Franz Dietrich

What is the relationship between degrees of belief and binary beliefs? Can the latter be expressed as a function of the former—a so-called “belief-binarization rule”—without running into difficulties such as the lottery paradox? We show that this problem can be usefully analyzed from the perspective of judgment-aggregation theory. Although some formal similarities between belief binarization and judgment aggregation have been noted before, the connection between the two problems has not yet been studied in full generality. In this paper, we seek to fill this gap. The paper is organized around a baseline impossibility theorem, which we use to map out the space of possible solutions to the belief-binarization problem. Our theorem shows that, except in limiting cases, there exists no belief-binarization rule satisfying four initially plausible desiderata. Surprisingly, this result is a direct corollary of the judgment-aggregation variant of Arrow’s classic impossibility theorem in social choice theory.

We present a new “reason-based” approach to the formal representation of moral theories, drawing on recent decision-theoretic work. We show that any moral theory within a very large class can be represented in terms of two parameters: (i) a specification of which properties of the objects of moral choice matter in any given context, and (ii) a specification of how these properties matter. Reasonbased representations provide a very general taxonomy of moral theories, as di↵erences among theories can be attributed to di↵erences in their two key parameters. We can thus formalize several distinctions, such as between consequentialist and non-consequentialist theories, between universalist and relativist theories, between agent-neutral and agent-relative theories, between monistic and pluralistic theories, between atomistic and holistic theories, and between theories with a teleological structure and those without. Reason-based representations also shed light on an important but under-appreciated phenomenon: the “underdetermination of moral theory by deontic content”

How can several individuals’ probability functions on a given σσ -algebra of events be aggregated into a collective probability function? Classic approaches to this problem usually require ‘event-wise independence’: the collective probability for each event should depend only on the individuals’ probabilities for that event. In practice, however, some events may be ‘basic’ and others ‘derivative’, so that it makes sense first to aggregate the probabilities for the former and then to let these constrain the probabilities for the latter. We formalize this idea by introducing a ‘premise-based’ approach to probabilistic opinion pooling, and show that, under a variety of assumptions, it leads to linear or neutral opinion pooling on the ‘premises’.

The contemporary theory of epistemic democracy often draws on the Condorcet Jury Theorem to formally justify the `wisdom of crowds'. But this theorem is inapplicable in its current form, since one of its premises -- voter independence -- is notoriously violated. This premise carries responsibility for the theorem's misleading conclusion that `large crowds are infallible'. We prove a more useful jury theorem: under defensible premises, `large crowds are fallible but better than small groups'. This theorem rehabilitates the importance of deliberation and education, which appear inessential in the classical jury framework. Our theorem is related to Ladha's (1993) seminal jury theorem for interchangeable (`indistinguishable') voters based on de Finetti's Theorem. We prove a more general and simpler version of such a theorem.

If a group is modelled as a single Bayesian agent, what should its beliefs be? I propose an axiomatic model that connects group beliefs to beliefs of the group members. The group members may have different information, different prior beliefs and even different domains (algebras) within which they hold beliefs, accounting for differences in awareness and conceptualisation. As is shown, group beliefs can incorporate all information spread across individuals without individuals having to explicitly communicate their information (that may be too complex or personal to describe, or not describable in principle in the language). The group beliefs derived here take a simple multiplicative form if people's information is independent (and a more complex form if information overlaps arbitrarily). This form contrasts with familiar linear or geometric opinion pooling and the (Pareto) requirement of respecting unanimous beliefs.

In the framework of judgment aggregation, we assume that some formulas of the agenda are singled out as premisses, and that both Independence (formula-wise aggregation) and Unanimity Preservation hold for them. Whether premiss-based aggregation thus defined is compatible with conclusion-based aggregation, as defined by Unanimity Preservation on the non-premisses, depends on how the premisses are logically connected, both among themselves and with other formulas. We state necessary and sufficient conditions under which the combination of both approaches leads to dictatorship (resp. oligarchy), either just on the premisses or on the whole agenda. Our analysis is inspired by the doctrinal paradox of legal theory and is arguably relevant to this field as well as political science and political economy. When the set of premisses coincides with the whole agenda, a limiting case of our assumptions, we obtain several existing results in judgment aggregation theory.

Many theorists tie social norms to attitudes, such as expectations towards others, perhaps along with conforming practices. Challenging this view, we instead ground social norms in a social norming process, an often non-verbal social communication process that ‘makes’ the norm through mutual expressions of support. We present the process-based account of social norms and social normativity, and distinguish social norms from social pressures, social practices and Lewisian conventions. The process-based view brings social norms closer to legal norms, by tying them to ‘expressive acts’, just as laws and contracts arise through acts of voting or signing, not through mere attitudes.

