Combinatoric and Geometric Aspects of Some Probabilistic Choice Models — A Review

  • Jean-Paul Doignon
  • Jean-Claude Falmagne
  • Michel Regenwetter
Part of the Theory and Decision Library book series (TDLB, volume 40)


Two recent developments in random utility theory are reviewed, with special attention devoted to their combinatoric and geometric underpinnings. One concerns a new class of stochastic models describing the evolution of preferences, and the other some probabilistic models for approval voting. After recalling various commonly used preference relations, we discuss the fundamental property of ‘wellgradedness’ which is satisfied by certain important families of relations, such as the semiorder and the biorder families. The wellgradedness property plays a crucial role in the design of recent stochastic models of preference. Social choice, and approval voting in particular, provide natural arenas for the application of probabilistic models. We examine some partial results regarding the so-called ‘approval voting polytope’ which can be used for the characterization of a particular model of subset choices. We review several families of facets of this polytope and list some unsolved problems. An example illustrates how these geometric results help understand competing models of subset choice.

This paper reviews recent developments in two areas of random utility theory. One concerns a new class of stochastic models describing the evolution of preferences, and the other some probabilistic models for approval voting (cf. Doignon and Falmagne, 1994, Doignon and Falmagne, 1997, Doignon and Regenwetter, 1997, Doi-gnon and Regenwetter, in preparation, Falmagne, 1997, Falmagne and Doignon, 1997, Falmagne and Regenwetter, 1996, Falmagne, Regenwetter and Grofman, 1997, Regenwetter, 1996, Regenwetter, 1997, Regenwetter and Doignon, 1998, Regenwetter, Falmagne and Grofman, 1998, Regenwetter and Grofman, 1998a, Regenwetter and Grofman, 1998b, Regenwetter, Marley and Joe, 1998). Sections 1, 2 and 3 are devoted to the stochastic models and their combinatoric structure, and Sections 4 to 7 review some results on the geometric underpinnings of the approval voting model. Section 8 reviews related recent geometric structures, and the last section provides a conclusion and outlook.


