We study the problem of aggregating individual geometric discounting (GD) preferences over infinite streams of consumption profiles into a system specifying, for each consumption history, a social ranking of the streams that follow that history. Such a system is time-consistent if and only if it is generated by a single underlying preference ordering over the lifetime streams (Lemma 1). Pareto Indifference and Neutrality lead to evaluating a lifetime stream according to the profile of range-normalized GD utilities it yields (Lemma 2). The Weak Pareto Principle, Anonymity and Discounting Irrelevance (individual discount factors do not affect the ranking of constant streams) imply that the ordering of the range-normalized GD utility profiles is monotonic, symmetric, and independent of the profile of discount factors (Theorem 1). History Independence singles out the range-normalized utilitarian rule (Theorem 2) whereas the range-normalized Nash rule is characterized by Independence of Infeasible Consumptions (the ranking of streams delivering consumption profiles in an interval from the origin does not depend on individual valuations outside that interval) or Valuation Irrelevance (the ranking of pure timing streams is independent of individual valuations) (Theorem 3).
A rational weighted tournament x specifies for every pair of social alternatives i,j the proportion xij of voters (with linear preference orderings) who prefer i to j. An extension rule assigns to each x a random choice function y (specifying a collective choice probability distribution for each subset of alternatives) that chooses i from {i,j} with probability xij.
We analyze extension rules axiomatically. We impose that the random choice function y be stochastically rational. We show that there exist multiple neutral and stochastically rational extension rules. Linearity (requiring that y be a linear function of x) and Independence of Irrelevant Comparisons (asking that the probability distribution on a subset of alternatives depend only on the restriction of the weighted tournament to that subset) are incompatible with very weak properties implied by stochastic rationality.
We identify a class of domains, which we call sequentially binary, guaranteeing that a weighted tournament arising from a population of voters with preferences in such a domain has a unique stochastically rational extension.
Call a choice function C more rational than C’ if C is rational on every collection of agendas where C’ is rational, and on some collection where C’ is not; call the two choice functions equally rational if they are rational on the same collections. We analyze the quasi ordering induced by that definition.
The broken Borda rule and other refinements of approval ranking (with G. Barokas)
We study the social aggregation problem in the preference-approval model of Brams and Sanver (2009). Each voter reports a linear ordering of the alternatives and an acceptability threshold. A rule transforms every profile of such "opinions" into a social ordering. The approval rule ranks the alternatives according to the number of voters who find them acceptable. The broken Borda rule ranks them according to the total score they receive; the scores assigned by a voter follow the standard Borda scale except that a large break is introduced between the score of her worst acceptable alternative and the score of her best unacceptable alternative. We offer an axiomatization of this rule and other lexicographic combinations of the approval rule and a fixed social welfare function.
We show that the triangular inequalities are sufficient to extend a system of binary probabilities to a regular random choice function.
Strategy-proof choice under monotonic additive preferences (with E. Bahel)
We describe the class of strategy-proof mechanisms for choosing sets of objects when preferences are additive and monotonic.
Two-Stage Majoritarian Choice (with S. Horan)
We propose a class of decisive collective choice rules that rely on an exogenous linear ordering to partition the majority relation into two acyclic relations. The first relation is used to obtain a shortlist of the feasible alternatives while the second is used to make a final choice.
Rules in this class are characterized by four properties: two classical rationality requirements (Sen's expansion consistency and Manzini and Mariotti's weak WARP); and adaptations of two well-known collective choice conditions (Arrow's independence of irrelevant alternatives and Saari and Barney's no preference reversal bias). These rules also satisfy a number of other desirable properties including May's positive responsiveness.
Strategyproof choice of social acts: Revised version --June 2019 (with E. Bahel)
We model uncertain social prospects as acts mapping states of nature to (social) outcomes. A social choice function (or SCF) assigns an act to each profile of subjective expected utility preferences over acts. A SCF is strategyproof if no agent ever has an incentive to misrepresent her beliefs about the states of nature or her valuation of the outcomes. It is unanimous if it picks the feasible act that all agents find best whenever such an act exists. We offer a characterization of the class of strategyproof and unanimous SCFs in two settings. In the setting where all acts are feasible, the chosen act must yield the favorite outcome of some (possibly different) agent in every state of nature. The set of states in which an agent's favorite outcome is selected may vary with the reported belief profile; it is the union of all states assigned to her by a collection of constant, bilaterally dictatorial, or bilaterally consensual assignment rules. In a setting where each state of nature defines a possibly different subset of available outcomes, bilaterally dictatorial or consensual rules can only be used to assign control rights over states characterized by identical sets of available outcomes.
Non-dictatorial Paretian aggregation of subjective expected utility preferences is possible under binary uncertainty when differences in tastes are small compared to differences in beliefs.
In the context of uncertainty, belief-weighted relative utilitarianism consists in comparing acts according to a weighted sum of the (0,1)-normalized subjective expected utilities they yield. The weights may change with the profile of beliefs but do not depend upon the profile of individual utilities for the outcomes. This class of social welfare functions is characterized by the Pareto principle, the sure-thing principle, a continuity condition, and an independence condition requiring that the social ranking of two acts is unaffected by the addition of an outcome that leaves everyone's best and worst outcomes unchanged. The weights must be constant if the social ranking of constant acts is independent of individual beliefs. Anonymity then pins down plain relative utilitarianism: acts are compared according the sum of (0,1)-normalized subjective expected utilities they generate.
We study two informational simplicity conditions for aggregating von Neumann-Morgenstern preferences. When the best relevant alternative for each individual cannot be ascertained with confidence (as when allocating an uncertain endowment of goods), Independence of Harmless Expansions requires that the social ranking of lotteries be unaffected by the addition of any alternative that every agent deems at least as good as the one she originally found worst. This axiom, along with the Weak Pareto Principle and Anonymity, characterizes bottom-calibrated Nash welfarism: utilities are calibrated so that the worst alternative is worth zero and lotteries are ranked according to the product of such bottom-calibrated utilities. When the worst relevant alternatives are difficult to identify, replacing Independence of Harmless Expansions by the dual axiom of Independence of Useless Expansions yields a characterization of top-calibrated Nash welfarism: lotteries are ranked according to the opposite of the product of the absolute values of top-calibrated utilities. The distributive implications of our two informational simplicity axioms are thus drastically different: while bottom-calibrated Nash welfarism recommends randomizing between two alternatives that it deems equally good, top-calibrated Nash welfarism is randomization-averse.
The belief-weighted Nash social welfare functions are methods for aggregating Savage preferences defined over a set of acts. Each such method works as follows. Fix a 0-normalized subjective expected utility representation of every possible preference and assign a vector of individual weights to each profile of beliefs. To compute the social preference at a given preference profile, rank the acts according to the weighted product of the individual 0-normalized subjective expected utilities they yield, where the weights are those associated with the belief profile generated by the preference profile. We show that these social welfare functions are characterized by the weak Pareto principle, a continuity axiom, and the following informational robustness property: the social ranking of two acts is unaffected by the addition of any outcome that every individual deems at least as good as the one she originally found worst. This makes the belief-weighted Nash social welfare functions appealing in contexts where the best relevant outcome for an individual is difficult to identify.
This working paper version of the SCW 2013 article entitled ''Relative Egalitarianism'' contains additional results and different proofs.
This working paper version of the IJGT 2008 article with the same title contains full proofs.