Abstract. We consider a model where an agent is must choose between alternatives that each provide only an imprecise description of the world (e.g. linguistic expressions). The set of alternatives is closed under logical conjunction and disjunction, but not necessarily negation. (Formally: it is a distributive lattice, but not necessarily a Boolean algebra). In our main result, each alternative is identified with a subset of an (endogenously defined) state space, and two axioms characterize maximin decision making. This means: from the agent's preferences over alternatives, we derive a preference order on the endogenous state space, such that alternatives are ranked in terms of their worst outcomes. 

Abstract. One agent may encounter many sources of uncertainty and many menus of outcomes, which can be combined together into many different decision problems. There may be analogies between different uncertainty sources (or different outcome menus). Some uncertainty sources (or outcome menus) may exhibit internal symmetries. The agent may also have different levels of awareness. In some situations, the state spaces and outcome spaces have additional mathematical structure (e.g. a topology, metric, or differentiable structure), and feasible acts must respect this structure (i.e. they must be continuous, short, or smooth functions). In other situations, the agent might only be aware of a set of abstract “acts”, and be unable to specify explicit state spaces and outcome spaces. We introduce a modelling framework that addresses all of these issues. We then define and axiomatically characterize a subjective expected utility representation that is “global” in two senses. First: it posits probabilistic beliefs for all uncertainty sources and utility functions over all outcome menus, which simultaneously rationalize the agents’ preferences across all possible decision problems, and which are consistent with the aforementioned analogies, symmetries, and awareness levels.  Second: it applies in many mathematical environments (i.e. categories), making it unnecessary to develop a new theory for each one.

• 1. • Slides and Video recording of presentation at the 7th International Workshop on Applied Category Theory (Oxford, June 2024).

Abstract. To adequately describe notions like "autonomy" or "responsibility" within the standard economic model of rational choice, it is necessary to introduce metapreferences ---that is, preferences over preferences. This paper develops two models of metapreferences: a "hierarchical" model, involving second-order preferences, third-order preferences, etc., and a "recursive" model, involving a type space where each type determines preferences over type-outcome pairs. For each model, the paper introduces concepts of internal coherence and rational choice. It then studies the relationship between the two models. 

Abstract: Give a set X of “outcomes” and a set T of “types”, a recursive preference structure (RPS) is a function that assigns a continuous partial order over T x X to every element of T . This describes an agent who has preferences not only over the outcomes in X , but also over her own preferences (as encoded by the types). We prove the existence of a universal RPS —one into which any other RPS can be mapped in a unique way. Formally, this universal RPS is a terminal coalgebra of a suitably defined endofunctor on the category of compact Hausdorff spaces.

• 1. •  Slides and Video recording of presentation at the 7th International Workshop on Applied Category Theory (Oxford, June 2024).

Abstract. Let X be a topological space. We consider two spaces of continuous preference relations on X. A local continuous quasiorder is a continuous, complete, quasitransitive binary relation defined on a closed subset of X. Let Q(X) be the set of all such relations. A local continuous strict partial order is a continuous partial order (with no indifference) defined on a closed subset of X. Let P(X) be the set of all such relations. There is a natural bijection ("duality") between Q(X) and P(X). We endow both sets with the Fell topology (also called the topology of closed convergence, the Vietoris topology, or the Hausdorff metric topology, depending on context), and show that under mild conditions, they are compact Hausdorff spaces, compact metrizable spaces, continua, or even contractible continua. (In contrast, most "preference spaces" in the literature have none of these nice properties.) Furthermore, any continuous function 𝜑 : X → Y induces continuous functions Q𝜑 : Q(X) → Q(Y) and P𝜑 : P(X) → P(Y). We thus obtain two endofunctors Q and P on the category of compact Hausdorff spaces, which are naturally isomorphic to each other. Finally, we show that these endofunctors are "continuous": if X is the limit of a chain X_1 ← X_2 ← X_3 ← .... of compact Hausdorff spaces, then Q(X) is the limit of the corresponding chain Q(X_1) ← Q(X_2) ← Q(X_3) ← ..... (and likewise for P(X)). 

Abstract:  In many decisions under uncertainty, there are technological constraints on the acts an agent can perform and on the events she can observe. To model this, we assume that the set S of possible states of the world and the set X of possible outcomes each have a topological structure. The only feasible acts are continuous functions from S to X, and the only observable events are regular open subsets of S. We axiomatically characterize Subjective Expected Utility (SEU) representations of conditional preferences over acts, involving a continuous utility function on X (unique up to positive affine transformations), and a unique Borel probability measure on S, along with an auxiliary apparatus called a "liminal structure", which describes the agent’s imperfect perception of events. We also give other SEU representations, which use residual probability charges or compactifications of the state space. 

Abstract. In recent work, I have developed a general category-theoretic framework for decision theory. This paper applies this to the category of orthomodular lattices. Every Boolean algebra is an orthomodular lattice, so this yields a new ("syntactic") model of decision-making with classical uncertainty. The lattice of closed subspaces of a Hilbert space is also an orthomodular lattice, so this also yields a new model of decision-making with quantum uncertainty.

Let X be a connected metric space, and let > be a weak order defined on a suitable subset of XN. We characterize when > has a Cesàro average utility representation. This means that there is a continuous real-valued function u on X such that, for all sequences x = (xn)∞n=1 and y = (yn)∞n=1 in the domain of >, we have x > y if and only if the limit as N → ∞ of the average value of u(x1), u(x2),….,u(xN) is higher than limit as N → ∞ of the average value of u(y1), u(y2),….,u(yN).  This has applications to decision theory, game theory, and intergenerational social choice.

Abstract We consider a model of intertemporal choice where time is a continuum, the set of instantaneous outcomes (e.g. consumption bundles) is a topological space, and where intertemporal plans (e.g. consumption streams) must be continuous functions of time. We assume the agent can form preferences over plans defined on open time intervals. We axiomatically characterize the intertemporal preferences that admit a representation via discounted utility integrals. In this representation, the utility function is continuous and unique up to positive affine transformations, and the discount structure is represented by a unique Riemann-Stieltjes integral plus a unique linear functional measuring the long-run asymptotic utility. 

