A Qualitative Comparison of the Suitability of Four Theorem Provers for Basic Auction Theory

  • Christoph Lange
  • Marco B. Caminati
  • Manfred Kerber
  • Till Mossakowski
  • Colin Rowat
  • Makarius Wenzel
  • Wolfgang Windsteiger
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7961)

Abstract

Novel auction schemes are constantly being designed. Their design has significant consequences for the allocation of goods and the revenues generated. But how to tell whether a new design has the desired properties, such as efficiency, i.e. allocating goods to those bidders who value them most? We say: by formal, machine-checked proofs. We investigated the suitability of the Isabelle, Theorema, Mizar, and Hets/CASL/TPTP theorem provers for reproducing a key result of auction theory: Vickrey’s 1961 theorem on the properties of second-price auctions. Based on our formalisation experience, taking an auction designer’s perspective, we give recommendations on what system to use for formalising auctions, and outline further steps towards a complete auction theory toolbox.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aspinall, D.: Proof General: A Generic Tool for Proof Development. In: Graf, S. (ed.) TACAS 2000. LNCS, vol. 1785, pp. 38–43. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  2. 2.
  3. 3.
    Bancerek, G.: Information Retrieval and Rendering with MML Query. In: Borwein, J.M., Farmer, W.M. (eds.) MKM 2006. LNCS (LNAI), vol. 4108, pp. 266–279. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  4. 4.
    Caminati, M.B., Rosolini, G.: Custom automations in Mizar. Automated Reasoning 50(2) (2013)Google Scholar
  5. 5.
  6. 6.
    Conitzer, V., Sandholm, T.: Self-interested automated mechanism design and implications for optimal combinatorial auctions. In: Conference on Electronic Commerce. ACM (2004)Google Scholar
  7. 7.
    Cramton, P., Shoham, Y., Steinberg, R. (eds.): Combinatorial auctions. MIT Press (2006)Google Scholar
  8. 8.
    Farmer, W.M.: The seven virtues of simple type theory. Applied Logic 6(3) (2008)Google Scholar
  9. 9.
    Geanakoplos, J.D.: Three brief proofs of Arrow’s impossibility theorem. Discussion Paper 1123RRR. Cowles Foundation (2001)Google Scholar
  10. 10.
    Geist, C., Endriss, U.: Automated search for impossibility theorems in social choice theory: ranking sets of objects. Artificial Intelligence Research 40 (2011)Google Scholar
  11. 11.
    Grabowski, A., Korniłowicz, A., Naumowicz, A.: Mizar in a Nutshell. Formalized Reasoning 3(2) (2010)Google Scholar
  12. 12.
    Griffioen, D., Huisman, M.: A comparison of PVS and isabelle/HOL. In: Grundy, J., Newey, M. (eds.) TPHOLs 1998. LNCS, vol. 1479, pp. 123–142. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  13. 13.
    Initiative for Computational Economics, http://ice.uchicago.edu
  14. 14.
  15. 15.
    Kerber, M., Lange, C., Rowat, C.: An economist’s guide to mechanized reasoning (2012), http://cs.bham.ac.uk/research/projects/formare/
  16. 16.
    Kerber, M., Rowat, C., Windsteiger, W.: Using Theorema in the Formalization of Theoretical Economics. In: Davenport, J.H., Farmer, W.M., Urban, J., Rabe, F. (eds.) Calculemus/MKM 2011. LNCS (LNAI), vol. 6824, pp. 58–73. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  17. 17.
    Kirkegaard, R.: A Mechanism Design Approach to Ranking Asymmetric Auctions. Econometrica 80(5) (2012)Google Scholar
  18. 18.
    Klemperer, P.: Auctions: theory and practice. Princeton Univ. Press (2004)Google Scholar
  19. 19.
    Klemperer, P.: The product-mix auction: a new auction design for differentiated goods. European Economic Association Journal 8(2-3) (2010)Google Scholar
  20. 20.
    Korniłowicz, A.: On Rewriting Rules in Mizar. Automated Reasoning 50(2) (2013)Google Scholar
  21. 21.
    Lamport, L., Paulson, L.C.: Should your specification language be typed? ACM TOPLAS 21(3) (1999)Google Scholar
  22. 22.
    Lange, C., Rowat, C., Kerber, M.: The ForMaRE Project – Formal Mathematical Reasoning in Economics. In: Carette, J., Aspinall, D., Lange, C., Sojka, P., Windsteiger, W. (eds.) CICM 2013. LNCS (LNAI), vol. 7961, pp. 330–334. Springer, Heidelberg (2013)Google Scholar
