Franz Dietrich

A probability aggregation rule assigns to each profile of probability functions across a group of individuals (representing their individual probability assignments to some propositions) a collective probability function (representing the group's probability assignment). The rule is “non-manipulable” if no group member can manipulate the collective probability for any proposition in the direction of his or her own probability by misrepresenting his or her probability function (“strategic voting”). We show that, except in trivial cases, no probability aggregation rule satisfying two mild conditions (non-dictatorship and consensus preservation) is non-manipulable.

People reason not only in beliefs, but also in intentions, preferences, and other attitudes. They form preferences from existing preferences, or intentions from existing beliefs and intentions, and so on. This often involves choosing between rival conclusions. Building on Broome (Rationality through reasoning, Hoboken, Wiley. https://doi.org/10.1002/9781118609088, 2013) and Dietrich et al. (J Philos 116:585–614. https://doi.org/10.5840/jphil20191161138, 2019), we present a philosophical and formal analysis of reasoning in attitudes, with or without facing choices in reasoning. We give different accounts of choosing, in terms of a conscious activity or a partly subconscious process. Reasoning in attitudes differs fundamentally from reasoning _about_ attitudes, a form of theoretical reasoning in which one discovers rather than forms attitudes. We show that reasoning in attitudes has standard formal properties (such as monotonicity), but is indeterministic, reflecting choice in reasoning. Like theoretical reasoning, it need not follow logical entailment, but for a more radical reason, namely indeterminism. This makes reasoning in attitudes harder to model logically than theoretical reasoning. But it can be studied abstractly, using indeterministic consequence operators.

Does pre-voting group deliberation increase majority competence? To address this question, we develop a probabilistic model of opinion formation and deliberation. Two new jury theorems, one pre-deliberation and one post-deliberation, suggest that deliberation is beneficial. Successful deliberation mitigates three voting failures: (1) overcounting widespread evidence, (2) neglecting evidential inequality, and (3) neglecting evidential complementarity. Formal results and simulations confirm this. But we identify four systematic exceptions where deliberation reduces majority competence, always by increasing Failure 1. Our analysis recommends deliberation that is 'participatory', 'neutral', but not necessarily 'equal', i.e., that involves substantive sharing, privileges no evidences, but might privilege some persons.

We present an abstract model of rationality that focuses on structural properties of attitudes. Rationality requires coherence between your attitudes, such as your beliefs, values, and intentions. We define three 'logical' conditions on attitudes: consistency, completeness, and closedness. They parallel the familiar logical conditions on beliefs, but contrast with standard rationality conditions like preference transitivity. We establish a formal correspondence between our logical conditions and standard rationality conditions. Addressing John Broome's programme 'rationality through reasoning', we formally characterize how you can (not) become more logical by reasoning. Our analysis connects rationality with logic, and enables logical talk about multi-attitude psychology.

Jury theorems are mathematical theorems about the ability of collectives to make correct decisions. Several jury theorems carry the optimistic message that, in suitable circumstances, ‘crowds are wise’: many individuals together (using, for instance, majority voting) tend to make good decisions, outperforming fewer or just one individual. Jury theorems form the technical core of epistemic arguments for democracy, and provide probabilistic tools for reasoning about the epistemic quality of collective decisions. The popularity of jury theorems spans across various disciplines such as economics, political science, philosophy, and computer science. This entry reviews and critically assesses a variety of jury theorems. It first discusses Condorcet's initial jury theorem, and then progressively introduces jury theorems with more appropriate premises and conclusions. It explains the philosophical foundations, and relates jury theorems to diversity, deliberation, shared evidence, shared perspectives, and other phenomena. It finally connects jury theorems to their historical background and to democratic theory, social epistemology, and social choice theory.

