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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 tha

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.

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.