1. Introduction

Utilitarianism is the view that a social planner should choose options which maximise the social utility of the resulting social outcome. The central object in utilitarianism is the social utility function $u:S\to R$ which assigns a real value $u\left(s\right)\in R$ to each social outcome $s\in S$. This function typically involves variables such as the well-being, preferences, and mental states of individuals, distributional factors like inequality, and other relevant factors such as justice, social cohesion, and freedoms. Utilitarianism is a broad class of social choice principles, one corresponding to each function $u:S\to R$.

In my previous article, I introduced aggregative principles, which state that a social planner should make decisions as if they will face the aggregated personal outcomes of every individual in the population. The central object in aggregativism is the function $\zeta :S\to P$, represented with the the greek letter zeta, which assigns a personal outcome $\zeta \left(s\right)\in P$ to each social outcome $s\in S$. This function typically aggregates the collection of personal outcomes facing the entire population into a single personal outcome. Aggregativism is a broad class of social choice principles, one corresponding to each function $\zeta :S\to P$.

We examined three well-known aggregative principles:

1. Live Every Life Once (LELO), where $\zeta \left(s\right)$ is the concatenation of every individual's life.
2. Harsanyi's Lottery (HL), where $\zeta \left(s\right)$ is a uniform lottery over every individual's life.
3. Rawls' Original Position (ROI), where $\zeta \left(s\right)$ is Knightian uncertainty over every individual's life.

I'm interested in aggregative principles because they avoid many theoretical pitfalls of utilitarian principles. Unlike utilitarianism, aggregativism doesn't require specifying a social welfare function, which is notoriously intractable. Moreover, it seems less prone to counterintuitive conclusions such as the repugnant conclusion or the violation of moral side constraints.^[1] In this article, I will show that, under natural conditions of human rationality, aggregative principles approximate utilitarian principles. Therefore, even though aggregativism avoids these theoretical pitfalls, we should nonetheless expect aggregativism to generate roughly-utilitarian recommendations in practical social contexts, and thereby retain the most appealing insights from utilitarianism.

The rest of the article is organized as follows. Section 2 formalises social choice principles as functions of type $(X\to S)\to P\left(X\right)$. Section 3 demonstrates the structural similarity between two strategies to specifying such principles, namely the aggregative and utilitarian strategies. Section 4 proves that under natural conditions about human rationality, the aggregative and utilitarian principles are mathematically equivalent. This theorem is the key contribution of the article. Sections 5, 6, and 7 applies the theorem to LELO, HL, and ROI respectively.

2. Social choice principles

Suppose you are a social planner choosing from a set of options $X=\{x_1,\dots ,x_n\}$. The set $X$ might be the set of available tax rates, environmental policies, military actions, political strategies, neural network parameters, or whatever else is being chosen by the social planner. Now, your choice will presumably depend on the social consequences of the options, even if you also consider non-consequentialist factors. We can model the social consequences with a function $f:X\to S$, where $S$ is the set of social outcomes. In particular, if you choose an option $x\in X$, then the resulting social outcome would be $f\left(x\right)\in S$.

We call $f:X\to S$ the "social context". As a concrete example, suppose the options are different tax rates (say 10%, 20%, and 30%), and the social outcomes are characterized by variables like total tax revenue, income inequality, and unemployment rate. Then the social context is the function $f:[0,1]\to S$ which maps each tax rate $x\in [0,1]$ to the resulting values of these social outcome variables.

A social choice principle should say, for each social context, which options are acceptable. Formally, a social choice principle is characterised by some function $\Psi :(X\to S)\to P\left(X\right)$, which takes a social context $f:X\to S$ as input and returns a subset of the options $\Psi \left(f\right)\subseteq X$ as output. Specifically, $\Psi \left(f\right)\subseteq X$ consists of exactly those options which satisfy the principle in the social context $f:X\to S$.

Note that $(X\to S)$ denotes the set of all functions from $X$ to $S$, so $\Psi$ is a higher-order function, meaning it receives another function as input. Additionally $P\left(X\right)$ denotes the powerset of $X$, i.e. the set of subsets of $X$. We use the powerset $P$ to allow for the fact that multiple options may satisfy a principle: if a principle $\Psi$ permits only options $x_1$ and $x_2$ in a context $f:X\to S$ then $\Psi \left(f\right)=\{x_1,x_2\}$. Finally, the powerset $P\left(X\right)$ includes the empty set $\varnothing$, which allows for the case $\Psi \left(f\right)=\varnothing$. Informally, $\Psi \left(f\right)=\varnothing$ means that the social planner, following principle $\Psi$ and faced with context $f:X\to S$, has no acceptable options, which allows for principles that aren't universally satisfiable.