Abstract. In many decisions under uncertainty, there are constraints on both the available information and the feasible actions. The agent can only make certain observations of the state space, and she cannot make them with perfect accuracy — she has imperfect perception. Likewise, she can only perform acts that transform states continuously into outcomes, and perhaps satisfy other regularity conditions. To incorporate such constraints, we modify the Savage decision model by endowing the state space S and outcome space X with topological structures. We axiomatically characterize a Subjective Expected Utility (SEU) representation of conditional preferences, involving a continuous utility function on X (unique up to positive affine transformations), and a unique probability measure on a Boolean algebra B of regular open subsets of S. We also obtain SEU representations involving a Borel measure on the Stone space of B — a “subjective” state space encoding the agent’s imperfect perception. 

Abstract.  An agent faces a decision under uncertainty with the following structure. There is a set A of “acts”; each will yield an unknown real-valued payoff. Linear combinations of acts are feasible; thus, A is a vector space. But there is no pre-specified set of states of nature. Instead, there is a Boolean algebra J describing information the agent could acquire. For each element of J, she has a conditional preference order on A. I show that if these conditional preferences satisfy certain axioms, then there is a unique compact Hausdorff space S such that elements of A correspond to continuous real-valued functions on S, elements of J correspond to regular closed subsets of S, and the conditional preferences have a subjective expected utility (SEU) representation given by a Borel probability measure on S and a continuous utility function. I consider two settings; in one, A has a partial order making it a Riesz space or Banach lattice, and J is the Boolean algebra of bands in A. In the other, A has a multiplication operator making it a commutative Banach algebra, and J is the Boolean algebra of regular ideals in A. Finally, given two such vector spaces A1 and A2 with SEU representations on topological spaces S1 and S2, I show that a preference-preserving homomorphism A2 --> A1 corresponds to a probability-preserving continuous function S1 --> S2. I interpret this as a model of changing awareness. 

Abstract. A difference preorder is a (possibly incomplete) preorder on a space of state changes (rather than the states themselves); it encodes information about preference intensity, in addition to ordinal preferences. We find necessary and sufficient conditions for a difference preorder to be representable by a family of cardinal utility functions which take values in linearly ordered abelian groups. We also discuss the sense in which this cardinal utility representation is unique up to affine transformations, and under what conditions it is real-valued. This has applications to interpersonal comparisons, social welfare, and decisions under uncertainty.

Abstract. Let X be a set of outcomes, and let I be an infinite indexing set.  Our first main result states that any separable, permutation-invariant preference order (≽) on X^I admits an additive representation.   That is: there exists a linearly ordered abelian group R and a `utility function' u:X ⟶ R such that, for any x,y in X^I which differ in only finitely many coordinates, we have x≽y if and only if the sum of [u(x_i)-u(y_i)] over all i in I is positive.

   Our second result states: If (≽) also satisfies a weak continuity condition, then, for any x,y in X^I, we have  x≽y  if and only if the `hypersum' of [u(x_i)-u(y_i)] over all i in I  is positive.  The `hypersum' is an infinite summation operator defined using methods from nonstandard analysis.  Like an integration operator or series summation operator, it allows us to define the sum of an infinite set of values.  However, unlike these operations, the hypersum does not depend on some form of convergence (X has no topology) ---it is always well-defined.  Also, unlike an integral, the hypersum does not depend upon a sigma-algebra or measure on the indexing set I.  The hypersum takes values in a linearly ordered abelian group *R, which is an ultrapower extension of R. 

    Applications include infinite-horizon intertemporal choice, choice under uncertainty, and variable-population social choice.

Abstract. A judgement aggregation rule takes the views of a collection of voters over a set of interconnected issues and yields a logically consistent collective view. The median rule is a judgement aggregation rule that selects the logically consistent view which minimizes the average distance to the views of the voters (where the “distance” between two views is the number of issues on which they disagree). In the special case of preference aggregation, this is called the Kemeny rule. We show that, under appropriate regularity conditions, the median rule is the unique judgement aggregation rule which satisfies three axioms: Ensemble Supermajority Efficiency, Reinforcement, and Continuity. Our analysis covers aggregation problems in which the consistency restrictions on input and output judgements may differ. We also allow for issues to be weighted, and provide numerous examples in which issue weights arise naturally.

Abstract. Which is the best, impartially most plausible consensus view to serve as the basis of democratic group decision when voters disagree? Assuming that the judgement aggregation problem can be framed as a matter of judging a set of binary propositions (“issues”), we develop a multi-issue majoritarian approach based on the criterion of supermajority efficiency (SME). SME reflects the idea that smaller supermajorities must yield to larger supermajorities so as to obtain better supported, more plausible group judgments. As it is based on a partial ordering, SME delivers unique outcomes only in special cases. In general, one needs to make cardinal, not just ordinal, trade-offs between different supermajorities. Hence we axiomatically characterize the class of additive majority rules, whose (generically unique) outcome can be interpreted as the “on balance most plausible” consensus judgement. 

Abstract. Sequential majority voting over interconnected binary decisions can lead to the overruling of unanimous consensus. We characterize, within the general framework of judgement aggregation, when this happens for some sequence of decisions. The large class of aggregation spaces for which this vulnerability is present includes the aggregation of preference orderings over at least four alternatives, the aggregation of equivalence relations over at least four objects, resource allocation problems, and most committee selection problems. 

    We also ask whether it is possible to design respect for unanimity by choosing appropriate decision sequences (independently from the ballot). Remarkably, while this is not possible in general, it can be accomplished in some interesting special cases. Generalizing and sharpening a classic result by Shepsle and Weingast, we show that respect for unanimity can indeed be thus guaranteed in the cases of the aggregation of weak preference orderings, linear preference orderings, and equivalence relations. By contrast, impossibility results can be obtained for the aggregation of acyclic relations and separable preference orderings. As a key technical tool, we introduce the notion of a covering fragment that serves as a counterpart and generalization of the notions of covering relation/uncovered set in voting theory. 

Abstract.  Judgement aggregation is a model of social choice in which the space of social alternatives is the set of consistent evaluations (‘views’) on a family of logically interconnected propositions, or yes/no issues. However, simply complying with the majority opinion in each issue often yields a logically inconsistent collective view. Thus, we consider the Condorcet set: the set of logically consistent views which agree with the majority on a maximal subset of issues. The elements of this set turn out to be exactly those that can be obtained through sequential majority voting, according to which issues are sequentially decided by simple majority unless earlier choices logically force the opposite decision. We investigate the size and structure of the Condorcet set for several important classes of judgement aggregation problems. While the Condorcet set verifies a version of McKelvey's (1979) celebrated ‘chaos theorem’ in a number of contexts, in others it is shown to be very regular and well-behaved.

