Observable interpersonal utility comparisons

Mononen, Lasse Article — Published Version Observable interpersonal utility comparisons Social Choice and Welfare Provided in Cooperation with: Springer Nature Suggested Citation: Mononen, Lasse (2025) : Observable interpersonal utility comparisons, Social Choice and Welfare, ISSN 1432-217X, Springer, Berlin, Heidelberg, Vol. 65, Iss. 3, pp. 629-644, https://doi.org/10.1007/s00355-025-01584-z This Version is available at: https://hdl.handle.net/10419/330231 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ Social Choice and Welfare (2025) 65:629–644 https://doi.org/10.1007/s00355-025-01584-z ORIGINAL PAPER Observable interpersonal utility comparisons Lasse Mononen1 Received: 23 February 2024 / Accepted: 31 January 2025 / Published online: 15 March 2025 © The Author(s) 2025 Abstract Harsanyi’s seminal aggregation theorem axiomatized weighted utilitarianism based on expected utility theory. However, the weights assigned to each individual cannot be separated from the individual’s utility. We show that once we depart from the expected utility framework, it is possible to uniquely identify the utilities and the weights. Specifically, we show that in the min-of-means social welfare function if each individual has a cardinal utility, unique up to a positive affine transformation, and any redistribution of utilities changes the social welfare for some initial allocation, then we can uniquely identify the utilities of the individuals and the weights of the social welfare function. 1 Introduction Harsanyi’s seminal aggregation theorem (Harsanyi 1955) axiomatized weighted utilitarianism based on expected utility theory. However, the result has been criticised since the weights attributed to individuals are not meaningful because they cannot be separated from the individuals’ utilities (Sen 1976; Broome 1987;Weymark1991). To overcome this identification issue, Harsanyi (1977) used direct interpersonal utility comparisons. However, interpersonal utility comparisons are difficult to make and have remained controversial in the literature (Elster and Roemer 1991; Greaves and Lederman 2018). Additionally, the assumption of expected utility theory in Harsanyi’s aggregation theorem has been criticised (e.g. Diamond (1967); Sen (1970); Broome (1987)). We show that once we depart from Harsanyi’s expected utility framework, the weights and utilities are can be meaningful even without direct interpersonal utility comparisons. Specifically, we show that in the min-of-means social welfare function if each individual has a cardinal utility, unique up to a positive affine transformaThe author thanks Niels Boissonnet, Arthur Dolgopolov, Larry Epstein, Marc Fleurbaey, Dominik Karos, Frank Riedel and the anonymous reviewers for useful comments. This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)-Project-ID 317210226-SFB 1283. BLasse Mononen [email protected] 1Center for Mathematical Economics, Bielefeld University, PO Box 10 01 31, 33 501 Bielefeld, Germany 123 630 L. Mononen tion, and for every utility redistribution, there is some initial allocation such that the redistribution changes the social welfare, then we can uniquely identify the utilities of the individuals and the weights of the social welfare function. This shows that the social observer is behaving as if making interpersonal utility comparisons that we can observe indirectly together with the fairness of the society. Our result formalizes Kaneko’s (1984) suggestion for observing interpersonal utility comparisons from the social welfare function. Here, we identify the interpersonal utility comparisons from the non-linearities of the social welfare function. For example, in the case of the Rawlsian social welfare function (Rawls 1971), the non-linearities capture the change of the worst-off individual that allows us to identify utilities across individuals. We generalize this identification strategy beyond the Rawlsian social welfare function. We study the identification of individuals’ utilities and weights in the min-of-means social welfare function. This has been considered as capturing the ignorant observer in Gajdos and Kandil (2008). Additionally, it has been considered in the context of income inequality in Ben-Porath et al. (1997), Gajdos and Maurin (2004), Crès et al. (2011), and recently in Mongin and Pivato (2021). This representation includes utilitarianism and the Rawlsian social welfare function (Rawls 1971) as special