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Topological connectedness and behavioral assumptions on preferences: a two-way relationship

  • M. Ali Khan
  • Metin UyanıkEmail author
Research Article

Abstract

This paper offers a comprehensive treatment of the question as to whether a binary relation can be consistent (transitive) without being decisive (complete), or decisive without being consistent, or simultaneously inconsistent or indecisive, in the presence of a continuity hypothesis that is, in principle, non-testable. It identifies topological connectedness of the (choice) set over which the continuous binary relation is defined as being crucial to this question. Referring to the two-way relationship as the Eilenberg–Sonnenschein (ES) research program, it presents four synthetic, and complete, characterizations of connectedness and its natural extensions and two consequences that stem from it. The six theorems make connections that have eluded decision theory and thereby generalize the pioneering work of Eilenberg, Sonnenschein, Schmeidler and Sen. We draw the relevance to several applied contexts, as well as to ongoing theoretical work.

Keywords

k-Connected k-Non-triviality Complete Transitive Fragile Flimsy 

Mathematics Subject Classification

91B55 37E05 

JEL Classification

C00 D00 D01 

Notes

References

  1. Alcantud, J., Gutiérrez, J.: Preference through indifference: a topological approach. J. Math. Econ. 31(4), 543–551 (1999)CrossRefGoogle Scholar
  2. Anand, P.: Are the preference axioms really rational? Theory Decis. 23(2), 189–214 (1987)CrossRefGoogle Scholar
  3. Anand, P.: The philosophy of intransitive preference. Econ. J. 103(417), 337–346 (1993)CrossRefGoogle Scholar
  4. Arrow, K.J., Debreu, G.: Existence of an equilibrium for a competitive economy. Econometrica 22(3), 265–290 (1954)CrossRefGoogle Scholar
  5. Arrow, K.J., Sen, A.K., Suzumura, K.: Social Choice Re-examined. vol. 1 and 2. St. Martin’s Press, New York (1995, 1996)Google Scholar
  6. Arrow, K.J., Sen, A.K., Suzumura, K.: Handbook of Social Choice and Welfare. vol. 1 and 2. Elsevier, North Holland (2002, 2011)Google Scholar
  7. Asheim, G.B., Bossert, W., Sprumont, Y., Suzumura, K.: Infinite-horizon choice functions. Econ. Theory 43(1), 1–21 (2010).  https://doi.org/10.1007/s00199-008-0423-z CrossRefGoogle Scholar
  8. Balasko, Y., Tvede, M.: General equilibrium without utility functions: how far to go? Econ. Theory 45(1–2), 201–225 (2010).  https://doi.org/10.1007/s00199-009-0496-3 CrossRefGoogle Scholar
  9. Barten, A.P., Böhm, V.: Consumer theory. In: Arrow, K., Intriligator, M. D. (eds.) Handbook of Mathematical Economics, vol. 2, pp. 381–429. North Holland (1982)Google Scholar
  10. Bergstrom, T.C., Parks, R.P., Rader, T.: Preferences which have open graphs. J. Math. Econ. 3(3), 265–268 (1976)CrossRefGoogle Scholar
  11. Bernheim, B.D., Rangel, A.: Beyond revealed preference: choice-theoretic foundations for behavioral welfare economics. Q. J. Econ. 124(1), 51–104 (2009)CrossRefGoogle Scholar
  12. Bleichrodt, H., Li, C., Moscati, I., Wakker, P.P.: Nash was a first to axiomatize expected utility. Theory Decis. 81(3), 309–312 (2016)CrossRefGoogle Scholar
  13. Bridges, D.S., Mehta, G.B.: Representations of Preference Orderings. Springer, Berlin (1995)CrossRefGoogle Scholar