Judgment-aggregation theory has always focused on the attainment of rational collective judgments. But so far, rationality has been understood in static terms: as coherence of judgments at a given time, defined as consistency, completeness, and/or deductive closure. This paper asks whether collective judgments can be dynamically rational, so that they change rationally in response to new information. Formally, a judgment aggregation rule is dynamically rational with respect to a given revision operator if, whenever all individuals revise their judgments in light of some information (a learnt proposition), then the new aggregate judgments are the old ones revised in light of this information, i.e., aggregation and revision commute. We prove an impossibility theorem: if the propositions on the agenda are non-trivially connected, no judgment aggregation rule with standard properties is dynamically rational with respect to any revision operator satisfying some basic conditions on revision. Our theorem is the dynamic-rationality counterpart of some well-known impossibility theorems for static rationality. We also explore how dynamic rationality might be achieved by relaxing some of the conditions on the aggregation rule and/or the revision operator. Notably, premise-based aggregation rules are dynamically rational with respect to so-called premise-based revision operators.

Can a group be an orthodox rational agent? This requires the group's aggregate preferences to follow expected utility (static rationality) and to evolve by Bayesian updating (dynamic rationality). Group rationality is possible, but the only preference aggregation rules which achieve it (and are minimally Paretian and continuous) are the linear-geometric rules, which combine individual values linearly and combine individual beliefs geometrically. Linear-geometric preference aggregation contrasts with classic linear-linear preference aggregation, which combines both values and beliefs linearly, but achieves only static rationality. Our characterisation of linear-geometric preference aggregation has two corollaries: a characterisation of linear aggregation of values (Harsanyi's Theorem) and a characterisation of geometric aggregation of beliefs.

While ordinary decision theory focuses on empirical uncertainty, real decision-makers also face normative uncertainty: uncertainty about value itself. From a purely formal perspective, normative uncertainty is comparable to (Harsanyian or Rawlsian) identity uncertainty in the 'original position', where one's future values are unknown. A comprehensive decision theory must address twofold uncertainty -- normative and empirical. We present a simple model of twofold uncertainty, and show that the most popular decision principle -- maximising expected value (`Expectationalism') -- has different formulations, namely Ex-Ante Expectationalism, Ex-Post Expectationalism, and hybrid theories. These alternative theories recommend different decisions, reasoning modes, and attitudes to risk. But they converge under an interesting (necessary and sufficient) condition.

John Broome has developed an account of rationality and reasoning which gives philosophical foundations for choice theory and the psychology of rational agents. We formalize his account into a model that differs from ordinary choice-theoretic models through focusing on psychology and the reasoning process. Within that model, we ask Broome’s central question of whether reasoning can make us more rational: whether it allows us to acquire transitive preferences, consistent beliefs, non-akratic intentions, and so on. We identify three structural types of rationality requirements: consistency requirements, completeness requirements, and closedness requirements. Many standard rationality requirements fall under this typology. Based on three theorems, we argue that reasoning is successful in achieving closedness requirements, but not in achieving consistency or completeness requirements. We assess how far our negative results reveal gaps in Broome's theory, or deficiencies in choice theory and behavioral economics.

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 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.

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.

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: whether a premise is justi…ed depends on the notion of probability considered; none of the notions renders both premises simultaneously justi…ed. Under the perhaps most interesting notions, the independence assumption should be weakened.

In the emerging literature on judgment aggregation over logically connected proposi- tions, expert rights or liberal rights have not been investigated yet. A group making collective judgments may assign individual members or subgroups with expert know- ledge on, or particularly affected by, certain propositions the right to determine the collective judgment on those propositions. We identify a problem that generalizes Sen's 'liberal paradox'. Under plausible conditions, the assignment of rights to two or more individuals or subgroups is inconsistent with the unanimity principle, whereby unanimously accepted propositions are collectively accepted. The inconsistency can be avoided if individual judgments or rights satisfy special conditions.

Agents are often assumed to have degrees of belief (“credences”) and also binary beliefs (“beliefs simpliciter”). How are these related to each other? A much-discussed answer asserts that it is rational to believe a proposition if and only if one has a high enough degree of belief in it. But this answer runs into the “lottery paradox”: the set of believed propositions may violate the key rationality conditions of consistency and deductive closure. In earlier work, we showed that this problem generalizes: there exists no local function from degrees of belief to binary beliefs that satisfies some minimal conditions of rationality and non-triviality. “Locality” means that the binary belief in each proposition depends only on the degree of belief in that proposition, not on the degrees of belief in others. One might think that the impossibility can be avoided by dropping the assumption that binary beliefs are a function of degrees of belief. We prove that, even if we drop the “functionality” restriction, there still exists no local relation between degrees of belief and binary beliefs that satisfies some minimal conditions. Thus functionality is not the source of the impossibility; its source is the condition of locality. If there is any non-trivial relation between degrees of belief and binary beliefs at all, it must be a “holistic” one. We explore several concrete forms this “holistic” relation could take.

