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

put it together to get the best action

d_z : U_z \rightarrow A_z where d_z(w) satisfies s i.e. \mu_z(w,d_z(a))=s_z(w)

or the satisficing version

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.

All of these are genuine and noble responses to the muddiness of the equations and I don’t want to detract from that urge. It’s just that the muddiness is inherent to the project and the rhetoric around it does not appear to grant this. Comparison of ethical systems is best served by acknowledging that they’re all heuristics and then examining which distortions each structure is alternately vulnerable to or guarded against.