Oligarchy and soft incompleteness

Piggins, Ashley and Duddy, Conal (2016): Oligarchy and soft incompleteness.

Abstract

The assumption that the social preference relation is complete is demanding. We distinguish between “hard” and “soft” incompleteness, and explore the social choice implications of the latter. Under soft incompleteness, social preferences can take values in the unit interval. We motivate interest in soft incompleteness by presenting a version of the strong Pareto rule that is suited to the context of a [0, 1]-valued social preference relation. Using a novel approach to the quasi-transitivity of this relation we prove a general oligarchy theorem. Our framework allows us to make a distinction between a “strong” and a “weak” oligarchy, and our theorem identifies when the oligarchy must be strong and when it can be weak. Weak oligarchy need not be undesirable.

• Item Type: MPRA Paper
• Language: English
• Keywords: Oligarchy; Gibbard’s theorem; Incompleteness; Max-star transitivity
• Subjects: D - Microeconomics > D7 - Analysis of Collective Decision-Making > D71 - Social Choice ; Clubs ; Committees ; Associations

References

Arrow, K.J., 1951, second ed. 1963. Social choice and individual values. Wiley, New York.

Aumann, R.J., 1962. Utility theory without the completeness axiom, Econometrica, 30, 445-462.

Banerjee, A., 1993. Rational choice under fuzzy preferences: the Orlovsky choice function, Fuzzy Sets and Systems 53, 295-299.

Banerjee, A., 1994. Fuzzy preferences and Arrow-type problems in social choice, Social Choice and Welfare 11, 121-130.

Banks, J.S., 1995. Acyclic social choice from finite sets, Social Choice and Welfare 12, 293-310.

Barberá, S., 1980. Pivotal voters: a new proof of Arrow’s theorem, Economics Letters 6, 13-16.

Barrett, C.R., Pattanaik, P.K. and M. Salles, 1986. On the structure of fuzzy social welfare functions, Fuzzy Sets and Systems 19, 1-10.

Barrett, R. and M. Salles, 2011. Social choice with fuzzy preferences in: K.J. Arrow, A. Sen and K. Suzumura, eds., Handbook of Social Choice and Welfare, vol. 2, North-Holland, Amsterdam.

Basu, K., 1984. Fuzzy revealed preference, Journal of Economic Theory 32, 212-227.

Billot, A., 1995. Economic theory of fuzzy equilibria. Springer, Berlin. 42

Blair, D.H. and R.A. Pollak, 1982. Acyclic collective choice rules, Economet- rica 50, 931-943.

Bossert, W. and K. Suzumura, 2008. A characterization of consistent collective choice rules, Journal of Economic Theory 138, 311-320.

Bossert, W. and K. Suzumura, 2010. Consistency, choice, and rationality. Harvard University Press, Cambridge.

Broome, J., 1997. Is incommensurability vagueness? in: R. Chang, ed., Incommensurability, incomparability, and practical reason, Harvard University Press, Cambridge.

Brown, D.J., 1975. Aggregation of preferences, Quarterly Journal of Eco- nomics 89, 456-469.

Campbell, D.E. and J.S. Kelly, 2002. Impossibility theorems in the Arrovian framework in: K.J. Arrow, A. Sen and K. Suzumura, eds., Handbook of Social Choice and Welfare, vol. 1, North-Holland, Amsterdam.

Chang, R., ed., 1997. Incommensurability, incomparability, and practical rea- son. Harvard University Press, Cambridge.

Dasgupta, M. and R. Deb, 1991. Fuzzy choice functions, Social Choice and Welfare 8, 171-182.

Dasgupta, M. and R. Deb, 1996. Transitivity and fuzzy preferences, Social Choice and Welfare 13, 305-318.

Dasgupta, M. and R. Deb, 1999. An impossibility theorem with fuzzy prefer- ences in: H. de Swart, ed., Logic, game theory and social choice: proceedings 43 of the international conference, LGS ’99, May 13-16, 1999, Tilburg University Press.

Dasgupta, M. and R. Deb, 2001. Factoring fuzzy transitivity, Fuzzy Sets and Systems 118, 489-502.

Duddy, C., Perote-Peña, J. and A. Piggins, 2010. Manipulating an aggregation rule under ordinally fuzzy preferences, Social Choice and Welfare 34, 411-428.

Duddy, C., Perote-Peña, J. and A. Piggins, 2011. Arrow’s theorem and max-star transitivity, Social Choice and Welfare 36, 25-34.

Duddy, C. and A. Piggins, 2013. Many-valued judgment aggregation: characterizing the possibility/impossibility boundary, Journal of Economic Theory 148: 793-805.