In this paper, we identify a new and mathematically well-defined sense in which the coherence of a set of hypotheses can be truth-conducive. Our focus is not, as usual, on the probability but on the confirmation of a coherent set and its members. We show that, if evidence confirms a hypothesis, confirmation is “transmitted” to any hypotheses that are sufficiently coherent with the former hypothesis, according to some appropriate probabilistic coherence measure such as Olsson’s or Fitelson’s measure. Our findings have implications for scientific methodology, as they provide a formal rationale for the method of indirect confirmation and the method of confirming theories by confirming their parts.

This symposium in the overlap of philosophy and decision theory is described well by its title “Beliefs in Groups”. Each word in the title matters, with one intended ambiguity. The symposium is about beliefs rather than other attitudes such as preferences; these beliefs take the form of probabilities in the first three contributions, binary yes/no beliefs (‘judgments’) in the fourth contribution, and qualitative probabilities (‘probability grades’) in the fifth contribution. The beliefs occur in groups, which is ambiguous between beliefs of groups as a whole and beliefs of group members. The five contributions – all of them interesting, we believe – address several aspects of this general theme

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.

A group is often construed as one agent with its own probabilistic beliefs (credences), which are obtained by aggregating those of the individuals, for instance through averaging. In their celebrated “Groupthink”, Russell et al. (2015) require group credences to undergo Bayesian revision whenever new information is learnt, i.e., whenever individual credences undergo Bayesian revision based on this information. To obtain a fully Bayesian group, one should often extend this requirement to non-public or even private information (learnt by not all or just one individual), or to non-representable information (not representable by any event in the domain where credences are held). I pro- pose a taxonomy of six types of ‘group Bayesianism’. They differ in the information for which Bayesian revision of group credences is required: public representable information, private representable information, public non-representable information, etc. Six corre- sponding theorems establish how individual credences must (not) be aggregated to ensure group Bayesianism of any type, respectively. Aggregating through standard averaging is never permitted; instead, different forms of geometric averaging must be used. One theorem—that for public representable information—is essentially Russell et al.’s central result (with minor corrections). Another theorem—that for public non-representable information—fills a gap in the theory of externally Bayesian opinion pooling.

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: a specification of which properties of the objects of moral choice matter in any given context, and a specification of how these properties matter. Reason-based representations provide a very general taxonomy of moral theories, as differences among theories can be attributed to differences 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”.

We give a review and critique of jury theorems from a social-epistemology perspective, covering Condorcet’s (1785) classic theorem and several later refinements and departures. We assess the plausibility of the conclusions and premises featuring in jury theorems and evaluate the potential of such theorems to serve as formal arguments for the ‘wisdom of crowds’. In particular, we argue (i) that there is a fundamental tension between voters’ independence and voters’ competence, hence between the two premises of most jury theorems; (ii) that the (asymptotic) conclusion that ‘huge groups are infallible’, reached by many jury theorems, is an artifact of unjustified premises; and (iii) that the (nonasymptotic) conclusion that ‘larger groups are more reliable’, also reached by many jury theorems, is not an artifact and should be regarded as the more adequate formal rendition of the ‘wisdom of crowds’.

In a single framework, I address the question of the informational basis for evaluating social states. I particularly focus on information about individual welfare, individual preferences and individual (moral) judgments, but the model is also open to any other informational input deemed relevant, e.g. sources of welfare and motivations behind preferences. In addition to proving some possibility and impossibility results, I discuss objections against using information about only one aspect (e.g. using only preference information). These objections suggest a multi-aspect informational basis for aggregation. However, the multi-aspect approach faces an impossibility result created by a lack of inter-aspect comparability. The impossibility could be overcome by measuring information on non-cardinal scales.