Here are some examples of social choice principle:

1. Context-independence
   Let $X_0\subseteq X$ be any fixed subset of the options, and consider the principle $\Psi :f\mapsto X_0$, which returns $X_0$ regardless of the input $f$. Whether an option $x\in X$ satisfies this principle depends only on whether $x\in X_0$, and is independent of the social context. At one extreme, there's a trivial principle $\Psi :f\mapsto X$ which never constrains the social planner, and at the other extreme, there's a principle $\Psi :f\mapsto \varnothing$ which is always unsatisfiable. When $X_0=\left\{x_0\right\}$ consists of a single option, the principle $\Psi$ states that the social planner must choose $x_0\in X$ regardless of the context.^[2]
2. Targets
   Let $S_{\text{target}}\subseteq S$ be any fixed subset of the social outcomes, whose elements we'll call targets. There is a principle which says that the social planner should choose an option which achieves a target. This principle is characterised by the function $\Psi :f\mapsto f^{-1}\left(S_{\text{target}}\right)$, where $f^{-1}\left(S_{\text{target}}\right)\subseteq X$ denotes the preimage of $S_{\text{target}}$, i.e. $f^{-1}\left(S_{\text{target}}\right):=\{x\in X:f(x)\in S_{\text{target}}\}$. Note that if $f\left(x\right)\notin S_{\text{target}}$ for all $x\in X$ then $\Psi \left(f\right)=\varnothing$, i.e. if no option would achieve a target then the principle says all options are unacceptable.
3. Impact minimisation
   We can also characterise more unusual principles as functions $(X\to S)\to P\left(X\right)$. For example, consider the principle that says a social planner should choose an option if most other options would've led to the same social outcome. Intuitively, this captures some notion of impact minimisation. Formally, this principle is characterised by the function $\Psi \left(f\right):=\{x\in X\mid \#[x{]}_f>\frac{\#X}{2}\}$, where $\#S$ denote the cardinality of a set and $[x{]}_f\subseteq X$ denotes the $f$-equivalency class of $x\in X$, i.e. $[x{]}_f=\{y\in X\mid f\left(x\right)=f\left(y\right)\}$.

These examples illustrate the diversity of conceivable social choice principles. The key point is that they can all be represented by functions $\Psi :(X\to S)\to P\left(X\right)$. I've found this a productive way to think about principles of decision-making, and agency more generally.^[3] Finding compelling social choice principle is the central problem in social ethics, and different normative frameworks will propose different principles.

3. Two strategies for specifying principles

3.1. Utilitarian principles

Utilitarianism and aggregativism are two strategies for specifying a social choice principle $\Psi :(X\to S)\to P\left(X\right)$. The utilitarian strategy specifies a social choice principle using two components:

1. A social utility function $u:S\to R$ that assigns a real-valued utility $u\left(s\right)\in R$ to each social outcome $s\in S$.
2. The ${\text{argmax}}_X$ operator, which maps a real-valued function $r:X\to R$ to the set of points that maximize it. Formally, ${\text{argmax}}_X\left(r\right):=\{x\in X\mid \forall x^{\prime }\in X:r(x^{\prime })\le r(x\left)\right\}$. Note that ${\text{argmax}}_X\left(r\right)$ is a subset of $X$, possibly containing multiple points in case of ties, or no points in the case of unbounded functions.

Given the social utility function $u:S\to R$ and the operator ${\text{argmax}}_X:(X\to R)\to P\left(X\right)$, the utilitarian principle is defined by $\Psi \left(f\right):={\text{argmax}}_X(u\circ f)$. Note that if $f:X\to S$ is the social context, then the composition $u\circ f:X\to R$ calculates the social utility resulting from each option, thereby providing a real-valued function $r:X\to R$. The utilitarian principle $f\mapsto {\text{argmax}}_X(u\circ f)$ says that the social planner should choose an option that maximizes this function.

As a simplistic example, consider a social utility function $u:S\to R$ that measures the gross world product of a social outcome. The resulting utilitarian principle $f\mapsto {\text{argmax}}_X(u\circ f)$ would oblige maximizing gross world product. In practice, utilitarians typically endorse more nuanced utility functions that account for factors like individual well-being, fairness, and existential risk.

