Success Maximization: An Alternative to Expected Utility Theory and a Generalization of Maxipok to Moral Uncertainty

Mahendra Prasad

Standard expected utility theory (EUT) assumes moral certainty, but also embeds epistemic/ontological uncertainty about the state of the world that may occur as a result of our actions. Harsanyi expected utility theory (HEUT) allows us to assign probabilities to our potential moral viewpoints, and thus gives us a mechanism by which to handle moral uncertainty.

Unfortunately, there are several problems with EUT and HEUT. First, the St. Petersburg paradox shows that unbounded utility valuations can justify almost any action, even if the probability of a good outcome is almost zero. For example, a banker may be in a situation where the probability of a bank run is nearly one, but because potential returns of being overleveraged in a near zero probability world are so high, the banker may foolishly still choose to be overleveraged to maximize expected utility. Second, diminishing returns typically force us to produce or consume more in order to realize the same amounts of utility; this is usually a recipe for us to consume and produce in unsustainable ways. Third, as Herbert Simon noted, optimizing expected utility is often computationally intractable.

A response of early effective altruism research to these problems was maxipok (i.e., maximizing probabilities of an okay outcome). Under this construct, constraints of an okay outcome are identified, a probability of satisfying those constraints is assigned to each action, and the action that maximizes the probability of satisfying the constraints is adopted.

The problem with maxipok is that it assumes moral certainty about the constraints of what constitutes an okay outcome. For example, if we believe a trolley problem is inevitable, one might infer it is an okay outcome for someone to die, given its unavoidability. On the other hand, if a trolley problem is avoidable, one may infer that someone dying is not okay. Thus in that overall scenario, what constitutes an okay outcome is contingent on what probabilities we assign to the inevitability of a trolley problem.

Success maximization is a mechanism by which to generalize maxipok for moral uncertainty. Let a_i be an action i from the set of m actions A = {a₁, a₂, …, a_m}. Let s_x be a definition of moral success, namely x, from S = {s₁, s₂, …, s_n}. The probability π that i satisfies the constraints of s_x is 0 ≤ π_i(s_x) ≤ 1. Let p(s_x) be the estimated probability that x is the correct definition of moral success, where p(s₁) + p(s₂) + … + p(s_n) = 1. Thus, the expected success of action i is 0 ≤ π_i(s₁)p(s₁) + π_i(s₂)p(s₂) + … + π_i(s_n)p(s_n) ≤ 1. A success maximizing agent will choose an action a_j є A such that π_j(s₁)p(s₁) + π_j(s₂)p(s₂) + … + π_j(s_n)p(s_n) ≥π_i(s₁)p(s₁) + π_i(s₂)p(s₂) + … + π_i(s_n)p(s_n) for all a_i є A where i ≠ j.

Success maximization resolves many of the problems of von Neumann-Morgenstern and Harsanyi expected utility theories. First, because success valuations are bounded between 0 and 1, it is much less likely we will encounter St. Petersburg paradox situations where any action is justified by extremely high utility valuations despite near zero probabilities of occurrence. Second, unsustainable behaviors produced by chasing diminishing returns is much less likely in the world of maximizing probabilities of constraint satisfaction than it is in the world of maximizing unbounded expected utilities. Third, because probabilities of success are bounded between zero and one, terms of the linear combination (where p(s_x) is relatively low) can often be ignored to make for quicker calculations, making calculations more tractable.

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Jonas MossNov 26 20222

If I understand you correctly, what you're proposing is essentially a subset of classical decision theory with bounded utility functions. Recall that, under classical decision theory, we choose our action according to where $X$ is a random state of nature and $A$ an action space.

Suppose there are $N$ (infinitely many works too) moral theories $s_{1}, s_{2}, \dots, s_{N}$ , each with probability $p (s_{i})$ and associated utility $u_{i}$ . Then we can define $u (a, X) = N \sum i = 1 p (s_{i}) u_{i} (a, X) .$ This step gives us (moral) uncertainty in our utility function.

Then, as far as I understand you, you want to define some component utility functions as $\begin{matrix} u_{i} (a, X) & = & {\begin{matrix} 1, & if (a, X) is acceptable under theory s_{i}, 0, & if (a, X) is unacceptable under theory s_{i} . \end{matrix} \end{matrix}$ As then $0 \leq E u_{i} (a, X) \leq 1$ is the probability of an acceptable outcome under $s_{i}$ . And since we're taking the expected value of these bounded component utilities to construct $u$ , we're in classical bounded utility function land.

That said, I believe that

This post would benefit from a rewrite of the paragraph starting with "Success maximization is a mechanism by which to generalize maxipok". It states " Let $a_{i}$ be an action $i$ from the set of $m$ actions $A = a_{1}, a_{2}, \dots, a_{m}$ . " Is $i$ and action, $a_{i}$ and action, or both? I also don't understand what $π$ is. Are there states of nature in this framework? You say that $s$ is a moral theory, so it cannot be $s$ ?
You should add concrete examples. If you add one or two it might become easier to understand what you're doing despite the formal definition not being 100% clear.

deepNov 26 20222

Speaking as a non-expert: This is an interesting idea, but I'm confused as to how seriously I should take it. I'd be curious to hear:

Your epistemic status on this formalism. My guess is you're at "seems like a good cool idea; others should explore this more", but maybe you want to make a stronger statement, in which case I'd want to see...
Examples! Either a) examples of this approach working well, especially handling weird cases that other approaches would fail at. Or, conversely, b) examples of this approach leading to unfortunate edge cases that suggest directions for further work.

I'm also curious if you've thought about the parliamentary approach to moral uncertainty, as proposed by some FHI folks. I'm guessing there are good reasons they've pushed in that direction rather than more straightforward "maxipok with p(theory is true)", which makes me think (outside-view) that there are probably some snarls one would run into here.

deepNov 26 20221

Inside-view, some possible tangles this model could run into:

Some theories care about the morality of actions rather than states. But I guess you can incorporate that into 'states' if the history of your actions is included in the world-state -- it just makes things a bit harder to compute in practice, and means you need to track "which actions I've taken that might be morally meaningful-in-themselves according to some of my moral theories." (Which doesn't sound crazy, actually!)
the obvious one: setting boundaries on "okay" states is non-obvious, and is basically arbitrary for some moral theories. And depending on where the boundaries are set for each theory, theories could increase or decrease in influence on one's actions. How should we think about okayness boundaries?
- One potential desideratum is something like "honest baragaining." Imagine each moral theory as an agent that sets its "okayness level" independent of the others, and acts to maximize good from its POV. Then the our formalism should lead to each agent being incentivized to report its true views. (I think this is a useful goal in practice, since I often do something like weighing considerations by taking turns inhabiting different moral views).
  - I think this kind of thinking naturally leads to moral parliament models -- I haven't actually read the relevant FHI work, but I imagine it says a bunch of useful things, e.g. about using some equivalent of quadratic voting between theories.
- I think there's an unfortunate tradeoff here, where you either have arbitrary okayness levels or all the complexity of nuanced evaluations. But in practice maybe success maximization could function as the lower level heuristic (or middle level, between easier heuristics and pure act-utilitarianism) of a multi-level utilitarianism approach.