McCarthy, David and Mikkola, Kalle and Thomas, Teruji (2017): Aggregation for potentially infinite populations without continuity or completeness.
This is the latest version of this item.
Preview 
PDF
MPRA_paper_96751.pdf Download (493kB)  Preview 
Abstract
We present an abstract social aggregation theorem. Society, and each individual, has a preorder that may be interpreted as expressing values or beliefs. The preorders are allowed to violate both completeness and continuity, and the population is allowed to be infinite. The preorders are only assumed to be represented by functions with values in partially ordered vector spaces, and whose product has convex range. This includes all preorders that satisfy strong independence. Any Pareto indifferent social preorder is then shown to be represented by a linear transformation of the representations of the individual preorders. Further Pareto conditions on the social preorder correspond to positivity conditions on the transformation. When all the Pareto conditions hold and the population is finite, the social preorder is represented by a sum of individual preorder representations. We provide two applications. The first yields an extremely general version of Harsanyi's social aggregation theorem. The second generalizes a classic result about linear opinion pooling.
Item Type:  MPRA Paper 

Original Title:  Aggregation for potentially infinite populations without continuity or completeness 
Language:  English 
Keywords:  Social aggregation; discontinuous preferences and comparative likelihood relations; incomplete preferences and comparative likelihood relations; infinite populations; Harsanyi's social aggregation theorem; linear opinion pooling; partially ordered vector spaces 
Subjects:  D  Microeconomics > D6  Welfare Economics > D60  General D  Microeconomics > D6  Welfare Economics > D63  Equity, Justice, Inequality, and Other Normative Criteria and Measurement D  Microeconomics > D7  Analysis of Collective DecisionMaking > D70  General D  Microeconomics > D8  Information, Knowledge, and Uncertainty > D81  Criteria for DecisionMaking under Risk and Uncertainty D  Microeconomics > D8  Information, Knowledge, and Uncertainty > D83  Search ; Learning ; Information and Knowledge ; Communication ; Belief ; Unawareness 
Item ID:  96751 
Depositing User:  Dr David McCarthy 
Date Deposited:  05 Nov 2019 17:30 
Last Modified:  05 Nov 2019 17:30 
References:  Alon, S. and Lehrer, E., 2014. Subjective multiprior probability: a representation of a partial likelihood relation. Journal of Economic Theory 151: 476492. Anscombe, F., Aumann, R., 1963. A definition of subjective probability. The Annals of Mathematical Statistics 34: 199205. Armstrong, T. and Prikry, K., 1981. Liapounoff's theorem for nonatomic, finitelyadditive, bounded, finitedimensional, vectorvalued measures. Transactions of the American Mathematical Society 266: 499514. Aumann, R., 1962. Utility theory without the completeness axiom. Econometrica 30: 455462. Baucells, M. and Shapley, L. 2008. Multiperson utility. Games and Economic Behavior 62: 329347. Benci, V., Horsten, L., Wenmackers, S., 2018a. Infinitesimal probabilities. British Journal for the Philosophy of Science 69:509552. Benci, V., Horsten, L., and Wenmackers, S., 2018b. NonArchimedean probability. arXiv:1106.1524 [math.PR] Bewley, T., 2002. Knightian decision theory. Part I. Decisions in Economics and Finance 25(2): 79110. [Originally appeared in 1986 as Cowles Foundation Discussion Paper No. 807, Yale University.] Blackwell, D. and Girshick, M., 1954. Theory of Games and Statistical Decisions. New York, John Wiley. Blume, L., Brandenburger, A., and Dekel, E., 1989. An overview of lexicographic choice under uncertainty. Annals of Operations Research 19: 231246. Blume, L., Brandenburger, A., and Dekel, E., 1991a. Lexicographic probabilities and choice under uncertainty. Econometrica 59(1): 6179. Blume, L., Brandenburger, A. and Dekel, E., 1991b. Lexicographic probabilities and equilibrium refinements. Econometrica 59(1): 8198. Borie, D., 2016. Lexicographic expected utility without completeness. Theory and Decision 81, 167176. Brandenburger, A., Friedenberg, A., and Keisler, H. J., 2008. Admissibility