Abstract.  We consider the degree of indeterminacy which can arise in a judgement aggregation problem when propositions are decided one at a time through majority vote, with earlier decisions placing logical constraints on later decisions. The outcome of such “sequential majority voting” can be highly  path-dependent. A judgement aggregation problem is “globally indeterminate” if the truth value of every proposition is path-dependent. We show that,  for several common classes of judgement aggregation problems, such global  indeterminacy is not only possible, but fairly commonplace. We then consider problems which are “fully indeterminate”, meaning that every possible  assignment of truth-values can arise from some path.

Abstract.  Judgement aggregation is a model of social choice where the space of social alternatives is the set of consistent truth-valuations (‘judgements’) on a family of logically interconnected propositions. It is well known that propositionwise majority voting can yield logically inconsistent judgements. We show that, for a variety of spaces, propositionwise majority voting can yield any possible judgement. By considering the geometry of sub-polytopes of the Hamming cube, we also estimate the number of voters required to achieve all possible judgements. These results generalize the classic results of McGarvey (1953)  and Stearns (1959).

Abstract.  Given a set of propositions with unknown truth values, a ‘judgement aggregation function’ is a way to aggregate the personal truth-valuations of a group of voters into some ‘collective’ truth valuation. We introduce the class of ‘quasima- joritarian’ judgement aggregation functions, which includes majority vote, but also includes some functions which use different voting schemes to decide the truth of different propositions. We show that if the profile of individual beliefs satisfies a condition called ‘value restriction’, then the output of any quasimajoritarian function is logically consistent; this directly generalizes the recent work of Dietrich and List (Majority voting on restricted domains, JET, 2010). We then provide two sufficient conditions for value-restriction, defined geometrically in terms of a lattice ordering or a metric structure on the set of individuals and propositions. Finally, we introduce another sufficient condition for consistent majoritarian judgement aggregation, called ‘convexity’. We show that convexity is not logically related to value-restriction.

Abstract. We consider collective decisions under uncertainty, when agents have "generalized Hurwicz" preferences, a broad class allowing many different ambiguity attitudes, including subjective expected utility preferences. We consider sequences of acts that are “almost-objectively uncertain” in the sense that asymptotically, all agents almost-agree about the probabilities of the underlying events. We introduce a Pareto axiom which applies only to asymptotic preferences along such almost-objective sequences. This axiom implies that the social welfare function is utilitarian, but it does not impose any constraint on collective beliefs. On the other hand, a Pareto axiom for “dichotomous” acts implies that collective beliefs are contained in the closed convex hull of individual beliefs, but imposes no constraints on the social welfare function. Neither axiom entails any link between individual and collective ambiguity attitudes. 

Abstract. We consider social decisions under uncertainty. We show that the ex ante social preference order satisfies a Pareto axiom with respect to ex ante individual preferences, along with an axiom of Statewise Dominance, if and only if all agents admit subjective expected utility (SEU) representations, and furthermore the social planner is a utilitarian. The social utility function is the sum of the individual utility functions. In these SEU representations, the utility functions take values in an ordered abelian group, and probabilities are represented by order-preserving automorphisms of this group. This group may be non-Archimedean; this allows the SEU representations to encode lexicographical preferences and/or infinitesimal probabilities. 

Abstract.   How should we aggregate the ex ante preferences of Bayesian agents with hetero- geneous beliefs? Suppose the state of the world is described by a random process that unfolds over time. Different agents have different beliefs about the probabilistic laws governing this process. As new information is revealed over time by the pro- cess, agents update their beliefs and preferences via Bayes rule. Consider a Pareto principle that applies only to preferences which remain stable in the long run under these updates. I show that this “eventual Pareto” principle implies that the social planner must be a utilitarian. But it does not impose any relationship between the beliefs of the individuals and those of the planner, except for a weak compatibility condition.

(See also the  Presentation slides and Recording of presentation in the COMSOC video seminar, 10 September 2020.)

Abstract.  We investigate the conflict between the ex ante and ex post criteria of social welfare in a novel axiomatic framework of individual and social decisions, which distinguishes between a subjective and an objective source of uncertainty. This framework permits us to endow the individuals and society not only with ex ante and ex post preferences, as is classically done, but also with interim preferences of two kinds, and correspondingly, to introduce interim forms of the Pareto principle. After characterizing the ex ante and ex post criteria, we present a first solution to their conflict that amounts to extending the former as much possible in the direction of the latter. Then, we present a second solution, which goes in the opposite direction, and is our preferred one. This solution combines the ex post criterion with an objective interim Pareto principle, which avoids the pitfalls of the ex ante Pareto principle, and especially the problem of "spurious unanimity" discussed in the literature. Both solutions translate the assumed Pareto conditions into weighted additive utility representations, and both attribute common individual probability values only to the objective source of uncertainty.

Abstract. This handbook chapter covers the existing theoretical literature on social preference and social welfare under risk (i.e., when probability values enter the data of the situation) and uncertainty (i.e., when this is not the case and only subjective probability assessments can be formed). Section 1 sets the stage historically by contrasting classical social choice theory and welfare economics, which are restricted to the certainty case, with Harsanyi's pathbreaking attempt at extending these fields to the risk case. Section 2 reviews the work, both ancient and recent, stemming from Harsanyi's Impartial Observer Theorem. Section 3 does the same job for Harsanyi's Social Aggregation Theorem and discusses Sen's objections against the utilitarian relevance of either theorem. Section 4 explains why the Social Aggregation Theorem does not carry through from risk to uncertainty, a major conundrum that can also be expressed as a clash between ex ante and ex post welfare assessments; the proposed solutions are covered, including some very recent ones. Section 5 explains that equality, like social welfare, can be defined either ex ante or ex post, and using a basic example by Diamond, that these two definitions clash with each other. Section 6 covers the main solutions that egalitarian writers have given to this problem, again including some very recent ones.

