The Formalization of Vickrey Auctions: A Comparison of Two Approaches in Isabelle and Theorema

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10383)


In earlier work presented at CICM, four theorem provers (Isabelle, Mizar, Hets/CASL/TPTP, and Theorema) were compared based on a case study in theoretical economics, the formalization of the landmark Theorem of Vickrey in auction theory. At the time of this comparison the Theorema system was in a state of transition: The original Theorema system (Theorema 1) had been shut down by the Theorema group and the successor system Theorema 2.0 was just about to be launched. Theorema 2.0 participated in the competition, but only parts of the system were ready for use. In particular, the new reasoning engines had not been set up, so that some of the results in the system comparison had to be extrapolated from experience we had with Theorema 1. In this paper, we now want to compare a complete formalization of Vickrey’s Theorem in Theorema 2.0 with the original formalization in Isabelle. On the one hand, we compare the mathematical setup of the two theories and, on the other hand, we also give an overview on statistical indicators, such as number of auxiliary lemmas and the total number of proof steps needed for all proofs in the theory. Last but not least, we present a shorter version of proof of the main theorem in Isabelle.


  1. 1.
    Bancerek, G., Byliński, C., Grabowski, A., Korniłowicz, A., Matuszewski, R., Naumowicz, A., Pa̧k, K., Urban, J.: Mizar: state-of-the-art and beyond. In: Kerber, M., Carette, J., Kaliszyk, C., Rabe, F., Sorge, V. (eds.) CICM 2015. LNCS, vol. 9150, pp. 261–279. Springer, Cham (2015). doi: 10.1007/978-3-319-20615-8_17 CrossRefGoogle Scholar
  2. 2.
    Buchberger, B., Jebelean, T., Kutsia, T., Maletzky, A., Windsteiger, W.: Theorema 2.0: computer-assisted natural-style mathematics. JFR 9(1), 149–185 (2016)MathSciNetGoogle Scholar
  3. 3.
    Caminati, M.B., Kerber, M., Lange, C., Rowat, C.: VCG - Combinatorial Vickrey-Clarke-Groves Auctions. Archive of Formal Proofs, April 2015Google Scholar
  4. 4.
    Grabowski, A., Korniłowicz, A., Naumowicz, A.: Mizar in a nutshell. J. Formal. Reason. 3(2), 153–245 (2010)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Kerber, M., Lange, C., Rowat, C., Windsteiger, W.: Developing an auction theory toolbox. In: Kerber, M., Lange, C., Rowat, C. (eds.) AISB 2013, pp. 1–4 (2013). Proceedings available online.
  6. 6.
    Lange, C., Caminati, M.B., Kerber, M., Mossakowski, T., Rowat, C., Wenzel, M., Windsteiger, W.: A qualitative comparison of the suitability of four theorem provers for basic auction theory. In: Carette, J., Aspinall, D., Lange, C., Sojka, P., Windsteiger, W. (eds.) CICM 2013. LNCS, vol. 7961, pp. 200–215. Springer, Heidelberg (2013). doi: 10.1007/978-3-642-39320-4_13 CrossRefGoogle Scholar
  7. 7.
    Maskin, E.: The unity of auction theory. J. Econ. Lit. 42(4), 1102–1115 (2004)CrossRefGoogle Scholar
  8. 8.
    Milgrom, P.: Putting Auction Theory to Work. Cambridge University Press, Cambridge (2004)CrossRefGoogle Scholar
  9. 9.
    Mossakowski, T., Haxthausen, A.E., Sannella, D., Tarlecki, A.: CASL - the common algebraic specification language. In: Bjørner, D., Henson, M.C. (eds.) Logics of Specification Languages. Monographs in Theoretical Computer Science, pp. 241–298. Springer, Heidelberg (2008)Google Scholar
  10. 10.
    Mossakowski, T., Maeder, C., Codescu, M.: Hets user guide. Technical report. version 0.98, DFKI Bremen (2013)Google Scholar
  11. 11.
    Paulson, L.C.: Isabelle: the next 700 theorem provers. In: Odifreddi, P. (ed.) Logic and Computer Science, pp. 361–386. Academic Press (1990)Google Scholar
  12. 12.
    Sutcliffe, G.: The TPTP problem library and associated infrastructure: the FOF and CNF parts, v3.5.0. J. Autom. Reason. 43(4), 337–362 (2009)CrossRefzbMATHGoogle Scholar
  13. 13.
    Vickrey, W.: Counterspeculation, auctions, and competitive sealed tenders. J. Financ. XVI, 8–37 (1961)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Wenzel, M.: Isabelle/Isar Reference Manual (2017)Google Scholar
  15. 15.
    Windsteiger, W.: Theorema 2.0: a graphical user interface for a mathematical assistant system. In: Kaliszyk, C., Lueth, C. (eds.) Proceedings of the 10th International Workshop UITP. EPTCS, vol. 118, pp. 72–82. Open Publishing Association (2012). doi10.4204/EPTCS.118.5.

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Research Institute for Symbolic Computation (RISC)Johannes Kepler University Linz (JKU)HagenbergAustria

Personalised recommendations