Préférences et rationalité stochastiques

Bouyaux, Pascal

Mathématiques informatique et sciences humaines, Tome 109 (1990), pp. 17-40.

Le but de cet article est de procéder à une présentation pédagogique d'un concept étendu de rationalité, la rationalité stochastique. Dans une première partie, nous exposons le problème à l'aide d'un exemple simple et posons un ensemble de définitions préliminaires. Puis, dans une seconde partie, nous présentons le résultat fondamental de Falmagne (1978) s'appliquant aux situations de choix multiples ; l'approche ensembliste de cet auteur est formalisée à partir du concept de polynômes de Block-Marschak (1960). Dans une troisième partie, nous traitons du cas particulier des choix binaires, développant ainsi l'approche géométrique de Mac-Fadden-Richter (1970) reposant sur le concept de permutoèdre ; les applications potentielles de cette approche en théorie du vote, optimisation combinatoire et analyse des données ordinales sont brièvement évoquées. La dernière partie conclut.

The aim of this article is to undertake a pedagogical presentation of a wide-spread concept of rationality, the stochastic rationality. In a first part, we set the problem with the help of a simple example and we put a set of preliminary definitions. Then, in a second part, we expound the basic result of Falmagne (1978) applied to multiple choices situations ; the set approach of this author is formalized with the Block-Marshak (1960) polynomials. In a third part, we deal with the particular case of binary choices, expanding the geometrical approach of Mac-Fadden-Richter (1970) based on a permutation polytope ; the potentials applications of this approach in voting theory, combinatorial optimization and ordinal data analysis are briefly described. The last part concludes.

[1].Aizerman M.A. (1985), "New Problems in General Choice Theory" ; Social choice and welfare, Vol 2.,1985, 235-282. | MR | Zbl

[2].Arrow K.J. (1959), "Rational choice function and ordering", Economica, 26, (1959a), 121-127. Social choice and Individual Values, New York, Wiley, 1959b.

[3].Barbera S. (1985), "Rationalizable Stochastic Choice on Restricted domains", 5th world congress of the econometric society, Cambridge, Mass, 1985.

[4].Barbera S., Pattanaik P.K. (1986), "Falmagne and the Rationalizability of Stochastic Choices in Terms of Random Orderings", Econometrica, 54,1986, 707-715. | MR | Zbl

[5].Barbera S., Sonnenschein H. (1978), "Preference Aggregation with Randomized Social Orderings", Journal of Economic Theory, 18, 1978, 244-154. | MR | Zbl

[6].Barbera S., Valenciano F. (1983), "Collective Probabilistic Judgements", Econometrica, 51,1983,1033-1046. | MR | Zbl

[7].Barret C., Pattanaik P., Salles M. (1987), On Choosing Rationally when preferences arefuzzy,. Mimeograph, University of Binnimgham and CREMERC, 1987.

[8].Barthélemy J.P., Monjardet B. (1981), "The Median Procedure in Cluster Analysis and Social Choice Theory", Mathematical Social Sciences, 1, 1981, 235-267. | MR | Zbl

[9].Barthélemy J.P., Monjardet B. (1988), "The Median Procedure in Data Analysis : new results and open problems", in classification and related methods of data analysis, HH Bock Editor, Elsevier Science Publishers, North-Holland, 1988, 309-316. | MR

[10].Block H., Marschak J. (1960), "Random Orderings and Stochastic Theories of Response ". Chapter 10 in contributions to probability and statistics, edited by J. Olkin, Stanford: Stanford University Press, 1960. | MR | Zbl

[11].Bowman V.J. (1972), "Permutation Polyhedra ", Siam Journal of Applied Mathematics, 22, n°4,1972, 580-589. | MR | Zbl

[12]. Bouyaux P., "Une difficulté d'interprétation de l'Approche Logit : l'exemple de l'économie des transports", Économie et prévision n°91, à paraître.

[13].Chipman J., Hurwicz L., Richter M., Sonnenschein H. (1971), Preferences, utility and demand : a minnesota symposium, Harcourt Brace Jovanovich, New York, Chicago, San-Francisco, Atlanta, 1971. | MR | Zbl

[14].Cohen M., Falmagne J.C. (1978), Random Scale Représentation of Binary Choice Probabilities : a counterexample to a conjecture of Marschak, Mimeograph, department of Psychology, New York University, 1978.

