# Toy model piece #4: partial preferences, re-re-visited

## Two failed attempts

I initially defined [partial preferences](https://www.lesswrong.com/posts/CiB3myyeEhFRgmKPL/partial-preferences-and-models) in terms of foreground variables $Y$ and background variables $Z$.

Then a partial preference would be defined by $y_+$ and $y_-$ in $Y$, such that, for any $z\in Z$, the world described by $(y_+,z)$ would be better than the world described by $(y_-,z)$. The idea being that, everything else being equal (ie the same $z$), a world with $y_+$ was better than a world with $y_-$. The other assumption is that, within mental models, human preferences can be phrased as one or many binary comparisons. So if we have a partial preference like $P_1$: "I prefer a chocolate ice-cream to getting kicked in the groin", then $(y_+,z)$ and $(y_-,z)$ are otherwise identical worlds with a chocolate ice-cream and a groin-kick, respectively.

Note that in this formalism, there are two subsets of the set of worlds, $y_+ \times Z$ and $y_- \times Z$, and map $l$ between them (which just sends $(y_+,z)$ to $(y_-,z)$).

In a [later post](https://www.lesswrong.com/posts/BydQtwfN97pFwEWtW/toy-model-piece-1-partial-preferences-revisited), I realised that such a formalism can't capture seemingly simple preferences, such as $P_2$: "$n+1$ people is better than $n$ people". The problem is that that preferences like that don't talk about just two subsets of worlds, but many more.

Thus a partial preference was defined as a [preorder](https://en.wikipedia.org/wiki/Preorder). Now, a preorder is certainly rich enough to include preferences like $P_2$, but its allows for far too many different types of structures, needing a complicated energy-minimisation procedure to turn a preorder into a utility function.

This post presents another formalism for partial preferences, that keeps the initial intuition but can capture preferences like $P_2$.

## The formalism

Let $\mathcal{W}$ be the (finite) set of all worlds, seen as universes with their whole history.

Let $\mathcal{X}$ be a subset of $\mathcal{W}$, and let $l$ be an injective (one-to-one) map from $\mathcal{X}$ to $\mathcal{W}$. Define $\mathcal{Y}=l(\mathcal{X})$, the image of $l$, and $l^{-1}: \mathcal{Y} \to \mathcal{X}$ as the inverse.

Then the preference is determined by:

* For all $x\in\mathcal{X}$, $x > l(x)$.

If $\mathcal{X}$ and $\mathcal{Y}$ are disjoint, this just reproduces the original definition, with $\mathcal{X} = y_+\times Z$ and $\mathcal{Y} = y_-\times Z$.

But it also allows preferences like $P$, defining $l(x)$ as something like "the same world as $x$, but with one less person". In that case, $l$ maps some parts of $\mathcal{X}$ to itself[^cp].

[^cp]: Note that there is a similarity with [CP-nets](https://arxiv.org/pdf/1107.0023.pdf), if we consider this as expressing a preference over population size while keeping other variables constant.

Then for any element $x\in X$, we can construct its upwards and downwards chain:

* $\ldots, l^{-3}(x), l^{-2}(x), l^{-1}(x), x, l(x), l^2(x), l^3(x), \ldots$.

These chains end when they cycle: so there is an $n$ and an $m$ so that $l^{-n}(x)=l^{m}(x)$ (equivalently, $l^{m+n}(x)=x$).

If they don't cycle, the upwards chain ends when there is an $l^{-n}(x)$ which is not an element of $\mathcal{Y}$ (hence $l^{-1}$ is not defined on in), and the downward chain ends when there is an $l(x)$ which is not in $\mathcal{X}$ (and hence $l$ is not defined on it).

So, for example, for $P_1$, all the chains contain two elements only: $x$ and $l(x)$. For $P_2$, there are no cycles, and the lower chain ends when the population hits zero, while the upper chain ends when the population hits some maximal value.

## Utilities difference between clearly comparable worlds

Since the worlds of $\mathcal{X} \cup \mathcal{Y}$ decompose either into chains or cycles via $l$, there is not need for the full machinery for utilities constructed in [this post](https://www.lesswrong.com/posts/BydQtwfN97pFwEWtW/toy-model-piece-1-partial-preferences-revisited).

One thing we can define unambiguously, is the relative utility between two elements of the same chain/cycle:

* If $x$ and $y=l^n(x)$ are in the same cycle, then $U_l(x)=U_l(y)$.
* Otherwise, if $x$ and $y=l^n(x)$ are in the same chain, then $U_l(x)-U_l(y)=n$.

Currently, lets normalise these relative utilities to $\widehat{U}_l$, by normalising each chain individually; note that if every world in the chain is reachable, this is the same as the [mean-max normalisation](https://www.lesswrong.com/posts/hBJCMWELaW6MxinYW/intertheoretic-utility-comparison) on each chain:

* If $x$ and $y=l^n(x)$ are in the same cycle, then $U_l(x)=U_l(y)$.
* Otherwise, if $x$ and $y=l^n(x)$ are in the same chain with $m$ total elements in the chain, then $U_l(x)-U_l(y)=n/(m-1)$.

We we could try and extend $\widehat{U}_l$ to a global utility function which compares different chains and compares values in chains with values outside of $\mathcal{X}\cup\mathcal{Y}$. But as we [shall see in the next post](https://www.lesswrong.com/posts/ucG63yw3JgP3rW7tH/toy-model-piece-5-combining-partial-preferences), this doesn't work when combining different partial preferences.

## Interpretation of $l$

The interpretation of $l$ is something like "this is the key difference in features that causes the difference in world-rankings". So, for $P_1$, the $l$ switches out a chocolate ice-cream and substitutes a groin-kick. While for $P_2$, the $l$ simply removes one person from the world.

This means that, **locally**, we can express $\mathcal{X} \cup \mathcal{Y}$ in the same $Y \times Z$ formalism as in the [first post](https://www.lesswrong.com/posts/CiB3myyeEhFRgmKPL/partial-preferences-and-models). Here the $Z$ are the background variables, while $Y$ is a discrete variable that $l$ operates on.

We cannot necessarily express this $Y\times  Z$ product **globally**. Consider, for $P_2$, a situation where $z_0$ is an idyllic village, $z_1$ is an Earthbound human population, and $z_2$ a [star-spanning civilization](http://www.fhi.ox.ac.uk/wp-content/uploads/intergalactic-spreading.pdf) with extensive use of [human uploads](https://www.fhi.ox.ac.uk/brain-emulation-roadmap-report.pdf).

And if $Y$ denotes the number of people in each world, it's clear that $Y$ hits a low maximum for $z_0$ (thousands?), can rise much higher for $z_1$ (trillions?), and even higher for $z_2$ (need to use scientific notation). So though $(10^{20},z_2)$ makes sense, $(10^{20},z_0)$ is nonsense. So there is no global decomposition of these worlds as $Y\times Z$.