1. Introduction

1.1. Three aggregative principles

This article examines aggregative principles of social justice. These principles state that a social planner should make decisions as if they will face the aggregated personal outcomes of every individual in the population. Different conceptions of aggregation generate different aggregative principles.

Aggregative principles avoid many theoretical pitfalls of utilitarian principles. Unlike utilitarianism, aggregative principles do not require specifying a social welfare function, which is notoriously intractable. Moreover, they seem less prone to counterintuitive conclusions such as the repugnant conclusion or the violation of moral side constraints.^[1]

There are three well-known aggregative principles:

1. Live every life once (LELO)
2. Harsyani's Lottery (HL)
3. Rawls' Original Position (ROI)

By the end of this article, we will see that these three aggregative principles are instances of an vast family of similar principles.

1.2. Living Every Life Once

The idea, as articulated below by William MacAskill, is that a social planner should make decisions as if they will live out every individual's life (past, present, and future) in sequence. We will call this principle of social justice "Live Every Life Once" (LELO).^[2]

Imagine living the life of every human being who has ever existed — in order of birth. Your first life begins about 300,000 years ago in Africa. After living that life and dying, you travel back in time to be reincarnated as the second-ever person, born slightly later than the first, then the third-ever person, and so on.

[...] If you knew you were going to live all these future lives, what would you hope we do in the present? How much carbon dioxide would you want us to emit into the atmosphere? How careful would you want us to be with new technologies that could destroy, or permanently derail, your future? How much attention would you want us to give to the impact of today’s actions on the long term?

• William MacAskill (2022), "The Case for Longtermism"

MacAskill's hope is that the social planner, following LELO, would choose policies benefiting each individual because they anticipate living each individual's life, and they would avoid policies harming any individual for the same reason. For example, the social planner wouldn't choose to emit dangerous pollution that will harm the health of future generations, because the social planner anticipates suffering the consequences themselves, although delayed by a many millennia.

MacAskill's thought experiment bares a striking similarity to two other thought experiments in social ethics — namely, Harsanyi's Lottery and Rawls' Original Position.

1.3. Harsanyi's Lottery

The economist John C. Harsanyi offers a different principle of social justice: a social planner should make decisions as if they faced a hypothetical lottery over the personal outcomes of each individual in society. This lottery would assign a likelihood to each individual, so the social planner wouldn't be sure which individual's life they will face. For example, they may face a 20% chance of being individual A, a 35% chance of being B, and so on. The ignorance is meant to force an impartial perspective for making decisions, a feature of social justice. We will call this principle of social justice "Harsanyi's Lottery" (HL).^[3]

Harsanyi's hope is that the social planner, following HL, would choose policies benefiting each individual because there is some nonzero probability that they face that individual's life. They would also avoid policies harming any individual for the same reason. For instance, they would not choose to impoverish the majority of society for a small gain to a minority, because the expected value of the corresponding lottery of outcomes is negative.

Typically, the hypothetical lottery is taken to be uniform over all individuals in society. This uniformity assumption is crucial for ensuring impartiality: the social planner would not rationally prioritize any one individual over another if they have an equal probability of being each person.

1.4. Rawls' Original Position

The philosopher John Rawls' offers a third principle of social justice, similar to Harsanyi's Lottery.^[4] His principle states that a social planner should make decisions as if they were ignorant about which individual in society they will be. We will call this principle of social justice "Rawls' Original Position" (ROI).^[5]

Rawls' hope is that the social planner, following ROI, would choose policies benefiting each individual because they must consider the possibility that they could be any individual. They would also avoid policies harming any individual for the same reason. For example, the social planner wouldn't choose to torture someone, even to greatly benefit the rest of society, because the social planner must consider the possibility that they will end up being that person.

HL and ROI share obvious similarities: both principles ask the social planner to imagine themselves in a state of ignorance about which individual's personal outcome they will face. However, they understand this ignorance in different ways. Under HL, the ignorance is probabilistic, with likelihoods attached to the alternatives. By contrast, under ROI, the ignorance is possibilistic, meaning the planner considers it possible that they could be any individual, without assigning probabilities to those possibilities. This situation (i.e. having no basis for assigning probabilities to the possible alternatives) is sometimes called Knightian uncertainty. Moreover, note that HL proposes a physical mechanism by which the individual is selected, namely, a random lottery. By contrast, ROI merely states that each individual might be selected, without specifying any physical mechanism.

1.5. Structural similarities

The similarity between HL and ROI was apparent to Harsanyi and Rawls.^[6] On the other hand, HL and ROI seem, at first glance, quite distinct from LELO. Firstly, Harsanyi's and Rawls' principles both begin with a planner in a state of ignorance about which individual's personal outcome they will face, whereas LELO posits no such uncertainty: so long as the planner knows the personal outcomes of each individual and the ordering of the individual's births, then their hypothetical fate is certain. Moreover, LELO asks the social planner to contemplate an abnormal prospect, i.e. a lifetime spanning millennia, whereas HL and ROI involve prospects that actual individuals in society will face.

However, these three principles are structurally similar: LELO, HL and ROI each involve aggregating the prospects faced by the individuals into a single hypothetical prospect faced by the social planner. They differ in the mode of aggregation they employ: LELO aggregates via a concatenation, HL via a lottery, and ROI via a disjunction. This common aggregative structure appears to be underexplored in the existing literature.

I will call principles of this general form — defining social justice in terms of an aggregation of individual prospects — aggregative principles of social justice. LELO, HL, and ROI are three examples, but they do not exhaust the space of aggregative principles. In fact, for any well-defined mode of aggregation, we can generate a corresponding aggregative principle. The space of aggregative principles is large and underexplored.

The rest of the article is organized as follows. Section 2 formalises LELO, HL, and ROI in parallel, highlighting the structural similarity. Section 3 formalises the informal notion of a "mode of aggregation" with the mathematical concept of monads, and presents a full characterization of the space of aggregative principles. This is the key contribution of the article. Section 4 explores examples of the algebraic structures on personal outcomes that are necessary for the aggregative principles to be well-defined.

2. Formalising LELO, HL, and ROI

2.1. Personal and social outcomes

Each of LELO, HL, and ROI attempt to extend the planner's self-interested attitudes towards personal outcomes to moral attitudes towards social outcomes. They achieve this by assigning to each social outcome $s$ a hypothetical personal outcome $p$, and then stating that the social planner should treat $s$ as they would treat $p$. In other words, a social outcome $s$ is deemed socially desirable if the corresponding personal outcome $p$ is personally desirable.

