3 research outputs found
The ForMaRE Project - Formal Mathematical Reasoning in Economics
The ForMaRE project applies formal mathematical reasoning to economics. We
seek to increase confidence in economics' theoretical results, to aid in
discovering new results, and to foster interest in formal methods, i.e.
computer-aided reasoning, within economics. To formal methods, we seek to
contribute user experience feedback from new audiences, as well as new
challenge problems. In the first project year, we continued earlier game theory
studies but then focused on auctions, where we are building a toolbox of
formalisations, and have started to study matching and financial risk.
In parallel to conducting research that connects economics and formal
methods, we organise events and provide infrastructure to connect both
communities, from fostering mutual awareness to targeted matchmaking. These
efforts extend beyond economics, towards generally enabling domain experts to
use mechanised reasoning.Comment: Conference on Intelligent Computer Mathematics, 8--12 July, Bath, UK.
Published as number 7961 in Lecture Notes in Artificial Intelligence,
Springe
Budget Imbalance Criteria for Auctions: A Formalized Theorem
We present an original theorem in auction theory: it specifies general
conditions under which the sum of the payments of all bidders is necessarily
not identically zero, and more generally not constant. Moreover, it explicitly
supplies a construction for a finite minimal set of possible bids on which such
a sum is not constant. In particular, this theorem applies to the important
case of a second-price Vickrey auction, where it reduces to a basic result of
which a novel proof is given. To enhance the confidence in this new theorem, it
has been formalized in Isabelle/HOL: the main results and definitions of the
formal proof are re- produced here in common mathematical language, and are
accompanied by an informal discussion about the underlying ideas.Comment: 6th Podlasie Conference on Mathematics 2014, 11 page
Set Theory or Higher Order Logic to Represent Auction Concepts in Isabelle?
When faced with the question of how to represent properties in a formal proof
system any user has to make design decisions. We have proved three of the
theorems from Maskin's 2004 survey article on Auction Theory using the
Isabelle/HOL system, and we have produced verified code for combinatorial
Vickrey auctions. A fundamental question in this was how to represent some
basic concepts: since set theory is available inside Isabelle/HOL, when
introducing new definitions there is often the issue of balancing the amount of
set-theoretical objects and of objects expressed using entities which are more
typical of higher order logic such as functions or lists. Likewise, a user has
often to answer the question whether to use a constructive or a
non-constructive definition. Such decisions have consequences for the proof
development and the usability of the formalization. For instance, sets are
usually closer to the representation that economists would use and recognize,
while the other objects are closer to the extraction of computational content.
In this paper we give examples of the advantages and disadvantages for these
approaches and their relationships. In addition, we present the corresponding
Isabelle library of definitions and theorems, most prominently those dealing
with relations and quotients.Comment: Preprint of a paper accepted for the forthcoming CICM 2014 conference
(cicm-conference.org/2014): S.M. Watt et al. (Eds.): CICM 2014, LNAI 8543,
Springer International Publishing Switzerland 2014. 16 pages, 1 figur