Michael St Jules 🔸 comments on Causes and Uncertainty: Rethinking Value in Expectation

[Michael St Jules 🔸]

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.