Let $P$ be the space of personal outcomes and $S$ be the space of social outcomes. By "personal outcome", I mean a full description of the state-of-affairs for a single individual, and by "social outcome", I mean a full description of the state-of-affairs for society as a whole. Each aforementioned principle of social justice proposes a function $\zeta :S\to P$ assigning to each social outcome $s\in S$ a hypothetical personal outcome $\zeta \left(s\right)\in P$. However, they differ on the function they propose:

LELO uses the function ${\zeta }_{LELO}:S\to P$ where ${\zeta }_{LELO}\left(s\right)\in P$ is the personal outcome of facing a concatenation of the lives of the individuals facing social outcome $s\in S$. For instance, if $s$ consists of three individuals facing personal outcomes $p_1$, $p_2$, and $p_3$ then ${\zeta }_{LELO}\left(s\right)$ is the personal outcome of first facing $p_1$, then $p_2$, then $p_3$ in sequence. We'll denote this outcome by ${\zeta }_{LELO}\left(s\right)=p_1▹p_2▹p_3$.

HL uses the function ${\zeta }_{HL}:S\to P$ where ${\zeta }_{HL}\left(s\right)\in P$ is the personal outcome of facing a lottery among the personal outcomes of the individuals in social outcome $s\in S$. For instance, if $s$ consists of three individuals facing personal outcomes $p_1$, $p_2$, and $p_3$ then ${\zeta }_{HL}\left(s\right)$ is the personal outcome of facing each outcome $p_i$ with equal likelihood $\frac{1}{3}$. We'll denote this outcome by ${\zeta }_{HL}\left(s\right)=⟨p_1:\frac{1}{3}\mid p_2:\frac{1}{3}\mid p_3:\frac{1}{3}⟩$.

ROI uses the function ${\zeta }_{ROI}:S\to P$ where ${\zeta }_{ROI}\left(s\right)\in P$ is the personal outcome of possibly facing the personal outcome of any individual in social outcome $s$. For instance, if $s$ consists of three individuals facing personal outcomes $p_1$, $p_2$, and $p_3$ respectively, then ${\zeta }_{ROI}\left(s\right)$ is the personal outcome of facing either $p_1$, $p_2$, or $p_3$, but without any probabilities attached to these possibilities. We'll denote this outcome by ${\zeta }_{ROI}=p_1\oplus p_2\oplus p_3$.

It remains to define these three functions, ${\zeta }_{LELO}$, ${\zeta }_{HL}$, and ${\zeta }_{ROI}$. For simplicity, let's assume that all social outcomes share a fixed, finite population of individuals, represented by the set $I=\{i_1,\dots ,i_n\}$. This assumption could be relaxed in future work, to handle populations that vary across social outcomes. Moreover, let's assume that the personal outcome for each individual $i\in I$ is fully determined by the social outcome. Formally, there exists a function $\gamma :I\times S\to P$ such that, if the social outcome $s\in S$ obtains, then each individual $i\in I$ faces the personal outcome $\gamma (i,s)\in P$. We will treat this as a global assumption, not localised to any particular principle of social justice.

For example, $S$ might be the set of all possible physical configurations of the universe across time, while $P$ might be the set of an individual's possible health and economic outcomes. However, the precise definitions of $S$ and $P$ are not crucial for the present analysis. Conceptually, $P$ represents the domain of personal, self-interested preferences, while $S$ represents the domain over which we seek to define social or ethical preferences. In Section 4, we will give concrete examples of these two spaces.

If $P$ is already well-understood, there is a simple way to define $S$ and $\gamma$. We could define $S=P^I$ to be the space of functions from individuals to personal outcomes, and let $\gamma :I\times P^I\to P$ be the standard evaluation function, mapping the pair $(i,f)$ to $f\left(i\right)$. Intuitively, a social outcome is just a vector specifying each individual's personal outcome, and $\gamma$ simply looks up individual $i$'s outcome in this vector. However, I have opted for a more general presentation in which social outcomes are not entirely characterized by the personal outcomes of individuals. This allows for the possibility that $S$ contains information beyond just the vector of personal outcomes. Consequently, there may exist distinct social outcomes $s,s^{\prime }\in S$ such that $\gamma (i,s)=\gamma (i,s^{\prime })$ for all $i\in I$.

If we are provided with the function $\gamma :I\times S\to P$, how might we construct the target function $\zeta :S\to P$? As we will see, the key to doing this is to assume some additional algebraic structure on the space $P$, beyond it just being an abstract set. I will explain how this construction occurs in each aggregative principle, starting with Live Every Life Once.

2.1. Formalising LELO

Informally, this whole procedure can be summarized as follows: (1) The population is represented by a list of individuals. (2) Each social outcome provides a function from individuals to their personal outcomes. This function can be lifted to a function from lists of individuals to the corresponding lists of personal outcomes, and then applied to the list representing the population. Hence each social outcome provides a list of personal outcomes. (3) Any list of personal outcomes can be concatenated into a single personal outcome. (4) Therefore, each social outcome provides a single personal outcome.

I'll now spell out the details.

(1) For any set $X$, a list over $X$ is a finite sequence $[x_1,\dots ,x_n]$ where all entries $x_1,\dots ,x_n$ are elements of $X$. The set of all lists over $X$ is denoted by $\text{List}\left(X\right)$. This includes the empty list, denoted by [], or lists with repeated entries. Note that a list is more than just a set, it also imposes an ordering on the individuals.^[7]

LELO must assume that the population is represented by a distinguished list of individuals $l\in \text{List}\left(I\right)$. Typically, this list consists of all humans ordered by their birth, although alternative orderings could be considered.

(2) As discussed before, there exists a function $\gamma :I\times S\to P$ such that, if the social outcome $s\in S$ obtains, then each individual $i\in I$ faces the personal outcome $\gamma (i,s)\in P$. It follows that each social outcome $s\in S$ provides a function $\gamma (-,s):I\to P$ from individuals to their personal outcomes, where $\gamma (-,s)$ denotes the function $i\mapsto \gamma (i,s)$.

Now, any function $f:I\to P$ from individuals to their personal outcomes can be lifted to a function $f^{\text{List}}:\text{List}\left(I\right)\to \text{List}\left(P\right)$ from lists of individuals to the corresponding lists of personal outcomes. Concretely, $f^{\text{List}}$ sends a list $[i_1,\dots ,i_n]$ to the list $\left[f\right(i_1),\dots ,f(i_n\left)\right]$, by applying $f$ componentwise. This lifting operation is a general feature of lists.

Hence, each social outcome $s$ provides a list of personal outcomes $\gamma (-,s{)}^{\text{List}}(\pi )$, obtained by lifting $\gamma (-,s):I\to P$ to a function $\gamma (-,s{)}^{\text{List}}:\text{List}(I)\to \text{List}(P)$ and then applying to the distinguished list of individuals $l\in \text{List}\left(I\right)$.

(3) LELO assumes that any list of personal outcomes can be concatenated into a single personal outcome. Formally, there exists a function $\text{conc}:\text{List}\left(P\right)\to P$ which reduces any list of personal outcomes $[p_1,\dots ,p_n]\in \text{List}\left(P\right)$ into a single personal outcome $\text{conc}\left(\right[p_1,\dots ,p_n\left]\right)\in P$.