Which rules for aggregating judgments on logically connected propositions are manipulable and which not? In this paper, we introduce a preference-free concept of non-manipulability and contrast it with a preference-theoretic concept of strategy-proofness. We characterize all non-manipulable and all strategy-proof judgment aggregation rules and prove an impossibility theorem similar to the Gibbard--Satterthwaite theorem. We also discuss weaker forms of non-manipulability and strategy-proofness. Comparing two frequently discussed aggregation rules, we show that “conclusion-based voting” is less vulnerable to manipulation than “premise-based voting”, which is strategy-proof only for “reason-oriented” individuals. Surprisingly, for “outcome-oriented” individuals, the two rules are strategically equivalent, generating identical judgments in equilibrium. Our results introduce game-theoretic considerations into judgment aggregation and have implications for debates on deliberative democracy.

Decision making typically requires judgments about causal relations: we need to know both the causal e¤ects of our actions and the causal relevance of various environmental factors. Judgments about the nature and strength of causal relations often differ, even among experts. How to handle such diversity is the topic of this paper. First, we consider the possibility of aggregating causal judgments via the aggregation of probabilistic ones. The broadly negative outcome of this investigation leads us to look at aggregating causal judgments independently of probabilistic ones. We do so by transcribing causal claims into the formal judgment-aggregation framework and applying some recent results in this field. Finally, we look at the implications for probability aggregation when it is constrained by prior aggregation of qualitative causal judgments.

Social epistemology studies knowledge in social contexts. Knowledge is ‘social’ when its holder communicates with or learns from others (Epistemology in groups), or when its holder is a group as a whole, literally or metaphorically (Epistemology of groups). Group knowledge can emerge explicitly, through aggregation procedures like voting, or implicitly, through institutions like deliberation or prediction markets. In the truth-tracking paradigm, group beliefs aim at truth, and group decisions at ‘correctness’ – in virtue of external facts that are empirical or normative, real or constructed, universal or relativistic, etc. Procedures and institutions are evaluated by epistemic performance: Are they truth-conducive? Do groups become ‘wiser’ than their members? We review several procedures and institutions, discussing epistemic successes and failures. Jury theorems provide formal arguments for epistemic success. Some jury theorems misleadingly conclude that ‘huge groups are infallible’ – an artefact of inappropriate premises. Others have defensible premises, and still conclude that groups outperform individuals, without being infallible.

Standard impossibility theorems on judgment aggregation over logically connected propositions either use a controversial systematicity condition or apply only to agendas of propositions with rich logical connections. Are there any serious impossibilities without these restrictions? We prove an impossibility theorem without requiring systematicity that applies to most standard agendas: Every judgment aggregation function (with rational inputs and outputs) satisfying a condition called unbiasedness is dictatorial (or effectively dictatorial if we remove one of the agenda conditions). Our agenda conditions are tight. When applied illustratively to (strict) preference aggregation represented in our model, the result implies that every unbiased social welfare function with universal domain is effectively dictatorial.

Suppose a committee, expert panel, or other group is making judgments on some issues, where these may be not just yes/no-questions, such as whether a defendant is guilty, but also variables with many possible values, such as macroeconomic or meteorological variables or travel directions. Furthermore, there may be interconnections between different issues, as in the case of economic or climate variables. How can the group arrive at “intelligent” collective judgments, based on the group members’ individual judgments? We investigate three challenges raised by this judgment-aggregation problem. First, reasonable methods of aggregation (such as defining the collective judgment for each issue as the average or median judgment) can produce inconsistent collective judgments. Secondly, many methods of aggregation are manipulable by strategic voting. Finally, not all methods of aggregation are conducive to tracking the truth on the issues in question. We prove new impossibility or possibility theorems on all three challenges, identifying what it takes to produce collective judgments in a consistent, non-manipulable, and truth-tracking manner and thereby to achieve collective intelligence through aggregation. Overall, the median method, though imperfect, performs reasonably well. We also note the relevance of our analysis for non-human group decisions. (The manuscript has been accepted for publication in Philosophical Transactions of the Royal Society B and is to appear in 2026.)