  14. Broome, J.: Approximate equilibrium in economies with indivisible commodities. J. Econ. Theory 5(2), 224–249 (1972)CrossRefGoogle Scholar
  15. Brown, R.: Groupoids, the Phragmen–Brouwer property, and the Jordan curve theorem. J. Homotopy Relat. Struct. 1(1), 175–183 (2006)Google Scholar
  16. Brown, R., Camarena, O.A.: Erratum to: Groupoids, the Phragmen–Brouwer property, and the Jordan curve theorem. J. Homotopy Relat. Struct. 10(3), 669–672 (2015)CrossRefGoogle Scholar
  17. Carbonell-Nicolau, O., McLean, R.P.: Nash and Bayes–Nash equilibria in strategic-form games with intransitivities. Econ. Theory (2018).  https://doi.org/10.1007/s00199-018-1151-7 CrossRefGoogle Scholar
  18. Carmona, G.: Symposium on: existence of Nash equilibria in discontinuous games. Econ. Theory 48(1), 1–4 (2011).  https://doi.org/10.1007/s00199-010-0576-4 CrossRefGoogle Scholar
  19. Cerreia-Vioglio, S., Giarlotta, A., Greco, S., Maccheroni, F., Marinacci, M.: Rational preference and rationalizable choice. Econ. Theory (2018).  https://doi.org/10.1007/s00199-018-1157-1 CrossRefGoogle Scholar
  20. Chateauneuf, A.: Continuous representation of a preference relation on a connected topological space. J. Math. Econ. 16(2), 139–146 (1987)CrossRefGoogle Scholar
  21. Dasgupta, P., Maskin, E.: The existence of equilibrium in discontinuous economic games, I: theory, II: applications. Rev. Econ. Stud. 53(1), 1–41 (1986)CrossRefGoogle Scholar
  22. Debreu, G.: A social equilibrium existence theorem. Proc. Natl. Acad. Sci. 38(10), 886–893 (1952)CrossRefGoogle Scholar
  23. Debreu, G.: Chapter 11: Representation of a preference ordering by a numerical function. In: Thrall, M., Davis, R., Coombs, C. (eds.) Decision Processes, pp. 159–165. Wiley, New York (1954)Google Scholar
  24. Debreu, G.: Topological methods in cardinal utility theory. In: Arrow, K., Karlin, S., Suppes, P. (eds.) Mathematical Methods in the Social Sciences, pp. 16–26. Stanford University Press, California (1960)Google Scholar
  25. Debreu, G.: Existence of competitive equilibrium. In: Arrow, K., Intriligator, M. D. (eds.) Handbook of Mathematical Economics, vol. 2, pp. 697–743. North Holland (1982)Google Scholar
  26. Dickman Jr., R.F.: A strong form of the Phragmen–Brouwer theorem. Proc. Am. Math. Soc. 90(2), 333–337 (1984)Google Scholar
  27. Dierker, E.: Equilibrium analysis of exchange economies with indivisible commodities. Econometrica 39(6), 997–1008 (1971)CrossRefGoogle Scholar
  28. Dubra, J.: Continuity and completeness under risk. Math. Soc. Sci. 61(1), 80–81 (2011)CrossRefGoogle Scholar
  29. Dubra, J., Maccheroni, F., Ok, E.A.: Expected utility theory without the completeness axiom. J. Econ. Theory 115(1), 118–133 (2004)CrossRefGoogle Scholar
  30. Duggan, J.: Uncovered sets. Soc. Choice Welf. 41(3), 489–535 (2013)CrossRefGoogle Scholar
  31. Dugundji, J.: Topology. Allyn and Bacon, Boston (1966)Google Scholar
  32. Edelman, P.H., Jamison, R.E.: The theory of convex geometries. Geom. Dedic. 19(3), 247–270 (1985)CrossRefGoogle Scholar
  33. Eilenberg, S.: Ordered topological spaces. Am. J. Math. 63(1), 39–45 (1941)CrossRefGoogle Scholar
  34. Evren, Ö.: On the existence of expected multi-utility representations. Econ. Theory 35(3), 575–592 (2008).  https://doi.org/10.1007/s00199-007-0252-5 CrossRefGoogle Scholar
  35. Evren, Ö.: Scalarization methods and expected multi-utility representations. J. Econ. Theory 151, 30–63 (2014)CrossRefGoogle Scholar
