Utilitarianism with and without expected utility

McCarthy, David and Mikkola, Kalle and Thomas, Teruji (2016): Utilitarianism with and without expected utility.

Abstract

We give two social aggregation theorems under conditions of risk, one for constant population cases, the other an extension to variable populations. Intra and interpersonal welfare comparisons are encoded in a single `individual preorder'. The individual preorder then uniquely determines a social preorder. The social preorders described by these theorems have features that may be considered characteristic of Harsanyi-style utilitarianism, such as indifference to ex ante and ex post equality. However, the theorems are also consistent with the rejection of all of the expected utility axioms, completeness, continuity, and independence, at both the individual and social levels. In that sense, expected utility is inessential to Harsanyi-style utilitarianism. In fact, the variable population theorem imposes only a mild constraint on the individual preorder, while the constant population theorem imposes no constraint at all. We then derive further results under the assumption of our basic axioms. First, the individual preorder satisfies the main expected utility axiom of strong independence if and only if the social preorder has a vector-valued expected total utility representation, covering Harsanyi's utilitarian theorem as a special case. Second, stronger utilitarian-friendly assumptions, like Pareto or strong separability, are essentially equivalent to strong independence. Third, if the individual preorder satisfies a `local expected utility' condition popular in non-expected utility theory, then the social preorder has a `local expected total utility' representation. Although our aggregation theorems are stated under conditions of risk, they are valid in more general frameworks for representing uncertainty or ambiguity.

• Item Type: MPRA Paper
• Language: English
• Keywords: Harsanyi, utilitarianism, expected and non-expected utility, incompleteness, uncertainty, variable populations.
• 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 Decision-Making > D71 - Social Choice ; Clubs ; Committees ; Associations; D - Microeconomics > D8 - Information, Knowledge, and Uncertainty > D81 - Criteria for Decision-Making under Risk and Uncertainty

References

Adler, M., Fleurbaey, M. (Eds.), 2016. The Oxford Handbook of Well-Being and Public Policy. Oxford University Press.

Allais, M., 1979. The foundations of a positive theory of choice involving risk and a criticism of the postulates and axioms of the American school. In M. Allais and O. Hagen (Eds.), Expected Utility Hypothesis and the Allais Paradox. Dordrecht, Reidel, pp. 27-145.

Alon, S., Gayer, G., 2016. Utilitarian preferences with multiple priors. Econometrica 84: 1181-1201.

Anscombe, F., Aumann, R., 1963. A definition of subjective probability. The Annals of Mathematical Statistics 34: 199-205.

Arrhenius, G., Rabinowicz, W., 2015. Value superiority. In I. Hirose and J. Olson (eds.), The Oxford Handbook of Value Theory. Oxford University Press.

d'Aspremont, C., Gevers, L., 1977. Equity and the informational basis of collective choice. Review of Economic Studies 44, 199-209.

Aumann, R., 1962. Utility theory without the completeness axiom. Econometrica 30, 455-462.

Barry, B., 1977. Rawls on average and total utility: a comment. Philosophical Studies 31, 317-325.

Ben-Porath, E., Gilboa, I., Schmeidler, D., 1977. On the measurement of inequality under uncertainty. Journal of Economic Theory 75, 194-204.

Bewley, T., 2002. Knightian decision theory: part I. Decisions in Economics and Finance, 25, 79-110.

Billot, A., Vergopoulos, V., 2016. Aggregation of Paretian preferences for independent individual uncertainties. Social Choice and Welfare published online 24 October 2016.

Blackorby, C., Donaldson, D., 1980. A theoretical treatment of indices of absolute equality. International Economic Review 21, 107-136.

Blackorby, C., Donaldson, D., 1984. Social criteria for evaluating population change. Journal of Public Economics 25, 13-33.

Blackorby, C., Bossert, W., Donaldson, D., 1995. Intertemporal population ethics: critical-level utilitarian principles. Econometrica 65, 1303-1320.

Blackorby, C., Bossert, W., Donaldson, D., 1998. Uncertainty and critical-level population principles. Journal of Population Economics 11, 1-20.

Blackorby, C., Bossert, W., Donaldson, D., 2005. Population Issues in Social Choice Theory, Welfare Economics, and Ethics. New York, Cambridge University Press.

Blackorby, C., Bossert, W., Donaldson, D., 2007. Variable-population extensions of social aggregation theorems. Social Choice and Welfare 28, 567-589.