It is often claimed that opinions are more likely to be correct if they are held independently by many individuals. But what does it mean to hold independent opinions? To clarify this condition, we distinguish four notions of probabilistic opinion independence. Which notion applies depends on environmental factors such as commonly perceived evidence. More formally, it depends on the causal network that determines how people interact and form their opinions. In a general theorem, we identify conditions on this network that guarantee the four notions of opinion independence. Our results have implications for ‘wisdom of crowds’ arguments, as we illustrate with old and new jury theorems.

Standard impossibility theorems on judgment aggregation over logically connected propositions either use a controversial systematicity condition or apply only to agendas of propositions with rich logical connections. Are there any serious impossibilities without these restrictions? We prove an impossibility theorem without requiring systematicity that applies to most standard agendas: Every judgment aggregation function (with rational inputs and outputs) satisfying a condition called unbiasedness is dictatorial (or effectively dictatorial if we remove one of the agenda conditions). Our agenda conditions are tight. When applied illustratively to (strict) preference aggregation represented in our model, the result implies that every unbiased social welfare function with universal domain is effectively dictatorial.

If a group is modelled as a single Bayesian agent, what should its beliefs be? I propose an axiomatic model that connects group beliefs to beliefs of the group members. The group members may have different information, different prior beliefs and even different domains (algebras) within which they hold beliefs, accounting for differences in awareness and conceptualisation. As is shown, group beliefs can incorporate all information spread across individuals without individuals having to explicitly communicate their information (that may be too complex or personal to describe, or not describable in principle in the language). The group beliefs derived here take a simple multiplicative form if people's information is independent (and a more complex form if information overlaps arbitrarily). This form contrasts with familiar linear or geometric opinion pooling and the (Pareto) requirement of respecting unanimous beliefs.

How can several individuals’ probability assignments to some events be aggregated into a collective probability assignment? Classic results on this problem assume that the set of relevant events—the agenda—is a σ-algebra and is thus closed under disjunction (union) and conjunction (intersection). We drop this demanding assumption and explore probabilistic opinion pooling on general agendas. One might be interested in the probability of rain and that of an interest-rate increase, but not in the probability of rain or an interest-rate increase. We characterize linear pooling and neutral pooling for general agendas, with classic results as special cases for agendas that are σ-algebras. As an illustrative application, we also consider probabilistic preference aggregation. Finally, we unify our results with existing results on binary judgment aggregation and Arrovian preference aggregation. We show that the same kinds of axioms (independence and consensus preservation) have radically different implications for different aggregation problems: linearity for probability aggregation and dictatorship for binary judgment or preference aggregation.

Economists routinely measure individual welfare by (von-Neumann-Morgenstern) utility, for instance when analysing welfare intensity, social welfare, or welfare inequality. Is this welfare measure justified? Natural working hypotheses turn out to imply a different measure. It overcomes familiar problems of utility, by faithfully capturing non-ordinal welfare features, such as welfare intensity -- despite still resting on purely ordinal evidence, such as revealed preferences or self-reported welfare comparisons. Social welfare analysis changes when based on this new individual welfare measure rather than utility. For instance, Harsanyi's `utilitarian theorem' now supports prioritarianism. We compare the standard utility-based versions of utilitarianism and prioritarianism with new versions based on our welfare measure. We show that utility is a hybrid object, determined by two rival influences: welfare, and the attitude to intrinsic risk, i.e., to risk in welfare. A new version of Harsanyi's theorem shows that Harsanyi implicitly makes the questionable assumption that society is neutral to intrinsic risk, even when all individuals are averse to intrinsic risk.

In solving judgment aggregation problems, groups often face constraints. Many decision problems can be modelled in terms the acceptance or rejection of certain propositions in a language, and constraints as propositions that the decisions should be consistent with. For example, court judgments in breach-of-contract cases should be consistent with the constraint that action and obligation are necessary and sufficient for liability; judgments on how to rank several options in an order of preference with the constraint of transitivity; and judgments on budget items with budgetary constraints. Often more or less demanding constraints on decisions are imaginable. For instance, in preference ranking problems, the transitivity constraint is often contrasted with the weaker acyclicity constraint. In this paper, we make constraints explicit in judgment aggregation by relativizing the rationality conditions of consistency and deductive closure to a constraint set, whose variation yields more or less strong notions of rationality. We review several general results on judgment aggregation in light of such constraints.