[This version of the paper has been superseded by "Reason-based choice and context-dependence: An explanatory framework", forthcoming in Economics & Philosophy.] We introduce a “reason-based” way of rationalizing an agent’s choice behaviour, which explains choices by specifying which properties of the options or choice context the agent cares about (the “motivationally salient properties”) and how he or she cares about these properties (the “fundamental preference relation”). Reason-based rationalizations can explain non-classical choice behaviour, including boundedly rational and sophisticated rational behaviour, and predict choices in unobserved contexts, an issue neglected in standard choice theory. We characterize the behavioural implications of different reason-based models and distinguish two kinds of context-dependent motivation: “context-variant” motivation, where the agent cares about different properties in different contexts, and “context-regarding” motivation, where the agent cares not only about properties of the options, but also about properties relating to the context.

Several recent results on the aggregation of judgments over logically connected propositions show that, under certain conditions, dictatorships are the only propositionwise aggregation functions generating fully rational (i.e., complete and consistent) collective judgments. A frequently mentioned route to avoid dictatorships is to allow incomplete collective judgments. We show that this route does not lead very far: we obtain oligarchies rather than dictatorships if instead of full rationality we merely require that collective judgments be deductively closed, arguably a minimal condition of rationality, compatible even with empty judgment sets. We derive several characterizations of oligarchies and provide illustrative applications to Arrowian preference aggregation and Kasher and Rubinsteinís group identification problem.

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 group members, who are themselves modelled as Bayesian agents, possibly with different priors and different information. Group beliefs are proven to take a simple multiplicative form if people’s information is independent, and a more complex form if information overlaps arbitrarily. This shows that group beliefs can incorporate all information spread over the individuals without the individuals having to communicate their (possibly complex and hard-to-describe) private information; communicating prior and posterior beliefs sufices. JEL classification: D70, D71..

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, non-dictatorship and propositionwise independence. But it is only partially known (e.g., only in the monotonic case) 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), Dietrich and List (2007a) and Dokow and Holzman (2010a).

The new field of judgment aggregation aims to merge many individual sets of judgments on logically interconnected propositions into a single collective set of judgments on these propositions. Judgment aggregation has commonly been studied using classical propositional logic, with a limited expressive power and a problematic representation of conditional statements ("if P then Q") as material conditionals. In this methodological paper, I present a simple unified model of judgment aggregation in general logics. I show how many realistic decision problems can be represented in it. This includes decision problems expressed in languages of classical propositional logic, predicate logic (e.g. preference aggregation problems), modal or conditional logics, and some multi-valued or fuzzy logics. I provide a list of simple tools for working with general logics, and I prove impossibility results that generalise earlier theorems.

The new …eld of judgment aggregation aims to …nd collective judgments on logically interconnected propositions. Recent impossibility results establish limitations on the possibility to vote independently on the propositions. I show that, fortunately, the impossibility results do not apply to a wide class of realistic agendas once propositions like “if a then b” are adequately modelled, namely as subjunctive implications rather than material implications. For these agendas, consistent and complete collective judgments can be reached through appropriate quota rules (which decide propositions using acceptance thresholds). I characterise the class of these quota rules. I also prove an abstract result that characterises consistent aggregation for arbitrary agendas in a general logic.

How can different 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 sigma-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 sigma-algebras. As an illustrative application, we also consider probabilistic preference aggregation. Finally, we compare our results with existing results on binary judgment aggregation and Arrovian preference aggregation. This paper is the first of two self-contained, but technically related companion papers inspired by binary judgment-aggregation theory.

Imagine a group that faces a decision problem but does not agree on which decision procedure is appropriate. In that case, can a decision be reached that respects the procedural concerns of the group? There is a sense in which legitimate decisions are possible even if people disagree on which procedure to use. I propose to decide in favour of an option which maximizes the number of persons whose judged-right procedure happens to entail this decision given the profile. This decision rule is based not only ona profile in the standard sense, but in addition on a profile of judged-right procedures. To justify this decision rule, I present a set of simple axioms leading to it as the only solution.