Bogachev, V., 2007 Measure Theory. Springer.

Border, K., 1981. Notes on von Neumann-Morgenstern social welfare functions. Unpublished manuscript, California Institute of Technology, Department of Economics.

Borie, D., 2016. Lexicographic expected utility without completeness. Theory and Decision 81, 167-176.

Bostrom, N., 2011. Infinite ethics. Analysis and Metaphysics 10, 9-59.

Broome, J., 1989. What's the good of equality?. In J. D. Hey (Ed.), Current Issues in Microeconomics. Macmillan, pp. 236-262.

Broome, J., 1990. Bolker-Jeffrey expected utility theory and axiomatic utilitarianism. Review of Economic Studies 57: 477-502.

Broome, J., 1991. Weighing Goods. Blackwell, Oxford.

Broome, J., 1993. A cause of preference is not an object of preference. Social Choice and Welfare 10: 57-68.

Broome, J., 1999. Ethics out of Economics. Cambridge University Press.

Broome, J., 2004. Weighing Lives. New York, Oxford University Press.

Brunnermeier, M., Simsek, A., Xiong, W., 2014. A welfare criterion for models with distorted beliefs. The Quarterly Journal of Economics 129(4), 1753-1797.

Buchak, L., 2013. Risk and Rationality. Oxford University Press.

Cerreia-Vioglio, S., Maccheroni, F., Marinacci, M., 2016. Stochastic dominance analysis without the independence axiom. Management Science Published online: May 23, 2016 10.1287/mnsc.2015.2388

Chambers, C., Hayashi, T., 2006. Preference aggregation under uncertainty: Savage vs. Pareto. Games and Economic Behavior 54, 430-440.

Chambers, C., Hayashi, T., 2014. Preference aggregation with incomplete information. Econometrica 82, 589-599.

Chateauneuf, A., Cohen, M., 1994. Risk-seeking with diminishing marginal utility in a non-expected utility model. Journal of Risk and Uncertainty 9, 77-91.

Chew, S., Karni, E., Safra, Z., 1987. Risk aversion in the theory of expected utility with rank dependent probabilities. Journal of Economic Theory 42, 370-381.

Chew, S., Mao, M., 1995. A Schur concave characterization of risk-aversion for non-expected utility preferences. Journal of Economic Theory 67, 402-435.

Chew, S., Nishimura, N., 1992. Differentiability, comparative statics, and non-expected utility preferences. Journal of Economic Theory 56, 294-312.

Conrad, P., 1953. Embedding theorems for abelian groups with valuations. American Journal of Mathematics 75, 1-29.

Coulhon, T., Mongin, P., 1989. Social choice theory in the case of von Neumann Morgenstern utilities. Social Choice and Welfare 6, 175-187.

Crès, A., Gilboa, I., Vieille, N., 2011. Aggregation of multiple prior opinions. Journal of Economic Theory 146, 2563-2582.

Danan, E., Gajdos, T., Hill, B., Tallon, J.-M., 2016. Robust social decisions. American Economic Review 106: 2407-25.

Danan, E., Gajdos, T., Tallon, J-M., 2015. Harsanyi's aggregation theorem with incomplete preferences. American Economic Journal: Microeconomics 7, 61-69.

d'Aspremont, C., Gevers, L., 2002. Social welfare functionals and interpersonal comparability. In Arrow, K., Sen, A., and Suzumara, K., (Eds.) Handbook of Social Choice and Welfare 1: 459-541.

De Meyer, B., Mongin, P., 1995. A note on affine aggregation. Economics Letters 47, 177-183.

Diamond, P., 1967. Cardinal welfare, individualistic ethics, and interpersonal comparisons of utility: comment. Journal of Political Economy 75, 765-66.

Dietrich, F., List, C., 2016. Probabilistic opinion pooling. In Hàjek, A. and Hitchcock, C., (Eds.) The Oxford Handbook of Philosophy and Probability, Oxford University Press, 519-541.

Domotor, Z., 1979. Ordered sum and tensor product of linear utility structures. Theory and Decision 11, 375-399.

Dravecký, J., 1975. Spaces with measurable diagonal. Matematický Časopis 25, 3-9. Dubra, J., 2011. Continuity and completeness under risk. Mathematical Social Sciences 61, 80-81.