3.2. Aggregative principles

Aggregativism offers an alternative strategy to specifying social choice principle. Like utilitarianism, it defines the principle $\Psi :(X\to S)\to P\left(X\right)$ using two components:

1. A function $\zeta :S\to P$ that assigns a personal outcome $\zeta \left(s\right)\in P$ to each social outcome $s\in S$. We call $\zeta$ the social zeta function.
2. A model of a self-interested human, characterised by a function $\Pi :(X\to P)\to P\left(X\right)$, explained below.

The function $\Pi :(X\to P)\to P\left(X\right)$ should model a self-interested human in the following sense: for each personal context $g:X\to P$ the subset $\Pi \left(f\right)\subseteq X$ should contain the options that the hypothetical human might choose in that context. A personal context $g:X\to P$ is an assignment of a personal outcome to each of the options, analogously to a social context. For example, if $g:X\to P$ maps some options to finding a dollar and the remaining options to drowning in a swamp, then presumably $\Pi \left(g\right)$ contains only the former options.

Given the social zeta function $\zeta :S\to P$ and a model of self-interested human $\Pi :(X\to P)\to P\left(X\right)$, the aggregationist principle is defined by $\Psi \left(f\right):=\Pi (\zeta \circ f).$ Note that if $f:X\to S$ is the social context, then the composition $\zeta \circ f:X\to P$ calculates the hypothetical personal outcome resulting from each option, thereby providing a personal context $g:X\to P$. The aggregative principle $f\mapsto \Pi (\zeta \circ f)$ says that the social planner should choose an option a self-interested human might choose in this personal context.

For example, consider a social zeta function $\zeta :S\to P$ that maps each social outcome $s$ to the personal outcome of living every individual's life in sequence, starting with the earliest-born humans. The resulting aggregative principle $f\mapsto \Pi (\zeta \circ f)$ obliges affecting society such that living the concatenated lives is personally desirable.

3.3. Structural similarity between the two strategies

This comparison reveals the structural similarity between utilitarianism and aggregativism. Both strategies specify the principle $\Psi$ using a two components:

1. A function mapping social outcomes to a different space, either $R$ (in the case of utilitarianism) or $P$ (in the case of aggregativism).
2. A choice principle in that different space, either maximization (in the case of utilitarianism) or a model of a self-interested human (in the case of aggregativism).

Both $\Pi$, the model of a self-interested human, and the ${\text{argmax}}_X$ operator are choice principles: $\Pi$ is a personal choice principle, it 'chooses' one the options based on their associated personal outcomes, and ${\text{argmax}}_X$ is a real choice principle, it 'chooses' one of the options based on their associated real value. (Of course, ${\text{argmax}}_X$ doesn’t literally choose anything, it’s simply a mathematical operator, but so too is $\Pi$.)

In general, for any space $R$, let's say an $R$-context is any function with type-signature $X\to R$, and an $R$-choice principle is any function with type-signature $(X\to R)\to P\left(X\right)$. That is, an $R$-choice principle $\Phi$, when provided with an $R$-context $r:X\to R$, identifies some subset $\Phi \left(f\right)\subseteq X$ of the options which are 'acceptable'.

How might one use an $R$-choice principle $\Phi$ to specify a social choice principle $\Psi$? Well, what's needed is some function $\sigma :S\to R$ from social outcomes to elements of $R$. This function $\sigma$ will extends any social context $f:X\to S$ to an $R$-context $\sigma \circ f:X\to R$, which can then be provided to the $R$-choice principle to identify the acceptable options. Formally, $\Psi :f\mapsto \Phi (\sigma \circ f)$. This is how utilitarianism and aggregativism succeed in defining social choice principles. The key difference is that utilitarianism uses real numbers while aggregativism uses personal outcomes.

4. Equivalence between aggregativism and utilitarianism

4.1. Three conditions for equivalence

Despite their differences, there are natural conditions under which the utilitarian and aggregative principles are equivalent, in the sense that a social planner is permitted to choice an option, under the utilitarian principle, if and only if they are permitted to choice the same option under the aggregative principle.

Formally, let ${\Psi }_u$ denote the utilitarian principle ${\Psi }_u:f\mapsto {\text{argmax}}_X(u\circ f)$ and let ${\Psi }_a$ denote the aggregative principle ${\Psi }_a:f\mapsto \Pi (\zeta \circ f)$; under what conditions does ${\Psi }_u\left(f\right)={\Psi }_a\left(f\right)$ for all social contexts $f:X\to S$?