cases. The min-of-means representation consists of a (von Neumann–Morgenstern) utility function uifor each member i∈I={1,...,n}and a set of weights for each member ⊆(I)such that the societal value of an alternative xis min λ∈ i∈I λiui(x). We show that the set of weights and the utility functions are identified if and only if any utility redistribution from one member to another changes the welfare in some situation. That is if (vi)i∈I∈Rnis a utility redistribution such that there exist members iand jwith vi>0>v j, then there exists an initial utility allocation (wi)i∈I∈Rn such that redistributing the utility by (vi)i∈Ichanges the social welfare, min λ∈ i∈I λiwi= min λ∈ i∈I λi(wi+vi). (1) This condition captures in terms of the social welfare function that the societal value of every utility redistribution depends on the context: Formally, for each utility redistribution (vi)i∈I∈Rnsuch that there exist members iand jwith vi>0>v j, there exist weights λ,λ∈such that  i∈I λivi=  i∈I λ ivi. Technically, this condition is equivalent to the set of weights having a non-empty interior. This result shows that once we move away from the expected utility framework and the societal value of every utility redistribution depends on the current utility distribution, interpersonal utility comparisons and weights assigned to individuals can be observable. Especially we show that in the min-of-means representation with affine utilities, our identifying condition Eq. (1) is equivalent to the identification of the set of Pareto weights and equivalent to the identification of interpersonal utility 123 Observable interpersonal utility… 631 Mathematically, the separation of weights and utility functions is symmetrical to the separation of probabilities and state dependent utilities in choice under uncertainty. Here, the min-of-means social welfare function corresponds to the state dependent maxmin expected utility. Our identification result follows as a corollary from the identification of probabilities and state dependent utilities in a state dependent maxmin expected utility in Mononen (2024). Our second contribution is that we characterize the existence of the min-of-means social welfare function by relaxing Harsanyi’s assumption that the societal preferences satisfy the expected utility theory. Instead, we allow for violations of expected utility theory when the alternatives involve trade-offs across the members and only assume that the societal preferences satisfy expected utility theory when there are no trade-offs across the members.1 Our results are closely related to Gajdos and Kandil (2008). They study when an impartial observer’s extended preferences have a min-of-means representation. In this extended setting, the observer is especially able to make direct interpersonal utility comparisons and Harsanyi’s utilitarianism is fully identified. We instead study observable social preferences that do not include direct interpersonal utility comparisons. This allows us to substantially simplify the axiomatization of Gajdos and Kandil (2008) and clarify further the difference between the min-of-means social welfare function and utilitarianism. We follow the single-profile formalism pioneered by Harsanyi (1955) that studied preference aggregation over fixed preferences. This approach has been used for example in Mongin (1995), Gilboa et al. (2004), Chambers and Hayashi (2006) and Gajdos et al. (2008). This approach is in contrast to the multi-profile formalism studying preference aggregation over varying preferences as pioneered by Arrow (1951) and Sen (1970) and has been summarized in d’Aspremont and Louis (2002). In choice under uncertainty, Mongin (1995) and Mongin (1998) study the aggregation of individuals under subjective expected utility theory. Mongin shows that if there is sufficient preference diversity, then under state independent utility and Pareto monotonicity, this leads to a dictatorial choice rule. However, under state dependent utility, non-dictatorial aggregation is possible. Amarante and Ghossoub (2021)shows the possibility of non-dictatorial aggregation when the aggregated preferences do not follow subjective expected utility theory. The solution in the literature for the lack of identification in Harsanyi (1955) has been to consider non-observable extended lotteries that allow for direct interpersonal utility comparisons. This approach was pioneered in Harsanyi (1977) and used in Karni and Weymark (1998), Gajdos and Kandil (2008), Grant et al. (2010), and discussed in Adler (2014) and Greaves and Lederman (2018). Another solution to the lack of identification has been to consider relative utilitarianism that was introduced by Dhillon and Mertens (1999) and Segal (2000) and preference satisfaction. Börgers and Choo (2017) and Karni and Weymark (2024) study the elicitation of Pareto weights in relative utilitarianism. These approaches are discussed later in Sect.2.3. 