Abstract.  We introduce a ranking of multidimensional alternatives, including uncertain prospects as a particular case, when these objects can be given a matrix form. This ranking is separable in terms of rows and columns, and continuous and monotonic in the basic quantities. Owing to the theory of additive separability developed here, we derive very precise numerical representations over a large class of domains (i.e., typically not of the Cartesian product form). We apply these representations to (1) streams of commodity baskets through time, (2) uncertain social prospects, (3) uncertain individual prospects. Concerning (1), we propose a finite horizon variant of Koopmans's (1960)  axiomatization of infinite discounted utility sums. The main results concern (2). We push the classic comparison between the ex ante and ex post social welfare criteria one step further by avoiding any expected utility assumptions, and as a consequence obtain what appears to be the strongest existing form of Harsanyi's (1955)  Aggregation Theorem. Concerning (3), we derive a subjective probability for Anscombe and Aumann's (1963)  finite case by merely assuming that there are two epistemically independent sources of uncertainty.  

Abstract.  Geometric belief aggregation has four serious drawbacks. First, on uncountably infinite spaces, its standard definition has a major domain restriction: it requires all beliefs to be absolutely continuous with respect to a common “reference measure”. Second, the standard definition cannot be applied to beliefs defined on abstract Boolean algebras, without specifying the underlying state space. Third, it can only be applied to profiles that are coherent, meaning that the beliefs of the agents have overlapping support. Finally, geometric aggregation does not satisfy the marginalization property. So it is sensitive to the way that one individuates states in the state space. This paper defines and axiomatically characterizes versions of geometric aggregation that resolve the first three problems, and mitigate the fourth one.

Abstract.  We study the effects of deliberation on epistemic social choice, in two settings. In the first setting, the group faces a binary epistemic decision analogous to the Condorcet Jury Theorem. In the second setting, group members have probabilistic beliefs arising from their private information, and the group wants to aggregate these beliefs in a way that makes optimal use of this information. During deliberation, each agent discloses private information to persuade the other agents of her current views. But her views may also evolve over time, as she learns from other agents. This process will improve the performance of the group, but only under certain conditions; these involve the nature of the social decision rule, the group size, and also the presence of “neutral agents” whom the other agents try to persuade.

Abstract.  Many collective decisions depend upon questions about objective facts or probabilities. Several theories in social choice and political philosophy suggest that democratic institutions can obtain accurate answers to such questions. But these theories are founded on assumptions and modelling paradigms that are both implausible and incompatible with one another. I will propose a roadmap for a more realistic and unified approach to this problem.

Abstract.  We develop a general theory of epistemic democracy in large societies, which subsumes the classical Condorcet Jury Theorem, the Wisdom of Crowds, and other similar results. We show that a suitably chosen voting rule will converge to the correct answer in the large-population limit, even if there is significant correlation amongst voters, as long as the average covariance between voters becomes small as the population becomes large. Finally, we show that these hypotheses are consistent with models where voters are correlated via a social network, or through the DeGroot model of deliberation.

Abstract.  We adopt an ‘epistemic’ interpretation of social decisions: there is an objectively correct choice, each voter receives a ‘noisy signal’ of the correct choice, and the social objective is to aggregate these signals to make the best possible guess about the correct choice. One epistemic method is to fix a probability model and compute the maximum likelihood estimator (MLE), maximum a posteriori (MAP) estimator or expected utility maximizer (EUM), given the data provided by the voters. We first show that an abstract voting rule can be interpreted as MLE or MAP if and only if it is a scoring rule. We then specialize to the case of distance-based voting rules, in particular, the use of the median rule in judgement aggregation. Finally, we show how several common ‘quasiutilitarian’ voting rules can be interpreted as EUM.

Abstract. We briefly review Condorcet and Young's epistemic interpretations of preference aggregation rules as maximum likelihood estimators. We then develop a general framework for interpreting epistemic social choice rules as maximum likelihood estimators, maximum a posteriori estimators, or expected utility maximizers. We illustrate this framework with several examples. Finally, we critique this program.

Abstract.  Given a large enough population of voters whose utility functions satisfy certain statistical regularities, we show that voting rules such as the Borda rule, approval voting, and evaluative voting have a very high probability of selecting the social alternative which maximizes the utilitarian social welfare function. We also characterize the speed with which this probability approaches one as the population grows.

Abstract. Given a sufficiently large population satisfying certain statistical regularities, we show that it is often possible to accurately estimate the utilitarian social welfare func- tion and identify the welfare-maximizing social alternative, even if we only have very noisy data about individual utility functions and interpersonal utility comparisons, and even if the individuals can be strategically dishonest.

Abstract.  We show that if the statistical distribution of utility functions in a population satisfies a certain condition, then a Condorcet winner will not only exist, but will also maximize the utilitarian social welfare function. We also show that, if people’s utility functions are generated according to certain plausible random processes, then in a large population, this condition will be satisfied with very high probability. Thus, in a large population, the utilitarian outcome will be selected by any Condorcet consistent voting rule.

Abstract: This article surveys the recent literature on infinite-horizon intergenerational social welfare and infinite-population ethics, reviewing the negative and positive results about the existence or constructibility of social preference relations and social welfare functions for infinite populations. Impossibility results primarily refer to the tension between Pareto and Anonymity (or inequality aversion). Positive results include characterizations of core social preference relations with which any relation satisfying desirable properties must be compatible, as well as overtaking criteria, asymptotic criteria, averaging criteria, hyperreal criteria, and criteria that focus on the worst-off.

Abstract. Widely accepted theories in modern cosmology say that spacetime is probably infinite. This raises the question how to define a social welfare order (SWO) for an infinite population of people dispersed throughout time and space. Any such SWO should be Lorentz invariant: it should yield the same value independent of the position and velocity of the social observer. I define and axiomatically characterize spatiotemporal Cesàro average utility as a solution to this problem.

Abstract. If Rawls’s A Theory of Justice has achieved fame among economists, this is due to his Difference Principle, which says that inequalities of resources should be to the benefit of the less fortunate, or more operationally, that allocations of resources should be ranked by the maximin criterion. We extend the Rawlsian maximin in two ways: first, by resorting to the more general min-of-means formula of decision theory, second, by addressing the case where the resources accruing to each individual are uncertain to society. For the latter purpose, we resort to the ex ante versus ex post distinction of welfare economics. The paper axiomatically characterizes the ex ante and ex post forms of the Rawlsian maximin and compares them in terms of egalitarian criteria. It finally recommends and axiomatizes a compromise egalitarian theory that mixes the two forms. 