Choice Model Linear Order Social Choice Probabilistic Choice Random Utility 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Arrow, K.J. (1951), Social Choice and Individual Values. Wiley, New-York Brams, S J., and Fishburn, P.C. (1983), Approval Voting. Birkhäuser, Boston.Google Scholar
  2. Christof, T. (software), “PORTA - A polyhedron representation transformation algorithm”. Available from Scholar
  3. Doignon, J.-P. (1988), “Sur les représentations minimales des semiordres et des ordres d’intervalles”, Mathématiques, Informatique et Sciences Humaines 101, 49–59.Google Scholar
  4. Doignon J.-P., Ducamp, A., and Falmagne, J.-C. (1984), “On Realizable Biorders and the Biorder Dimension of a Relation”, Journal of Mathematical Psychology 28, 73–109.CrossRefGoogle Scholar
  5. Doignon, J.-P., and Falmagne, J.-C. (1994), “A polynomial time algorithm for unidimensional unfolding representations”, Journal of Algorithms 16, 218–233.CrossRefGoogle Scholar
  6. Doignon, J.-P., and Falmagne, J.-C. (1997), “Well-graded families of relations”, Discrete Mathematics 173, 35–44.CrossRefGoogle Scholar
  7. Doignon, J.-P. and Falmagne, J.-C. (1998), Knowledge Spaces. Springer-Verlag, Heidelberg.Google Scholar
  8. Doignon, J.-P., and Regenwetter, M. (1997), “An approval-voting polytope for linear orders”, Journal of Mathematical Psychology 41, 171–188.CrossRefGoogle Scholar
  9. Doignon, J.-P., and Regenwetter, M. (in preparation), “On the combinatorial structure of the approval-voting polytope”, Draft.Google Scholar
  10. Ducamp, A., and Falmagne, J.-C. (1969), “Composite Measurement”, Journal of Mathematical Psychology 6, 359–390.CrossRefGoogle Scholar
  11. Dukhovny, A., Falmagne, J-C. and Ovchinnikov, S. (in preparation), “Media Theory”. Draft.Google Scholar
  12. Falmagne, J.-C. (1994), “The monks’ vote: A dialogue on unidimensional probabilistic geometry”. In Humphreys, P. (Ed.), Patrick Suppes, Mathematical Philosopher (Vol. 1, pp. 239–254 ). Kluwer, Amsterdam.Google Scholar
  13. Falmagne, J.-C. (1996), “A stochastic theory for the emergence and the evolution of preference structures”, Mathematical Social Sciences 31, 63–84.CrossRefGoogle Scholar
  14. Falmagne, J.-C. (1997), “Stochastic token theory”, Journal of Mathematical Psychology 41, 129–143.CrossRefGoogle Scholar
  15. Falmagne, J.-C., and Doignon, J.-P. (1997), “Stochastic evolution of rationality”, Theory and Decision 43, 107–138.CrossRefGoogle Scholar
  16. Falmagne, J.-C., and Regenwetter, M. (1996), “Random utility models for ap- proval voting”, Journal of Mathematical Psychology 40, 152–159.CrossRefGoogle Scholar
  17. Falmagne, J.-C., Regenwetter, M., and Grofman, B. (1997), “A stochastic model for the evolution of preferences”. In Marley, A. A. J. (Ed.), Choice, Decision and Measurement: Essays in Honor of R. Duncan Luce (pp. 113–131 ). Lawrence Erlbaum, Mahwah, NJ.Google Scholar
  18. Fiorini, S. (1997), “Le polytope des ordres partiels”. Mémoire de Licence, Université Libre de Bruxelles.Google Scholar
  19. Fishburn, P.C. (1970), “Intransitive indifference with unequal indifference intervals”, Journal of Mathematical Psychology 7, 144–149.CrossRefGoogle Scholar
  20. Fishburn, P. C. (1992), “Induced binary probabilities and the linear ordering polytope: A status report”, Mathematical Social Sciences 23, 67–80.CrossRefGoogle Scholar
  21. Fishburn, P.C., and Monjardet, B. (1992), “Norbert Wiener on the theory of measurement (1914, 1915, 1921)”, Journal of Mathematical Psychology 36, 165–184.CrossRefGoogle Scholar
  22. Grötschel, M., Jünger, M., and Reinelt, G. (1985), “Facets of the linear ordering polytope”, Mathematical Programming 33, 43–60.CrossRefGoogle Scholar
  23. Grünbaum, B. (1967), Convex Polytopes. Wiley, New York.Google Scholar
  24. Guttman, L. (1944), “A basis for scaling quantitative data”, American Sociological Review 9, 139–150.CrossRefGoogle Scholar
  25. Koppen, M. (1995), “Random utility representation of binary choice probabilities: Critical graphs yielding critical necessary conditions”, Journal of Mathematical Psychology 39, 21–39.CrossRefGoogle Scholar
  26. Luce, R. D. (1959), Individual Choice Behavior: A Theoretical Analysis. John Wiley, New York.Google Scholar
  27. Marley, A. A. J. (1993), “Aggregation theorems and the combination of probabilistic rank orders”. In Critchlow, D. E. and Fligner, M. A. and Verducci, J. S. (Eds.), Probability Models and Statistical Analyses for Ranking Data (pp. 216–240 ). Springer, New York.CrossRefGoogle Scholar