  23. 23.
    Lange, C., et al.: Auction Theory Toolbox (2013), http://cs.bham.ac.uk/research/projects/formare/code/auction-theory/
  24. 24.
    Maskin, E.: The unity of auction theory: Milgrom’s master class. Economic Literature 42(4) (2004)Google Scholar
  25. 25.
    Milgrom, P.: Putting auction theory to work. Cambridge Univ. Press (2004)Google Scholar
  26. 26.
  27. 27.
    Mossakowski, T.: Hets: the Heterogeneous Tool Set, http://dfki.de/cps/hets
  28. 28.
    Mossakowski, T., Maeder, C., Codescu, M.: Hets User Guide. Tech. rep. Version 0.98. DFKI Bremen (2013), http://informatik.uni-bremen.de/agbkb/forschung/formal_methods/CoFI/hets/UserGuide.pdf
  29. 29.
    Mosses, P.D. (ed.): CASL Reference Manual. LNCS, vol. 2960. Springer, Heidelberg (2004)MATHGoogle Scholar
  30. 30.
    Nipkow, T.: Social choice theory in HOL: Arrow and Gibbard-Satterthwaite. Automated Reasoning 43(3) (2009)Google Scholar
  31. 31.
    Rudnicki, P., Urban, J., et al.: Escape to ATP for Mizar. In: Workshop Proof eXchange for Theorem Proving (2011)Google Scholar
  32. 32.
    Sutcliffe, G.: The TPTP Problem Library and Associated Infrastructure: The FOF and CNF Parts, v3.5.0. Automated Reasoning 43(4) (2009)Google Scholar
  33. 33.
    Sutcliffe, G., Schulz, S., Claessen, K., Baumgartner, P.: The TPTP Typed First-order Form with Arithmetic. In: Bjørner, N., Voronkov, A. (eds.) LPAR-18. LNCS (LNAI), vol. 7180, pp. 406–419. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  34. 34.
  35. 35.
    Tadjouddine, E.M., Guerin, F., Vasconcelos, W.: Abstracting and Verifying Strategy-Proofness for Auction Mechanisms. In: Baldoni, M., Son, T.C., van Riemsdijk, M.B., Winikoff, M. (eds.) DALT 2008. LNCS (LNAI), vol. 5397, pp. 197–214. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  36. 36.
    Tang, P., Lin, F.: Computer-aided proofs of Arrow’s and other impossibility theorems. Artificial Intelligence 173(11) (2009)Google Scholar
  37. 37.
    Tang, P., Lin, F.: Discovering theorems in game theory: two-person games with unique pure Nash equilibrium payoffs. Artificial Intelligence 175(14-15) (2011)Google Scholar
  38. 38.
    Urban, J.: MizarMode—an integrated proof assistance tool for the Mizar way of formalizing mathematics. Applied Logic 4(4) (2006)Google Scholar
  39. 39.
    Wenzel, M.: Isabelle/jEdit – a Prover IDE within the PIDE framework. In: Jeuring, J., Campbell, J.A., Carette, J., Dos Reis, G., Sojka, P., Wenzel, M., Sorge, V. (eds.) CICM 2012. LNCS (LNAI), vol. 7362, pp. 468–471. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  40. 40.
    Wiedijk, F.: De Bruijn factor, http://cs.ru.nl/~freek/factor/
  41. 41.
    Wiedijk, F.: Formal proof sketches. In: Berardi, S., Coppo, M., Damiani, F. (eds.) TYPES 2003. LNCS, vol. 3085, pp. 378–393. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  42. 42.
    Wiedijk, F.: Formalizing Arrow’s theorem. Sādhanā 34(1) (2009)Google Scholar
  43. 43.
    Wiedijk, F.: The QED Manifesto Revisited. Studies in Logic, Grammar and Rhetoric 10(23) (2007)Google Scholar
  44. 44.
    Wiedijk, F. (ed.): The Seventeen Provers of the World. LNCS (LNAI), vol. 3600. Springer, Heidelberg (2006)Google Scholar
  45. 45.
  46. 46.
    Windsteiger, W.: Theorema 2.0: A Graphical User Interface for a Mathematical Assistant System. In: UITP Workshop at CICM (2012)Google Scholar
  47. 47.
    Woodcock, J., et al.: Formal method: practice and experience. ACM Computing Surveys 41(4) (2009)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Christoph Lange
    • 1
  • Marco B. Caminati
    • 2
  • Manfred Kerber
    • 1
  • Till Mossakowski
    • 3
  • Colin Rowat
    • 4
  • Makarius Wenzel
    • 5
  • Wolfgang Windsteiger
    • 6
  1. 1.Computer ScienceUniversity of BirminghamUK
  2. 2.Italy
  3. 3.University of Bremen and DFKI GmbH BremenGermany
  4. 4.EconomicsUniversity of BirminghamUK
  5. 5.Laboratoire LRI, UMR8623Univ. Paris-SudOrsayFrance
  6. 6.RISCJohannes Kepler University Linz (JKU)Austria

Personalised recommendations