Testing Consumer Rationality Using Perfect Graphs and Oriented Discs

  • Shant Boodaghians
  • Adrian Vetta
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9470)


Given a consumer data-set, the axioms of revealed preference proffer a binary test for rational behaviour. A natural (non-binary) measure of the degree of rationality exhibited by the consumer is the minimum number of data points whose removal induces a rationalisable data-set. We study the computational complexity of the resultant consumer rationality problem in this paper. This problem is, in the worst case, equivalent (in terms of approximation) to the directed feedback vertex set problem. Our main result is to obtain an exact threshold on the number of commodities that separates easy cases and hard cases. Specifically, for two-commodity markets the consumer rationality problem is polynomial time solvable; we prove this via a reduction to the vertex cover problem on perfect graphs. For three-commodity markets, however, the problem is NP-complete; we prove this using a reduction from planar 3-sat that is based upon oriented-disc drawings.


  1. 1.
    Afriat, S.: The construction of a utility function from expenditure data. Int. Econ. Rev. 8, 67–77 (1967)CrossRefzbMATHGoogle Scholar
  2. 2.
    Afriat, S.: On a system of inequalities in demand analysis: an extension of the classical method. Int. Econ. Rev. 14, 460–472 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Apesteguia, J., Ballester, M.: A measure of rationality and welfare. Journal of Political Economy (2015, to appear)Google Scholar
  4. 4.
    Berge, C.: Färbung von Graphen deren sämtliche beziehungsweise deren ungerade Kreise starr sind (Zusammenfassung). Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg Math.-Natur. Reihe 10, 114–115 (1961)Google Scholar
  5. 5.
    Chudnovsky, M., Robertson, N., Seymour, P., Thomas, R.: The strong perfect graph theorem. Ann. Math. 164, 51–229 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Dean, M., Martin, D.: Measuring rationality with the minimum cost of revealed preference violations. Review of Economics and Statistics (2015, to appear)Google Scholar
  7. 7.
    Deb, R., Pai, M.: The geometry of revealed preference. J. Math. Econ. 50, 203–207 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Dinur, I., Safra, S.: On the hardness of approximating minimum vertex cover. Ann. Math. 162(1), 439–485 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Earl, R.: Geometry II: 3.1 Stereographic Projection and the Riemann Sphere (2007).
  10. 10.
    Echenique, F., Lee, S., Shum, M.: The money pump as a measure of revealed preference violations. J. Polit. Econ. 119(6), 1201–1223 (2011)CrossRefGoogle Scholar
  11. 11.
    Gross, J.: Testing data for consistency with revealed preference. Rev. Econ. Stat. 77(4), 701–710 (1995)CrossRefGoogle Scholar
  12. 12.
    Grötschel, M., Lovász, L., Schrijver, A.: Polynomial algorithms for perfect graphs. Ann. Discret. Math. 21, 325–356 (1984)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Grötschel, M., Lovász, L., Schrijver, A.: Geometric Algorithms and Combinatorial Optimisation. Springer-Verlag, Berlin (1988)CrossRefzbMATHGoogle Scholar
  14. 14.
    Guruswami, V., Hastad, J., Manokaran, R., Raghavendra, P., Charikar, M.: Beating the random ordering is hard: every ordering CSP is approximation resistant. SIAM J. Comput. 40(3), 878–914 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Famulari, M.: A household-based, nonparametric test of demand theory. Rev. Econ. Stat. 77, 372–383 (1995)CrossRefGoogle Scholar
  16. 16.
    Heufer, J.: A geometric approach to revealed preference via Hamiltonian cycles. Theor. Decis. 76(3), 329–341 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Houthakker, H.: Revealed preference and the utility function. Economica New Ser. 17(66), 159–174 (1950)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Houtman, M., Maks, J.: Determining all maximal data subsets consistent with revealed preference. Kwantitatieve Methoden 19, 89–104 (1950)Google Scholar
  19. 19.
    Karp, R.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations, pp. 85–103. Plenum, New York (1972)CrossRefGoogle Scholar
  20. 20.
    Khot, S.: On the power of unique 2-prover 1-round games. In: Proceedings of STOC, pp 767–775 (2002)Google Scholar
  21. 21.
    Koo, A.: An emphirical test of revealed preference theory. Econometrica 31(4), 646–664 (1963)CrossRefGoogle Scholar
  22. 22.
    Lovász, L.: Normal hypergraphs and the perfect graph conjecture. Discret. Math. 2(3), 253–267 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Rose, H.: Consistency of preference: the two-commodity case. Rev. Econ. Stud. 25, 124–125 (1958)CrossRefGoogle Scholar
  24. 24.
    Samuelson, P.: A note on the pure theory of consumer’s behavior. Economica 5(17), 61–71 (1938)CrossRefGoogle Scholar
  25. 25.
    Samuelson, P.: Consumption theory in terms of revealed preference. Economica 15(60), 243–253 (1948)CrossRefGoogle Scholar
  26. 26.
    Seymour, P.: Packing directed circuits fractionally. Combinatorica 15(2), 281–288 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Svensson, O.: Hardness of vertex deletion and project scheduling. In: Gupta, A., Jansen, K., Rolim, J., Servedio, R. (eds.) APPROX 2012 and RANDOM 2012. LNCS, vol. 7408, pp. 301–312. Springer, Heidelberg (2012) CrossRefGoogle Scholar
  28. 28.
    Swafford, J., Whitney, G.: Nonparametric test of utility maximization and weak separability for consumption, leisure and money. Rev. Econ. Stat. 69, 458–464 (1987)CrossRefGoogle Scholar
  29. 29.
    Varian, H.: Revealed preference. In: Szenberg, M., et al. (eds.) Samulesonian Economics and the 21st Century, pp. 99–115. Oxford University Press, New York (2005)Google Scholar
  30. 30.
    Varian, H.: Goodness-of-fit in optimizing models. J. Econometrics 46, 125–140 (1990)CrossRefGoogle Scholar
  31. 31.
    Wang, D., Kuo, Y.: A study on two geometric location problems. Inf. Process. Lett. 28(6), 281–286 (1988)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Open Access This chapter is distributed under the terms of the Creative Commons Attribution Noncommercial License, which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsMcGill UniversityMontrealCanada
  2. 2.Department of Mathematics and Statistics, and School of Computer ScienceMcGill UniversityMontrealCanada

Personalised recommendations