  36. Fishburn, P.C.: Utility theory. Manag. Sci. 14(5), 335–378 (1968)CrossRefGoogle Scholar
  37. Fishburn, P.C.: Intransitive indifference in preference theory: a survey. Oper. Res. 18(2), 207–228 (1970)CrossRefGoogle Scholar
  38. Fishburn, P.C.: Mathematics of Decision Theory. Mouton, The Hauge (1972)Google Scholar
  39. Fleurbaey, M., Blanchet, D.: Beyond GDP: Measuring Welfare and Assessing Sustainability. Oxford University Press, Oxford (2013)CrossRefGoogle Scholar
  40. Fleurbaey, M., Salles, M., Weymark, J.: Justice, Political Liberalism, and Utilitarianism. Oxford University Press, Oxford (2006)Google Scholar
  41. Galaabaatar, T., Karni, E.: Subjective expected utility with incomplete preferences. Econometrica 81(1), 255–284 (2013)CrossRefGoogle Scholar
  42. Galaabaatar, T., Khan, M.A., Uyanık, M.: Completeness and transitivity of preferences on mixture sets. Math. Soc. Sci. 99, 49–62 (2019)CrossRefGoogle Scholar
  43. Gerasimou, G.: Consumer theory with bounded rational preferences. J. Math. Econ. 46(5), 708–714 (2010)CrossRefGoogle Scholar
  44. Gerasimou, G.: On continuity of incomplete preferences. Soc. Choice Welf. 41(1), 157–167 (2013)CrossRefGoogle Scholar
  45. Gerasimou, G.: (Hemi) continuity of additive preference preorders. J. Math. Econ. 58, 79–81 (2015)CrossRefGoogle Scholar
  46. Gerasimou, G.: Partially dominant choice. Econ. Theory 61(1), 127–145 (2016).  https://doi.org/10.1007/s00199-015-0869-8 CrossRefGoogle Scholar
  47. Gerasimou, G.: Indecisiveness, undesirability and overload revealed through rational choice deferral. Econ. J. 128, 2450–2479 (2017)CrossRefGoogle Scholar
  48. Gerasimou, G.: Dominance solvable multicriteria games with incomplete preferences. Econ. Theory Bull. (2018).  https://doi.org/10.1007/s40505-018-0159-2 CrossRefGoogle Scholar
  49. Gilboa, I.: Theory of Decision Under Uncertainty. Cambridge University Press, Cambridge (2009)CrossRefGoogle Scholar
  50. Gilboa, I., Lapson, R.: Aggregation of semiorders: intransitive indifference makes a difference. Econ. Theory 5(1), 109–126 (1995).  https://doi.org/10.1007/BF01213647 CrossRefGoogle Scholar
  51. Gorman, W.M.: The structure of utility functions. Rev. Econ. Stud. 35(4), 367–390 (1968)CrossRefGoogle Scholar
  52. Gorno, L.: The structure of incomplete preferences. Econ. Theory 66(1), 159–185 (2018).  https://doi.org/10.1007/s00199-017-1057-9 CrossRefGoogle Scholar
  53. Harsanyi, J.C.: Cardinal welfare, individualistic ethics, and interpersonal comparisons of utility. J. Political Econ. 63(4), 309–321 (1955)CrossRefGoogle Scholar
  54. He, W., Yannelis, N.C.: Existence of Walrasian equilibria with discontinuous, non-ordered, interdependent and price-dependent preferences. Econ. Theory 61(3), 497–513 (2016).  https://doi.org/10.1007/s00199-015-0875-x CrossRefGoogle Scholar
  55. Herstein, I.N., Milnor, J.: An axiomatic approach to measurable utility. Econometrica 21(2), 291–297 (1953)CrossRefGoogle Scholar
  56. Hervés-Beloso, C., del Valle-Inclán Cruces, H.: Continuous preference orderings representable by utility functions. J. Econ. Surv. 33(1), 179–194 (2019)CrossRefGoogle Scholar
  57. Ioannides, Y.M.: Topologies of social interactions. Econ. Theory 28(3), 559–584 (2006).  https://doi.org/10.1007/s00199-005-0646-1 CrossRefGoogle Scholar
  58. Jackson, M.O.: Socal and Economic Networks. Princeton University Press, Princeton (2008)Google Scholar