Group decisions must often obey exogenous constraints. While in a preference aggregation problem constraints are modelled by restricting the set of feasible alternatives, this paper discusses the modelling of constraints when aggregating individual yes/no judgments on interconnected propositions. For example, court judgments in breach-of-contract cases should respect the constraint that action and obligation are necessary and sufficient for liability, and judgments on budget items should respect budgetary constraints. In this paper, we make constraints in judgment aggregation explicit by relativizing the rationality conditions of consistency and deductive closure to a constraint set, whose variation yields more or less strong notions of rationality. This approach of modelling constraints explicitly contrasts with that of building constraints as axioms into the logic, which turns compliance with constraints into a matter of logical consistency and thereby conflates requirements of ordinary logical consistency and requirements dictated by the environment . We present some general impossibility results on constrained judgment aggregation; they are immediate corollaries of known results on judgment aggregation.

I propose a general collective decision problem consisting in many issues that are interconnected in two ways: by mutual constraints and by connections of relevance. Aggregate decisions should respect the mutual constraints, and be based on relevant information only. This general informational constraint has many special cases, including premise-basedness and Arrow''s independence condition; they result from special notions of relevance. The existence and nature of (non-degenerate) aggregation rules depends on both types of connections. One result, if applied to the preference aggregation problem and adopting Arrow's notion of (ir)relevance, becomes Arrow's Theorem, without excluding indifferences unlike in earlier generalisations.

Democratic decision-making is often defended on grounds of the ‘wisdom of crowds’: decisions are more likely to be correct if they are based on many independent opinions, so a typical argument in social epistemology. But what does it mean to have independent opinions? Opinions can be probabilistically dependent (threatening the ‘wisdom of crowds’) even if individuals form their opinion in causal isolation from each other. We distinguish four probabilistic notions of opinion independence. Which of them holds depends on how individuals are causally affected by environmental factors such as commonly perceived evidence. In a general theorem, we identify causal conditions guaranteeing each kind of opinion independence. These results have implications for whether and how ‘wisdom of crowds’ arguments are possible, and how truth-conducive institutions can be designed.

In the theory of judgment aggregation, it is known for which agendas of propositions it is possible to aggregate individual judgments into collective ones in accordance with the Arrow-inspired requirements of universal domain, collective rationality, unanimity preservation, nondictatorship and propositionwise independence. But it is only partially known for which agendas it is possible to respect additional requirements, notably non-oligarchy, anonymity, no individual veto power, or implication preservation. We fully characterize the agendas for which there are such possibilities, thereby answering the most salient open questions about propositionwise judgment aggregation. Our results build on earlier results by Nehring and Puppe (2002), Nehring (2006) and Dietrich and List (2007a).

Economic models describe individuals in terms of underlying characteristics, such as taste for some good, sympathy level for another player, time discount rate, risk attitude, and so on. In real life, such characteristics change through experiences: taste for Mozart changes through listening to it, sympathy for another player through observing his moves, and so on. Models typically ignore change, not just for simplicity but also because it is unclear how to incorporate change. I introduce a general axiomatic framework for defining, analysing and comparing rival models of change. I show that seemingly basic postulates on modelling change together have strong implications, like irrelevance of the order in which someone has his experiences and ‘linearity’ of change. This is a step towards placing the modelling of change on solid axiomatic grounds and enabling non-arbitrary incorporation of change into economic models.

Condorcet's famous jury theorem reaches an optimistic conclusion on the correctness of majority decisions, based on two controversial premises about voters: they are competent and vote independently, in a technical sense. I carefully analyse these premises and show that: (i) whether a premise is justi…ed depends on the notion of probability considered; (ii) none of the notions renders both premises simultaneously justi…ed. Under the perhaps most interesting notions, the independence assumption should be weakened.

In judgment aggregation, unlike preference aggregation, not much is known about domain restrictions that guarantee consistent majority outcomes. We introduce several conditions on individual judgments su¢ - cient for consistent majority judgments. Some are based on global orders of propositions or individuals, others on local orders, still others not on orders at all. Some generalize classic social-choice-theoretic domain conditions, others have no counterpart. Our most general condition gen- eralizes Sen’s triplewise value-restriction, itself the most general classic condition. We also prove a new characterization theorem: for a large class of domains, if there exists any aggregation function satisfying some democratic conditions, then majority voting is the unique such function. Taken together, our results provide new support for the robustness of majority rule.

All existing impossibility theorems on judgment aggregation require individ- ual and collective judgment sets to be consistent and complete (in some recent results with completeness relaxed to deductive closure), arguably a demand- ing rationality requirement. They do not carry over to aggregation functions mapping pro…les of (merely) consistent individual judgment sets to (merely) consistent collective ones. We prove that, whenever the agenda of propositions under consideration exhibits mild interconnections, any such aggregation func- tion that is "neutral" between the acceptance and rejection of each proposition is dictatorial. We relate this theorem to the literature.