[15].Cohen M., Falmagne J.C. (1989), Random Utility Représentation of Binary Choice Probabilities : a new class of necessary conditions. Mimeograph, department of Psychology, New York University, 1989.

[16].Debord B. (1987), "Caractérisation des matrices des préférences nettes et méthodes d'agrégation associées", Mathématiques et Sciences humaines, n°97, 1987, 5-17. | Numdam | MR | Zbl

[17].Debreu G. (1958), "Stochastic Choice and Cardinal Utility", Econometrica, 26, 1958, 440-444. | Zbl

[18] De Palma A., et Trisse J. (1989), "Les modèles de choix discrets, Annales d'Economie et de Statistique, vol.9,1989,151-190. | MR

[19].Dridi T. (1980), "Sur les Distributions Binaires Associées à des Distributions Ordinales", Mathématiques et Sciences humaines, n°69,1980,15-31. | Numdam | MR | Zbl

[20] Doignon J.P., Monjardet B., Roubens M., Vincke Ph., (1986), "Biorder Families, Valued Relations, and Préférence Modelling", Journal of Mathematical Psychology, 30, n°4,1986, 435-480. | MR | Zbl

[21].Falmagne J.C. (1978), "A représentation theorem for finite random scale systems", Journal of Mathematical Psychology, 18, 1978, 52-72. | MR | Zbl

[22].Fechner G. (1859), Elemente der Psychophysik, Leipzig, Breitkopf und Härtel, 1859.

[23].Fishburn P. (1973), "Binary Choice Probabilities : on the varieties of stochastic transitivity", Journal of Mathematical Psychology, 10, 1973, 327-352. | MR | Zbl

[24].Fishburn P. (1978), "Choice Probabilities and choice functions", Journal of Mathematical Psychology, 18, 1978, 205-219. | MR | Zbl

[25].Fishburn P. (1987), "Decomposing Weighted Digraphs into Sums of Chains", Discrete Applied Mathematics, 16, 1987, 223-238. | MR | Zbl

[26].Fishburn P.(1988), Binary Probabilities induced par rankings, Mimeograph, Bell Laboratories, Murray Hill, New-Jersey and New York University, 1988.

[27].Fishburn P., Falmagne J.C. (1989), "Binary choice probabilities and rankings", Economic Letters, 31,1989,113-117. | MR

[28].Georgescu-Roegen N. (1958), "Threshold in choice and the theory of demand", Econometrica, 26, 1958, 157-168. | MR | Zbl

[29].Gilboa I. (1985), A necessary but insufficient condition for the stochastic order problem, Mineograph, Faculty of Social Sciences, Tel Aviv University, 1985.

[30].Gblboa I. (1989), A necessary but insufficient condition for the stochastic binary choice problem, Mimeograph, North Western University, 1989.

[31].Guilbaud G.Th. (1953), "Sur une difficulté de la théorie du risque" in Colloques internationaux du CNRS, Paris, 12-17 mai 1952, Editions du CNRS, 1953, 19-28. | MR | Zbl

[32].Guilbaud G.Th. (1970), "Préférences Stochastiques ", Mathématiques et Sciences humaines, n°32, 1970, 45-56. | Numdam | MR | Zbl

[33].Hausman J., Wise D. (1978), "A conditional probit model for qualitative choice : discrete décisions recognizing interdépendance ans heterogeneous preferences", Econometrica, 46, 1978, 403-426. | MR | Zbl

[34].Houthakker H. (1950), "Revealed préférence and the utility functions", Economica, 17, 1950,159-174. | MR

[35].Luce R.D. (1959), Individual choice behavior, New York, Wiley,1959. | MR | Zbl

[36].Luce R.D., Suppes P. (1965), "Preference, utility and subjective probability ", Chapter 19 in Handbook of Mathematical Psychology, edited by Luce, Bush, and Galanter, Wiley, 1965.

[37].Mac-Fadden D. (1974), "Conditional logit analysis of qualitative choice behavior", Chapter 4 in Frontiers in Econometrics, edited by P. Zarembka, New York, Academic Press, 1974.

[38].Mac-Fadden D. (1976), "Quantal choice analysis : a survey", Annals of Economic and Social Measurement, 1976, 363-390.