Abstract. The class of rank-additive social welfare orders (RA SWOs) includes rank-weighted utilitarian, generalized utilitarian, and rank-discounted generalized utilitarian rules; it is a flexible framework for population ethics. This paper axiomatically characterizes RA SWOs and studies their properties in two frameworks: the actualist framework (which only tracks the utilities of people who actually exist) and the possibilist framework (which also assigns zero utilities to people who don’t exist). The axiomatizations and properties are quite different in the two frameworks. For example, actualist RA SWOs can simultaneously evade the Repugnant Conclusion and promote equality, whereas in the possibilist framework, there is a trade-off between these two desiderata. On the other hand, possibilist RA SWOs satisfy the Positive expansion and Negative expansion axioms, whereas the actualist ones don’t.

We propose a new voting mechanism in which voters simultaneously report their von Neumann-Morgenstern (vNM) utility functions across multiple decision problems, each of which has a finite number of alternatives. Each voter must report a real-valued "valuation" for each alternative of each decision. Each voter's valuation vector is rescaled to have unit magnitude (where this magnitude is measured using a specially constructed quadratic form). We show that it is a dominant strategy for each voter to reveal her true vNM utility function. With very high probability, the mechanism selects the alternative that maximizes a utilitarian social welfare function. The mechanism does not use money, and does not assume quasilinear utilities.

Abstract. We propose a new system of democratic representation. Any voter can choose any legislator as her representative; thus, different legislators can represent different numbers of voters. Decisions in the legislature are made by weighted majority voting, where the weight of each legislator is determined by the number of voters she represents. We show that, if the size of the electorate is very large, then with very high probability, the decisions obtained in the legislature agree with those which would have been reached by a popular referendum decided by simple majority vote. 

Abstract.  We propose a new solution to the problem of strategic voting for large electorates. For any deterministic voting rule, we design a stochastic rule that asymptotically approximates it in the following sense: for a sufficiently large population of voters, the stochastic voting rule (i) incentivizes every voter to reveal her true preferences and (ii) produces the same outcome as the deterministic rule, with arbitrarily high probability. We then apply these results to obtain an implementation in Bayesian Nash equilibrium.

Abstract.  In formal utilitarian voting, each voter assigns a numerical value to each alternative, and society chooses the alternative with the highest total value. Range voting is similar, except that each voter’s values are constrained to lie in the interval [0,1]. We characterize these rules via the axioms of anonymity, neutrality, reinforcement, overwhelming majority, and two novel conditions: maximal expressiveness, and an absence of “minority overrides”.

Abstract. Let X be a set of social alternatives, and let V be a set of ‘votes’ or ‘signals’. (We do not assume any structure on X or V.) A variable population voting rule F takes any number of anonymous votes drawn from V as input, and produces a nonempty subset of X as output. The rule F satisfies reinforcement if, whenever two disjoint sets of voters independently select some subset Y ⊆ X, the union of these two sets will also select Y. We show that F satisfies reinforcement if and only if F is a balance rule. If F satisfies a form of neutrality, then F satisfies reinforcement if and only if F is a scoring rule (with scores taking values in an abstract linearly ordered abelian group R); this generalizes a result of Myerson (1995).

Abstract. Given a bargaining problem, the relative utilitarian (RU) solution maximizes the sum total of the bargainer’s utilities, after having first renormalized each utility function to range from zero to one. We show that RU is “optimal” in two very different senses. First, RU is the maximal element (over the set of all bargaining solutions) under any partial ordering which satisfies certain axioms of fairness and consistency; this result is closely analogous to the result of Segal (2000).  Second, RU offers each person the maximum expected utility amongst all rescaling-invariant solutions, when it is applied to a random sequence of future bargaining problems generated using a certain class of distributions; this is recalls the results of Harsanyi (1953) and Karni (1998).

Abstract.  Suppose it is possible to make approximate interpersonal comparisons of welfare gains and losses. Thus, if w, x, y and z are personal states (each encoding all factors which influence the well-being of an individual), then it is sometimes possible to say “The welfare gain of the state change w⇝ x is greater than the welfare gain of the state change y ⇝ z”. We can represent this by the formula “(w ⇝ x) ≽ (y ⇝ z)”, where (≽) is a difference preorder: an incomplete preorder on the space of all possible personal state changes. A social state change is a bundle of personal state changes. A social difference preorder (SDP) is an incomplete preorder on the space of social state changes, which satisfies Pareto and Anonymity axioms. We characterize a family of SDPs which are roughly utilitarian in nature. We also apply the model to redistributive taxation.

Abstract.  Let X be a set of “personal states”; any person, in any circumstance, is at some point in X. A social state assigns an element of X to every person in society. Suppose it is sometimes possible to make ordinal interpersonal comparisons of well-being. We represent this with a (possibly incomplete) preorder on X.  From this, we can derive a (possibly incomplete) preorder on the set of social states, which ranks them in terms of their aggregate welfare. We define the appropriate analogs of the maximin and leximin social welfare orders in this framework, and axiomatically characterize them.

Abstract. In this study, we develop a model of social choice over lotteries, where people’s psychological characteristics are mutable, their preferences may be incomplete, and incomplete or noisy interpersonal comparisons of well-being are possible. Formally, we suppose individual preferences are described by a von Neumann-Morgenstern (vNM) preference order on a space of lotteries over psychophysical states; the social planner must construct a vNM preference order on lotteries over social states. First, we consider a model where the individual vNM preference order is incomplete (so not all interpersonal comparisons are possible). Then, we consider a model where the individual vNM preference order is complete, but unknown to the planner, and thus modelled by a random variable. In both cases, we obtain characterizations of a utilitarian social welfare function.

Abstract. In this paper we study the co-existence of two well known trading protocols, bargaining and price-posting. To do so we consider a frictional environment where buyers and sellers play price-posting and bargaining games infinitely many times. Sellers switch from one market to the other at a rate that is proportional to their payoff differentials. Given the different informational requirements associated with these two trading mechanisms, we examine their possible co-existence in the context of informal and formal markets. Other than having different trading protocols, we also consider other distinguishing features. We find a unique stable equilibrium where price-posting (formal markets) and bargaining (informal markets) co-exist. In a richer environment where both sellers and buyers can move across markets, we show that there exists a unique stable dynamic equilibrium where formal and informal activities also co-exist whenever sellers’ and buyers’ net costs of trading in the formal market have opposite signs.