1However, since we do not observe interpersonal utility comparisons, we make a more general assumption and assume there are two different lotteries that satisfy the expected utility theory. 123 632 L. Mononen Technically, our results are related to the literature on income inequality measurement Weymark (1981), Yaari (1988), Ben-Porath et al. (1997). However, here we focus on the more general welfare inequality measurement with subjective utility for each member. The remainder of the paper proceeds as follows: Sect.1.1 offers a simple example highlighting the intuition for our identification result. Section2studies the identifications of the min-of-means social welfare function, Sect.2.3 compares the identification to relative utilitarianism and utilitarianism. Section3axiomatically characterizes the existence of the representations and Sect.4concludes. The Appendix proves all the results. 1.1 An example of identification We begin with a simple example illustrating that with the weighted utilitarian social welfare function individuals’ utilities and weights cannot be separated. However, this is only an unidentified special case. In the second part of the example, we show that with the min-of-means social welfare function these can be separated and identified from the violations of the independence axiom. First, we illustrate the lack of identification in weighted utilitarianism. Consider a society consisting of two individuals 1 and 2 that have preferences over some set of lotteries (X)over social alternatives X. Each of the individuals has an affine von Neumann–Morgenstern utilities u1,u2:(X)→R. These are aggregated into weighted utilitarian social welfare with equal weight for both of the individuals: The social value of alternative p∈(X)is 0.5u1(p)+0.5u2(p). Now these preferences have an alternative weighted utilitarian representation with any weight λ∈(0,1)for individual 1 since 0.5u1(p)+0.5u2(p)=λ0.5 λu1(p)+(1−λ)0.5 1−λu2(p)=λu1(p)+(1−λ)u2(p), where the terms inside the parentheses define new utility functions u1,u2. In this alternative representation, we have replaced the weight of the individual for the intensity of preferences. This highlights the impossibility of identification in weighted utilitarianism since the intensities of preferences are inseparable from the weights. This violates our identification condition (1) since for utility redistribution (v1,v 2)=(1 λ,−1 1−λ), we have for all (w1,w 2)∈R2, λw1+(1−λ)w2=λ(w1+v1)+(1−λ)(w2+v2). Next, we move on to min-of-means social welfare over lotteries pdefined by two weights λ∗<λ ∗for individual 1 and affine von Neumann–Morgenstern utilities u1,u2 min λ∈[λ∗,λ∗]λu1(p)+(1−λ)u2(p). We show that utilities are identifiable across individuals from changes in the Pareto weights, that is violations of the independence axiom.2For this, let pand qbe two 2The independence axiom from von Neumann and Morgenstern (1947) and Harsanyi (1955) characterizes 123 Observable interpersonal utility… 633 Fig. 1 An example of identifying when the utilities across individuals are equal from the non-linearities. Lotteries pand qare such that u1(p)+u1(q)=u2(p)+u2(q)and u1(q)<u2(q). The x-axis changes p to qwith convex combinations. The y-axis is the min-of-means social welfare of αp+(1−α)q.Atα=0.5 utilities across the individuals are equal that is observable as a violation of the independence axiom lotteries such that u1(p)+u1(q)=u2(p)+u2(q)and u1(q)<u2(q).Next,weshow that there is a violation of the independence axiom at 0.5p+0.5q. We focus on the min-of-means social welfare of αp+(1−α)qwhen αchanges from 0 to 1 as in Fig.1. First, between 0 and 0.5, by the affine utilities, u1(α p+ (1−α)q)<u2(α p+(1−α)q). So the min-of-means welfare uses the weight λ∗ and it changes linearly at the rate λ∗u1(p)−u1(q)+(1−λ∗)u2(p)−u2(q). Second, between 0.5 and 1, u1(α p+(1−α)q)>u2(α p+(1−α)q). So the minof-means welfare uses the weight λ∗and this time it changes linearly at the rate λ∗u1(p)−u1(q)+(1−λ∗)u2(p)−u2(q). Since u1(p)−u1(q)>u2(p)−u2(q), the rate of