Abstract
Revealed preference techniques are used to test whether a data set is compatible with rational behaviour. They are also incorporated as constraints in mechanism design to encourage truthful behaviour in applications such as combinatorial auctions. In the auction setting, we present an efficient combinatorial algorithm to find a virtual valuation function with the optimal (additive) rationality guarantee. Moreover, we show that there exists such a valuation function that both is individually rational and is minimum (that is, it is component-wise dominated by any other individually rational, virtual valuation function that approximately fits the data). Similarly, given upper bound constraints on the valuation function, we show how to fit the maximum virtual valuation function with the optimal additive rationality guarantee. In practice, revealed preference bidding constraints are very demanding. We explain how approximate rationality can be used to create relaxed revealed preference constraints in an auction. We then show how combinatorial methods can be used to implement these relaxed constraints. Worst/best-case welfare guarantees that result from the use of such mechanisms can be quantified via the minimum/maximum virtual valuation function.
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- 2.
It is not necessary to present the model in terms of “time". We do so because this best accords with the combinatorial auction application.
- 3.
Local non-satiation states that for any bundle \({\varvec{x}}\) there is a more preferred bundle arbitrarily close to \({\varvec{x}}\). A monotonic utility function is locally non-satiated, but the converse need not hold.
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For example, in a bandwidth auction there are at most a few hundred rounds.
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Indeed, several schools of thought in the field of bounded rationality argue that people utilize simple (but often effective) heuristics rather than attempt to optimize; see, for example, [10].
References
Afriat, S.: The construction of a utility function from expenditure data. Int. Econ. Rev. 8, 67–77 (1967)
Akerlof, G., Yellen, J.: Can small deviations from rationality make significant differences to economic equilibria? Am. Econ. Rev. 75(4), 708–720 (1985)
Ausubel, L., Baranov, O.: Market design and the evolution of the combinatorial clock auction. Am. Econ. Rev. 104(5), 446–451 (2014)
Ausubel, L., Cramton, P., Milgrom, P.: The clock-proxy auction: a practical combinatorial auction design. In: Cramton, P., Shoham, Y., Steinberg, R. (eds.) Combinatorial Auctions, pp. 115–138. MIT Press, Cambridge (2006)
Brown, D., Echenique, F.: Supermodularity and preferences. J. Econ. Theor. 144(3), 1004–1014 (2009)
Cramton, P.: Spectrum auction design. Rev. Ind. Organ. 42(2), 161–190 (2013)
Echenique, F., Golovin, D., Wierman, A.: A revealed preference approach to computational complexity in economics. In: Proceedings of EC, pp 101–110 (2011)
Edelman, B., Ostrovsky, M., Schwarz, M.: Internet advertising and the generalized second-price auction: selling billions of dollars worth of keywords. Am. Econ. Rev. 97(1), 242–259 (2007)
Fostel, A., Scarf, H., Todd, M.: Two new proofs of Afriat’s theorem. Econ. Theor. 24, 211–219 (2004)
Gigerenzer, G., Selten, R. (eds.): Bounded Rationality: the Adaptive Toolbox. MIT Press, Cambridge (2001)
Gross, J.: Testing data for consistency with revealed preference. Rev. Econ. Stat. 77(4), 701–710 (1995)
Gul, F., Stacchetti, E.: Walrasian equilibrium with gross substitutes. J. Econ. Theor. 87, 95–124 (1999)
Harsha, P., Barnhart, C., Parkes, D., Zhang, H.: Strong activity rules for iterative combinatorial auctions. Comput. Oper. Res. 37(7), 1271–1284 (2010)
Houthakker, H.: Revealed preference and the utility function. Economica New Ser. 17(66), 159–174 (1950)
Karp, R.: A characterization of the minimum cycle mean in a digraph. Discrete Math. 23(3), 309–311 (1978)
Kelso, A., Crawford, P.: Job matching, coalition formation, and gross substitutes. Econometrica 50(6), 1483–1504 (1982)
Milgrom, P.: Putting auction theory to work: the simultaneous ascending auction. J. Polit. Econ. 108, 245–272 (2000)
Samuelson, P.: A note on the pure theory of consumer’s behavior. Economica 5(17), 61–71 (1938)
Samuelson, P.: Consumption theory in terms of revealed preference. Economica 15(60), 243–253 (1948)
Varian, H.: Revealed preference. In: Szenberg, M., Ramrattand, L., Gottesman, A. (eds.) Samulesonian Economics and the \(21\)st Century, pp. 99–115. Oxford University Press, New York (2005)
Varian, H.: The nonparametric approach to demand analysis. Econometrica 50(4), 945–973 (1982)
Varian, H.: Goodness-of-fit in optimizing models. J. Econometrics 46, 125–140 (1990)
Varian, H.: Position auctions. Int. J. Ind. Organ. 25(6), 1163–1178 (2007)
Varian, H.: Revealed preference and its applications. Working paper (2011)
Vohra, R.: Mechanism Design: A Linear Programming Approach. Cambridge University Press, Cambridge (2011)
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Boodaghians, S., Vetta, A. (2015). The Combinatorial World (of Auctions) According to GARP. In: Hoefer, M. (eds) Algorithmic Game Theory. SAGT 2015. Lecture Notes in Computer Science(), vol 9347. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48433-3_10
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