Dubra, J., Maccheroni, F., Ok, E., 2004. Expected utility theory without the completeness axiom. Journal of Economic Theory 115, 118-133.

Dudley, R., 2002. Real Analysis and Probability 2nd. ed. New York, Cambridge University Press.

Evren, Ö., 2008. On the existence of expected multi-utility representations. Economic Theory 35, 575-592.

Evren, Ö., 2014. Scalarization methods and expected multi-utility representations. Journal of Economic Theory 151: 30-63.

Fishburn, P., 1970. Utility Theory for Decision Making. New York, Wiley.

Fishburn, P., 1984. On Harsanyi's utilitarian cardinal welfare theorem. Theory and Decision 17: 21-28.

Fishburn, P., 1988. Nonlinear Preference and Utility Theory. The John Hopkins University Press, Baltimore and London.

Fleurbaey, M., 2003. On the informational basis of social choice. Social Choice and Welfare 21, 347-384.

Fleurbaey, M., 2009. Two variants of Harsanyi's aggregation theorem. Economics Letters 105, 300-302.

Fleurbaey, M., 2010. Assessing risky social situations. Journal of Political Economy, 118, 649-680.

Fleurbaey, M., 2018. Welfare economics, risk and uncertainty. Canadian Journal of Economics, 51.

Fleurbaey, M., Voorhoeve, A., 2016. Priority or equality for possible people? Ethics 126, 929-954.

Fleurbaey, M., Zuber, S., 2017. Fair management of social risk. Journal of Economic Theory 169: 666-706.

Gajdos, T., Tallon, J.-M. Vergnaud, J.-C., 2008. Representation and aggregation of preferences under uncertainty. Journal of Economic Theory 141: 68-99.

Ghirardato, P., Maccheroni, F., Marinacci, M., Siniscalchi, M., 2003. A subjective spin on roulette wheels. Econometrica 71: 1897-1908.

Gilboa, I., Samet, D., Schmeidler, D., 2004. Utilitarian aggregation of beliefs and tastes. Journal of Political Economy 112: 932-938.

Gilboa, I., Samuelson, L., Schmeidler, D., 2014. No-betting Pareto dominance. Econometrica, 82: 1405-1442.

Gilboa, I., Schmeidler, D., 1989. Maxmin expected utility with non-unique prior. Journal of Mathematical Economics 18: 141-153.

Gorno, L., 2017. A strict expected multi-utility theorem. Journal of Mathematical Economics 71: 92-95.

Grant, S., Kajii, A., Polak, B., Safra, Z., 2010. Generalized utilitarianism and Harsanyi's impartial observer theorem. Econometrica 78, 1939-1971.

Grant, S., Kajii, A., Polak, B., Safra, Z., 2012. A generalized representation theorem for Harsanyi's (`impartial') observer. Social Choice and Welfare 39, 833-846.

Grant, S., Kajii, A., Polak, B., Safra, Z., 2012. Equally-distributed equivalent utility, ex post egalitarianism and utilitarianism. Journal of Economic Theory 147: 1545-1571.

Greaves, H., 2017. A reconsideration of the Harsanyi-Sen-Weymark debate on utilitarianism. Utilitas 29(2): 175-213.

Halpern, J., 2003. Reasoning about Uncertainty M.I.T. Press.

Hammond, P., 1988. Consequentialist demographic norms and parenting rights. Social Choice and Welfare 25, 127-145.

Hammond, P., 1991. Interpersonal comparisons of utility: why and how they are and should be made. In J. Elster, J., Roemer, J. (Eds.), Interpersonal Comparisons of Well-Being. Cambridge University Press, pp. 200-254.

Hammond, P., 1998. Objective expected utility: a consequentialist perspective. In Barberà, S., Seidel, C., Hammond, P. (Eds.), Handbook of Utility Theory, Vol. 1. Dordrecht, Kluwer, pp. 145-211.

Hara, K., Ok, E., Riella, G., 2016. Coalitional expected multi-utility theory. http://economics.mit.edu/files/12659

Harsanyi, J., 1953. Cardinal utility in welfare economics and in the theory of risk-taking. Journal of Political Economy 61, 434-435.

Harsanyi, J., 1955. Cardinal welfare, individualistic ethics, and interpersonal comparisons of utility. Journal of Political Economy 63, 309-21.

