with utility theory. [0] https://www.princeton.edu/~wbialek/rome/refs/kelly_56.pdf https://www.princeton.edu/~wbialek/rome/refs/kelly_56.pdf [1] http://www-stat.wharton.upenn.edu/~steele/Resources/FTSResources/KellyBreiman/Breiman61.pdf http://www-stat.wharton.upenn.edu/~steele/Resources/FTSResou... [2] https://pubsonline.informs.org/doi/abs/10.1287/moor.5.2.161 https://pubsonline.informs.org/doi/abs/10.1287/moor.5.2.161
  - rssoconnor 5y ago
    Exactly. The Kelly strategy is the strategy that, with probability 1, eventually and permanently beats any other strategy. This isn't true for any other strategy and this criterion has nothing to do with utility.
- jsw97 5y ago
  Paul Samuelson wrote an article in 1979 on this. It contains only one-syllable words, except for the last word, which is "syllable". Title: "Why We Should Not Make Mean Log of Wealth Big though Years to Act Are Long." His 1971 paper, "The 'Fallacy' of Maximizing the Geometric Mean in Long Sequences of Investing or Gambling" is more readable.
- LudwigNagasena 5y ago
  That’s wrong. The assumptions behind the Kelly criterion lead to the log utility only in a particular case. What if each week you can bet $1 that in 50% of cases will triple your bet and in 50% of cases will give you $0? According to the log utility whether you should bet depends on your current wealth. According to the assumptions behind the Kelly criterion you should take the bet every week.
  - kqr 5y ago
    Technically the Kelly criterion suggests you should take this bet if your wealth is greater than $1.23, so it does depend on your current wealth. However, this might still look dumb, and it's because my calculation assumes you could pick a smaller stake for the same bet, which isn't necessarily the case. If you can't, the calculation gets a little bit more complicated, but sure, the Kelly criterion can deal with that too.
    - LudwigNagasena 5y ago
      According to the assumptions behind the Kelly criterion (maximizing long-term growth rate) you should always take the bet. Kelly criterion can’t be applied here because we have a different setup (you can’t choose your bet size), that’s my entire point!
paulpauper 5y ago
the entire field of martingale pricing
- arthurcolle 5y ago
  I can hear your sigh in answering this all the way from here :D

ltbarcly3 5y ago

This so much. Like anything that is 'optimal' it is optimal with respect to some criterion. For the Kelly Criterion it is to maximize the logarithm of the weighted sum of the expected value across all outcomes. This is probably not what you actually want in almost any situation. The one time it actually makes sense to bring it in is if you are forced to make a certain number of wagers in some game AND you have good-enough knowledge of the odds of the game AND your payout is proportional to how much money you have left at the end of the game, AND the wager size does not effect the outcome of the game in any way. This scenario never happens. Even when you meet many of the necessary prerequisites to use Kelly, it still doesn't make sense at all. For example, Blackjack tournaments. You are given a set amount of starting money, you play for a set number of hands (or time), and you know the odds perfectly. However, the payout structure isn't proportional to your final amount of money, so using Kelly has more or less no chance of winning any money in the tournament. They usually pay for the top N results, which means you have to go with a very high variance strategy AND win consistently to place. Poker: Nope, not even close, bet size is a direct input into the dynamics of the game. Predictable betting of any kind is a maximally bad strategy. Stock Market: utility of money isn't logarithmic, so it is not worth maximizing, even if you knew probabilities and were forced to make wagers, which you don't and aren't. If you could even approximate probabilities you could use that power to basically print money on derivatives, so even if Kelly applied, the prerequisites are too strong and would make far far better strategies available. The only valid use case for bringing in the Kelly Criteria is for gamblers to feel better about burning money at the tables by improperly applying it.

arthurcolle 5y ago
Kelly must be the name of the wife in the WSB "wife's boyfriend" memes. More seriously, great analysis. Couldn't have said it better myself.
thenaturalist 5y ago
Hi there, you seem to know a good deal about the intricacies of where the Kelly Criterion lies within the range of possible options to calculate outcomes. Could you point me to any good resource you know of to learn more about the range itself and other options?

ArtWomb 5y ago

I believe it's come into fashion again with the rise of legal sportsbook in the US (DraftKings, etc). Particulaly, with the fascination over long shot "parlays" or chain bets that can achieve very high payouts with very little outlays. Better odds than buying a lottery ticket by far, or so it is perceived. I think you have to have a few jokers short of a full deck to get into sports gaming and expect any outcome other than ruin. But the parlays are interesting. And the simplicity one could devise a winning strategy is enticing. Consider a typical NBA season: 120 game nights, about 8 games per night. Let's say you constructed a parlay strategy in which you pick the under to hit on every game played that night. If the payout is large, say $1000 for a $3 bet. That's $360 for the season risk. And a high probability that eventually it'll cash ;)

confidantlake 5y ago
If the odds of an under happening is 50% then that is indeed a great bet! The odds of winning would be (0.5)^8 or 1 in 256. So the EV of the bet would be E(x) = (1/256 * 1000) - (255/256 * 3) E(x) = .91796875 So you are expected to make nearly 92 cents on every 3 dollars bet. Over a 30% return! Any bookie offering these odds would quickly go broke.
- nly 5y ago
  In practice they'd simply flag you as a sharp and ban your account before you made them broke

lordnacho 5y ago

When I was a young guy who happened to be partner in a hedge fund, I used to ask candidates about this. I kinda regret it now. I'm not sure what I was trying to find out from people. I guess in some way it's a cultural question masquerading as a technical question, because it reveals whether you've heard of it and whether you have heard of the standard stuff that's said about it: - Don't do full Kelly because if you're on the wrong side (too big bets) that's definitely bad. - Depends on the probabilities being static. - You can find a continuous form of it. You can also find the implied leverage from the Sharpe ratio. I wonder if my memory is even correct on these. But the point remains that it's not terribly useful as a thing to evaluate people with. I guess the question "How do you size your positions?" needs to start somewhere.

sohei 5y ago
> I guess the question "How do you size your positions?" needs to start somewhere. Maybe the question itself is a good place to start. If someone asked me about Kelly, my mind would immediately drift from "statistical models and dynamic programming" to something else. The real engineering problem is threefold: (1) how do I model my return generating process, (2) what is my utility function, and (3) do the first two steps yield a tractable Bellman equation? If the problem is framed right, most people will have something to say about 1 and 2. The third part is tricky. A real world solution will involve finite element methods. But in an interview, people may be hesitant to bring up approaches that don't yield closed form solutions. That's what's special about Kelly. It's assumptions for steps 1 and 2 that produce a closed form solution in step 3.

pradn 5y ago

Bloom filters are the equivalent in computer science. Just beyond the basics, but ubiquitous enough to be annoying when it’s on the front page every other week.