Automated Search for Impossibility Theorems in Choice Theory: Ranking Sets of Objects
Christian Geist
Abstract:
In the subarea of (social) choice theory commonly referred to as
ranking sets of objects the question arises whether, given preferences
over some domain, there is a preference relation on the power set of
this domain that is compatible with certain axioms. The Kannai-Peleg
Theorem (Journal of Economic Theory, 1984) gives a negative answer to
this question for the case of six or more elements in the domain in
combination with the two (very intuitive) axioms called dominance and
independence. It is the main (and earliest) impossibility theorem in
this particular area.
Our initial goal was to find a suitable formulation of the
Kannai-Peleg Theorem in logic that would facilitate an automatic
verification of the result. It turned out, however, that developing
a first-order formulation of this problem and feeding it to a
first-order theorem prover was not very effective. Therefore, we
successfully tried an inductive proof consisting of a manual in-
ductive step together with a formalization of the base case in
propositional logic (in order to use a SAT solver). This particular
way of using automated theorem proving in social choice theory is due
to Tang and Lin (Artificial Intelligence Journal, 2009), who have in
this way proved major impossibility results, like Arrow’s Theorem, the
Muller-Satterthwaite Theorem, and Sen’s Theorem. We extended their
technique to be able to treat the Kannai-Peleg Theorem and were able
to verify it with a verification time of less than ten seconds.
With our initial objective met, we then developed a further extension
of the method in order to discover new impossibility results. Using
tools from model theory, a universal form of the previous inductive
step (now applicable to a large class of axioms) allows for a fully
automated theorem search, which has produced 84 impossibility theorems
on a space of 20 axioms from the literature. Many of these results
are variations of known impossibilities, but also a few new results
were obtained. Interestingly, one of the impossibility theorems found
by our program had even wrongly been published as a possibility result
earlier (Economic Theory, 2000, 2003). Finally, we give some manual
proofs of the new results to underline the fruitfulness of this
computer-aided method of searching for impossibility results in the
field of ranking sets of objects.