4 ms·

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

58 points by stared 3y ago

edpichler 3y ago
I am so glad I found this video. This can be applied to so many things.
abetusk 3y ago
The Kelly Criterion [0] tells what strategy to employ if you're given fair odds on an unfair coin. For example, if a bet can be made where you get 1:1 payout for a coin that's weighted heads with probability 0.8 and tails with probability 0.2, how much should you bet to maximize wins, under a suitable formulation, if you're allowed to play repeatedly. Betting all your wealth at each round nearly guarantees you'll lose everything the more you play, so it's not as straight forward as one might think. If you decide to bet a fixed amount of your current wealth at every turn, that's the condition that the Kelly criterion tells you to bet: K = (bp - q) / b Where p=1-q is the probability of heads and b is the odds of winning (that is, the coin gives b:1 odds). There's some SO posts about it [1] and I've also made a short blog post, with minimal explanation, which derives it [2]. Note that this has wider applicability than just "how to bet on a horse race you know is fixed". If you believe the stock market is not zero sum and is increasing by a certain amount, how do you bet part of your fortune to maximize your returns? If you have a cohort of people founding startups with a certain failure rate and return, how much of your funds wealth do you invest in these ventures? The real world is complex but the Kelly criterion gives a starting point to talk about these things. I heard somewhere that Warren Buffet uses a factor of 0.9 on top of the Kelly criterion calculation to account for increased risk. [0] https://en.wikipedia.org/wiki/Kelly_criterion https://en.wikipedia.org/wiki/Kelly_criterion [1] https://math.stackexchange.com/questions/4384904/geometric-growth-rate-and-kellys-criterion-question https://math.stackexchange.com/questions/4384904/geometric-g... [2] https://mechaelephant.com/dev/Kelly-Criterion.html https://mechaelephant.com/dev/Kelly-Criterion.html
suoinguon 3y ago
[dead]
bitshiftfaced 3y ago
There's a related idea, centered on the St. Petersburg paradox. It's the idea that the median outcome is sometimes a much better predictor of human behavior than the mean outcome: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3811154/ https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3811154/
- chriswarbo 3y ago
  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.
Robotbeat 3y ago
I found this easier to just explain as a factor instead of speaking of percentages. If you win, your money gets bigger by a factor of 1.8. If you lose, your money gets smaller by a factor of 2 or multiplied by a factor of 1/2. So you find your end value by just multiplying all these factors up. Obviously if the number of winnings and losings is the same, you’ll be multiplying by 1.8 as often as dividing by 2, so on average your money will get smaller by like 0.9 each round. Not counter-intuitive if you think in terms of factors.
- kgwgk 3y ago
  Agreed. Most people understand quickly that after a 50% loss a 50% gain is not enough to break even.
- drivers99 3y ago
  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.
cubefox 3y ago
The author of the video says that the Kelly criterion is used in investing, but that's not really the case. Kelly should only be used when the probabilities of winning/losing are known exactly. When they are only estimated, like in investing, the Kelly criterion is still far too risky. Here is a bit more about this problem: https://youtube.com/watch?v=-9JM9suCIHs https://youtube.com/watch?v=-9JM9suCIHs Unfortunately there doesn't yet seem to exist an equivalent to the Kelly formula which would apply to subjective probabilities. I haven't even found an explanation of why uncertainty about probabilities requires betting more conservatively than Kelly would recommend.
- proamdev123 3y ago
  > 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.
  - cubefox 3y ago
    I guess something like that is indeed the reason. However, as stated (in terms of "risk of ruin") the argument doesn't quite work. Because even if the probabilities are known with certainty, betting the Kelly optimum would increase the risk of ruin compared to betting less. But Kelly betting would nonetheless be better.
    - cochne 3y ago
      The thing about Kelly is that it assumes risk of ruin is zero because you never bet your entire bankroll. It assumes you can always bet a fraction of a fraction of what you have remaining.
      - 3y ago
chriswarbo 3y ago
Scott Garrabrant wrote a nice series on Less Wrong which explains/derives the Kelly Criterion, and relates it to other areas (Nash bargaining, Thompson sampling, Bayesian updating), and how they all follow the same underlying logic (optimising the geometric mean) https://www.lesswrong.com/s/4hmf7rdfuXDJkxhfg https://www.lesswrong.com/s/4hmf7rdfuXDJkxhfg
jshprentz 3y ago
I recommend Fortune’s Formula, a book about the Kelly criterion by William Poundstone. The book's blurb claims it "... traces how the Kelly formula sparked controversy even as it made fortunes at race tracks, casinos, and trading desks. It reveals the dark side of this alluring scheme, which is founded on exploiting an insider’s edge." The book is light on math and strong on stories about and interviews with people using the Kelly criterion.
1-more 3y ago
Here's a fun game where you have 10 minutes to flip an unfair coin. In my experience I get the best results when I just move really fast and don't worry about matching Kelly exactly. https://elmwealth.com/coin-flip/ https://elmwealth.com/coin-flip/

58 points by stared 3y ago

I am so glad I found this video. This can be applied to so many things.

