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.
This work has been supported by EPSRC grant EP/J007498/1. We would like to thank Peter Cramton and Elizabeth Baldwin for sharing their auction designer’s point, and Christian Maeder for his recent improvements to Hets.
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References
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)
Auctions: The Past, Present and Future, http://realestateauctionglobalnetwork.blogspot.co.uk/2011/11/auctions-past-present-and-future.html
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)
Caminati, M.B., Rosolini, G.: Custom automations in Mizar. Automated Reasoning 50(2) (2013)
CASL, http://informatik.uni-bremen.de/cofi/wiki/index.php/CASL
Conitzer, V., Sandholm, T.: Self-interested automated mechanism design and implications for optimal combinatorial auctions. In: Conference on Electronic Commerce. ACM (2004)
Cramton, P., Shoham, Y., Steinberg, R. (eds.): Combinatorial auctions. MIT Press (2006)
Farmer, W.M.: The seven virtues of simple type theory. Applied Logic 6(3) (2008)
Geanakoplos, J.D.: Three brief proofs of Arrow’s impossibility theorem. Discussion Paper 1123RRR. Cowles Foundation (2001)
Geist, C., Endriss, U.: Automated search for impossibility theorems in social choice theory: ranking sets of objects. Artificial Intelligence Research 40 (2011)
Grabowski, A., Korniłowicz, A., Naumowicz, A.: Mizar in a Nutshell. Formalized Reasoning 3(2) (2010)
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)
Initiative for Computational Economics, http://ice.uchicago.edu
Isabelle, http://isabelle.in.tum.de
Kerber, M., Lange, C., Rowat, C.: An economist’s guide to mechanized reasoning (2012), http://cs.bham.ac.uk/research/projects/formare/
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)
Kirkegaard, R.: A Mechanism Design Approach to Ranking Asymmetric Auctions. Econometrica 80(5) (2012)
Klemperer, P.: Auctions: theory and practice. Princeton Univ. Press (2004)
Klemperer, P.: The product-mix auction: a new auction design for differentiated goods. European Economic Association Journal 8(2-3) (2010)
Korniłowicz, A.: On Rewriting Rules in Mizar. Automated Reasoning 50(2) (2013)
Lamport, L., Paulson, L.C.: Should your specification language be typed? ACM TOPLAS 21(3) (1999)
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)
Lange, C., et al.: Auction Theory Toolbox (2013), http://cs.bham.ac.uk/research/projects/formare/code/auction-theory/
Maskin, E.: The unity of auction theory: Milgrom’s master class. Economic Literature 42(4) (2004)
Milgrom, P.: Putting auction theory to work. Cambridge Univ. Press (2004)
Mizar manuals (2011), http://mizar.org/project/bibliography.html
Mossakowski, T.: Hets: the Heterogeneous Tool Set, http://dfki.de/cps/hets
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
Mosses, P.D. (ed.): CASL Reference Manual. LNCS, vol. 2960. Springer, Heidelberg (2004)
Nipkow, T.: Social choice theory in HOL: Arrow and Gibbard-Satterthwaite. Automated Reasoning 43(3) (2009)
Rudnicki, P., Urban, J., et al.: Escape to ATP for Mizar. In: Workshop Proof eXchange for Theorem Proving (2011)
Sutcliffe, G.: The TPTP Problem Library and Associated Infrastructure: The FOF and CNF Parts, v3.5.0. Automated Reasoning 43(4) (2009)
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)
System on TPTP, http://cs.miami.edu/~tptp/cgi-bin/SystemOnTPTP
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)
Tang, P., Lin, F.: Computer-aided proofs of Arrow’s and other impossibility theorems. Artificial Intelligence 173(11) (2009)
Tang, P., Lin, F.: Discovering theorems in game theory: two-person games with unique pure Nash equilibrium payoffs. Artificial Intelligence 175(14-15) (2011)
Urban, J.: MizarMode—an integrated proof assistance tool for the Mizar way of formalizing mathematics. Applied Logic 4(4) (2006)
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)
Wiedijk, F.: De Bruijn factor, http://cs.ru.nl/~freek/factor/
Wiedijk, F.: Formal proof sketches. In: Berardi, S., Coppo, M., Damiani, F. (eds.) TYPES 2003. LNCS, vol. 3085, pp. 378–393. Springer, Heidelberg (2004)
Wiedijk, F.: Formalizing Arrow’s theorem. Sādhanā 34(1) (2009)
Wiedijk, F.: The QED Manifesto Revisited. Studies in Logic, Grammar and Rhetoric 10(23) (2007)
Wiedijk, F. (ed.): The Seventeen Provers of the World. LNCS (LNAI), vol. 3600. Springer, Heidelberg (2006)
Wikipedia (ed.): Vickrey auction (2012), http://en.wikipedia.org/w/index.php?title=Vickrey_auction&oldid=523230741
Windsteiger, W.: Theorema 2.0: A Graphical User Interface for a Mathematical Assistant System. In: UITP Workshop at CICM (2012)
Woodcock, J., et al.: Formal method: practice and experience. ACM Computing Surveys 41(4) (2009)
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Lange, C. et al. (2013). 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) Intelligent Computer Mathematics. CICM 2013. Lecture Notes in Computer Science(), vol 7961. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39320-4_13
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DOI: https://doi.org/10.1007/978-3-642-39320-4_13
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