Using Theorema in the Formalization of Theoretical Economics

Abstract

Theoretical economics makes use of strict mathematical methods. For instance, games as introduced by von Neumann and Morgenstern allow for formal mathematical proofs for certain axiomatized economical situations. Such proofs can—at least in principle—also be carried through in formal systems such as Theorema. In this paper we describe experiments carried through using the Theorema system to prove theorems about a particular form of games called pillage games. Each pillage game formalizes a particular understanding of power. Analysis then attempts to derive the properties of solution sets (in particular, the core and stable set), asking about existence, uniqueness and characterization.

Concretely we use Theorema to show properties previously proved on paper by two of the co-authors for pillage games with three agents. Of particular interest is some pseudo-code which summarizes the results previously shown. Since the computation involves infinite sets the pseudo-code is in several ways non-computational. However, in the presence of appropriate lemmas, the pseudo-code has sufficient computational content that Theorema can compute stable sets (which are always finite). We have concretely demonstrated this for three different important power functions.

The first author would like to thank the ‘Bridging The Gap’ initiative under EPSRC grant EP/F033087/1 for supporting this work. The first and second author would like to thank the JKU Linz for its hospitality.