change switches at α=0.5 and there is a non-linearity at that point. This represents a violation of the independence axiom at the lottery 0.5p+0.5qwhere both of the individuals have the same utility. Finally, there can be violations of the independence axiom only if the utilities for both individuals are the same. The only situations where there can be non-linearities as in Fig.1are when the used Pareto weight changes. However, the min-of-means social welfare function with two individuals always maximizes the weight for the individual with a lower utility. Thus, the change in the used Pareto weight means that the utility order of the individuals changed. Especially, in here, the utilities for both of the individuals are exactly the same. In summary, the lotteries where the utilities for both individuals are equal are characterized by the violations of the independence axiom and especially they are observable. The min-of-means social welfare function rules out the previous violations of our identification condition (1). First, for many redistributions, the axiom holds trivially. If (v1,v 2)is a utility redistribution such that v1<0<v2and minλ∈[λ∗,λ∗]λv1+(1−λ)v2= the linearity of weighted utilitarianism. It states that for all lotteries p,q,rand α∈(0,1), pq⇐⇒ αp+(1−α)rαq+(1−α)r. 123 634 L. Mononen 0, then the condition holds for (˜w1,˜w2)=(0,0). Second, we focus on redistributions that preserve the welfare when redistributing from (0,0)and show that in another context when redistributing from (w1,w 2)=(−0.5v1,−0.5v2), the redistribution affects the welfare. Now, we have since v1<0<v 2and the redistribution does not affect welfare when redistributing from (0,0), min λ∈[λ∗,λ∗]λ(w1+v1)+(1−λ)(w2+v2)=λ∗(0.5v1)+(1−λ∗)(0.5v2)=0. Next, since λ∗<λ ∗and v1<0<v 2,wehave 0<λ ∗(0.5v1)+(1−λ∗)(0.5v2). Hence, 0>λ ∗(−0.5v1)+(1−λ∗)(−0.5v2)=min λ∈[λ∗,λ∗]λw1+(1−λ)w2. This shows that the identification condition (1) holds also in this case. This illustrates that for most of the utility redistributions, the identifying condition holds trivially. However, the identifying condition assumes that utility redistributions that are welfare preserving in one context affect welfare in some other context. This identification example is generalized in our main result, Theorem 1, to finitely many individuals. There we show that if the social value of every redistribution depends on the context, then the individuals’ utilities are observable. 2 Identification 2.1 Preliminaries and notation We follow the setting from Harsanyi (1955,1977). Society consists of members I= {1,...,n}.Xis a set of social-alternatives. Each member i∈Ihas preferences i over (simple) social-alternative lotteries (X)and additionally, we observe societal preferences 0over (simple) social-alternative lotteries (X). (Normalized) weights for the members are probability distributions on the members (I).(I)is equipped with the Euclidean topology. We consider the min-of-means social welfare function following (Ben-Porath et al. 1997; Gajdos and Kandil 2008) over expected utility members as in Harsanyi (1955, 1977). Definition Affine utilities ui:(X)→Rfor each i∈Iand a convex and closed set of Pareto weights ⊆(I)is a min-of-means representation for ((i)i∈I,0) if the following two conditions hold: 1. for each i∈Iand p,q∈(X),wehave piq⇐⇒ ui(p)≥ui(q). 2. for all p,q∈(X),wehave p0q⇐⇒ min λ∈ i∈I λiui(p)≥min λ∈ i∈I λiui(q). We offer an axiomatic characterization for the min-of-means social welfare function later on in Sect.3. We focus especially on min-of-means representations with the smallest possible set of Pareto weights as defined next. 123 Observable interpersonal utility… 635 Definition Affine utilities ui:(X)→Rfor each i∈Iand a convex and closed set of Pareto weights ⊆(I)is a minimal min-of-means representation for ((i)i∈I,0) if for any other min-of-means representation with the same utilities (ui)i∈Iand a set of Pareto weights ˜ ,wehave⊆˜ . The next example shows the significance of minimal representations since there can be weights that the social welfare function never uses. We connect general and minimal representations in the next section. Remark (Non-minimal example) If n=2 and for all p∈(X),u1(p)<u2(p), then the set of Pareto weights ={(λ, 1−λ)|λ∈[0,1 2]} is not minimal since for all p∈(X) min λ∈[0,1 2] λu1(p)+(1−λ)u2(p)=1 2u1(p)+1 2u2(p). Next, we define when the minimal set of weights and the utility functions are identified. Definition The set of weights in the minimal min-of-means