My previous approach to combining preferences went like this: from the partial preferences $P_i$, create a normalised utility function ${\hat{U}}_i$ that is defined over all worlds (and which is indifferent to the information that didn't appear in the partial model). Then simply add these utilities, weighted according to the weight/strength of the preference.

But this method fails. Consider for example the following partial preferences, all weighted with the same weight of $1$:

• $P_1:A>C$.
• $P_2:A>B$.
• $P_3:B>C$.
• $P_4:B>D$.
• $P_5:B>E$.

If we follow the standard normalisation approach, then the normalised utility ${\hat{U}}_1$ will be defined^[1] as:

• ${\hat{U}}_1\left(A\right)=0.5$, ${\hat{U}}_1\left(C\right)=-0.5$, and otherwise ${\hat{U}}_1(-)=0$.

Then adding together all five utility functions would give:

• $U\left(\begin{array}{c}A\\ B\\ C\\ D\\ E\end{array}\right)=\left(\begin{array}{c}1\\ 1\\ -1\\ -0.5\\ -0.5\end{array}\right)$.

There are several problems with this utility. Firstly, the utility of $A$ and the utility of $B$ are the same, even though in the only case where there is a direct comparison between them, $A$ is ranked higher. We might say that we are missing the comparisons between $A$ and $D$ and $E$, and could elicit these preferences using one-step hypotheticals. But what if comparing $A$ to $D$ is a complex preference, and all that happens is that the agent combines $A>B$ and $B>D$? If we added another partial preference that said $B>F$, then $B$ would end up ranked above $A$!

Another, more subtle point, is that the difference between $A$ and $C$ is too large. Simply having $A>B$ and $B>C$ would give $U\left(A\right)-U\left(C\right)=1$. Adding in $A>C$ moves this difference to $2$. But note that $A>C$ is already implicit in $A>B$ and $B>C$, so adding it shouldn't make the difference larger.

In fact, if the difference in utility between $A$ and $C$ were larger than $1$, adding in $A>C$ should make the difference between $U\left(A\right)$ and $U\left(C\right)$ smaller: because having $A>C$ weighted at $1$ means that the agent's preference of $A$ over $C$ is not that strong.

Energy minimising between utilities

So, how should we combine these preferences otherwise? Well, if I have a preference $P_i$, of weight $w_i$, that ranks outcome $G$ below outcome $H$ (write this as $G{<}_{i}H$), then, if these outcomes appear nowhere else in any partial preference, $U\left(G\right)-U\left(H\right)$ will be $w_i$.

So in a sense, that partial preference is trying to set the distance between those two outcomes to $w_i$. Call this the energy-minimising condition for $P_i$.

Then for a utility function $U$, we can define the energy of $U$, as compared with the (partially defined) normalised utility ${\hat{U}}_i$ corresponding to $P_i$. It is:

• ${\sum }_{G{<}_{i}H}(w_i\left({\hat{U}}_i\right(H)-{\hat{U}}_i(G\left)\right)-\left(U\right(H)-U(G\left)\right){)}^2$.

This is the difference between the weighted distance between the outcomes that $w_i{\hat{U}}_i$, and the one that $U$ actually gives.

Because different partial preferences have different number of elements to compare, we can compute the average energy of $U$:

• $E(U,P_i)=\frac{{\sum }_{G{<}_{i}H}(w_i\left({\hat{U}}_i\right(H)-{\hat{U}}_i(G\left)\right)-\left(U\right(H)-U(G\left)\right){)}^2}{{\sum }_{G{<}_{i}H}1}$.

Global energy minimising condition

But weights have another role to play here; they measure not only how much $H$ is preferred to $G$, but how important it is to reach that preference. So, for humans, "$G<H$ with weight $ϵ$" means both:

• $H$ is not much preferred to $G$.
• The humans isn't too fussed about the ordering of $G$ and $H$.

For general agents, these two could be separate phenomena; but for humans, they generally seem to be the same thing. So we can reuse the weights to compute the global energy for $U$ as compared to all partial preferences, which is just the weighted sum of its average energy for each partial preference:

• $E(U,\{P_i\left\}\right)={\sum }_{P_i}{w}_{i}E(U,P_i)={\sum }_{P_i}{w}_i\frac{{\sum }_{G{<}_{i}H}(w_i\left({\hat{U}}_i\right(H)-{\hat{U}}_i(G\left)\right)-\left(U\right(H)-U(G\left)\right){)}^2}{{\sum }_{G{<}_{i}H}1}$.

Then the actual ideal $U$ is defined to be the $U$ that minimises this energy term.

Solutions

Now, it's clear this expression is convex. But it need not be strictly convex (which would imply a single solution): for example, if $P_1$ ($A>C$) and $P_4$ ($B>D$) were the only partial preferences, then there would be no conditions on the relative utilities of $\{A,C\}$, $\{B,D\}$ and $\left\{E\right\}$.

Say that $H$ is linked to $G$, by defining a link as "there exists a $P_i$ with $G{\le }_{i}H$ or $H{\le }_{i}G$", and then making this definition transitive and reflexive (it's automatically symmetric). In the example above, with $P_i$, $1\le i\le 5$, all of $\{A,B,C,D,E\}$ are linked.

Being linked is an equivalence relation. And within a class of linked worlds, if we fix the utility of one world, then the energy minimisation equation becomes strictly convex (and hence has a single solution). Thus, within a class of linked worlds, the energy minimisation equation has a single solution, up to translation.

So if we want a single $U$, translate the solution for each linked class so that the average utility in that class is equal to the average of every other linked class. And this would then define $U$ uniquely (up to translation).

For example, if we only had $P_1$ ($A>C$) and $P_4$ ($B>D$), this could set $U$ to be:

• $U\left(\begin{array}{c}A\\ B\\ C\\ D\\ E\end{array}\right)=\left(\begin{array}{c}0.5\\ 0.5\\ -0.5\\ -0.5\\ 0\end{array}\right)$

Here, the average utility in each linked class ($\{A,C\}$, $\{B,D\}$ and $\left\{E\right\}$) is $0$.

Applying this to the example

So, applying this approach to the full set of the $P_i$, $1\le i\le 5$ above (and fixing $U\left(B\right)=0$), we'd get:

• $U\left(\begin{array}{c}A\\ B\\ C\\ D\\ E\end{array}\right)=\left(\begin{array}{c}2/3\\ 0\\ -2/3\\ -1\\ -1\end{array}\right)$.

Here $B$ is in the middle of $A$ and $C$, as it should be, while the utilities of $D$ and $E$ are defined by their distance from $B$ only. The distance between $A$ and $C$ is $4/3\approx \mathrm{1.33333...}$. This is between $2$ (which would be given by $A>B$ and $B>C$ only) and $1$ (which would be given by $A>C$ only).

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1. I've divided the normalisation from that post by $2$, to fit better with the methods of this post. Dividing everything in a sum by the same constant gives the same equivalence class of utility functions. ↩︎