To align with MacAskill's intended interpretation, we should view $\text{conc}\left(\right[p_1,\dots ,p_n\left]\right)$ as the personal outcome of facing each $p_i$ in order, starting with $p_1$ and ending with $p_n$. Perhaps after each life $p_i$ ends, one is instantaneously transported to the beginning of the life $p_{i+1}$ with one's memories of the proceeding life wiped. The process of living a life, dying, memory wiping, and moving to the next life is repeated until the full list of outcomes is exhausted.

It is worth noting that the concatenation operator $\text{conc}:\text{List}\left(P\right)\to P$ can equivalently be presented by a binary operator $▹:P\times P\to P$ and a constant element $ϵ\in P$, provided $▹$ and $ϵ$ satisfy the monoid axioms of associativity and identity.^[8] Specifically, given $\text{conc}$, define $▹$ as $p▹p^{\prime }:=\text{conc}\left(\right[p,p^{\prime }\left]\right)$ and define $ϵ$ as $ϵ:=\text{conc}\left(\right[\left]\right)$. Conversely, given $▹$ and $ϵ$, define $\text{conc}$ as $\text{conc}\left(\right[p_1,\dots ,p_n\left]\right):=ϵ▹p_1▹p_2▹\cdots ▹p_n$, evaluating the products left-to-right.

The monoid axioms are:

• Associativity: $(p▹p^{\prime })▹p^{\prime \prime }=p▹(p^{\prime }▹p^{\prime \prime })$
• Identity: $p▹ϵ=p$ and $ϵ▹p=p$

(4) Putting it all together, each social outcome $s$ provides a single personal outcome, namely ${\zeta }_{LELO}\left(s\right):=\text{conc}\left(\gamma \right(-,s{)}^{\text{List}}\left(l\right))\in P$, obtained by applying the concatenation operator $\text{conc}:\text{List}\left(P\right)\to P$ to the list of personal outcomes $\gamma (-,s{)}^{\text{List}}(l)\in \text{List}(P)$ generated by $s$. This defines the LELO aggregation function ${\zeta }_{LELO}:S\to P$, which assigns to each social outcome the concatenated personal outcome.

To illustrate, suppose the population $I=\{i_1,\dots ,i_n\}$ is represented by the list $l=[i_1,\dots ,i_n]$, and suppose the social outcome $s$ assigns personal outcome $p_k$ to individual $i_k$ for each $k$, i.e. $\gamma (i_k,s)=p_k$. Then the concatenated personal outcome is ${\zeta }_{LELO}\left(s\right)=p_1▹\cdots ▹p_n$. As a sanity check, consider the trivial case where $I=\left\{i\right\}$ consists of a single individual. Here the population list is just $\left[i\right]\in \text{List}\left(I\right)$, and the concatenated outcome is simply ${\zeta }_{LELO}\left(s\right)=\gamma (i,s)$, i.e. the personal outcome assigned to the sole individual $i$.

The ordering of the distinguished list affects the structure of the concatenated outcome, due to the non-commutativity of the binary concatenation operator $▹$. In general, $p▹p^{\prime }$ and $p^{\prime }▹p$ yield different personal outcomes. The choice of ordering has substantive implications for the resulting principle of social justice. For example, suppose the social planner has a positive rate of time preference, i.e. they discount the value of future experiences. This is a realistic assumption about human preferences. A LELO principle using a chronological ordering of individuals (from earliest-born to latest-born) will prioritize the interests of earlier generations compared to a principle using the reverse-chronological ordering, all else being equal. More formally, suppose the social planner has a utility function $u:P\to R$ over personal outcomes and a discount factor $\beta >0$. Then the utility of a concatenated outcome $p_1▹p_2$ is given by $u(p_1▹p_2)=u\left(p_1\right)+{\beta }^{\text{duration}\left(p_1\right)}\cdot u\left(p_2\right)$ where $\text{duration}:P\to R^{\ge 0}$ maps each personal outcome to its duration. This discounting formula places more weight on the first outcome $p_1$ than the second outcome $p_2$, and the difference grows exponentially with the duration of $p_1$. Thus, the social planner's time preferences, combined with the ordering of the list, can lead to a "tyranny of the earlier" in the resulting principle of social justice.

Next, I will turn to Harsanyi's Lottery, the earliest of the three aggregative principles of social justice.

2.2. Formalising HL

The procedure is similar to LELO: (1) The population is represented by a distribution of individuals. (2) Each social outcome provides a function from individuals to their personal outcomes. This function can be lifted to a function from distributions of individuals to the corresponding distributions of personal outcomes, and then applied to the distribution representing the population. Hence each social outcome provides a distribution of personal outcomes. (3) Any distribution of personal outcomes can be interpolated into a single personal outcome. (4) Therefore, each social outcome provides a single personal outcome.

I'll now spell out the details.

(1) For any set $X$, a distribution over $X$ is a function $\pi :X\to [0,1]$ such that the support set $\text{supp}\left(\pi \right):=\{x\in X:\pi (x)>0\}$ is finite and ${\sum }_{x\in X}\pi \left(x\right)=1$. We will sometimes use the notation $⟨x_1:{\lambda }_1\mid \cdots \mid x_n:{\lambda }_n⟩$ to denote a distribution $\pi :X\to [0,1]$ satisfying $\pi \left(x\right)={\sum }_{k:x_k=x}{\lambda }_k$ for each $x\in X$. For example, $⟨x:0.1\mid x:0.3\mid y:0.6⟩$ and $⟨x:0.4\mid y:0.6⟩$ denote the same distribution $\pi :X\to [0,1]$ satisfying $\pi \left(x\right)=0.4$ and $\pi \left(y\right)=0.6$.

The set of all distributions over $X$ is denoted by $\Delta \left(X\right)$. This includes the point-mass distributions, denoted by $⟨x:1⟩$ for each $x\in X$, or uniform distributions $⟨x_1:\frac{1}{n}\mid \dots \mid x_n:\frac{1}{n}⟩$. Note that a distribution is more than just a set, it also imposes a weighting on the individuals.

HL must assume that the population is represented by a distinguished distribution over individuals $\pi \in \Delta \left(I\right)$. Typically, the distinguished distribution is taken to be uniform over the entire population. That is, if there are $n$ individuals in total, then $\pi =⟨i_1:\frac{1}{n}\mid \cdots \mid i_n:\frac{1}{n}⟩$. However, alternative weightings could be considered.

(2) As with LELO, there exists a function $\gamma :I\times S\to P$ such that, if the social outcome $s\in S$ obtains, then each individual $i\in I$ faces the personal outcome $\gamma (i,s)\in P$. It follows that each social outcome $s\in S$ provides a function $\gamma (-,s):I\to P$ from individuals to their personal outcomes, where $\gamma (-,s)$ denotes the function $i\mapsto \gamma (i,s)$.

