4 ms·

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!

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!