Moral Hope
Laplace wrote this great essay on probability1. This was back in 18th c. when mathematicians prose was the language of math instead of formulas, and there was no differentiation between philosophy and the practical application of math from its theory.
He defines this concept of “hope”, which is really just expectation. We say we are hopeful when our advantage is positive.
He then goes on to show that this classic definition of expectation doesn’t match our common sense knowledge.
Consider a game2 where you get paid $2 if the first H appears on the first toss, $4 if first H appears on the second, $8… so on. The expectation is infinite since the probabilities of H appearing on the nth toss is (1/2)^n, which means that the expectation is 1+1+… ad infinitum. Yet this is contrary to our common sense; nobody would pay an infinite amount to play this game.
So Laplace comes up with “moral hope” which, instead of taking the arithmetic mean in expectation, takes the geometric mean3. Intuitively if you have more fortune, the same profit means less to you; $100 is more valuable if you only have $10 vs if you have $1M. So moral expectation is function of both fortune and expected profit. It is the expected utility of the outcome rather than the net gain.
Most importantly, moral hope converges to a finite value for our game. Particularly, if you have $200, we should expect one to be willing to pay $8.714.
This is both cool and useful - nonlinear expectation is currently how we model human behavior!5
If only SBF knew…
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https://en.wikipedia.org/wiki/A_Philosophical_Essay_on_Probabilities ↩
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https://en.wikipedia.org/wiki/St._Petersburg_paradox ↩
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By AM-GM inequality, this implies moral hope is always less than or equal to mathematical hope. ↩
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See problem and derivation below ↩
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https://en.wikipedia.org/wiki/Expected_utility_hypothesis While Bernoulli’s equation is similar to Laplace’s, it’s not quite the same, and interestingly I haven’t been able to find Laplace’s formulation of moral hope anywhere online ↩