Abstract
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.
J.-P.D.’s work was partially conducted during a stay at the Institute of Mathematical Behavioral Sciences of the University of California, Irvine. This stay was supported by NSF grant No. SBR 93-07423 to J.-C.F.
J.-C.F.’s work in this area is supported by NSF grant No. SBR 93-07423 to the University of California, Irvine.
M.R. thanks NSF for grant No. SBR 97-30076 which partially supported this work.
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References
Arrow, K.J. (1951), Social Choice and Individual Values. Wiley, New-York Brams, S J., and Fishburn, P.C. (1983), Approval Voting. Birkhäuser, Boston.
Christof, T. (software), “PORTA - A polyhedron representation transformation algorithm”. Available from ftp://ftp.zib-berlin.de/pub/mathprog/polyth/porta/index.html.
Doignon, J.-P. (1988), “Sur les représentations minimales des semiordres et des ordres d’intervalles”, Mathématiques, Informatique et Sciences Humaines 101, 49–59.
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.
Doignon, J.-P., and Falmagne, J.-C. (1994), “A polynomial time algorithm for unidimensional unfolding representations”, Journal of Algorithms 16, 218–233.
Doignon, J.-P., and Falmagne, J.-C. (1997), “Well-graded families of relations”, Discrete Mathematics 173, 35–44.
Doignon, J.-P. and Falmagne, J.-C. (1998), Knowledge Spaces. Springer-Verlag, Heidelberg.
Doignon, J.-P., and Regenwetter, M. (1997), “An approval-voting polytope for linear orders”, Journal of Mathematical Psychology 41, 171–188.
Doignon, J.-P., and Regenwetter, M. (in preparation), “On the combinatorial structure of the approval-voting polytope”, Draft.
Ducamp, A., and Falmagne, J.-C. (1969), “Composite Measurement”, Journal of Mathematical Psychology 6, 359–390.
Dukhovny, A., Falmagne, J-C. and Ovchinnikov, S. (in preparation), “Media Theory”. Draft.
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.
Falmagne, J.-C. (1996), “A stochastic theory for the emergence and the evolution of preference structures”, Mathematical Social Sciences 31, 63–84.
Falmagne, J.-C. (1997), “Stochastic token theory”, Journal of Mathematical Psychology 41, 129–143.
Falmagne, J.-C., and Doignon, J.-P. (1997), “Stochastic evolution of rationality”, Theory and Decision 43, 107–138.
Falmagne, J.-C., and Regenwetter, M. (1996), “Random utility models for ap- proval voting”, Journal of Mathematical Psychology 40, 152–159.
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.
Fiorini, S. (1997), “Le polytope des ordres partiels”. Mémoire de Licence, Université Libre de Bruxelles.
Fishburn, P.C. (1970), “Intransitive indifference with unequal indifference intervals”, Journal of Mathematical Psychology 7, 144–149.
Fishburn, P. C. (1992), “Induced binary probabilities and the linear ordering polytope: A status report”, Mathematical Social Sciences 23, 67–80.
Fishburn, P.C., and Monjardet, B. (1992), “Norbert Wiener on the theory of measurement (1914, 1915, 1921)”, Journal of Mathematical Psychology 36, 165–184.
Grötschel, M., Jünger, M., and Reinelt, G. (1985), “Facets of the linear ordering polytope”, Mathematical Programming 33, 43–60.
Grünbaum, B. (1967), Convex Polytopes. Wiley, New York.
Guttman, L. (1944), “A basis for scaling quantitative data”, American Sociological Review 9, 139–150.
Koppen, M. (1995), “Random utility representation of binary choice probabilities: Critical graphs yielding critical necessary conditions”, Journal of Mathematical Psychology 39, 21–39.
Luce, R. D. (1959), Individual Choice Behavior: A Theoretical Analysis. John Wiley, New York.
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.
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.
Pirlot, M. (1990), “Minimal representation of a semiorder”, Theory and Decision 28, 109–141.
Pirlot, M. (1991), “Synthetic description of a semiorder”, Discrete Applied Mathematics 31, 299–308.
Pirlot, M., and Vincke, Ph. (1997), Semiorders: Properties, Representations, Applications. Kluwer, Amsterdam.
Regenwetter, M. (1996), “Random utility representations of finite m-ary relations”, Journal of Mathematical Psychology 40, 219–234.
Regenwetter, M. (1997), “Probabilistic preferences and topset voting”, Mathematical Social Sciences, 34, 91–105.
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.
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.
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.
Regenwetter, M., and Grofman, B. (1998b), “Approval voting, Borda winners and Condorcet winners: evidence from seven elections”, Management Science 44, 520–533.
Regenwetter, M., Marley, A.A.J., and Joe, H. (1998), “Random utility threshold models of subset choice”, Australian Journal of Psychology 50, 175–185.
Reinelt, G. (1985), The Linear Ordering Problem: Algorithms and Applications. Research and Exposition in Mathematics No. 8, Heldermann Verlag, Berlin.
Riguet, J. (1951), “Les relations de Fermis”, Comptes Rendus des Séances de l’Académie des Sciences (Paris) 232, 1729–1730.
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.
Scott, D., and Suppes, P. (1958), “Foundational aspects of theories of measurement”, Journal of Symbolic Logic 23, 113–128.
Suck, R. (1992), “Geometric and combinatorial properties of the polytope of binary choice probabilities”, Mathematical Social Sciences 23, 81–102.
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.
Suck, R. (in preparation), “Random utility representations based on semi orders, interval orders, and partial orders”. Manuscript at Dept. of Psychology, Universität Osnabrück.
Trotter, W.T. (1992), Combinatorics and Partially Ordered Sets: Dimension Theory. The Johns Hopkins University Press, Baltimore, Maryland.
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.
Wiener, N. (1914), “A contribution to the theory of relative position”, Proceedings of the Cambridge Philosophical Society 17, 441–449.
Ziegler, G. (1995), Lectures on Polytopes. Springer-Verlag, Heidelberg.
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Doignon, JP., Falmagne, JC., Regenwetter, M. (1999). Combinatoric and Geometric Aspects of Some Probabilistic Choice Models — A Review. In: Machina, M.J., Munier, B. (eds) Beliefs, Interactions and Preferences in Decision Making. Theory and Decision Library, vol 40. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-4592-4_12
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