  59. Karni, E.: Axiomatic foundations of expected utility and subjective probability. In: Machina, M., Viscusi, K. (eds.) Handbook of the Economics of Risk and Uncertainty, vol. 1, pp. 1–39. Elsevier (2014)Google Scholar
  60. Khan, M.A., Sun, Y.: On a reformulation of Cournot–Nash equilibria. J. Math. Anal. Appl 146(2), 442–460 (1990)CrossRefGoogle Scholar
  61. Khan, M.A., Sun, Y.: Non-cooperative games with many players. In: Aumann, R., Hart, S. (eds.) Handbook of Game Theory with Economic Applications, vol. 3, pp. 1761–1808. North Holland (2002)Google Scholar
  62. Khan, M.A., Uyanık, M.: On the existence of continuous binary relations on a topological space. mimeo (2018)Google Scholar
  63. Khan, M.A., Uyanık, M.: The continuity postulate in economic theory: a deconstruction and an integration. mimeo (2019)Google Scholar
  64. Khan, M.A., Yamazaki, A.: On the cores of economies with indivisible commodities and a continuum of traders. J. Econ. Theory 24(2), 218–225 (1981)CrossRefGoogle Scholar
  65. Kim, T., Richter, M.K.: Nontransitive-nontotal consumer theory. J. Econ. Theory 38(2), 324–363 (1986)CrossRefGoogle Scholar
  66. Lorimer, P.: A note on orderings. Econometrica 35(3–4), 537–539 (1967)CrossRefGoogle Scholar
  67. Luce, R.D.: Semiorders and a theory of utility discrimination. Econometrica 24(2), 178–191 (1956)CrossRefGoogle Scholar
  68. Magyarkuti, G.: Revealed preferences: a topological approach. J. Math. Econ. 46(3), 320–325 (2010)CrossRefGoogle Scholar
  69. Majumdar, M., Sen, A.: A note on representing partial orderings. Rev. Econ. Stud. 43(3), 543–545 (1976)CrossRefGoogle Scholar
  70. Malinvaud, E.: Note on von Neumann–Morgenstern’s strong independence axiom. Econometrica 20(4), 679 (1952)CrossRefGoogle Scholar
  71. Marschak, J.: Rational behavior, uncertain prospects, and measurable utility. Econometrica 18(2), 111–141 (1950)CrossRefGoogle Scholar
  72. Mas-Colell, A.: An equilibrium existence theorem without complete or transitive preferences. J. Math. Econ. 1(3), 237–246 (1974)CrossRefGoogle Scholar
  73. Mas-Colell, A.: A model of equilibrium with differentiated commodities. J. Math. Econ. 2(2), 263–295 (1975)CrossRefGoogle Scholar
  74. Mas-Colell, A.: Indivisible commodities and general equilibrium theory. J. Econ. Theory 16(2), 443–456 (1977)CrossRefGoogle Scholar
  75. Mas-Colell, A., Zame, W.R.: Equilibrium theory in infinite dimensional spaces. In: Hildenbrand, W., Sonnenschein, H. (eds.) Handbook of Mathematical Economics, vol. 4, pp. 1835–1898 (1991)Google Scholar
  76. McGehee, R.: Attractors for closed relations on compact Hausdorff spaces. Indiana Univ. Math. J. 41(4), 1165–1209 (1992)CrossRefGoogle Scholar
  77. McKenzie, L.W.: The classical theorem on existence of competitive equilibrium. Econometrica 49(4), 819–841 (1981)CrossRefGoogle Scholar
  78. McKenzie, L.W.: Classical General Equilibrium Theory, vol. 1. The MIT Press, Cambridge (2005)Google Scholar
  79. Mehta, G.B.: Preference and utility. In: Barbera, S., Hammond, P. J., Seidl, C. (eds.) Handbook of Utility Theory: Principles, vol. 1, pp. 1–47. Kluwer Academic Publishers (1998)Google Scholar
  80. Moldau, J.H.: A simple existence proof of demand functions without standard transitivity. J. Math. Econ. 25(3), 325–333 (1996)CrossRefGoogle Scholar
  81. Narens, L.: Abstract Measurement Theory. MIT Press, Cambridge (1985)Google Scholar