Harsanyi, J., 1977a. Morality and the theory of rational behavior. Social Research 44, 623-56.

Harsanyi, J. 1977b. Rational Behaviour and Bargaining Equilibrium in Games and Social Situations. Cambridge, Cambridge University Press.

Harsanyi, J., 1978. Bayesian decision theory and utilitarian ethics. American Economic Review 68, 223-8.

Harsanyi, J., 1979. Bayesian decision theory, rule utilitarianism, and Arrow's impossibility result. Theory and Decision 11, 289-317.

Hausner, M., Wendel, J., 1952. Ordered vector spaces. Proceedings of the American Mathematical Society 3, 977-982.

Herstein, I., Milnor, J., 1953. An axiomatic approach to measurable utility. Econometrica 21: 291-297.

Karni, E., Zhou, N., 2016. Weighted utility theory with incomplete preferences. Mimeo.

Kavka, G. S., 1975. Rawls on average and total utility. Philosophical Studies 27, 237-253.

Kreps, D., 1988. Notes on the Theory of Choice. Westview Press, Boulder and London.

Luce, R., Raiffa, H., 1957. Games and Decisions. John Wiley and Sons, New York.

Machina, M., 1982. `Expected utility' analysis without the independence axiom. Econometrica 50, 277-323.

McCarthy, D., 2015. Distributive equality. Mind 124, 1045-1109.

McCarthy, D., 2017. The priority view. Economics and Philosophy 33, 215-257.

McCarthy, D., Mikkola, K., 2018. Continuity and completeness of strongly independent preorders. Mathematical Social Sciences 93: 141-145.

McCarthy, D., Mikkola, K., Thomas, T., 2017a. Representation of strongly independent preorders by vector-valued functions. MPRA Paper No. 80806 https://mpra.ub.uni-muenchen.de/80806/.

McCarthy, D., Mikkola, K., Thomas, T., 2017b. Aggregation for general populations without continuity or completeness. MPRA Paper No. 80820 https://mpra.ub.uni-muenchen.de/80820/.

McCarthy, D., Mikkola, K., Thomas, T., 2017c. Utilitarianism and uncertainty. Mimeo.

Mongin, P., 1994. Harsanyi's aggregation theorem: multi-profile version and unsettled questions. Social Choice and Welfare 11, 331-354.

Mongin, P., 1995. Consistent Bayesian aggregation. Journal of Economic Theory 66: 313-51.

Mongin, P., 1998. The paradox of the Bayesian experts and state-dependent utility theory. Journal of Mathematical Economics 29: 331-61.

Mongin, P., 2001a. The impartial observer theorem of social ethics. Economics and Philosophy 17: 147-79.

Mongin, P., 2001b. A note on mixture sets in decision theory Decisions in Economics and Finance 24: 59-69.

Mongin, P., 2016. Spurious unanimity and the Pareto principle. Economics and Philosophy 32: 511-532.

Mongin, P., d'Aspremont, C., 1998. Utility theory and ethics. In Barberà, S., Hammond, P., Seidl, C. (Eds.), Handbook of Utility Theory, vol. 1, Kluwer, Dordrecht, pp. 233-289.

Mongin, P., Pivato, M., 2015. Ranking multidimensional alternatives and uncertain prospects. Journal of Economic Theory 157, 146-171.

Mongin, P., Pivato, M., 2016a. Social choice under risk and uncertainty. In Adler, M., Fleurbaey, M. (Eds.), 2016. The Oxford Handbook of Well-Being and Public Policy, pp. 711-745.

Mongin, P., Pivato, M., 2016b. Social preference under twofold uncertainty. MPRA Paper No. 71776. https://mpra.ub.uni-muenchen.de/71776/

Myerson, R., 1981. Utilitarianism, egalitarianism, and the timing effect in social choice problems. Econometrica 49, 883-897.

Ng, Y., 1983. Some broader issues of social choice. In P. Pattanaik and M. Salles eds. Social Choice and Welfare. North-Holland, Amsterdam, pp. 151-173

Parfit, D., 1986. Reasons and Persons. New York, Oxford University Press.

Pivato, M., 2013. Risky social choice with incomplete or noisy interpersonal comparisons of well-being. Social Choice and Welfare 40, 123-39.

Pivato, M., 2014. Additive representation of separable preferences over infinite products. Theory and Decision 77, 31-83.

