Hayley Clatterbuck comments on Causes and Uncertainty: Rethinking Value in Expectation

[Hayley Clatterbuck]

Thanks for your comment, Michael. Our team started working through your super helpful recent post last week! We discuss some of these issues (including the last point you mention) in a document where we summarize some of the philosophical background issues. However, we only mention bounded utility very briefly and don’t discuss infinite cases at all. We focus instead on rounding down low probabilities, for two reasons: first, we think that’s what people are probably actually doing in practice, and second, it avoids the seeming conflict between bounded utility and theories of value. I’m sure you have answers to that problem, so let us know!

[MichaelStJules]

I got a bit more time to think about this.

I think there probably is no conflict between bounded utility (or capturing risk aversion with concave increasing utility functions) and theories of deterministic value, because without uncertainty/​risk, bounded utility functions can agree with unbounded ones on all rankings of outcomes. The utility function just captures risk attitudes wrt to deterministic value.

Furthermore, bounded and concave utility functions can be captured as weighting functions, much like WLU. Suppose you have a utility function $u$ of the value $v$, which is a function of outcomes. Then, whether $u$ is bounded or concave or whatever, we can still write:

$$u\left(v\right(x\left)\right)=\frac{u\left(v\right(x\left)\right)}{v\left(x\right)}v\left(x\right)=w\left(x\right)v\left(x\right)$$

where $w\left(x\right)=\frac{u\left(v\right(x\left)\right)}{v\left(x\right)}$.^[1] Then, for a random variable $X$ over outcomes:

$$E\left[u\right(v\left(X\right)\left)\right]=E\left[w\right(X\left)v\right(X\left)\right]$$

Compare to WLU, with some weighting function $w$ of outcomes:

$$WLU\left(X\right)=\frac{E\left[w\right(X\left)v\right(X\left)\right]}{E\left[w\right(X\left)\right]}$$

The difference is that WLU renormalizes.

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By the way, because of this renormalizing, WLU can also be seen as adjusting the probabilities in $X$ to obtain a new prospect. If $p$ is the original probability distribution (for $X\sim p$, i.e.$P(X\in A)=p\left(A\right)$ for each set of outcomes $A$), then we can define a new one by:^[2]

$$q\left(A\right)=\frac{1}{\int w\left(x\right)dp\left(x\right)}{\int }_{A}w\left(x\right)dp\left(x\right)=\frac{1}{E_{X\sim p}\left[w\right(X\left)\right]}{\int }_{A}w\left(x\right)dp\left(x\right)$$

so

$$WLU\left(X\right)=\frac{E\left[w\right(X\left)v\right(X\left)\right]}{E\left[w\right(X\left)\right]}=\frac{1}{\int w\left(x\right)dp\left(x\right)}{\int }_{A}w\left(x\right)dp\left(x\right)={\int }_{A}v\left(y\right)dq\left(y\right)=E_{Y\sim q}\left[v\right(Y\left)\right]$$

1. ^

   We can define $w\left(x\right)$ arbitrarily when $v\left(x\right)=0$ to avoid division by 0.

2. ^

   You can replace the integrals with sums for discrete distributions, but integral notation is more general in measure theory.