Now, any function $f:I\to P$ from individuals to their personal outcomes can be lifted to a function $f^{\Delta }:\Delta \left(I\right)\to \Delta \left(P\right)$ from distributions of individuals to the corresponding distributions of personal outcomes. Concretely, $f^{\Delta }$ sends a distribution $\pi =⟨i_1:{\lambda }_1\mid \cdots \mid i_n:{\lambda }_n⟩$ to the distribution $\rho =⟨f\left(i_1\right):{\lambda }_1\mid \cdots \mid f\left(i_n\right):{\lambda }_n⟩$. This lifting operation is a general feature of distributions.

Hence, each social outcome $s$ provides a distribution of personal outcomes $\gamma (-,s{)}^{\Delta }(\pi )$, obtained by lifting $\gamma (-,s):I\to P$ to a function $\gamma (-,s{)}^{\Delta }:\Delta (I)\to \Delta (P)$ and then applying to the distinguished distribution of individuals $\pi \in \Delta \left(I\right)$.

(3) HL assumes that any distribution of personal outcomes can be interpolated into a single personal outcome. Formally, there exists a function $E:\Delta \left(P\right)\to P$ which reduces any distribution of personal outcomes $\rho =⟨p_1:{\lambda }_1\mid \cdots \mid p_n:{\lambda }_n⟩\in \Delta \left(P\right)$ into a single personal outcome $E\left[\rho \right]\in P$.

To align with Harsanyi's intended interpretation, we should view $E\left[\rho \right]$ as the personal outcome of facing each $p_i\in P$ with probability ${\lambda }_i$. Perhaps a random outcome $p_i$ is sampled according to the distribution $\rho$ and the individual then faces that outcome. In contrast with LELO, the individual ultimately faces only a single human lifetime.

It is worth noting that the interpolation operator $E:\Delta \left(P\right)\to P$ can equivalently be presented by a family of binary operators ${+}_{\lambda }:P\times P\to P$, one for each $\lambda \in (0,1)$, provided ${+}_{\lambda }$ satisfies the convex space axioms of idempotence, skew-commutativity, and skew-associativity.^[9] Specifically, given $E$, define ${+}_{\lambda }$ as $p{+}_{\lambda }{p}^{\prime }:=E[⟨p:\lambda \mid p^{\prime }:1-\lambda ⟩]$. Conversely, given the family of operators $\{{+}_{\lambda }{\}}_{\lambda \in (0,1)}$, we can rather clumsily define $E$ by induction on $n$:

$E[⟨p_1:{\lambda }_1\mid \cdots \mid p_n:{\lambda }_n⟩]:=E[⟨p_1:\frac{{\lambda }_1}{1-{\lambda }_n}\mid \cdots \mid p_{n-1}:\frac{{\lambda }_{n-1}}{1-{\lambda }_n}⟩]{+}_{1-{\lambda }_n}{p}_n$

The convex space axioms are:

• Idempotence: $p{+}_{\lambda }p=p$
• Skew-commutativity: $p{+}_{\lambda }{p}^{\prime }=p^{\prime }{+}_{1-\lambda }p$
• Skew-associativity: $\left(p{+}_{\lambda }{p}^{\prime }\right){+}_{\mu }{p}^{\prime \prime }=p{+}_{\lambda \cdot \mu }\left(p^{\prime }{+}_{\kappa }{p}^{\prime \prime }\right)$ whenever $\lambda \cdot (1-\mu )=(1-\lambda \cdot \mu )\cdot \kappa$

(4) Putting it all together, each social outcome $s$ provides a single personal outcome, namely ${\zeta }_{HL}\left(s\right):=E\left(\gamma \right(-,s{)}^{\Delta }\left(\pi \right))\in P$, obtained by applying the interpolation operator $E:\Delta \left(P\right)\to P$ to the distribution of personal outcomes $\gamma (-,s{)}^{\Delta }(\pi )\in \Delta (P)$ generated by $s$. This defines the HL aggregation function ${\zeta }_{HL}:S\to P$, which assigns to each social outcome the interpolated personal outcome.

To illustrate, suppose the population $I=i_1,\dots ,i_n$ is represented by the uniform distribution $\pi =⟨i_1:\frac{1}{n}\mid \cdots \mid i_n:\frac{1}{n}⟩$. Suppose further that the social outcome $s$ assigns personal outcome $p_k$ to individual $i_k$ for each $k$, i.e. $\gamma (i_k,s)=p_k$. Then the interpolated personal outcome is ${\zeta }_{HL}\left(s\right)=E[⟨p_1:\frac{1}{n}\mid \cdots \mid p_n:\frac{1}{n}⟩]$. As a sanity check, consider the trivial case where $I=i$ consists of a single individual. Here the initial distribution is the point mass $⟨i:1⟩\in \Delta \left(I\right)$ and the interpolated outcome is simply ${\zeta }_{HL}\left(s\right)=\gamma (i,s)$, i.e. the personal outcome assigned to the sole individual $i$.

The weighting of the distinguished distribution affects the structure of the interpolated outcome, because $p{+}_{\lambda }{p}^{\prime }$ and $p{+}_{{\lambda }^{\prime }}{p}^{\prime }$ typically yield different personal outcomes when $\lambda \ne {\lambda }^{\prime }$. The choice of weighting has substantive implications for the resulting principle of social justice. If the HL principle uses a non-uniform distribution then the social planner with prioritize the individuals who are assigned a greater weighting, and this favoritism towards the higher-weighted group grows as the weighting distribution becomes more uneven.

Suppose there is a fixed amount of resources $R>0$ to be distributed among the population. Furthermore, suppose the resource yields diminishing marginal returns, i.e. the social planner's utility function over resources $u:R^{\ge 0}\to R$ is strictly concave. This is a realistic assumption about human preferences. Following HL, the social planner will allocate resources to maximise the expected value of the corresponding lottery. Formally, the social planner chooses $(r_1,\dots ,r_n)$ to maximize ${\lambda }_1\cdot u\left(r_1\right)+\cdots +{\lambda }_n\cdot u\left(r_n\right)$ subject to the constraint $r_1+\cdots +r_n=R$. In the optimal allocation, $(r_1^{\ast },\dots ,r_n^{\ast })$, the marginal utility of resources is inversely proportional to an individual's weight: $u^{\prime }\left(r_i^{\ast }\right)\propto 1/{\lambda }_i$. Therefore individuals with a larger weight will receive more resources: if ${\lambda }_i<{\lambda }_j$ then $1/{\lambda }_i>1/{\lambda }_j$ so the optimality condition implies $u^{\prime }\left(r_i^{\ast }\right)>u^{\prime }\left(r_j^{\ast }\right)$ and the strict concavity implies $r_i^{\ast }<r_j^{\ast }$. In the special case where $u$ is logarithmic, the resources allocated to an individual will be directly proportional to their weight. Thus, the social planner's preferences, combined with the weights of the distribution, can lead to a "tyranny of the majority" in the resulting principle of social justice.

