Wednesday, November 9, 2022

How to discount small probabilities

A very intuitive solution to a variety of problems in infinite decision theory is that “for possibilities that have very small probabilities of occurring, we should discount those probabilities down to zero” when making decisions (Monton).

Suppose throughout this post that ϵ > 0 counts as our threshold of “very small probabilities”. No doubt ϵ < 1/100.

In this post I want to offer a precise and friendly amendment to the solution of neglecting small probabilities. But first why we need an amendment. Consider a game where an integer K is randomly chosen between  − 1 and N for some large fixed positive N, so large that 1/(2+N) < ϵ, and you get K dollars. The game is clearly worth playing. But if you discount “possibilities that have very small probabilities”, you are left with nothing: every possibility has a very small probability!

Perhaps this is uncharitable. Maybe the idea is not that we discount to zero all possibilities with small probabilities, but that we discount such possibilities until the total discount hits the threshold ϵ. But while this sounds like a charitable interpretation of the suggestion, it leaves the theory radically underdetermined. For which possibilities do we discount? In my lottery case, do we start by discounting the possibilities at the low end ( − 1, 0, 1, ...) until we have hit the threshold? Or do we start at the high end (N, N − 1, N − 2, ...) or somewhere in the middle?

Here is my friendly proposal. Let U be the utility function we want to evaluate the value of. Let T be the smallest value such that P(U>T) ≤ ϵ/2. (This exists: T = inf {λ : P(U>λ) ≤ ϵ/2}.) Let t be the largest value such that P(U<t) ≤ ϵ/2 (i.e., t = sup {λ : P(U<λ) ≤ ϵ/2}). Take U and replace any values bigger than T with T and any values smaller than t with t, and call the resulting utility function Uϵ. We now replace U with Uϵ in our expected value calculations. (In the lottery example, we will be trimming from both ends at the same time.)

The result is a precise theory (given the mysterious threshold ϵ). It doesn’t neglect all possibilities with small probabilities, but rather it trims low-probability outliers. The trimming procedure respects the fact that often utility functions are defined up to positive affine transformations.

Moreover, the trimming procedure can yield an answer to what I think is the biggest objection to small-probability discounting, namely that in a long enough run—and everyone should think there is a non-negligible chance of eternal life—even small probabilities can add up. If you are regularly offered the same small chance of a gigantic benefit during an eternal future, and you turn it down each time because the chance is negligible, you’re almost surely missing out on an infinite amount of value. But we can apply the trimming procedure at the level of choice of policies rather than of individual decisions. Then if small chances are offered often enough, they won’t all be trimmed away.

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