Dutta, B., 1987. Fuzzy preferences and social choice, Mathematical Social Sciences 13, 215-229.

Dutta, B., Panda, S.C. and P.K. Pattanaik, 1986. Exact choice and fuzzy preferences, Mathematical Social Sciences 11, 53-68.

Espinoza, N., 2008. The small improvement argument, Synthese 165: 127-139.

Fono, L.A. and N.G. Andjiga, 2005. Fuzzy strict preference and social choice, Fuzzy Sets and Systems 155, 372-389.

García-Lapresta, J.L. and B. Llamazares, 2000. Aggregation of fuzzy preferences: some rules of the mean, Social Choice and Welfare 17: 673-690.

Geanakoplos, J., 2005. Three brief proofs of Arrow’s impossibility theorem, Economic Theory 26, 211-215.

Gibbard, A., 2014. Intransitive social indifference and the Arrow dilemma, Review of Economic Design 18, 3-10.

Gibilisco, M.B., A.M. Gowen, K.E. Albert, J.N. Mordeson, M.J. Wierman, T.D. Clark, 2014. Fuzzy social choice theory, Springer, Dordrecht.

Hájek, P., 1998. Metamathematics of fuzzy logic, Springer, Dordrecht.

Keefe, R., 2000. Theories of vagueness, Cambridge University Press, Cam- bridge.

Keefe, R. and P. Smith, eds., 1997. Vagueness: a reader, MIT Press, Cam- bridge.

Klement, E.P., Mesiar, R. and E. Pap, 2000. Triangular norms, Kluwer Aca- demic Publishers, Dordrecht.

Leclerc, B., 1984. Efficient and binary consensus functions on transitively valued relations, Mathematical Social Sciences 8, 45-61.

Leclerc, B., 1991. Aggregation of fuzzy preferences: a theoretic Arrow-like approach, Fuzzy Sets and Systems 43, 291-309.

Leclerc, B. and B. Monjardet, 1995. Lattical theory of consensus in: W. Barnett, H. Moulin, M. Salles and N. Schofield, eds., Social choice, welfare and ethics, Cambridge University Press, Cambridge.

Mas-Colell, A. and H. Sonnenschein, 1972. General possibility theorems for group decisions, Review of Economic Studies 39, 185-192.

Orlovsky, S.A., 1978. Decision-making with a fuzzy preference relation, Fuzzy Sets and Systems 1, 155-167.

Ovchinnikov, S.V., 1981. Structure of fuzzy binary relations, Fuzzy Sets and Systems 6, 169-195.

Ovchinnikov, S.V., 1991. Social choice and Lukasiewicz logic, Fuzzy Sets and Systems 43, 275-289.

Ovchinnikov, S.V. and M. Roubens, 1991. On strict preference relations, Fuzzy Sets and Systems 43, 319-326.

Ovchinnikov, S.V. and M. Roubens, 1992. On fuzzy strict preference, indif- ference and incomparability relations, Fuzzy Sets and Systems 49, 15-20.

Perote-Peña, J. and A. Piggins, 2007. Strategy-proof fuzzy aggregation rules, Journal of Mathematical Economics 43, 564-580.

Piggins, A. and M. Salles, 2007. Instances of indeterminacy, Analyse und Kritik 29, 311-328.

Plott, C.R., 1973. Path independence, rationality, and social choice, Econometrica 41, 1075-1091.

Ponsard, C., 1990. Some dissenting views on the transitivity of individual preference, Annals of Operations Research 23, 279-288.

Richardson, G., 1998. The structure of fuzzy preferences: social choice impli- cations, Social Choice and Welfare 15, 359-369.

Salles, M., 1998. Fuzzy utility in: S. Barbera, P.J. Hammond and C. Seidl, eds., Handbook of utility theory, vol.1: principles, Kluwer, Dordrecht.

Sen, A.K., 1969. Quasi-transitivity, rational choice and collective decisions, Review of Economic Studies 36, 381-393.

Sen, A.K., 1970a. Collective choice and social welfare. Holden-Day, San Fran- cisco.

Sen, A.K., 1970b. Interpersonal aggregation and partial comparability, Econo- metrica 38, 393-409.

Tang, F-F., 1994. Fuzzy preferences and social choice, Bulletin of Economic Research 46, 263-269.

Weymark, J.A., 1984. Arrow’s theorem with social quasi-orderings, Public Choice, 42, 235-246.

Weymark, J.A., 2014. An introduction to Allan Gibbard’s oligarchy theorem paper, Review of Economic Design, 18: 1-2.

Williamson, T., 1994. Vagueness. Routledge, London.