Finally, I will turn to Rawls' Original Position, the most famous aggregative principle of social justice.

2.3. Formalising ROI

The procedure is similar to LELO and HL: (1) The population is represented by a nonempty finite subset of individuals. (2) Each social outcome provides a function from individuals to their personal outcomes. This function can be lifted to a function from nonempty finite subsets of individuals to the corresponding nonempty finite subsets of personal outcomes, and then applied to the subset representing the population. Hence each social outcome provides a nonempty finite subset of personal outcomes. (3) Any nonempty finite subset of personal outcomes can be fused into a single personal outcome. (4) Therefore, each social outcome provides a single personal outcome.

I'll now spell out the details.

(1) ROI must assume that the population is represented by a nonempty finite subset of individuals $A\in P_f^{+}\left(I\right)$. For any set $X$, let $P_f^{+}\left(X\right)$ denote the nonempty finite subsets of $X$. This is a standard notation, where $P$ stands for powerset, the superscript ${}^{+}$ stands for nonempty and the subscript ${}_f$ stands for finite.

Note that $A$ carries no additional structure beyond being a set — unlike the list $l\in \text{List}\left(I\right)$ used in LELO, it carries no ordering, and unlike the distribution $\pi \in \Delta \left(I\right)$ used in HL, it carries no weightings. Typically, $A$ is assumed to be the universal set $I$ itself, representing all individuals. However, Rawls suggests that alternative subsets could be considered, such as the set of "Heads of Families" or "presently existing people".

(2) As with LELO and HL, there exists a function $\gamma :I\times S\to P$ such that, if the social outcome $s\in S$ obtains, then each individual $i\in I$ faces the personal outcome $\gamma (i,s)\in P$. It follows that each social outcome $s\in S$ provides a function $\gamma (-,s):I\to P$ from individuals to their personal outcomes, where $\gamma (-,s)$ denotes the function $i\mapsto \gamma (i,s)$.

Now, any function $f:I\to P$ from individuals to their personal outcomes can be lifted to a function $f^{P_f^{+}}:P_f^{+}\left(I\right)\to P_f^{+}\left(P\right)$ from nonempty finite subsets of individuals to the corresponding nonempty finite subsets of personal outcomes. Concretely, $f^{P_f^{+}}$ sends a subset $i_1,\dots ,i_n$ to the subset $\left\{f\right(i_1),\dots ,f(i_n\left)\right\}$, by applying $f$ elementwise. This lifting operation is a general feature of nonempty finite subsets.

Hence, each social outcome $s$ provides a distribution of personal outcomes $\gamma (-,s{)}^{P_f^{+}}(A)$, obtained by lifting $\gamma (-,s):I\to P$ to a function $\gamma (-,s{)}^{P_f^{+}}:P_f^{+}(I)\to P_f^{+}(P)$ and then applying to the distinguished subset of individuals $A\in P_f^{+}\left(I\right)$.

(3) ROI assumes that any nonempty finite subset of personal outcomes can be fused into a single personal outcome. Formally, there exists a function $⨁:P_f^{+}\left(P\right)\to P$ which reduces any nonempty finite subset of personal outcomes $p_1,\dots ,p_n\in P_f^{+}\left(P\right)$ into a single personal outcome $⨁\left(\right\{p_1,\dots ,p_n\left\}\right)\in P$.

To obtain Rawls' principle of social justice, we should interpret $⨁\left(\right\{p_1,\dots ,p_n\left\}\right)$ as the personal outcome where one might face any of the outcomes $p_1,\dots ,p_n$, but without any information about which outcome is more likely. That is, the fusion operator acts like a disjunction between the personal outcomes — for example, if $p_1$ is the outcome of eating vanilla ice cream and $p_2$ is the outcome of eating chocolate ice cream, then $p_1\oplus p_2$ is the outcome of eating either vanilla or chocolate ice cream, with no probabilities attached. One could imagine that the exact prospect is selected by a third-party, maybe an adversary who selects the worse option or a benefactor who selects the best option.

It is worth noting that the fusion operator $⨁:P_f^{+}\left(P\right)\to P$ can equivalently be presented by a binary operator $\oplus :P\times P\to P$, provided $\oplus$ satisfies the axioms of a semilattice.^[10] Specifically, given $⨁$, define $\oplus$ as $p\oplus p^{\prime }:=⨁\left(\right\{p,p^{\prime }\left\}\right)$. Conversely, given $\oplus$, define $⨁$ as $⨁\left(\right\{p_1,\dots ,p_n\left\}\right)=p_1\oplus \cdots \oplus p_n$.

The semilattice axioms are:

• Idempotence: $p\oplus p=p$
• Commutativity: $p\oplus p^{\prime }=p^{\prime }\oplus p$
• Associativity: $(p\oplus p^{\prime })\oplus p^{\prime \prime }=p\oplus (p^{\prime }\oplus p^{\prime \prime })$

(4) Putting it all together, each social outcome $s$ provides a single fused personal outcome, namely ${\zeta }_{ROI}\left(s\right):=⨁\left(\gamma \right(-,s{)}^{P_f^{+}}\left(A\right))\in P$, obtained by applying the fusion operator $⨁:P_f^{+}\left(P\right)\to P$ to the nonempty finite subset of personal outcomes $\gamma (-,s{)}^{P_f^{+}}(A)\in P_f^{+}(P)$ generated by $s$. This defines the ROI aggregation function ${\zeta }_{ROI}:S\to P$, which assigns to each social outcome the fused personal outcome.

To illustrate, suppose the population $I=i_1,\dots ,i_n$ is represented by the universal subset $A=\{i_1,\dots ,i_n\}$. Suppose further that the social outcome $s$ assigns personal outcome $p_k$ to individual $i_k$ for each $k$, i.e. $\gamma (i_k,s)=p_k$. Then the fused personal outcome is ${\zeta }_{ROI}\left(s\right)=p_1\oplus \cdots \oplus p_n$. As a sanity check, consider the trivial case where $I=i$ consists of a single individual. Here the population subset is the singleton $A=\left\{i\right\}\in P_f^{+}\left(I\right)$, and the fused outcome is simply ${\zeta }_{ROI}\left(s\right)=\gamma (i,s)$, i.e. the personal outcome assigned to the sole individual $i$.

The choice of the distinguished subset $A$ affects the structure of the concatenated outcome. For example, suppose the social planner is pessimistic, evaluating the fused outcome $p_1\oplus p_2$ as no better than the worst of the individual outcomes $p_1$ and $p_2$. Formally, if the planner's preferences are represented by a utility function $u:P\to R$, this means assuming $u(p_1\oplus p_2)=min\left\{u\right(p_1),u(p_2\left)\right\}$. This is a realistic assumption of decision-making under Knightian uncertainty, where the planner considers the worst-case scenario.^[11] Under this assumption, the resulting ROI principle will be sensitive to the worst-off individuals in the population subset $A$, leading to a 'tyranny of the unfortunate'.

