Automated Search for Impossibility Theorems in Social Choice Theory: Ranking Sets of Objects
This work addresses the challenge of identifying logical inconsistencies in preference extensions for social choice and decision-making under uncertainty, offering an automated method that is incremental in applying computational techniques to economic theory.
The authors tackled the problem of automatically verifying and discovering impossibility theorems in ranking sets of objects, a key area in social choice theory, by using SAT solvers to search for logical inconsistencies among 20 principles, resulting in 84 theorems including new findings.
We present a method for using standard techniques from satisfiability checking to automatically verify and discover theorems in an area of economic theory known as ranking sets of objects. The key question in this area, which has important applications in social choice theory and decision making under uncertainty, is how to extend an agents preferences over a number of objects to a preference relation over nonempty sets of such objects. Certain combinations of seemingly natural principles for this kind of preference extension can result in logical inconsistencies, which has led to a number of important impossibility theorems. We first prove a general result that shows that for a wide range of such principles, characterised by their syntactic form when expressed in a many-sorted first-order logic, any impossibility exhibited at a fixed (small) domain size will necessarily extend to the general case. We then show how to formulate candidates for impossibility theorems at a fixed domain size in propositional logic, which in turn enables us to automatically search for (general) impossibility theorems using a SAT solver. When applied to a space of 20 principles for preference extension familiar from the literature, this method yields a total of 84 impossibility theorems, including both known and nontrivial new results.