  82. Nash, J.F.: The bargaining problem. Econometrica 18(2), 155–162 (1950a)CrossRefGoogle Scholar
  83. Nash, J.F.: Equilibrium points in n-person games. Proc. Natl. Acad. Sci. 36(1), 48–49 (1950b)CrossRefGoogle Scholar
  84. Neuefeind, W., Trockel, W.: Continuous linear representability of binary relations. Econ. Theory 6(2), 351–356 (1995).  https://doi.org/10.1007/BF01212495 CrossRefGoogle Scholar
  85. Newman, M.E.J.: Modularity and community structure in networks. Proc. Natl. Acad. Sci. USA 103(23), 8577–8582 (2006)CrossRefGoogle Scholar
  86. Peleg, B.: Utility functions for partially ordered topological spaces. Econometrica 38(1), 93–96 (1970)CrossRefGoogle Scholar
  87. Pfanzagl, J.: Theory of Measurement. Springer, Berlin (1971)CrossRefGoogle Scholar
  88. Qin, D.: Partially dominant choice with transitive preferences. Econ. Theory Bull. 5(2), 191–198 (2017)CrossRefGoogle Scholar
  89. Quah, J.K.: Weak axiomatic demand theory. Econ. Theory 29(3), 677–699 (2006).  https://doi.org/10.1007/s00199-005-0017-y CrossRefGoogle Scholar
  90. Rader, T.: The existence of a utility function to represent preferences. Rev. Econ. Stud. 30(3), 229–232 (1963)CrossRefGoogle Scholar
  91. Rébillé, Y.: Continuous utility on connected separable topological spaces. Econ. Theory Bull. 7, 147–153 (2019)CrossRefGoogle Scholar
  92. Reny, P.J.: On the existence of pure and mixed strategy Nash equilibria in discontinuous games. Econometrica 67(5), 1029–1056 (1999)CrossRefGoogle Scholar
  93. Reny, P.J.: Introduction to the symposium on discontinuous games. Econ. Theory 61(3), 423–429 (2016).  https://doi.org/10.1007/s00199-016-0962-7 CrossRefGoogle Scholar
  94. Richter, M., Rubinstein, A.: Back to fundamentals: Equilibrium in abstract economies. Am. Econ. Rev. 105(8), 2570–94 (2015)CrossRefGoogle Scholar
  95. Riella, G.: On the representation of incomplete preferences under uncertainty with indecisiveness in tastes and beliefs. Econ. Theory 58(3), 571–600 (2015).  https://doi.org/10.1007/s00199-015-0860-4 CrossRefGoogle Scholar
  96. Samuelson, P.A.: Bergsonian welfare economics. In: Rosefields, S. (ed.) Economic welfare and the economics of Soviet socialism: essays in honor of Abram Bergson, pp. 223–266. Cambridge University Press, New York (1981)CrossRefGoogle Scholar
  97. Schmeidler, D.: Competitive equilibria in markets with a continuum of traders and incomplete preferences. Econometrica 37(4), 578–585 (1969)CrossRefGoogle Scholar
  98. Schmeidler, D.: A condition for the completeness of partial preference relations. Econometrica 39(2), 403–404 (1971)CrossRefGoogle Scholar
  99. Sen, A.: Quasi-transitivity, rational choice and collective decisions. Rev. Econ. Stud. 36(3), 381–393 (1969)CrossRefGoogle Scholar
  100. Sen, A.: Internal consistency of choice. Econometrica 61(3), 495–521 (1993)CrossRefGoogle Scholar
  101. Sen, A.: Collective Choice and Social Welfare: An Expanded Edition. Harvard University Press, Massachusetts (2017)CrossRefGoogle Scholar
  102. Shafer, W.: The nontransitive consumer. Econometrica 42(5), 913–919 (1974)CrossRefGoogle Scholar
  103. Shafer, W., Sonnenschein, H.: Equilibrium in abstract economies without ordered preferences. J. Math. Econ. 2(3), 345–348 (1975)CrossRefGoogle Scholar
  104. Sonnenschein, H.: The relationship between transitive preference and the structure of the choice space. Econometrica 33(3), 624–634 (1965)CrossRefGoogle Scholar