In the previous article, we showed that LELO, HL, and ROI each employ social zeta functions which aggregates the personal outcomes across all individuals in the population. Formally, $\zeta \left(s\right):=\alpha \left(\gamma \right(-,s{)}^M\left(\pi \right))$ where $I$ is a fixed set of individuals; $\gamma :I\times S\to P$ is a fixed function mapping a social outcome $s\in S$ and an individual $i\in I$ to the personal outcome $\gamma (i,s)\in P$ that $i$ faces when $s$ obtains; $M$ is the monad capturing a notion of 'collection'; $\pi \in M\left(I\right)$ be a fixed collection of individuals impartially representing the population; and $\alpha :M\left(P\right)\to P$ is an $M$-algebra specifying how to aggregate collections of personal outcomes into a single personal outcome.

Supposing $\zeta$ has the general form above, and the three conditions below are satisfied, then the utilitarian principle ${\Psi }_u$ and the aggregative principle ${\Psi }_a$ are mathematically equivalent:

1. A self-interested human maximises personal utility.
   
   Formally, the first condition states that the function $\Pi :(X\to P)\to P\left(X\right)$ has the form $\Pi \left(f\right)={\text{argmax}}_X(v\circ f)$ for some personal utility function $v:P\to R$ which assigns a real-valued utility $v\left(p\right)\in R$ to each personal outcome $p\in P$. Even by itself, this condition is quite strong. It implies that if, for some personal context $f:X\to P$, two options $x_1$ and $x_2$ result in the same personal outcome, i.e. $f\left(x_1\right)=f\left(x_2\right)$, then the human might choose $x_1$ if and only if they might choose $x_2$. Hence, this condition precludes nonconsequential considerations.
   
   Let's call this condition "Humans Maximise Personal Utility" (HMPU).
    
2. Personal utility is 'rational', in a technical sense defined below.
   
   Let $\alpha :M\left(P\right)\to P$ denote an $M$-algebra on personal outcomes, describing how to aggregate a collection of personal outcomes into a single personal outcome. Let $\beta :M\left(R\right)\to R$ denote an $M$-algebra on real numbers, describing how to aggregate a collection of real numbers into a single real number. The second condition states that $v\circ \alpha =\beta \circ v^M$. Informally, this condition means that the personal utility of an aggregate of personal outcomes is the aggregate of the personal utilities of each personal outcome being aggregated. In mathematical jargon, the personal utility function $v:P\to R$ must be a homomorphism between the $M$-algebras $(P,\alpha )$ and $(R,\beta )$, which means it preserves the algebraic structure on $P$ and $R$.
   
   Let's call this condition "Rationality of Personal Utility" (RUP).
    
3. Social utility is the aggregate of personal utilities across all individual in the population.
   
   Formally, the third condition states that $u\left(s\right)=\beta \left(\right(v\circ \gamma \left)\right(-,s{)}^M\left(\pi \right))$, where $v:P\to R$ is the personal utility function introduced in HMPU, $\gamma :I\times S\to P$ is the function assigning personal outcomes to each individual in each social outcome, $\pi \in M\left(I\right)$ is the distinguished collection of individuals representing the population, and $\beta :M\left(R\right)\to R$ is the $M$-algebra describing how to aggregate a collection of real numbers into a single real number. Informally, this condition states that the social utility of a social outcome is the aggregate of the personal utilities of the personal outcomes faced by all individuals in the population.
   
   Let's call this condition "Social Utility Aggregates Personal Utilities" (SUAPU).

The aggregative principle (when our model of a self-interested human is a rational personal utility maximiser) is equivalent to the utilitarian principle (when social utility is the impartial aggregation of personal utility over each individual). The full proof is elementary and uninsightful.^[4]

Now, these three conditions are only approximately true, and they fail in systematic ways. However, the theorem will help elucidate exactly the extent to which the aggregative principle approximates the corresponding utilitarian principle. Namely, the aggregative principle will approximate the utilitarian principle to the degree that these conditions hold.

Because RPU and SUAPU depend on the specific monad $M$ under discussion, I will spell out the details for three paradigm examples: the list monad $\text{List}$ (representing finite sequences), the distribution monad $\Delta$ (representing probability distributions), and the nonempty finite powerset monad $P_f^{+}$ (representing nonempty finite sets).

5. Equivalence between LELO and longtermist total utilitarianism

The previous section proved an equivalence, under certain conditions, between aggregative principles and utilitarian principles. This section will apply that theorem to the monad $\text{List}$, which is used to formalise Live Every Life Once (LELO). We will see that LELO is equivalent to longtermist total utilitarianism.