The Kelly Criterion [0] tells what strategy to employ if you're given fair odds on an unfair coin. For example, if a bet can be made where you get 1:1 payout for a coin that's weighted heads with probability 0.8 and tails with probability 0.2, how much should you bet to maximize wins, under a suitable formulation, if you're allowed to play repeatedly. Betting all your wealth at each round nearly guarantees you'll lose everything the more you play, so it's not as straight forward as one might think. If you decide to bet a fixed amount of your current wealth at every turn, that's the condition that the Kelly criterion tells you to bet: K = (bp - q) / b Where p=1-q is the probability of heads and b is the odds of winning (that is, the coin gives b:1 odds). There's some SO posts about it [1] and I've also made a short blog post, with minimal explanation, which derives it [2]. Note that this has wider applicability than just "how to bet on a horse race you know is fixed". If you believe the stock market is not zero sum and is increasing by a certain amount, how do you bet part of your fortune to maximize your returns? If you have a cohort of people founding startups with a certain failure rate and return, how much of your funds wealth do you invest in these ventures? The real world is complex but the Kelly criterion gives a starting point to talk about these things. I heard somewhere that Warren Buffet uses a factor of 0.9 on top of the Kelly criterion calculation to account for increased risk. [0] https://en.wikipedia.org/wiki/Kelly_criterion https://en.wikipedia.org/wiki/Kelly_criterion [1] https://math.stackexchange.com/questions/4384904/geometric-growth-rate-and-kellys-criterion-question https://math.stackexchange.com/questions/4384904/geometric-g... [2] https://mechaelephant.com/dev/Kelly-Criterion.html https://mechaelephant.com/dev/Kelly-Criterion.html

suoinguon 3y ago

[dead]

bitshiftfaced 3y ago

There's a related idea, centered on the St. Petersburg paradox. It's the idea that the median outcome is sometimes a much better predictor of human behavior than the mean outcome: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3811154/ https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3811154/

chriswarbo 3y ago
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.

Robotbeat 3y ago

I found this easier to just explain as a factor instead of speaking of percentages. If you win, your money gets bigger by a factor of 1.8. If you lose, your money gets smaller by a factor of 2 or multiplied by a factor of 1/2. So you find your end value by just multiplying all these factors up. Obviously if the number of winnings and losings is the same, you’ll be multiplying by 1.8 as often as dividing by 2, so on average your money will get smaller by like 0.9 each round. Not counter-intuitive if you think in terms of factors.

kgwgk 3y ago
Agreed. Most people understand quickly that after a 50% loss a 50% gain is not enough to break even.
drivers99 3y ago
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.

cubefox 3y ago

The author of the video says that the Kelly criterion is used in investing, but that's not really the case. Kelly should only be used when the probabilities of winning/losing are known exactly. When they are only estimated, like in investing, the Kelly criterion is still far too risky. Here is a bit more about this problem: https://youtube.com/watch?v=-9JM9suCIHs https://youtube.com/watch?v=-9JM9suCIHs Unfortunately there doesn't yet seem to exist an equivalent to the Kelly formula which would apply to subjective probabilities. I haven't even found an explanation of why uncertainty about probabilities requires betting more conservatively than Kelly would recommend.

proamdev123 3y ago
> 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.
- cubefox 3y ago
  I guess something like that is indeed the reason. However, as stated (in terms of "risk of ruin") the argument doesn't quite work. Because even if the probabilities are known with certainty, betting the Kelly optimum would increase the risk of ruin compared to betting less. But Kelly betting would nonetheless be better.
  - cochne 3y ago
    The thing about Kelly is that it assumes risk of ruin is zero because you never bet your entire bankroll. It assumes you can always bet a fraction of a fraction of what you have remaining.
    - 3y ago

chriswarbo 3y ago

Scott Garrabrant wrote a nice series on Less Wrong which explains/derives the Kelly Criterion, and relates it to other areas (Nash bargaining, Thompson sampling, Bayesian updating), and how they all follow the same underlying logic (optimising the geometric mean) https://www.lesswrong.com/s/4hmf7rdfuXDJkxhfg https://www.lesswrong.com/s/4hmf7rdfuXDJkxhfg

jshprentz 3y ago

I recommend Fortune’s Formula, a book about the Kelly criterion by William Poundstone. The book's blurb claims it "... traces how the Kelly formula sparked controversy even as it made fortunes at race tracks, casinos, and trading desks. It reveals the dark side of this alluring scheme, which is founded on exploiting an insider’s edge." The book is light on math and strong on stories about and interviews with people using the Kelly criterion.

1-more 3y ago

Here's a fun game where you have 10 minutes to flip an unfair coin. In my experience I get the best results when I just move really fast and don't worry about matching Kelly exactly. https://elmwealth.com/coin-flip/ https://elmwealth.com/coin-flip/