  28. Niederée, R., and Heyer, D. (1997), “Generalized random utility models and the representational theory of measurement: a conceptual link”. In Marley, A. A. J. (Ed.), Choice, Decision and Measurement: Essays in Honor of R. Duncan Luce (pp. 155–189 ). Lawrence Erlbaum, Mahwah, NJ.Google Scholar
  29. Pirlot, M. (1990), “Minimal representation of a semiorder”, Theory and Decision 28, 109–141.CrossRefGoogle Scholar
  30. Pirlot, M. (1991), “Synthetic description of a semiorder”, Discrete Applied Mathematics 31, 299–308.CrossRefGoogle Scholar
  31. Pirlot, M., and Vincke, Ph. (1997), Semiorders: Properties, Representations, Applications. Kluwer, Amsterdam.Google Scholar
  32. Regenwetter, M. (1996), “Random utility representations of finite m-ary relations”, Journal of Mathematical Psychology 40, 219–234.CrossRefGoogle Scholar
  33. Regenwetter, M. (1997), “Probabilistic preferences and topset voting”, Mathematical Social Sciences, 34, 91–105.CrossRefGoogle Scholar
  34. Regenwetter, M., and Doignon, J.-P. (1998), “The choice probabilities of the latent-scale model satisfy the size-independent model when n is small”, Journal of Mathematical Psychology 42, 102–106.CrossRefGoogle Scholar
  35. Regenwetter, M., Falmagne, J.-C., and Grofman, B. (1998), “A stochastic model of preference change and its application to 1992 presidential election panel data”. Psychological Review.Google Scholar
  36. Regenwetter, M., and Grofman, B. (1998a), “Choosing subsets: A size-independent random utility model and the quest for a social welfare ordering”, Social Choice and Welfare 15, 423–443.Google Scholar
  37. Regenwetter, M., and Grofman, B. (1998b), “Approval voting, Borda winners and Condorcet winners: evidence from seven elections”, Management Science 44, 520–533.CrossRefGoogle Scholar
  38. Regenwetter, M., Marley, A.A.J., and Joe, H. (1998), “Random utility threshold models of subset choice”, Australian Journal of Psychology 50, 175–185.Google Scholar
  39. Reinelt, G. (1985), The Linear Ordering Problem: Algorithms and Applications. Research and Exposition in Mathematics No. 8, Heldermann Verlag, Berlin.Google Scholar
  40. Riguet, J. (1951), “Les relations de Fermis”, Comptes Rendus des Séances de l’Académie des Sciences (Paris) 232, 1729–1730.Google Scholar
  41. Roberts, F. S. (1979), Measurement Theory, with applications to decision making, utility and the social sciences. Encyclopedia of Mathematics and its Applications, Gian-Carlo Rota (Ed.), Vol. 7: Mathematics and the Social Sciences. Addison-Wesley, Reading, Mass.Google Scholar
  42. Scott, D., and Suppes, P. (1958), “Foundational aspects of theories of measurement”, Journal of Symbolic Logic 23, 113–128.CrossRefGoogle Scholar
  43. Suck, R. (1992), “Geometric and combinatorial properties of the polytope of binary choice probabilities”, Mathematical Social Sciences 23, 81–102.CrossRefGoogle Scholar
  44. Suck, R. (1996), “The equivalence relation polytope and random classification and clustering”. In Diday, E., Lechevallier, Y., Opitz, O. (Eds.), Ordinal and Symbolic Data Analysis (pp. 351–358 ). Springer, Berlin, Heidelberg.CrossRefGoogle Scholar
  45. Suck, R. (in preparation), “Random utility representations based on semi orders, interval orders, and partial orders”. Manuscript at Dept. of Psychology, Universität Osnabrück.Google Scholar
  46. Trotter, W.T. (1992), Combinatorics and Partially Ordered Sets: Dimension Theory. The Johns Hopkins University Press, Baltimore, Maryland.Google Scholar
  47. Wahl, N. (1998), “Extension au cas infini de la théorie des familles bien graduées et des géométries convexes”. Mémoire de Licence, Université Libre de Bruxelles.Google Scholar
  48. Wiener, N. (1914), “A contribution to the theory of relative position”, Proceedings of the Cambridge Philosophical Society 17, 441–449.Google Scholar
  49. Ziegler, G. (1995), Lectures on Polytopes. Springer-Verlag, Heidelberg.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1999

Authors and Affiliations

  • Jean-Paul Doignon
    • 1
  • Jean-Claude Falmagne
    • 2
  • Michel Regenwetter
    • 3
  1. 1.Département de MathématiquesUniversité Libre de BruxellesBelgium
  2. 2.Department of Cognitive ScienceUniversity of CaliforniaIrvineUSA
  3. 3.Fuqua School of BusinessDuke UniversityUSA

Personalised recommendations