5.2. Monoidal rationality of personal utility?

The real numbers admit a concatenation operator in the obvious way, i.e., there exists a function $\text{sum}:\text{List}\left(R\right)\to R$ defined by $\text{sum}\left(\right[r_1,\dots ,r_k\left]\right):=0+r_1+\cdots +r_k$. This is simply the well-known summation operator, which sends a list of real values to their sum.

Let's unpack RPU, which formally states that $v\circ \text{conc}=\text{sum}\circ v^{\text{List}}$. In other words, for any list of personal outcomes $[p_1,\dots ,p_n]$, we have equality between $v\circ \text{conc}\left(\right[p_1,\dots ,p_n\left]\right)$ and $0+v\left(p_1\right)+\cdots +v\left(p_n\right)$. Informally, the personal utility of a concatenated outcome equals the sum of the personal utilities of the outcomes being concatenated. This 'monoidal' rationality condition constrains how humans must value the concatenation of different personal outcomes.

In the previous article, we saw that the concatenation operator $\text{conc}:\text{List}\left(P\right)\to P$ can be equivalently presented by a binary operator $▹$ and a constant $ϵ\in P$, with the intended interpretation $p▹p^{\prime }:=\text{conc}\left(\right[p_1,p_2\left]\right)$ and $ϵ:=\text{conc}\left(\right[\left]\right)$. We can restate the RPU condition in terms of $▹$ and $ϵ$ with two equations: $v\left(ϵ\right)=0$ and $v(p▹p^{\prime })=v\left(p\right)+v\left(p^{\prime }\right)$ for all $p,p^{\prime }\in P$.

How realistic is the RPU condition? That is, supposing humans do maximise a personal utility function, how monoidally rational is it? I think this condition is approximately true, but unrealistic in several ways. I'll assume here that $p▹p^{\prime }$ is interpreted as facing $p$ and then facing $p^{\prime }$ in sequence, rather than some exotic notion of concatenation.

Firstly, RPU rules out permutation-dependent values. It precludes a personal utility function $v:P\to R$ such that $v(p_1▹p_2)\ne v(p_2▹p_1)$. Informally, RPU assumes human values must be invariant to the ordering of experiences: they cannot value saving the best till last, nor saving the worse till last. In particular, RPU assumes that humans values are time-symmetric, which seems unrealistic, as illustrated by the following examples. Compare the process of learning, i.e. ending with better beliefs than one started with, with the process of unlearning, i.e. ending with worse beliefs than one started with. Humans seem to value learning above unlearning, but such time-asymmetric values are precluded by RPU. Similarly, humans seem to value a history of improvement over a history of degradation, even if both histories are different permutations of the same list of moments, but such values are precluded by RPU.

Secondly, RPU rules out time-discounted values. Under exponential time-discounting, a common assumption in economics, the personal utility function $v:P\to R$ obeys the equation $v(p_1▹p_2)=v\left(p_1\right)+(1+\delta {)}^{-\text{duration}\left(p_1\right)}\cdot v(p_2)$. Here $\text{duration}:P\to R^{\ge 0}$ gives the duration of each outcome and $\delta >0$ is the discount rate. This discounting formula weights the first outcome $p_1$ more than the second outcome $p_2$, with the difference growing exponentially with the duration of $p_1$. For instance, let $p_1$ and $p_1^{\prime }$ be equally valuable experiences lasting different durations, like a minute of ecstasy and a week of contentment respectively. Time-discounting implies that $v(p_1▹p_2)$ depends more on $v\left(p_2\right)$ than $v(p_1^{\prime }▹p_2)$ does. However, RPU precludes this possibility, as it requires that $\delta =0$, i.e. that humans are equally concerned with all life stages, not discounting future rewards relative to present ones

Thirdly, RPU rules out path-dependent values. Informally, whether I value a future $p$ more than a future $q$ must be independent of my past experiences. But this is an unrealistic assumption about human values, as illustrated in the following examples. If $p$ denotes reading Moby Dick and $q$ denotes reading Oliver Twist, then humans seem to value $p▹p$ less than $p▹q$ but value $q▹p$ more than $q▹q$. This is because humans value reading a book higher if they haven't already read it, due to an inherent value for novelty in reading material. Alternatively, if $p$ and $q$ denote being married to two different people, then humans seem to value $p▹p$ more than $p▹q$ but value $q▹p$ less than $q▹q$. This is because humans value being married to someone for a decade higher if they've already been married to them, due to an inherent value for consistency in relationships.^[5] But RPU would precludes such path-dependent values.

5.2. Social utility sums personal utility?

Now let's unpack SUAPU, which formally states that $u\left(s\right)=\text{sum}\left(\right(v\circ \gamma (-,$