Unlike HL, ROI is scope-insensitive due to the idempotence of the fusion operation $\oplus$, meaning $p\oplus p=p$. This implies that the fused outcome is insensitive to the number of individuals facing each personal outcome, and depends only on which personal outcomes are faced at all. For a stark illustration, suppose that a social outcome contains 100 individuals facing great wealth ($p$) and 1 facing abject poverty ($p^{\prime }$). ROI yields the same fused outcome $p\oplus p^{\prime }$ as a social outcome with 1 individual facing wealth and 100 facing poverty. The drastically different proportions of individuals are irrelevant; only the presence or absence of each outcome matters.

2.4. Analysis

LELO, HL, and ROI share a common structure, differing only in the specific mathematical objects used. They represent populations using some type of collection: lists for LELO, distributions for HL, and subsets for ROI. And they aggregate personal outcomes using some mode of aggregation: concatenation for LELO, interpolation for HL, and fusion for ROI. This suggest that LELO, HL, and ROI are instances of a general family of aggregative principles, obtained by varying the type collection and mode of aggregation. In the next section, I will show that this is true.

3. Monads and aggregative principles

The key difference between LELO, HL, and ROI lies in their mode of aggregation. In Section 3, we will formalise this informal notion of a "mode of aggregation", and thereby find the general family of aggregative principles.

3.1. Monads formalise collections

The concept of a monad originates in category theory, and has found extensive applications in functional programming languages like Haskell. While category theory lies beyond the scope of this article, monads can be understood concretely as formalising the notion of a "collection". The core idea is that monads allow for operations on the elements to be lifted to operations of the collections, in a way that preserves certain intuitive properties.

Formally, a monad $M$ consists of four components:

1. $M$ assigns to each set $X$ another set, denoted $M\left(X\right)$, which we interpret as the collections over $X$. This is called the construct operator.
2. $M$ assigns to each function $f:X\to Y$ between sets to another function, denoted $f^M:M\left(X\right)\to M\left(Y\right)$, between their corresponding collections. This lifting operation formalizes the idea that we can apply a function to each element within a collection independently. For example, if $f$ maps each child to their birthday, then $f^M$ maps each collection of children to the corresponding collection of birthdays. This is called the lift operator.
3. $\eta$ assigns to each set $X$ a function ${\eta }_X:X\to M\left(X\right)$. This encodes the idea that each element $x\in X$ can be viewed as a "trivial" or "singleton" collection ${\eta }_X\left(x\right)\in M\left(X\right)$. This is called the unit operator.
4. $\mu$ assigns to each set $X$ a function ${\mu }_X:M\left(M\right(X\left)\right)\to M\left(X\right)$. Intuitively, ${\mu }_X$ takes a 'collection of collections' $m\in M\left(M\right(X\left)\right)$ and 'flattens' it into a single collection ${\mu }_X\left(m\right)\in M\left(X\right)$. This is called the multiplication operator.

These components must satisfy certain coherence conditions, known as the monad laws:

• Associativity: ${\mu }_X\circ {\mu }_X^M={\mu }_X\circ {\mu }_{M\left(X\right)}$ for all objects $X$. This ensures that flattening a collection of collections of collections is the same, regardless of the order in which we do the flattening.
• Left unit: ${\mu }_X\circ {\eta }_{M\left(X\right)}=1_{M\left(X\right)}$ for all objects $X$. This ensures that wrapping a collection in a singleton and then flattening is the same as doing nothing.
• Right unit: ${\mu }_X\circ {\eta }_X^M=1_{M\left(X\right)}$ for all objects $X$. This ensures that mapping each element of a collection to a singleton and then flattening is the same as doing nothing.

For the full technical details, see Mac Lane (1971) "Categories for the Working Mathematician".

The three types of collections we've encountered so far — lists, distributions, and nonempty subsets — are formalised by monads.

For example, the list monad $\text{List}$ has these four components:^[12]

1. Construct operator $\text{List}\left(X\right)$, the set of finite lists with elements in $X$.
2. Lift operator, $f^M\left(\right[x_1,\dots ,x_n\left]\right)=\left[f\right(x_1),\dots ,f(x_n\left)\right]$, which applies $f$ to each component.
3. Unit operator ${\eta }_X\left(x\right)=\left[x\right]$ which creates singleton lists,
4. Multiplication operator ${\mu }_X\left(\right[x_1^{\left(1\right)}\dots ,x_{m_1}^{\left(1\right)}],\dots ,[x_1^{\left(n\right)},\dots ,x_{m_n}^{\left(n\right)}\left]\right)=[x_1^{\left(1\right)},\dots ,x_{m_1}^{\left(1\right)},\dots ,x_1^{\left(n\right)},\dots ,x_{m_n}^{\left(n\right)}]$ which concatenates a list of lists into a single list.

And the distribution monad $\Delta$ has these four components:^[13]

1. Construct operator $\Delta \left(X\right)$, the set of probability distributions over $X$, i.e. functions $\pi :X\to [0,1]$ such that ${\sum }_{x\in X}\pi \left(x\right)=1$.
2. Lift operator $f^{\Delta }\left(\pi \right):y\mapsto {\sum }_{x:f\left(x\right)=y}\pi \left(x\right)$, which takes a distribution $\pi$ on $X$ and returns the pushed-forward distribution on $Y$, i.e. the distribution of $f\left(x\right)$ when $x$ is sampled from $\pi$. Intuitively, it marginalizes out the randomness in $X$.
3. Unit ${\eta }_X\left(x\right):x^{\prime }\mapsto \{\begin{array}{cc}1& \text{if}x=x^{\prime }\\ 0& \text{otherwise}\end{array}$, which creates a point-mass distribution at $x$, i.e. the distribution that always returns $x$ with probability 1.
4. Multiplication ${\mu }_X\left(\Phi \right):x\mapsto {\sum }_{\pi \in \Delta \left(X\right)}\Phi \left(\pi \right)\cdot \pi \left(x\right)$, which takes a distribution $\Phi$ over distributions on $X$, and returns the average of those distributions weighted by $\Phi$. Intuitively, it collapses a two-stage sampling process (first sample a distribution $\pi$ from $\Phi$, then sample an element $x$ from $\pi$) into a single-stage sampling process.

And finally, the nonempty powerset monad $P_f^{+}$ has these four components:

1. Type constructor $P_f^{+}\left(X\right)=\{A\subseteq X:A\text{is finite and nonempty}\}$, the set of finite nonempty subsets of $X$.
2. Lift operation $f^{P_f^{+}}\left(A\right)=\left\{f\right(x):x\in A\}$, which applies $f$ to each element of the set $A$.
3. Unit ${\eta }_X\left(x\right)=\left\{x\right\}$, which creates a singleton set containing $x$.
4. Multiplication ${\mu }_X\left(A\right)={\bigcup }_{A\in A}A$, which takes a set of sets and returns their union.