in games. Econometrica 76(2): 307352. Brickhill, H. and Horsten, L., 2018. Popper functions, lexicographical probability, and nonArchimedean probability. arXiv:1608.02850 [math.LO] Broome, J., 1990. BolkerJeffrey expected utility theory and axiomatic utilitarianism. Review of Economic Studies 57: 477502. Chambers, C., 2007. An ordinal characterization of the linear opinion pool. Economic Theory 33: 457474. Danan, E., Gajdos, T., Tallon, J.M., 2013. Aggregating sets of von NeumannMorgenstern utilities. Journal of Economic Theory 148: 663688. Danan, E., Gajdos, T. and Tallon, J.M., 2015. Harsanyi's aggregation theorem with incomplete preferences. American Economic Journal: Microeconomics 7(1): 6169. De Meyer, B. and Mongin, P., 1995. A note on affine aggregation. Economics Letters 47: 177183. Diamond, P. 1965. The evaluation of infinite utility streams. Econometrica 33: 170177. Dietrich, F., and List, C., 2016. Probabilistic opinion pooling. In A. Hájek and C. Hitchcock eds. The Oxford Handbook of Philosophy and Probability, Oxford University Press, 2016, 51819. Dubra, J., 2011. Continuity and completeness under risk. Mathematical Social Sciences 61: 8081. Dubra, J., Maccheroni, F., Ok, E., 2004. Expected utility theory without the completeness axiom. Journal of Economic Theory 115: 118133. Evren, Ö., 2008. On the existence of expected multiutility representations. Economic Theory 35: 575592. Evren, Ö., 2014. Scalarization methods and expected multiutility representations. Journal of Economic Theory 151: 3063. Fine, T., 1973. Theories of Probability, New York, Academic Press. Fishburn, P., 1971. A study of lexicographic expected utility. Management Science 17: 672678. Fishburn, P., 1982. The foundations of expected utility. Dordrecht, Reidel. Fishburn, P., 1984. On Harsanyi's utilitarian cardinal welfare theorem. Theory and Decision 17: 2128. Fleurbaey, M., 2009. Two variants of Harsanyi's aggregation theorem. Economics Letters 105, 300302. Galaabaatar, T. and Karni, E., 2012. Expected multiutility representations. Mathematical Social Sciences 64: 242246. Galaabaatar, T. and Karni, E., 2013. Subjective expected utility with incomplete preferences. Econometrica 81: 255284. Ghirardato, P., Maccheroni, F., Marinacci, M., and Siniscalchi, M., 2003. A subjective spin on roulette wheels. Econometrica 71: 18971908. Gilboa, I., 2009. Theory of Decision under Uncertainty, Cambridge University Press. Gilboa, I., Maccheroni, F., Marinacci, M., and Schmeidler, D., 2010. Objective and subjective rationality in a multiple prior model. Econometrica 78: 755770. Gilboa, I. and Schmeidler, D., 1989. Maxmin expected utility with nonunique prior. Journal of Mathematical Economics 18: 141153. Halpern, J., 2003. Reasoning about Uncertainty, MIT Press. Halpern, J., 2010. Lexicographic probability, conditional probability, and nonstandard probability. Games and Economic Behavior 68: 155179. Hammond P., 1994a. Elementary nonArchimedean representations of probability for decision theory and games. In: Humphreys, P. (Ed.), Patrick Suppes: Scientific Philosopher, vol. 1. Kluwer, Dordrecht, pp. 2549. Hammond, P., 1994b. Consequentialism, nonArchimedean probabilities, and lexicographic expected utility. Operations Research ?93. A. Bachem, U. Derigs, M. Jünger and R. Schrader. Heidelberg, Physica: 219250. Hammond, P., 1999. NonArchimedean subjective probabilities in decision theory and games. Mathematical Social Sciences 38(2): 139156. Hara, K., Ok, E., Riella, G., 2016. Coalitional expected multiutility theory. http://economics.mit.edu/files/12659 Harsanyi, J., 1955. Cardinal welfare, individualistic ethics, and interpersonal comparisons of utility. Journal of Political Economy 63, 30921. Harsanyi, J., 19671968. Games with incomplete information played by ``Bayesian'' players, IIII. Management Science 14: 159182, 320334, 486502. Hausner, M., 1954. Multidimensional utilities. In Thrall, R., Coombs, C. and Davis, R. eds. Decision Processes John Wiley. Hausner, M., Wendel, J., 1952. Ordered vector spaces. Proceedings of the American Mathematical Society 3, 977982. Herzberg, F., 2009. Elementary nonArchimedean utility theory. Mathematical Social Sciences 58: 814. Herzberg, F., 2015. Aggregating infinitely