A comment by a referee of a recent paper of mine that one of my
results in decision theory didn’t actually depend on numerical
probabilities and hence could extend to social choice principles made me
realize that this may be true for some other things I’ve done.

For instance, in the past I’ve proved theorems on qualitative
probabilities. A qualitative probability is a relation ≼ on the subsets of some sample space *Ω* such that:

≼ is transitive and
reflexive.

⌀ ≼ *A*

if *A* ∩ *C* = *B* ∩ *C* = ⌀,
then *A* ≼ *B* iff *A* ∩ *C* ≼ *B* ∩ *C*
(additivity).

But need not think of *Ω* as
a space of possibilities and of ≼ as a
probability comparison. We could instead think of it as a set of people
who are candidates for getting some good thing, with *A* ≼ *B* meaning that it’s at
least as good for the good thing to be distributed to the members of
*B* as to the members of *A*. Axioms (1) and (2) are then
obvious. And axiom (3) is an independence axiom: whether it is at least
as good to give the good thing to the members of *B* as to the members of *A* doesn’t depend on whether we give
it to the members of a disjoint set *C* at the same time.

Of course, for a general social choice principle we need more than
just a decision whether to give one and the same good to the members of
some set. But we can still formalize those questions in terms of
something pretty close to qualitative probabilities. For a general
framework, suppose a population set *X* (a set of people or places in
spacetime or some other sites of value) and a set of values *V* (this could be a set of types of
good, or the set of real numbers representing values). We will suppose
that *V* comes with a transitive
and reflexive (preorder) preference relation ≤. Now let *Ω* = *X* × *V*. A value
distribution is a function *f*
from *X* to *V*, where *f*(*x*) = *v* means
that *x* gets something of value
*v*.

We want to generate a reflexive and transitive preference ordering
≼ on the set *V*^{X} of value
distributions.

Write *f* ≈ *g* when
*f* ≼ *g* and *g* ≼ *f*, and *f* ≺ *g* when *f* ≼ *g* but not *g* ≼ *f*. Similarly for values
*v* and *w*, write *v* < *w* if *v* ≤ *w* but not *w* ≤ *v*.

Here is a plausible axiom on value distributions:

- Sameness independence: if
*f*_{1}, *f*_{2}, *g*_{1}, *g*_{2}
are value distributions and *A* ⊆ *X* is such that
(a) *f*_{1} ≼ *f*_{2},
(b) *f*_{1}(*x*) = *f*_{2}(*x*)
and *g*_{1}(*x*) = *g*_{2}(*x*)
if *x* ∉ *A*, (c) *f*_{1}(*x*) = *g*_{1}(*x*)
and *f*_{2}(*x*) = *g*_{2}(*x*)
if *x* ∈ *A*.

In other words, the mutual ranking between two value distributions
does not depend on what the two distributions do to the people on whom
the distributions agree. If it’s better to give $4 to Jones than to give
$2 to Smith when Kowalski is getting $7, it’s still better to give $4 to
Jones than to give $2 to Smith when Kowalski is getting $3. There is
probably some other name in the literature for this property, but I know
next to nothing about social choice literature.

Finally, we want to have some sort of symmetries on the population.
The most radical would be that the value distributions don’t care about
permutations of people, but more moderate symmetries may be required.
For this we need a group *G* of
permutations acting on *X*.

- Strong
*G*-invariance: if
*g* ∈ *G* and *f* is a value distribution, then
*f* ∘ *g* ≈ *f*.

Here, *f* ∘ *g* is the
value distribution where site *x* gets *f*(*g*(*x*)).

Additionally, the following is plausible:

- Pareto: If
*f*(*x*) ≤ *g*(*x*)
for all *x* with *f*(*x*) < *g*(*x*)
for some *x*, then *f* ≺ *g*.

**Theorem:** Assume the Axiom of Choice. Suppose ≤ on *V* is reflexive, transitive and
non-trivial in the sense that it contains two values *v* and *w* such that *v* < *w*. There exists a
reflexive, transitive preference ordering ≼ on the value distributions satisfying
(4)–(6) if and only if there is such an ordering that is total if and
only if *G* has locally finite
action on *X*.