Abstract.  Academic research is a public good whose production is supported by the tuition-paying students that a faculty's research accomplishments attract. A professor's spot contribution to the university's revenues thus depends not on her spot research production, but rather on her entire cumulative research record. We show that, under a broad range of education market conditions, a profit-maximizing university will apply a “high” minimum retention standard to the production of a junior professor who has no record of past research, but a “zero” retention standard to the spot production of a more senior professor whose background includes accomplishments sufficient to have cleared the “high” probationary hurdle. But if and when those education market conditions change, tenure-based contracts may cease to be optimal.

Abstract: Population ethics studies the tradeoff between the total number of people who will ever live, and their quality of life. But widely accepted theories in modern cosmology say that spacetime is probably infinite. In this case, its population is also probably infinite, so the quantity/quality tradeoff of population ethics is no longer meaningful. Instead, we face the problem of how to ethically evaluate an infinite population of people dispersed throughout time and space. I propose spatiotemporal Cesàro average utility as a way to make this evaluation, and axiomatically characterize it. 

• 1. • Presentation slides

     • Recording of presentation in the Online Social Choice Seminar (11 August 2020)

Abstract. Scientists often think of the world (or some part of it) as a dynamical system, a stochastic process, or a generalization of such a system. Prominent examples of systems are (i) the system of planets orbiting the sun or any other classical mechanical system, (ii) a hydrogen atom or any other quantum-mechanical system, and (iii) the earth's atmosphere or any other statistical mechanical system. We introduce a simple and general framework for describing such systems and show how it can be used to examine some familiar philosophical questions, including the following: how can we define nomological possibility, necessity, determinism, and indeterminism; what are symmetries and laws; what regularities must a system display to make scientific inference possible; is there any metaphysical basis for invoking principles of parsimony such as Occam's Razor when we make such inferences; and what is the role of space and time in a system? Our framework is intended to serve as a toolbox for the formal analysis of systems that is applicable in several areas of philosophy. 

Abstract. We offer a new argument for the claim that there can be non-degenerate objective chance in a deterministic world. Using a formal model of the relationship between different levels of description of a system, we show how objective chance at a higher level can coexist with its absence at a lower level. Unlike previous arguments for the level-specificity of chance, our argument shows, in a precise sense, that higher-level chance does not collapse into epistemic probability, despite higher-level properties supervening on lower-level ones. We show that the distinction between objective chance and epistemic probability can be drawn, and operationalized, at every level of description. There is, therefore, not a single distinction between objective and epistemic probability, but a family of such distinctions.

Abstract.  I propose a decentralized, multilayered representative democracy, where citizens participate in deliberative policy formation after self-organizing into a pyramidal hierarchy of small groups. Each group elects a delegate, who expresses the deliberative consensus of that group at the next tier of the pyramid. The pyramid thus acts as a communications network which efficiently aggregates useful information and policy ideas. It is also a powerful meritocratic device, which channels legislative responsibility towards the most committed and competent citizens. This yields a practical implementation of deliberative democracy in a large polity.

Abstract.  Suppose R is a finite commutative ring of prime characteristic, A is a finite R-module, M := Z^D × N^E , and 𝛷 is an R-linear cellular automaton on A^M. If μ is a 𝛷-invariant measure which is multiply σ-mixing in a certain way, then we show that μ must be the Haar measure on a coset of some submodule shift of AM. Under certain conditions, this means that μ must be the uniform Bernoulli measure on A^M.

Abstract.  Let M = ND be the positive orthant of a D-dimensional lattice and let (G, +) be a finite abelian group. Let G ⊆ GM be a subgroup shift, and let µ be a Markov random field whose support is G. Let 𝛷 : G→G be a linear cellular automaton. Under broad conditions on G, we show that the Cesaro average of 𝛷-iterates ofµ) converges to a measure of maximal entropy for the shift action on G.

Abstract. Let M=Z^D be a D-dimensional lattice, and let A be an abelian group. A^M is then a compact abelian group; a `linear cellular automaton' (LCA) is a topological group endomorphism Φ:A^M --> A^M that commutes with all shift maps. Suppose μis a probability measure on A^M whose support is a subshift of finite type or sofic shift. We provide sufficient conditions (on Φand μ) under which Φ`asymptotically randomizes' μ, meaning that wk*lim_{J\ni j --> oo} Φ^j μ= η, where ηis the Haar measure on A^M, and J has Cesaro density 1. In the case when Φ=1+σ, we provide a condition on μthat is both necessary and sufficient. We then use this to construct an example of a zero-entropy measure which is asymptotically randomized by 1+σ(all previously known examples had positive entropy).

Abstract: For the action of an algebraic cellular automaton on a Markov subgroup, we show that the Cesàro mean of the iterates of a Markov measure converges to the Haar measure. This is proven by using the combinatorics of the binomial coefficients on the regenerative construction of the Markov measure. 

Abstract: A `right-sided, nearest neighbour cellular automaton' (RNNCA) is a continuous transformation F:A^Z-->A^Z determined by a local rule f:A^{0,1}-->A so that, for any a in A^Z and any z in Z, F(a)_z = f(a_{z},a_{z+1}) . We say that F is `bipermutative' if, for any choice of a in A, the map g:A-->A defined by g(b) = f(a,b) is bijective, and also, for any choice of b in A, the map h:A-->A defined by h(a)=f(a,b) is bijective. We characterize the invariant measures of bipermutative RNNCA. First we introduce the equivalent notion of a `quasigroup CA', to expedite the construction of examples. Then we characterize F-invariant measures when A is a (nonabelian) group, and f(a,b) = a*b. Then we show that, if F is any bipermutative RNNCA, and mu is F-invariant, then F must be mu-almost everywhere K-to-1, for some constant K . We use this to characterize invariant measures when A^Z is a `group shift' and F is an `endomorphic CA'.

Abstract: If M is a monoid (e.g. the lattice Z^D), and A is an abelian group, then A^M is a compact abelian group; a linear cellular automaton (LCA) is a continuous endomorphism F:A^M --> A^M that commutes with all shift maps. If F is diffusive, and mu is a harmonically mixing (HM) probability measure on A^M, then the sequence {F^N mu} (N=1,2,3,...) weak*-converges to the Haar measure on A^M, in density. Fully supported Markov measures on A^Z are HM, and nontrivial LCA on A^{Z^D} are diffusive when A=Z/p is a prime cyclic group. In the present work, we provide sufficient conditions for diffusion of LCA on A^{Z^D} when A=Z/n is any cyclic group or when A=[Z/(p^r)]^J (p prime). We show that any fully supported Markov random field on A^{Z^D} is HM (where A is any abelian group).

