MoL-2010-02:
Geist, Christian
(2010)
*Automated Search for Impossibility Theorems in Choice Theory: Ranking Sets of Objects.*
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## 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.

Item Type: | Report |
---|---|

Report Nr: | MoL-2010-02 |

Series Name: | Master of Logic Thesis (MoL) Series |

Year: | 2010 |

Date Deposited: | 12 Oct 2016 14:38 |

Last Modified: | 12 Oct 2016 14:38 |

URI: | https://eprints.illc.uva.nl/id/eprint/827 |

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