Whenever you encounter an informal concept of a collection, it will typically be formalizable as a monad. Let's take the finite multiset: intuitively, a multiset is a collection that allows multiple instances of each element, but where the order doesn't matter. Formally, for any set $X$, a finite multiset on $X$ is a function $\pi :X\to N$, where $\pi \left(x\right)$ represents the number of occurrences of element $x$. The multiset is finite if there are finitely many $x\in X$ with $\pi \left(x\right)>0$. The set of all finite multisets on $X$ is denoted $N\left[X\right]$. Now, elements of $N[-]$ are intuitively collections over $X$. Sure enough, the assignment $X\mapsto N\left[X\right]$ is a monad, which we call the finite multiset monad $N[-]$. The definitions of $f^{N[-]}$, $\eta$, and $\mu$ are similar to those for $\Delta$.^[14]

3.2. Algebras formalise aggregations.

Algebraic structures are ubiquitous in mathematics: monoids, groups, rings, vector spaces, lattices, and so on. Informally, an algebraic structure is a set equipped with some operations (like addition, multiplication, etc.) that satisfy certain axioms (like associativity, commutativity, etc.). A core insight from category theory is that each type of algebraic structure corresponds to a monad.

As discussed earlier, each monad $M$ captures a general notion of a "collection" of elements. An algebra of $M$ is a way to aggregate any collection of those elements into a single element. Formally, given a monad $(M,\eta ,\mu )$, an $M$-algebra is a set $X$ equipped with a function $\alpha :M\left(X\right)\to X$ satisfying two laws:

1. $\alpha \circ {\eta }_X={\text{id}}_X$ (unit law)
2. $\alpha \circ {\mu }_X=\alpha \circ {\alpha }^M$ (associativity law)

Intuitively, the unit law says that aggregating a singleton collection ${\eta }_X\left(x\right)\in M\left(X\right)$ should just return the element $x\in X$ itself. The associativity law says that aggregating a collection of collections $m\in M\left(M\right(X\left)\right)$ can be done in two equivalent ways: first flatten the nested collections using ${\mu }_X$ and then aggregate the resulting collection using $\alpha$; or first aggregate each inner collection using $\alpha$ (this is what ${\alpha }^M$ does), and then aggregate the resulting outer collection using $\alpha$ again.

• For example, an algebra for the $\text{List}$ monad is a set $X$ equipped with an operator $\text{conc}:\text{List}\left(X\right)\to X$ specifying how to aggregate any list of elements into a single element. A $\text{List}$ algebra $(X,\text{conc})$ is called a monoid.
• Similarly, an algebra for the $\Delta$ monad is a set $X$ equipped with an operator $E:\Delta \left(X\right)\to X$ specifying how to aggregate any distribution of element into a single element. A $\Delta$ algebra $(X,E)$ is called a convex space.
• Finally, an algebra for the $P_f^{+}$ monad is a set $X$ equipped with an operator $⨁:P_f^{+}\left(X\right)\to X$ specifying how to aggregate any nonempty finite subset of element into a single element. A $P_f^{+}$ algebra $(X,⨁)$ is called a semilattice.

Each algebraic structure corresponds to a monad. For example, consider the most important algebraic structure: the vector space. The relevant monad $M$ assigns to each set $X$ the set $V\left(X\right)$ of functions $v:X\to R$ with $v\left(x\right)\ne 0$ for only finitely many $x\in X$. For example, if $X$ is the set $\{\text{milk},\text{eggs},\text{sugar}\}$ then a typical element of $V\left(X\right)$ might look like $2\cdot \text{milk}+1\cdot \text{eggs}-3\cdot \text{sugar}$. An algebra for the monad $V\left(X\right)$ is precisely a vector space: a set $X$ equipped with a function $\alpha :V\left(X\right)\to X$ satisfying the appropriate unit and associativity laws. This definition captures the essence of a vector space — the ability to aggregate arbitrary linear combinations — with a single operation $\alpha :V\left(X\right)\to X$.

3.3. A general aggregative principle

As promised, we can now formulate a general family of aggregative principles using the language of monads and algebras. Each principle has the following form: a social planner should make decisions as if they will face the aggregate of the personal outcomes across all individuals in the population.

Informally, this whole procedure can be summarized as follows: (1) The population is represented by a distinguished collection of individuals. (2) Each social outcome provides a function from individuals to their personal outcomes. This function can be lifted to a function from collections of individuals to the corresponding collections of personal outcomes, and then applied to the collection representing the population. Hence each social outcome provides a collection of personal outcomes. (3) Any collection of personal outcomes can be aggregated into a single personal outcome. (4) Therefore, each social outcome provides a single personal outcome.

(1) Let $M$ be any monad, assigning to every set $X$ another set $M\left(X\right)$ of collections over $X$. We must assume that the population is represented by a distinguished collection $i\in M\left(I\right)$. Typically, $i$ is chosen to represent the entire population impartially, although non-impartial collections could also be considered.

(2) As discussed before, there exists a function $\gamma :I\times S\to P$ such that, if the social outcome $s\in S$ obtains, then each individual $i\in I$ faces the personal outcome $\gamma (i,s)\in P$. It follows that each social outcome $s\in S$ provides a function $\gamma (-,s):I\to P$ from individuals to their personal outcomes, where $\gamma (-,s)$ denotes the function $i\mapsto \gamma (i,s)$.

Now, any function $f:I\to P$ from individuals to their personal outcomes can be lifted to a function $f^M:M\left(I\right)\to M\left(P\right)$ from collections of individuals to the corresponding collections of personal outcomes. This lifting operation is a general feature of monads.

Hence, each social outcome $s$ provides a collection of personal outcomes $\gamma (-,s{)}^M(i)$, obtained by lifting $\gamma (-,s):I\to P$ to a function $\gamma (-,s{)}^M:M(I)\to M(P)$ and then applying to the distinguished collection of individuals $i\in M\left(I\right)$.

(3) We assume that any collection of personal outcomes can be aggregated into a single personal outcome. Formally, there exists a function $\alpha :M\left(P\right)\to P$ which reduces any collection of personal outcomes $p\in M\left(P\right)$ into a single personal outcome $\alpha \left(p\right)\in P$.

A key requirement for obtaining a normatively compelling principle of social justice is that the aggregation function $\alpha :M\left(P\right)\to P$ is "monotonic". That is, aggregating more desirable personal outcomes should yield a more desirable result than aggregating less desirable personal outcomes, as judged by the the self-interested social planner. This feature incentivizes the social planner to choose policies that benefit individuals in society, and to avoid policies that harm individuals, all else being equal.