Abstract. If M=Z^D, and B is a finite (nonabelian) group, then B^M is a compact group; a multiplicative cellular automaton (MCA) is a continuous transformation G: B^M --> B^M which commutes with all shift maps, and where nearby coordinates are combined using the multiplication operation of B. We characterize when MCA are group endomorphisms of B^M, and show that MCA on B^M inherit a natural structure theory from the structure of B. We apply this structure theory to compute the measurable entropy of MCA, and to study convergence of initial measures to Haar measure.

Abstract: Let M be a monoid (e.g. the lattice Z^D), and A an abelian group. A^M is then a compact abelian group; a linear cellular automaton (LCA) is a continuous endomorphism F:A^M --> A^M that commutes with all shift maps. Let mu be a (possibly nonstationary) probability measure on A^M; we develop sufficient conditions on mu and F so that the sequence {F^N mu} (N=1,2,3,...) weak*-converges to the Haar measure on A^M, in density (and thus, in Cesaro average as well). As an application, we show: if A=Z/p (p prime), F is any ``nontrivial'' LCA on A^{(Z^D)}, and mu belongs to a broad class of measures (including most Bernoulli measures (for D >= 1) and ``fully supported'' N-step Markov measures (when D=1), then F^N mu weak*-converges to Haar measure in density.

Abstract.  A cellular automaton (CA) exhibits ‘emergent defect dynamics’ (EDD) if generic initial conditions rapidly coalesce into large, homogeneous ‘domains’ (exhibiting some spatial pattern) separated by moving ‘defects’. There are many known examples of EDD in one-dimensional CA, but not in higher dimensions. We describe the results of an automated search for two-dimensional CA exhibiting EDD. We found a plethora of examples of EDD, but we also found that the proportion of CA with EDD declines rapidly with increasing neighbourhood size.

Abstract: Let A^Z be the Cantor space of bi-infinite sequences in a finite alphabet A, and let sigma be the shift map on A^Z. A `cellular automaton' is a continuous, sigma-commuting self-map Phi of A^Z, and a `Phi-invariant subshift' is a closed, (Phi,sigma)-invariant subset X of A^Z. Suppose x is a sequence in A^Z which is X-admissible everywhere except for some small region we call a `defect'. It has been empirically observed that such defects persist under iteration of Phi, and often propagate like `particles'. We characterize the motion of these particles, and show that it falls into several regimes, ranging from simple deterministic motion, to generalized random walks, to complex motion emulating Turing machines or pushdown automata. One consequence is that some questions about defect behaviour are formally undecidable.

Abstract: Let L:=Z^D be a D-dimensional lattice. Let A^L be the Cantor space of L-indexed configurations in a finite alphabet A, with the natural L-action by shifts. A `cellular automaton' is a continuous, shift-commuting self-map F:A^L-->A^L. An `F-invariant subshift' is a closed, F-invariant and shift-invariant subset X of A^L. Suppose x is an element of A^L that is X-admissible everywhere except for some small region of L which we call a `defect'. Such defects are analogous to `domain boundaries' in a crystalline solid. It has been empirically observed that these defects persist under iteration of F, and often propagate like `particles' which coalesce or annihilate on contact. We use spectral theory to explain the persistence of some defects under F, and partly explain the outcomes of their collisions.

Abstract: Let L:= Z^D be the D-dimensional lattice and let A^L be the Cantor space of L-indexed configurations in some finite alphabet A, with the natural L-action by shifts. A `cellular automaton' is a continuous, shift-commuting self-map F of A^L, and an `F-invariant subshift' is a closed, F-invariant and shift-invariant subset X of A^L. Suppose x is a configuration in A^L that is X-admissible everywhere except for some small region we call a `defect'. It has been empirically observed that such defects persist under iteration of F, and often propagate like `particles' which coalesce or annihilate on contact. We construct algebraic invariants for these defects, which explain their persistence under F, and partly explain the outcomes of their collisions. Some invariants are based on the cocycles of multidimensional subshifts; others arise from the higher-dimensional (co)homology/homotopy groups for subshifts, obtained by generalizing the Conway-Lagarias tiling groups and the Geller-Propp fundamental group.

Abstract.      Many dynamical systems can be naturally represented as Bratteli–Vershik (or adic) systems, which provide an appealing combinatorial description of their dynamics. If an adic system X is linearly recurrent, then we show how to represent X using a two-dimensional subshift of finite type Y; each ‘row’ in a Y-admissible configuration corresponds to an infinite path in the Bratteli diagram of X, and the vertical shift on Y corresponds to the ‘successor’ map of X. Any Y-admissible configuration can then be recoded as the space-time diagram of a one-dimensional cellular automaton 𝛷; in this way X is embedded in 𝛷(i.e. X is conjugate to a subsystem of 𝛷). With this technique, we can embed many odometers, Toeplitz systems, and constant-length substitution systems in one-dimensional cellular automata.

Abstract: We consider a left permutive cellular automaton Phi, with no memory and positive anticipation, defined on the space of all doubly infinite sequences with entries from a finite alphabet. For each such automaton that is not one-to-one, there is a dense set of points X (which is large in another sense too) such that the Phi-orbit closure of each x in X is topologically conjugate to an odometer (the ``+1'' map on a projective limit of finite cyclic groups). We identify this odometer in several cases.

Abstract.  Ergodic theory is the study of how a dynamical system transforms the information encoded in an invariant probability measure. This article reviews the major recent results in the ergodic theory of cellular automata.

Abstract.  A symbolic dynamical system is a continuous transformation Φ : X −→ X of closed subset X ⊆ A^V, where A is a finite set and V is countable (examples include subshifts, odometers, cellular automata, and automaton networks). The function Φ induces a directed graph (‘network’) structure on V, whose geometry reveals information about the dynamical system (X, Φ ). The dimension dim(V) is an exponent describing the growth rate of balls in this network as a function of their radius. We show that, if X has positive entropy and dim(V) > 1, and the system (A^V,X,Φ) satisfies minimal symmetry and mixing conditions, then (X,Φ) cannot be positively expansive; this generalizes a well-known result of Shereshevsky about multidimensional cellular automata. We also construct a counterexample to a version of this result without the symmetry condition. Finally, we show that network dimension is invariant under topological conjugacies which are Hölder- continuous.

