# Utility Calculations in Practice

I have issues with utilitarianism as usually stated. It encourages poor calculation and, as a tool, is not a good fit for the problem it is trying to solve. The topic is complicated and this post is not meant to stand against any argument in particular, merely to capture the particulars of the situation as they appear in my mind.

By my understanding, utilitarianism is about the following calculations.

$z$, a principle actor
$U_z$, the set of universes with $z$ as principle actor
$A_z$, the set of actions available to $z$
$\rho : U_z \times A_z \rightarrow \{ f:U_z \rightarrow \mathbf{R}, f \text{ a probability measure} \}$, a probabilistic state of the world after an action
$\nu : U_z \rightarrow \mathbf{R}$, a universal value function
$\mu :=(\int_u \nu(u)\rho(w,a)(u))-\nu(w)$, the expected value of the action
$d : U_z \rightarrow A_z$ where $d(w)$ satisfies $\emph{max}_{a\in A_z} \mu(w,a)$, the best action to take

Taking this framing as correct, there are four categories of objection that I end up playing out.

Objections to the mathematical structure. Can we apply a suitable measure to $U_z$ or is $\rho$ incoherent for this reason? Can we integrate over $U_z$ for the purposes of $\mu$? These are the weakest objections because they hinge on my math skills (and those have become quite rusty) but for completeness’ sake we cannot presume to use the machinery of analysis without first satisfying its prerequisites.

Objections from physics. Are $U_z, A_z$ well-defined? Does $\rho$ place requirements on cause-and-effect that are realistic or do we require more variables to capture the set of things that could happen?

Objections from humanity. Is $\nu$ well-defined? There’s been no end of argument to resolve the question “What is the good?” and any attempt to get a utility framework off the ground has to bootstrap off some answer here.

Objections from computer science. Is $d$ computable? Is $\mu$ computable? Is $\rho$ computable? Are $U_z,A_z$ encodable? Our string of constructs are useful even as a thought experiment if they cannot be converted into a procedure for people to resolve emergent moral quandaries.

Going from that last set of objections, practicality forces humans to accept a few constraints when approximating this ideal. Our simple structure of above should be better rendered with the following.

$e_z: U_z \rightarrow U_z'$, epistemology, worlds we know
$c_z: U_z' \rightarrow U_z''$, reductive encoding of a world into a model
$i_z: U_z'' \rightarrow \{\text{actions } z \text{ can imagine taking}\}\subset A_z$, imagination function for generating actions
$\rho_z : U_z'' \times A_z \rightarrow \{ f:U_z'\rightarrow \mathbf{R}, f \text{ a probability measure}, f \text{ non-zero on a finite set} \}$, a set of probabilities to hold in your head (replace finite with finite parameterization if that bothers you)
$\nu_z$ value function of $z$, surely $z$ has a utility function
$\displaystyle \mu_z := (\sum_{\rho_z(c_z(e_z(w)),a)(u)>0} \nu_z(u)\rho_z(c_z(e_z(w)),a)(u)) - \nu_z(w)$, calculable value of action
$s_z := \emph{max}_{a\in i_z(A_z)} \mu_z(w,a)$, the best change in world value

$d_z : U_z \rightarrow A_z$ where $d_z(w)$ satisfies $s$ i.e. $\mu_z(w,d_z(a))=s_z(w)$
$d_z : U_z \rightarrow \{ a \text{ s.t. } \mu_z(w,a) > s_z(w) - \epsilon\} \subset A_z$ where $\epsilon$ is some error term.
Notice how many invocations of $z$ now appear. So many opportunities for subjective judgment or simply errors in judgment to sneak in and for a program that aims to provide an objective measure that should be a problem.
Different people deal with the subjectiveness in different ways. Life hackers focus on expanding and optimizing $i_z$. A lot of arguments arise in what makes an appropriate $c_z$. There’s much academic and theological agonizing over the size of $\{ \nu_z \}$ and the apparent difficulty of extracting an agreeable $\nu$. Similar agonizing over the optimal amount of self-interest in each $\nu_z$. Human forecasting (dubious profession that it is) tries to refine $\rho_z$. The bedrock of science is refining $e_z$. This last is my particular favorite though strictly speaking it is only a small portion of the final equation.