[39].Mac-Fadden D. (1981), "Econometric models of probabilistic choice", Chapter 5, in Structural Analysis of Discrete Data, edited by C. Manski et D.Mac-Fadden, Cambridge, Mass, MIT Press, 1981.

[40].Mac-Fadden D., Richter (1970), Stochastic rationality and revealed stochastic preference, Mineograph, University of California,1970.

[41].Machina M. (1985), "Stochastic choice functions generated from deterministic preferences over lotteries", Economic Journal, n°95,1985, 575-594.

[42].Mac-Lennan A. (1980), "Randomized preference aggregation : additivity of power and strategy proofness", Journal of Economic Theory, 22, 1980,1-11. | MR | Zbl

[43].Mac-Lennan A. (1983), Binary stochastic choice, Mimeograph, University of Minnesota, 1983.

[44].Marley J.A.A. (1965), "The relation between the discard and regularity conditions for choice probabilities", Journal of Mathematical Psychology, 2, 1965, 242-253. | MR | Zbl

[45].Marschak J. (1960), "Binary choice constraints and random utility indicators", in Mathematical Methods in the Social Sciences, edited by K.J.Arrow et als, Stanford, California: Stanford University Press, 1960. | MR

[46].Megiddo N. (1977), "Mixtures of order matrices and generalized order matrices", Discrete Mathematics, 19, 1977, 177-181. | MR | Zbl

[47].Mimiague F., Rousseau J.M. (1977), "Effet Condorcet : typologie et calculs de fréquence", Mathématiques et Sciences humaines, 43,1977, .7-27. | Numdam | MR

[48].Monjardet B. (1973), "Tournois et ordres médians pour une opinion", Mathématiques et Sciences humaines, 43, 1973, 55-70. | Numdam | MR | Zbl

[49].Monjardet B. (1985), "Concordance et consensus d'ordre totaux : les coefficients K et W", Revue de Statistique Appliquée, 33, n°2,1985, 55-87. | Numdam | MR | Zbl

[50].Mossin A. (1985), "Element of a stochastic theory of consumption", Swedish Journal of Economics, 70, 1985, 200-220.

[51].Moulin H. (1985), "Choice functions over a finite set : a summary ", Social Choice and Welfare, 2, 1985, 147-160. | MR | Zbl

[52].Pattanaik P.K., Peleg B. (1986), "Distribution of power under stochastic choice rules", Econometrica, 54, 1986, 909-921. | MR | Zbl

[53].Quandt R. (1956), "A probabilistic theory of consumer behavior", Quaterly Journal of Economics, 70, 1956, 507-536.

[54].Reinelt G. (1985), The linear ordering problem : algorithms and applications, Heldermann verlag, Berlin,1985. | MR | Zbl

[55].Richter M.K. (1966), "Revealed perference theory ", Econometrica, 34, 1966, 635-645. | Zbl

[56] Roberts F., (1979), Measurement Theory, with Applications to Decision-making, Utility, and the Social Sciences, Encyclopedia of Mathematics and its Applications, Vol.7, Reading, MA, Addison-Wesley, 1979. | MR | Zbl

[57].Samuelson S. (1948), "Consumption theory in terms of revealed preferences", Economica, 15, 1948, 243-253.

[58].Sen S.(1971), "Choice functions and revealed preferences", Review of Economic Studies, 38, 1971, 307-317. | Zbl

[59].Souza F.M.C. (1978), "Mixed models, random utilities, and the triangle inequality", Journal of Mathematical Psychology, 27,1978,183-200. | MR | Zbl

[60].Stigler G.S., Becker G.S. (1977), "De Gustibus non est disputandum", American Economic Review, 67, 1977, 76-90.

[61].Thurstone L. (1927), "A law of comparative judgement", Psychological Review, 34, (1927), 273-286.

[62].Tversky A.(1972), "Elimination by aspects : a theory of choice", Psychological Review, 79, 1972, 281-289.

[63].Tversky A., Russo J.E. (1969), "Substituability and similarity in binary choices", Journal of Mathematical Psychology, 6,1969, .1-12.

[64].Young H.P. (1978), "On permutations and permutation polytopes" Mathematical Programming Study, 8, 1978, 128-140. | MR | Zbl