Abstract: Let A:={0,1}. A `cellular automaton' (CA) is a shift-commuting transformation of A^{Z^D} determined by a local rule. Likewise, a `Euclidean automaton' is a shift-commuting transformation of A^{R^D} determined by a local rule. `Larger than Life' (LtL) CA are long-range generalizations of J.H. Conway's Game of Life CA, proposed by K.M. Evans. We prove a conjecture of Evans: as their radius grows to infinity, LtL CA converge to a `continuum limit' Euclidean automaton, which we call `RealLife'. We also show that the `life forms' (fixed points, periodic orbits, and propagating structures) of LtL CA converge to life forms of RealLife. Finally we prove a number of existence results for fixed points of RealLife.

Abstract: If L=Z^D and A is a finite set, then A^L is a compact space. A cellular automaton (CA) is a continuous transformation F:A^L--> A^L that commutes with all shift maps. A quasisturmian (QS) subshift is a shift-invariant subset obtained by mapping the trajectories of an irrational torus rotation through a partition of the torus. The image of a QS shift under a CA is again QS. We study the topological dynamical properties of CA restricted to QS shifts, and compare them to the properties of CA on the full shift A^L. We investigate injectivity, surjectivity, transitivity, expansiveness, rigidity, fixed/periodic points, and invariant measures. We also study `chopping': how iterating the CA fragments the partition generating the QS shift.

Abstract: If X is a discrete abelian group and B a finite set, then a cellular automaton (CA) is a continuous map F:B^X-->B^X that commutes with all X-shifts. If g is a real-valued function on B, then, for any b in B^X, we define G(b) to be the sum over all x in X of g(b_x) (if finite). We say g is `conserved' by F if G is constant under the action of F. We characterize such `conservation laws' in several ways, deriving both theoretical consequences and practical tests, and provide a method for constructing all one-dimensional CA exhibiting a given conservation law.

Abstract: If A is a finite alphabet, Z^D is a D-dimensional lattice, U is a subset of Z^D, and mu_U is a probability measure on A^U that ``looks like'' the marginal projection of a stationary random field on A^(Z^D), then can we ``extend'' mu_U to such a field? Under what conditions can we make this extension ergodic, (quasi)periodic, or (weakly) mixing? After surveying classical work on this problem when D = 1, we provide some sufficient conditions and some necessary conditions for mu_U to be extendible for D > 1, and show that, in general, the problem is not formally decidable.

Abstract. A coupled cell system is a network of dynamical systems, or ‘cells’, coupled together. Such systems can be represented schematically by a directed graph whose nodes correspond to cells and whose edges represent couplings. A symmetry of a coupled cell system is a permutation of the cells and edges that preserves all internal dynamics and all couplings. It is well known that symmetry can lead to patterns of synchronized cells, rotating waves, multirhythms, and synchronized chaos. Recently, the introduction of a less stringent form of symmetry, the ‘symmetry groupoid’, has shown that global group-theoretic symmetry is not the only mechanism that can create such states in a coupled cell system. The symmetry groupoid consists of structure-preserving bijections between certain subsets of the cell network, the input sets. Here, we introduce a concept intermediate between the groupoid symmetries and the global group symmetries of a network: ‘interior symmetry’. This concept is closely related to the groupoid structure, but imposes stronger constraints of a group-theoretic nature. We develop the local bifurcation theory of coupled cell systems possessing interior symmetries, by analogy with symmetric bifurcation theory. The main results are analogues for ‘synchrony-breaking’ bifurcations of the Equivariant Branching Lemma for steady-state bifurcation, and the Equivariant Hopf Theorem for bifurcation to time-periodic states.

Abstract. A coupled cell system is a network of dynamical systems, or "cells," coupled together. Such systems can be represented schematically by a directed graph whose nodes correspond to cells and whose edges represent couplings. A symmetry of a coupled cell system is a permutation of the cells that preserves all internal dynamics and all couplings. Symmetry can lead to patterns of synchronized cells, rotating waves, multirhythms, and synchronized chaos. We ask whether symmetry is the only mechanism that can create such states in a coupled cell system and show that it is not.

The key idea is to replace the symmetry group by the symmetry groupoid, which encodes information about the input sets of cells. (The input set of a cell consists of that cell and all cells connected to that cell.) The admissible vector fields for a given graph---the dynamical systems with the corresponding internal dynamics and couplings---are precisely those that are equivariant under the symmetry groupoid. A pattern of synchrony is "robust" if it arises for all admissible vector fields. The first main result shows that robust patterns of synchrony (invariance of "polydiagonal" subspaces under all admissible vector fields) are equivalent to the combinatorial condition that an equivalence relation on cells is "balanced." The second main result shows that admissible vector fields restricted to polydiagonal subspaces are themselves admissible vector fields for a new coupled cell network, the "quotient network." The existence of quotient networks has surprising implications for synchronous dynamics in coupled cell systems.

Abstract. The regular open subsets of a topological space form a Boolean algebra, where the join of two regular open sets is the interior of the closure of their union. A credence is a finitely additive probability measure on this Boolean algebra, or on one of its subalgebras. We develop a theory of integration for such credences. We then explain the relationship between credences, residual charges, and Borel probability measures. We show that a credence can be represented by a normal Borel measure, augmented with a liminal structure, which specifies how two or more regular open sets share the probability mass of their common boundary. In particular, a credence on a locally compact Hausdorff space can be represented by a normal Borel measure and a liminal structure on the Stone-Čech compactification of that space. We also show how credences can be represented by Borel measures on the Stone space of the underlying Boolean algebra of regular open sets. Finally, we show that these constructions are functorial.  

Abstract. A new method is developed for estimating the spectral measure of a multivariate stable probability measure, by representing the measure as a sum of spherical harmonics.

Abstract. Recent statistical analysis of a number of financial databases is summarized. Increasing agreement is found that logarithmic equity returns show a certain type of asymptotic behavior of the largest events, namely that the probability density functions have power law tails with an exponent α≈3.0. This behavior does not vary much over different stock exchanges or over time, despite large variations in trading environments. The present paper proposes a class of multivariate distributions which generalizes the observed qualities of univariate time series. A new consequence of the proposed class is the “spectral measure” which completely characterizes the multivariate dependences of the extreme tails of the distribution. This measure on the unit sphere in M–dimensions, in principle completely general, can be determined empirically by looking at extreme events. If it can be observed and determined, it will prove to be of importance for scenario generation in portfolio risk management.

Abstract.