Abstract
When choosing an alternative that has multiple attributes, it is common to form a weighted sum ranking. In this paper, we provide an axiomatic analysis of the weighted sum criterion using a general choice framework. We show that a preference order has a weak weighted sum representation if it satisfies three basic axioms: Monotonicity, Translation Invariance, and Substitutability. Further, these three axioms yield a strong weighted sum representation when the preference order satisfies a mild condition, which we call Partial Representability. A novel form of non-representable preference order shows that partial representability cannot be dispensed in establishing our strong representation result. We consider several related conditions each of which imply a partial representation, and therefore a strong weighted sum representation when combined with the three axioms. Unlike many available characterizations of weighted sums, our results directly construct a unique vector of weights from the preference order, which makes them useful for economic applications.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Banerjee, K., Mitra, T.: On Wold’s approach to representation of preferences. J. Math. Econ. 79, 65–74 (2018)
Birkoff, G.: Lattice Theory. American Mathematical Society Publications, New York (1948)
Blackwell, D., Girshick, M.A.: Theory of Games and Statistical Decisions. Wiley, New York (1954)
Bridges, D.S., Mehta, G.B.: Representation of Preference Orderings. Springer, Berlin (1995)
Candeal, J.C.: Invariance axioms for preferences: applications to social choice theory. Soc. Choi. Welf. 41, 453–471 (2013)
Candeal, J., Indurain, E.: A note on linear utility. Econ. Theory 6, 519–522 (1995)
Chipman, J.S.: The foundations of utility. Econometrica 28, 193–224 (1960)
d’Aspremont, C.: Axioms for social welfare orderings. In: Hurwicz, L., Schmeidler, D., Sonnenschein, H. (eds.) Social Goals and Social Organization. Essays in Memory of Elisha Pazner, pp. 19–75. Cambridge University Press, Cambridge (1985)
d’Aspremont, C., Gevers, L.: Equity and the informational basis of collective choice. Rev. Econ. Stud. 44, 199–209 (1977)
d’Aspremont, C., Gevers, L.: Social welfare functionals and interpersonal comparability. In: Arrow, K., Sen, A., Suzumura, K. (eds.) Handbook of Social Choice and Welfare, pp. 459–541. Elsevier, Amsterdam (2002)
Debreu, G.: Representation of a preference ordering by a numerical function. In: Thrall, R.M., Coombs, C.H., Davis, R.L. (eds.) Decision Processes, pp. 159–165. Wiley, New York (1954)
Fleurbaey, M., Maniquet, F.: A Theory of Fairness and Social Welfare. Cambridge University Press, Cambridge (2011)
Gevers, L.: On interpersonal comparability and social welfare orderings. Econometrica 47, 75–89 (1979)
Hara, K., Ok, E.A., Riella, G.: Coalitional expected multi-utility theory. Econometrica 87, 933–980 (2019)
Hausner, M., Wendel, J.: Ordered vector spaces. Proc. Am. Math. Soc. 3, 977–982 (1952)
Howie, J.M.: Fundamentals of Semigroup Theory. Clarendon Press, Oxford (1995)
Kaplansky, I.: Set Theory and Metric Spaces. Chelsea, New York (1972)
Krause, U.: Essentially lexicographic aggregation. Soc. Choice Welf. 12, 233–244 (1995)
Mehta, G.B.: Preference and utility. In: Barbera, S., Hammond, P., Seidl, C. (eds.) Handbook of Utility Theory, vol. 1. Kluwer, Dordrecht (1998)
Mitra, T., Ozbek, M.K.: On Representation of monotone preference orders in a sequence space. Soc. Choice Welf. 41, 473–487 (2013)
Mongin, P., d’Aspremont, C.: Utility theory and ethics. In: Barberà, S., Hammond, P., Seidl, C. (eds.) Handbook of Utility Theory, pp. 371–481. Kluwer, Boston (1998)
Morandi, P.: Field and Galois Theory. Springer, Berlin (1996)
Moulin, H.: Axioms of Cooperative Decision Making. Cambridge University Press, Cambridge (1988)
Neuefeind, W., Trockel, W.: Continuous linear representability of binary relations. Econ. Theory 6, 351–356 (1995)
Roberts, K.W.S.: Interpersonal comparability and social choice theory. Rev. Econ. Stud. 47, 421–439 (1980)
Royden, H.L.: Real Analysis. Macmillan, New York (1988)
Segal, U., Sobel, J.: Min, max, and sum. J. Econ. Theory 106, 126–150 (2002)
Trockel, W.: An alternative proof for the linear utility representation theorem. Econ. Theory 2, 298–302 (1992)
Weibull, J.W.: Discounted value representations of temporal preferences. Math. Oper. Res. 10, 244–250 (1985)
Yoshihara, N., Veneziani, R.: The Measurement of Labour Content: A General Approach. Mimeo, New York (2018)
Author information
Authors and Affiliations
Corresponding author
Additional information
Earlier versions of this work were circulated under the titles “On Representation and Weighted Utilitarian Representation of Preference Orders on Finite Streams” and “Weighted Utilitarianism over Finite Streams”. Professor Tapan Mitra passed away on February 3, 2019. This final draft is dedicated to his memory.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Mitra, T., Ozbek, K. Ranking by weighted sum. Econ Theory 72, 511–532 (2021). https://doi.org/10.1007/s00199-020-01305-w
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00199-020-01305-w
Keywords
- Decomposition of the set of irrationals
- Partial representation
- Substitutability
- Weak representation
- Weighted sum