A group of symmetries *G* has
locally finite action a set *X*
provided that for each finite subset *H* of *G* and each *x* ∈ *X*, applying finite
combinations of members of *G*
to *x* generates only a finite
subset of *X*. (More precisely,
if ⟨*H*⟩ is the subgroup
generated by *G*, then ⟨*H*⟩*x* is finite.)

If *X* is finite, then local
finiteness of action is trivial. If *X* is infinite, then it will be
satisfies in some cases but not others. For instance, it will be
satisfied if *G* is permutations
that only move a finite number of members of *X* at a time. It will on the other
hand fail if *X* is a infinite
bunch of people regularly spaced in a line and *G* is shifts.

The trick to the proof of the Theorem is to reduce preferences
between distributions to comparisons of subsets of *X* × *V* and to reduce
comparisons of subsets of *X* to
preferences between binary distributions.

**Proof of Therem:** Suppose that *G* has locally finite action. Define
*Ω* = *X* × *V*.
By Theorem 2 of my invariance of non-classical probabilities
paper, there is a strongly *G*-invariant regular (i.e., ⌀ ≺ *A* if *A* is non-empty) qualitative
probability ≼ on *Ω*. Given a value distribution *f*, let *f*^{*} = {(*x*,*v*) : *v* ≤ *f*(*x*)}
be a subset of *Ω*. Define *f* ≼ *g* iff *f*^{*} ≼ *g*.

Totality, reflexivity, transitivity and strong *G*-invariance for value distributions
follows from the same conditions for subsets of *Ω*. Regularity of ≼ on the subsets of *Ω* and additivity implies that if
*A* ⊂ *B* then *A* ≺ *B*. The Pareto condition
for ≼ on the value distributions
follows since if *f* and *g* satisfy are such that *f*(*x*) ≤ *g*(*x*)
for all *x* with strict
inequality for some *x*, then
*f*^{*} ⊂ *g*^{*}.
Finally, the complicated sameness independence condition follows from
additivity.

Now suppose there is a (not necessarily total) strongly *G*-invariant reflexive and transitive
preference ordering ≼ on the value
distributions satisfying (4)–(6). Given a subset *A* of *X*, define *A*^{†} to be the value
distribution that gives *w* to
all the members of *A* and *v* to all the non-members, where
*v* < *w*. Define
*A* ≼ *B* iff *A*^{†} ≼ *B*^{†}.
This will be a strongly *G*-invariant reflexive and transitive
relation on the subsets of *X*.
It will be regular by the Pareto condition. Finally, additivity follows
from the sameness independence condition. Local finiteness of action of
*G* then follows from Theorem 2
of my paper. ⋄

Note that while it is natural to think of *X* has just a set of people or of
locations, inspired
by Kenny Easwaran one can also think of it as a set *Q* × *Ω* where *Ω* is a probability space and *Q* is a population, so that *f*(*x*,*ω*) represents
the value *x* gets at location
*ω*. In that case, *G* might be defined by symmetries of
the population and/or symmetries of the probability space. In such a
setting, we might want a weaker Pareto principle that supposes
additionally that *f*(*x*,*ω*) < *g*(*x*,*ω*)
for some *x* and *all*
*ω*. With that weaker Pareto
principle, the proof that the existence of a *G*-invariant preference of the right
sort on the distributions implies local finiteness of action does not
work. However, I think we can still prove local finiteness of action in
that case if the symmetries in *G* act only on the population (i.e.,
for all *x* and *ω* there is an *y* such that *g*(*x*,*ω*) = (*y*,*ω*)).
In that case, given a subset *A*
of the population *Q*, we define
*A*^{†} to be the
distribution that gives *w* to
all the persons in *A* with
certainty (i.e., everywhere on *Ω*) and gives *v* to everyone else, and the rest of
the proof should go through, but I haven’t checked the details.