(4) Putting it all together, each social outcome $s$ provides a single aggregated personal outcome, namely ${\zeta }_{M,\alpha ,i}\left(s\right):=\alpha \left(\gamma \right(-,s{)}^M\left(i\right))\in P$, obtained by applying the aggregation operator $\alpha :M\left(P\right)\to P$ to the collection of personal outcomes $\gamma (-,s{)}^M(i)\in M(P)$ generated by $s$. This defines the general aggregation function ${\zeta }_{M,\alpha ,i}:S\to P$, which assigns to each social outcome the aggregated personal outcome.

As a sanity check, consider the trivial case where $I=\left\{i\right\}$ consists of a single individual. Here the population collection is the singleton $i={\eta }_I\left(i\right)\in M\left(I\right)$, and the aggregated outcome is simply ${\zeta }_{M,\alpha ,i}\left(s\right)=\gamma (i,s)$, i.e. the personal outcome assigned to the sole individual $i$.

By varying the monad $M$, the distinguished collection $i$, and the aggregation function $\alpha$, one can capture a wide range of principles, including LELO, HL, and ROI as special cases.

4. Algebraic structures on personal outcomes

As we can see, the algebraic structures that exist on the personal outcomes constrain which aggregative principles are well-defined. In particular, the monad $M$ and aggregation function $\alpha$ must be compatible, in the sense that $\alpha$ defines an $M$-algebra on the set $P$ of personal outcomes. In this section, we will explore some concrete examples of algebraic structures on personal outcomes — including monoids, convex spaces, and semilattices, which are required for LELO, HL, and ROI respectively. Some of these examples will be exotic, thereby generating novel aggregative principles of social justice.

This section is not intended to be exhaustive. Indeed, there are countless possible algebraic structures one could consider, and the choice of algebraic structure will depend on the phenomena under investigation.

4.1. Personal outcomes as monoid

How might we model personal outcomes such that they form a monoid, as required by LELO? Recall that LELO requires a concatenation operator $\text{conc}:\text{List}\left(P\right)\to P$. Equivalently, we seek a binary operator $▹:P\times P\to P$ and a constant element $ϵ\in P$ satisfying the axioms of a monoid, as discussed in section 3.1.

Example 1

The simplest way to model personal outcomes as a monoid is for each personal outcome $p$ to be list over a fixed alphabet $A$, i.e. $P:=\Delta \left(A\right)$. We can think of elements of $A$ as the discrete moments which constitute a human life. For example, $A$ might be the set of minute-long experiences — then a human life of 80 years would be modelled as a list of 42 million elements from $A$.

Indeed $P:=\text{List}\left(A\right)$ has a monoid structure. In fact, this is the free monoid over $A$, meaning it is the 'most general' or 'least constrained' monoid containing $A$. The monoid operation $▹$ is given by concatenation of lists: if $p=[a_1,\dots ,a_n]$ and $p^{\prime }=[a_1^{\prime },\dots ,a_{n^{\prime }}^{\prime }]$ are two lists, then $p▹p^{\prime }=[a_1,\dots ,a_n,a_1^{\prime },\dots ,a_{n^{\prime }}^{\prime }]$. The identity element $ϵ$ is the empty list $\left[\right]$. This is the simplest type of monoid, and thus the natural starting point for modeling personal outcomes in the context of LELO.

Example 2

Alternatively, we can model personal outcomes in a more continuous way. Suppose each personal outcome $p$ is a pair $(d,f)$ where:

• $d\in R_{\ge 0}$ is a duration.
• $f:(0,d]\to A$ is a trajectory, assigning to each moment in time $0<t\le d$ an instantaneous experience $f\left(t\right)\in A$. Here $(0,d]$ denotes the left-open, right-closed real interval of length $d$, and $A$ is some fixed set of possible instantaneous experiences. We might use this model if we want to track variables that change continuously over time, such as an individual's location.

We can define a monoid operation $▹$ on $P$ by concatenating durations and 'switching' between trajectories. Formally, for $p=(d,f)$ and $p^{\prime }=(d^{\prime },f^{\prime })$, we define $p▹p^{\prime }$ to be the pair $(d+d^{\prime },\tilde{f})$ where $\tilde{f}:(0,d+d^{\prime }]\to A$ is given by $\tilde{f}\left(t\right)=\{\begin{array}{cc}f\left(t\right)& \text{if}t\le d\\ f^{\prime }(t-d)& \text{else}\end{array}$. The identity element $ϵ\in P$ is the pair $(0,{!}_A)$ where ${!}_A:(0,0]\to A$ is the empty function to $A$.^[15]

We could also restrict the trajectories $f:(0,d]\to A$ to be piecewise smooth, piecewise continuous, piecewise constant, or to satisfy any other reasonable piecewise condition. $P$ remains a monoid under these restrictions, because the concatenation of piecewise smooth (resp. continuous, constant) functions is again piecewise smooth (resp. continuous, constant).

Example 3

In the previous two examples, we've modelled personal outcomes as predetermined trajectories through some space of experiences, either discrete or continuous. However, these models assume that an individual's life trajectory is fixed in advance, which is often unrealistic. In reality, individuals make choices that shape the course of their lives over time. To capture this agency, we can model personal outcomes as environments that are actively guided by the individual's actions.

Suppose we model a personal outcome as an interactive environment consisting of:

• A set $A$ of actions the individual can produce.
• A set $O$ of observations the individual can receive.
• A function $\tau :A\to \Delta \left(O\right)$ assigning to each action a probability distribution over observations.

We can define a monoid operation on the set $P$ by running the two environments $p$ and $p^{\prime }$ in parallel, where the individual simultaneously chooses actions and receives observations in both environments. Concretely, given $p=(A,O,\tau )$ and $p^{\prime }=(A^{\prime },O^{\prime },{\tau }^{\prime })$, we define their product $p▹p^{\prime }$ to be the environment $(A\times A^{\prime },O\times O^{\prime },\tau \otimes {\tau }^{\prime })$ where:

• The set of actions is the Cartesian product $A\times A^{\prime }$
• The set of observations is the Cartesian product $O\times O^{\prime }$
• If $\tau :A\to \Delta \left(O\right)$ and ${\tau }^{\prime }:A^{\prime }\to \Delta \left(O^{\prime }\right)$ are transition functions, considered as functions $\tau :A\times O\to [0,1]$ and ${\tau }^{\prime }:A^{\prime }\times O^{\prime }\to [0,1]$, then $\tau \otimes {\tau }^{\prime }$ is their product, i.e. $\tau \otimes {\tau }^{\prime }:A\times A^{\prime }\to \Delta (O\times O^{\prime })$ is defined by $\tau (a,a^{\prime },o,o^{\prime })=\tau (a,o)\cdot {\tau }^{\prime }(a^{\prime },o^{\prime })$. Intuitively, $\tau \otimes {\tau }^{\prime }$ receives a pair of actions $(a,a^{\prime })$ and produces a pair observations $(o,o^{\prime })$ by independently sampling $o$ from $\tau \left(a\right)$ and $o^{\prime }$ from ${\tau }^{\prime }\left(a^{\prime }\right)$.