  105. Sonnenschein, H.: Reply to “A note on orderings”. Econometrica 35(3, 4), 540 (1967)CrossRefGoogle Scholar
  106. Sonnenschein, H.: Demand theory without transitive indifference with applications to the theory of competitive equilibrium. In: Chipman, J., Hurwicz, L., Richter, M., Sonenschein, H. (eds.) Preferences, Utility and Demand: A Minnesota Symposium, pp. 215–234. Harcourt Brace Jovanovich, New York (1971)Google Scholar
  107. Strzalecki, T.: Temporal resolution of uncertainty and recursive models of ambiguity aversion. Econometrica 81(3), 1039–1074 (2013)CrossRefGoogle Scholar
  108. Suzumura, K.: Choice, Preferences and Procedures. Harvard University Press, Cambridge (2016)CrossRefGoogle Scholar
  109. Temkin, L.S.: Rethinking the Good: Moral Ideals and the Nature of Practical Reasoning. Oxford University Press, Oxford (2015)Google Scholar
  110. Thomson, W.: Fair allocation rules. In: Arrow, K., Sen, A., Suzumura, K. (eds.) Handbook of Social Choice and Welfare, vol. 2, pp. 393–506. Elsevier (2011)Google Scholar
  111. Tourky, R., Yannelis, N.C.: Markets with many more agents than commodities: Aumann’s “hidden” assumption. J. Econ. Theory 101(1), 189–221 (2000)CrossRefGoogle Scholar
  112. Tullock, G.: The irrationality of intransitivity. Oxf. Econ. Pap. 16(3), 401–406 (1964)CrossRefGoogle Scholar
  113. Ullman-Margalit, E., Morgenbesser, S.: Picking and choosing. Soc. Res. 44(4), 757–785 (1977)Google Scholar
  114. Uyanık, M.: Maximality in games and economies. Unpublished manuscript (2014)Google Scholar
  115. Uzawa, H.: Preference and Rational Choice in the Theory of Consumption. Stanford University Press, Palo Alto (1960)Google Scholar
  116. Vilkas, É.: Utility theory. J. Sov. Math. 13(4), 532–550 (1980)CrossRefGoogle Scholar
  117. Vind, K.: Independence, Additivity, Uncertainty. With Contributions by B. Grodal. Springer, Berlin (2003)CrossRefGoogle Scholar
  118. von Neumann, J., Morgenstern, O.: Theory of Games and Economic Behavior, 2nd edn. Princeton University Press, New Jersey (1947)Google Scholar
  119. Wakker, P.: The algebraic versus the topological approach to additive representations. J. Math. Psychol. 32(4), 421–435 (1988a)CrossRefGoogle Scholar
  120. Wakker, P.: Continuity of preference relations for separable topologies. Int. Econ. Rev. 29(1), 105–110 (1988b)CrossRefGoogle Scholar
  121. Wakker, P.P.: Additive Representations of Preferences: A New Foundation of Decision Analysis. Kluwer Academic Publishers, Boston (1989)CrossRefGoogle Scholar
  122. Ward, L.: Partially ordered topological spaces. Proc. Am. Math. Soc. 5(1), 144–161 (1954)CrossRefGoogle Scholar
  123. Wilder, R.L.: Topology of Manifolds. American Mathematical Society Colloquium Publications XXXII, Berlin (1949)CrossRefGoogle Scholar
  124. Yannelis, N.C., Prabhakar, N.D.: Existence of maximal elements and equilibria in linear topological spaces. J. Math. Econ. 12(3), 233–245 (1983)CrossRefGoogle Scholar
  125. Yannelis, N.C., Zame, W.R.: Equilibria in Banach lattices without ordered preferences. J. Math. Econ. 15(2), 85–110 (1986)CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Department of EconomicsJohns Hopkins UniversityBaltimoreUSA
  2. 2.School of EconomicsUniversity of QueenslandBrisbaneUSA

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