representation is identified if for all minimal min-of-means representations ((ui)i∈I,) and (( ˜ui)i∈I,˜ ),we have =˜ . In contrast to the set of weights that are only identified for minimal representations, we identify the utility functions for all the min-of-means representations. Definition The utilities in the min-of-means representation are identified up to a common positive affine transformation if for all min-of-means representations ((ui)i∈I,) and (( ˜ui)i∈I,˜ ), there exist α>0 and β∈Rsuch that for each i∈Iand p∈(X) ui(p)=α˜ui(p)+β. 2.2 Uniqueness Our main result characterizes when the minimal min-of-means representation is fully identified. This identification is characterized by the following condition. Axiom 1 For any lotteries pand qsuch that there exist i,j∈Iwith piqand qjp, there exists a lottery rand α∈[0,1]such that αp+(1−α)r0αq+(1−α)r Here, we consider redistributing the utilities by (uk(q)−uk(p))k∈Ithat benefits the member jand makes the member iworse off. However, we do not make any restrictions on how the redistribution affects other members. Then the axiom assumes that there exists an allocation αp+(1−α)rwith utilities uk(α p+(1−α)r)k∈I such that performing the utility redistribution to change the utilities to uk(α p+(1−α)r)+α(uk(q)−uk(p))k∈I=uk(αq+(1−α)r)k∈I 123 636 L. Mononen changes the welfare. That is, for any utility redistribution there is some situation such that the redistribution changes the welfare. For this interpretation, it is crucial that the members have affine utilities. The next result shows that Axiom 1characterizes the identification of the minimal min-of-means representation. Theorem 1 Assume that ((i)i∈I,0)has a minimal min-of-means representation ((ui)i∈I,)such that int ui(p)i∈Ip∈(X)= ∅. Then the following five conditions are equivalent. (1) ((i)i∈I,0)satisfy Axiom 1. (2) int = ∅. (3) For all v∈RIsuch that there exist i,j∈Iwith vi>0>vj, there exist λ,λ∈ such that  i∈I λivi=  i∈I λ ivi. (4) The set of weights in the minimal min-of-means representation is identified. (5) The utilities in the min-of-means representation are identified up to a common positive affine transformation. First, the equivalency between (1), (4), and (5) shows the identification of weights assigned to members and interpersonal utility comparisons: When any redistribution changes welfare in some situation, then the minimal set of weights assigned to members and interpersonal utility comparisons can be identified in the min-of-means representation from the societal preferences. Here, the social observer is behaving as if making interpersonal utility comparisons that we can observe indirectly. This identification was illustrated in Sect.1.1. As in the example, Axiom 1guarantees that any redistribution violates the independence axiom and there is a change in the Pareto weight for any redistribution that is used for the identification. Second, the equivalency between conditions (1), (2), and (3) characterizes when the minimal min-of-means representation satisfies Axiom 1. This shows that our identifying condition is equivalent to the societal value of every utility redistribution depending on the context or to the set of weights that the social welfare function uses having a non-empty interior. For example, this result shows that if all the Pareto weights agree on the weight of member ior if all the Pareto weights agree that the weight of member iis twice as large as member j, then the min-of-means representation does not satisfy Axiom 1. Theassumptionthat int ui(p)i∈Ip∈(X)= ∅ is a standard identification condition in the literature. It has been used e.g. in Harsanyi (1955), Weymark (1991), and Fleurbaey and Mongin (2016). It is characterized by the independent prospects axiom assuming that for each i∈I, there exists lotteries pand qsuch that piqand for each j= i,p∼jq. The next example shows the identification for a convex combination of utilitarianism and the Rawlsian social welfare function as proposed by Gajdos and Kandil 123 Observable interpersonal utility… 643 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/. References Adler MD (2014) Extended preferences and interpersonal comparisons: a new account. Econ Philos 30(2):123–162 Alon S, Schmeidler D (2014) Purely subjective maxmin expected utility. J Econ Theory 152:382–412 Amarante M, Ghossoub M (2021) Aggregation of opinions and risk measures. J Econ Theory 196:105310 Arrow KJ (1951) Social choice and individual values. Wiley, New York Barrett J (2019) Interpersonal comparisons with preferences and desires. Polit Philos Econ 18(3):219–241 Ben-Porath E, Gilboa I, Schmeidler D (1997) On the measurement of inequality under uncertainty. J Econ Theory 75(1):194–204 Börgers T, Choo Y-M (2017) Revealed relative utilitarianism CESifo Working Paper Series No. 6613, Munich. Boyd S, Vandenberghe L (2004) Convex optimization. Cambridge University Press, Cambridge Broome J (1987) Utilitarianism and expected utility. J Philos 84(8):405–422 Broome J (1991) Weighing goods. Basil Blackwell, Cambridge Chambers CP, Hayashi T (2006) Preference aggregation under uncertainty: savage vs. pareto. Games Econ Behav 54(2):430–440 Chandrasekher M, Frick M, Iijima R, Le Yaouanq Y (2022) Dual-self representations of ambiguity preferences. Econometrica 90(3):1029–1061 Crès H, Gilboa I, Vieille N (2011) Aggregation of multiple prior opinions. J Econ Theory 146(6):2563–2582 d’Aspremont C, Gevers L (2002) Social welfare functionals and interpersonal comparability. In: Arrow KJ, Sen AK, Suzumura K (eds) Handbook of social choice and welfare, vol 1. Elsevier, Amsterdam, pp 459–541 Dhillon A, Mertens J-F (1999) Relative utilitarianism. Econometrica 67(3):471–498 Diamond PA (1967) Cardinal welfare, individualistic ethics, and interpersonal comparison of utility: comment. J Polit Econ 75(5):765–766 Elster J, Roemer JE (eds) (1991) Interpersonal comparisons of well-being. Cambridge University Press, Cambridge Fleurbaey M, Mongin P (2016) The utilitarian relevance of the aggregation theorem. Am Econ J Microecon 8(3):289–306 Gajdos T, Kandil F (2008) The ignorant observer. Soc Choice Welf 31(2):193–232 Gajdos T, Maurin E (2004) Unequal uncertainties and uncertain inequalities: an axiomatic approach. J Econ Theory 116(1):93–118 Gajdos T, Tallon J-M, Vergnaud J-C (2008) Representation and aggregation of preferences under uncertainty. J Econ Theory 141(1):68–99 Gilboa I, Samet D, Schmeidler D (2004) Utilitarian aggregation of beliefs and tastes. J Polit Econ 112(4):932–938 Grant S, Kajii A, Polak B, Safra Z (2010) Generalized utilitarianism and Harsanyi’s impartial observer theorem. Econometrica 78(6):1939–1971 Greaves H, Lederman H (2018) Extended preferences and interpersonal comparisons of well-being. Philos Phenomenol Res 96(3):636–667 Harsanyi JC (1955) Cardinal welfare, individualistic ethics, and interpersonal comparisons of utility. J Polit Econ 63(4):309–321 Harsanyi JC (1977) Rational behaviour and bargaining equilibrium in games and social situations. Cambridge University Press, Cambridge Hausman DM (1995) The impossibility of interpersonal utility comparisons. Mind 104(415):473–490 Hausman DM (2011) Preference, value, choice, and welfare. Cambridge University Press, Cambridge Herstein IN, Milnor J (1953) An axiomatic approach to measurable utility. Econometrica 21(2):291–297 Kaneko M (1984) On interpersonal utility comparisons. Soc Choice Welf 1(3):165–175 123 644 L. Mononen Karni E, Weymark JA (1998) An informationally parsimonious impartial observer theorem. Soc Choice Welf 15(3):321–332 Karni E, Weymark JA (2024) Impartiality and relative utilitarianism. Soc Choice Welf 63(1):1–18 Mongin P (1995) Consistent Bayesian aggregation. J Econ Theory 66(2):313–351 Mongin P (1998) The paradox of the Bayesian experts and state-dependent utility theory. J Math Econ 29(3):331–361 Mongin P, Pivato M (2021) Rawls’s difference principle and maximin rule of allocation: a new analysis. Econ Theor 71(4):1499–1525 Mononen L (2024) State dependent utility and ambiguity. Working Paper 24020, SFB 1283, Bielefeld University. Rawls J (1971) A theory of justice. Belknap Press of Harvard University Press, Cambridge Segal U (2000) Let’s agree that all dictatorships are equally bad. J Polit Econ 108(3):569–589 Sen A (1970) Collective choice and social welfare. Holden-Day, San Francisco Sen A (1976) Welfare inequalities and Rawlsian axiomatics. Theor Decis 7(4):243–262 von Neumann J, Morgenstern O (1947) Theory of games and economic behavior, 2nd edn. Princeton University Press, Princeton Weymark JA (1981) Generalized Gini inequality indices. Math Soc Sci 1(4):409–430 Weymark JA (1991) A reconsideration of the Harsanyi–Sen debate on utilitarianism. In: Elster J, Roemer JE (eds) Interpersonal comparisons of well-being. Cambridge University Press, Cambridge, pp 255–320 Yaari M (1988) A controversial proposal concerning inequality measurement. J Econ Theory 44(2):381–397 Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. 123