Qu, X., 2017. Separate aggregation of beliefs and values under ambiguity. Economic Theory 63, 1-17.

Quiggin, J., 1982. A theory of anticipated utility. Journal of Economic Behavior and Organization 3, 323-43.

Quiggan, J., 1993. Generalized Expected Utility Theory: The Rank-Dependent Model. Dordrecht, Kluwer.

Rabinowicz, W., 2009. Broome and the intuition of neutrality. Philosophical Issues 19: 389-411.

Rawls, J., 1971. A Theory of Justice. Belknap Press.

Richter, M., 1971. Rational choice. In Chipman, J., Hurwicz, L., Richter, M., Sonnenschein, H., (Eds.), Preferences, Utility, and Demand: A Minnesota Symposium, pp. 29-58. Harcourt Brace Jovanovich, New York.

Rudin, W., 1991. Functional Analysis, 2nd. edition. McGraw-Hill.

Saito, K., 2013. Social preferences under risk: equality of opportunity versus equality of outcome. American Economic Review 103, 3084-3101.

Savage, L., 1972. ]The Foundations of Statistics. New York:Dover Publications Inc.

Scanlon, T., 1982. Contractualism and utilitarianism. In Sen, A., Williams, B., (Eds.), Utilitarianism and Beyond, pp. 103-128. Cambridge, Cambridge University Press.

Schmidt, U., 2001. Lottery dependent utility: a reexamination. Theory and Decision 50, 35-58.

Schmidt, U., 2004. Alternatives to expected utility: some formal theories. In S. Barberà, C. Seidel and P. Hammond eds. Handbook of Utility Theory, Vol. 2. Dordrecht, Kluwer, pp. 757-838.

Sen, A., 1970. Collective Choice and Social Welfare. San Fransisco, Holden Day.

Sen, A., 1973. On Economic Inequality. Oxford, Clarendon Press.

Sen, A., 1976. Welfare inequalities and Rawlsian axiomatics. Theory and Decision 7, 243-62.

Sen A., 1977. Non-linear social welfare functions: a reply to Professor Harsanyi. In Butts, R., Hintikka, J. (Eds.), Foundational Problems in the Special Sciences. Dordrecht, Reidel, pp. 297-302.

Sen A., 1981. Plural utility. Proceedings of the Aristotelian Society, 81, 193-216.

Sen, A., 1985. Well-being, agency and freedom. Journal of Philosophy 82: 169-221.

Sen, A., 1986. Social choice theory. In Arrow, K. and Intriligator, M. (Eds). Handbook of Mathematical Economics Amsterdam, North Holland, vol. 3, 1073-1181.

Starmer, C., 2000. Developments in non-expected utility theory: the hunt for a descriptive theory of choice under risk. Journal of Economic Literature 38, 332-382.

Tungodden, B., 2003. The value of equality. Economics and Philosophy 19, 1-44.

Thomas, T., 2018. Some possibilities in population axiology. Mind 127, 807-832.

Vickrey, W., 1945. Measuring marginal utility by reactions to risk. Econometrica 13, 319–333.

Wakker, P., 1994. Separating marginal utility and probabilistic risk aversion. Theory and Decision 36, 1-44.

Wakker, P., 2010. Prospect Theory. Cambridge, Cambridge University Press.

Weymark, J., 1981. Generalized gini inequality indices. Mathematical Social Sciences 1, 409-430.

Weymark, J., 1991. A reconsideration of the Harsanyi-Sen debate on utilitarianism. In Elster, J., Roemer, J. (Eds.), Interpersonal Comparisons of Well-Being. Cambridge University Press, pp. 255-320.

Weymark, J., 1993. Harsanyi's social aggregation theorem and the weak Pareto principle. Social Choice and Welfare 10: 209-221.

Weymark, J., 1995. Further remarks on Harsanyi's social aggregation theorem and the weak Pareto principle. Social Choice and Welfare 12 (1995): 87-92.

Yaari, M., 1987. The dual theory of choice under risk. Econometrica 55, 95-115.

Zhou, L., 1997. Harsanyi's utilitarianism theorems: general societies. Journal of Economic Theory, 72, 198-207.

Zuber, S., 2016. Harsanyi's theorem without the sure-thing principle: On the consistent aggregation of Monotonic Bernoullian and Archimedean preferences. Journal of Mathematical Economics 63: 78-83.