Abstract
An observer attempts to infer the unobserved ranking of two ideal objects, A and B, from observed rankings in which these objects are `accompanied' by `noise' components, C and D. In the first ranking, A is accompanied by C and B is accompanied by D, while in the second ranking, A is accompanied by D and B is accompanied by C. In both rankings, noisy-A is ranked above noisy-B. The observer infers that ideal-A is ranked above ideal-B. This commonly used inference rule is formalized for the case in which A,B,C,D are sets. Let X be a finite set and let \(\succ\) be a linear ordering on 2X. The following condition is imposed on \(\succ\). For every quadruple (A,B,C,D)∈Y, where Y is some domain in (2X)4, if \(A \cup C \succ B \cup D\) and \(A \cup D \succ B \cup C \succ B \cup C\), then \(A \succ B\). The implications and interpretation of this condition for various domains Y are discussed.
Similar content being viewed by others
REFERENCES
Barbera, S. and Pattanaik, P. K. (1984), Extending an order on a set to the power set: Some remarks on Kannai and Peleg's approach, Journal of Economic Theory 32: 185-191.
Bossert, W., Pattanaik, P. K. and Xu, Y. (1994), Ranking opportunity sets: An axiomatic approach, Journal of Economic Theory 63: 326-345.
Bossert, W., Pattanaik, P. K. and Xu, Y. (2000), Choice under complete uncertainty, Economic Theory 16: 295-312.
Dekel, E., Lipman, B. and Rustichini, A. (2001), Representing preferences with a unique subjective state space, Econometrica, forthcoming.
Gärdenfors, P. (1976), Manipulability of social choice functions, Journal of Economic Theory 13: 217-218.
Gul, F. and Pesendorfer, W. (2001), Temptation and self-control, Econometrica, forthcoming.
Kannai, Y. and Peleg, B. (1984), A note on the extension of an order on a set to the power set, Journal of Economic Theory 32: 172-175.
Kreps, D. M. (1979), A representation theorem for 'preference for flexibility', Econometrica 47: 565-577.
Nehring, K. and Puppe, C. (1996), Continuous extensions of an order on a set to the power set, Journal of Economic Theory 68: 456-479.
Pattanaik, P. K. and Peleg, B. (1984), An axiomatic characterization of the lexicographic maximin extension of an ordering over a set to the power set, Social Choice and Welfare 1: 113-122.
Puppe, C. (1995), Freedom of choice and rational decisions, Social Choice and Welfare 12: 137-153.
Sen, A. K. (1991),Welfare, preference and freedom. Journal of Econometrics 50: 15-29.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Spiegler, R. Inferring a linear ordering over a power set. Theory and Decision 51, 31–49 (2001). https://doi.org/10.1023/A:1012400922478
Issue Date:
DOI: https://doi.org/10.1023/A:1012400922478