many probability measures. Theory and Decision 78: 319337. Herstein, I. and Milnor, J., 1953. An axiomatic approach to measurable utility. Econometrica 21: 291297. Insua, D. R., 1992. On the foundations of decision making under partial information. Theory and Decision 33: 83100. Koopmans, T., 1960. Stationary ordinal utility and impatience. Econometrica 28: 287309. Manzini, P. and Mariotti, M., 2008. On the representation of incomplete preferences over risky alternatives. Theory and Decision 65: 303323. McCarthy, D., Mikkola, K., 2018. Continuity and completeness of strongly independent preorders. Mathematical Social Sciences 93: 141145. McCarthy, D., Mikkola, K., Thomas, T., 2017a. Representation of strongly independent preorders by sets of scalarvalued functions. MPRA Paper No. 79284 https://mpra.ub.unimuenchen.de/79284/ McCarthy, D., Mikkola, K., Thomas, T., 2017b. Representation of strongly independent preorders by vectorvalued functions. MPRA Paper No. 80806. https://mpra.ub.unimuenchen.de/80806/ McCarthy, D., Mikkola, K. and Thomas, T., 2018. Utilitarianism with and without expected utility. MPRA Paper No. 90125. https://mpra.ub.unimuenchen.de/90125/ McConway, K., 1981. Marginalization and linear opinion pools. Journal of the American Statistical Association 76(374): 410414. Mongin, P., 1995. Consistent Bayesian aggregation. Journal of Economic Theory 66: 313351. Mongin, P., 1998. The paradox of the Bayesian experts and statedependent utility theory. Journal of Mathematical Economics 29: 33161. Mongin, P., 2001. A note on mixture sets in decision theory. Decisions in Economics and Finance 24: 5969. Mongin, P., Pivato, M., 2015. Ranking multidimensional alternatives and uncertain prospects. Journal of Economic Theory 157, 146171. Nau, R., 2006. The shape of incomplete preferences. The Annals of Statistics 34: 24302448. Nielsen, M., 2019. On linear aggregation of infinitely many finitely additive probability measures. Theory and Decision 86: 421436. Ok, E., 2007. Real Analysis with Economic Applications, Princeton University Press. Ok, E., Ortoleva, P. and Riella, G., 2012. Incomplete preferences under uncertainty: indecisiveness in beliefs vs. tastes. Econometrica 80: 17911808. Pivato, M., 2013. Risky social choice with incomplete or noisy interpersonal comparisons of wellbeing. Social Choice and Welfare 40: 123139. Pivato, M., 2014. Additive representation of separable preferences over infinite products. Theory and Decision 77, 3183. Ramsey, F., 1928. A mathematical theory of saving. Economic Journal 38: 543559. Rudin, W., 1991. Functional Analysis, 2nd. edition. McGrawHill. Schmeidler, D., 1971. A condition for the completeness of partial preference relations. Econometrica 39: 403404. Seidenfeld, T., Schervish, M. and Kadane, J., 1995. A representation of partially ordered preferences. Annals of Statistics 23: 21682217. Shapley, L., Baucells, M., 1998. Multiperson utility. University of California, Los Angeles (UCLA) Department of Economics Working Paper 779. Stinchcombe, M., 2016. Objective and subjective foundations for multiple priors. Journal of Economic Theory 165: 263291. Stone, M., 1961. The opinion pool. Annals of Mathematical Statistics 32, 4: 13391342. Villegas, C., 1964. On qualitative probability sigmaalgebras. Annals of Mathematical Statistics 35: 17871796. Weymark, J., 1993. Harsanyi's social aggregation theorem and the weak Pareto principle. Social Choice and Welfare 10: 209221. Weymark, J., 1995. Further remarks on Harsanyi's social aggregation theorem and the weak Pareto principle. Social Choice and Welfare 12: 8792. Zhou, L., 1997. Harsanyi's utilitarianism theorems: general societies. Journal of Economic Theory 72: 198207. Zuber, S., 2016. Harsanyi's theorem without the surething principle: On the consistent aggregation of Monotonic Bernoullian and Archimedean preferences. Journal of Mathematical Economics 63: 7883. 
URI:  https://mpra.ub.unimuenchen.de/id/eprint/96751 
Available Versions of this Item

Aggregation for general populations without continuity or completeness. (deposited 16 Aug 2017 15:55)
 Aggregation for potentially infinite populations without continuity or completeness. (deposited 05 Nov 2019 17:30) [Currently Displayed]