Online research in philosophy

How can di¤erent individuals’probability assignments to some events be aggregated into a collective probability assignment? Although there are several classic results on this problem, they all assume that the ‘agenda’ of relevant events forms a -algebra, an overly demanding assumption for many practical applications. We drop this assumption and explore probabilistic opinion pooling on general agendas. Our main theorems characterize linear pooling and neutral pooling for large classes of agendas, with standard results as special cases.

Journal of Theoretical Politics 19(4) (in press).

Economics and Philosophy 23(3) (in press).

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 di¤erent priors and di¤erent 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 su¢ ces. JEL classi…cation: D70, D71.. (shrink)

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

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

The new …eld 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 uni…ed 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. (shrink)

In the emerging literature on judgment aggregation over logically connected propositions, expert rights or liberal rights have not been investigated yet. A group making collective judgments may assign individual members or subgroups with expert knowledge on, or particularly a¤ected 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. (shrink)

A population’s level of terrorism depends on two factors: people’s preferences (would they like creating damage?) and the constraints under which people act (what damage could they create, and at what punishment?). Causerelated policies, e.g. improving social stability or education, aim at appeasing preferences, thereby reducing terrorism. Symptom-related policies, e.g. embargoes or wars, change the constraints (‘deterrence’), but may have side e¤ects on preferences (‘provocation’); terrorism increases if provocation overweighs deterrence. I model preferences for damage as endogenous and policy-dependent. I (...) argue that provocation by tough policies is easy to overlook, and show that provocation-neglect leads to toughness-exaggeration. (shrink)

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 indi¤erences unlike in earlier generalisations. (shrink)

In response to recent work on the aggregation of individual judgments on logically connected propositions into collective judgments, it is often asked whether judgment aggregation is a special case of Arrowian preference aggregation. We argue for the converse claim. After proving two impossibility theorems on judgment aggregation (using "systematicity" and "independence" conditions, respectively), we construct an embedding of preference aggregation into judgment aggregation and prove Arrow’s theorem (stated for strict preferences) as a corollary of our second result. Although we thereby (...) provide a new proof of Arrow’s theorem, our main aim is to identify the analogue of Arrow’s theorem in judgment aggregation, to clarify the relation between judgment and preference aggregation, and to illustrate the generality of the judgment aggregation model. JEL Classi…cation: D70, D71.. (shrink)

Assuming that votes are independent, the epistemically optimal procedure in a binary collective choice problem is known to be a weighted supermajority rule with weights given by personal log-likelihood-ratios. It is shown here that an analogous result holds in a much more general model. Firstly, the result follows from a more basic principle than expected-utility maximisation, namely from an axiom ("Epistemic Monotonicity") which requires neither utilities nor prior probabilities of the ‘correctness’ of alternatives. Secondly, a person’s input need not be (...) a vote for an alternative, it may be any type of input, for instance a subjective degree of belief or probability of the correctness of one of the alternatives. The case of a profile of subjective degrees of belief is particularly appealing, since here no parameters such as competence parameters need to be known. (shrink)

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 on a 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. (shrink)

The aggregation of individual judgments over interrelated propositions is a newly arising …eld of social choice theory. I introduce several independence conditions on judgment aggregation rules, each of which protects against a speci…c type of manipulation by agenda setters or voters. I derive impossibility theorems whereby these independence conditions are incompatible with certain minimal requirements. Unlike earlier impossibility results, the main result here holds for any (non-trivial) agenda. However, independence conditions arguably undermine the logical structure of judgment aggregation. I therefore (...) suggest restricting independence to "premises", which leads to a generalised premise-based procedure. This procedure is proven to be possible if the premises are logically independent. JEL Classi…cation Numbers: D70, D71, D79. (shrink)

We show that, when a group takes independent majority votes on interconnected propositions, the outcome is consistent once the pro…le of individual judgment sets respects appropriate structural conditions. We introduce several such conditions on pro…les, based on ordering the propositions or ordering the individuals, and we clarify the relations between these conditions. By restricting the conditions to appropriate subagendas, we obtain local conditions that are less demanding but still guarantee consistent majority judgments. By applying the conditions to agendas representing preference (...) aggregation problems, we show parallels of some conditions to existing social-choice-theoretic conditions, speci…cally to order restriction and intermediateness, restricted to triples of alternatives in the case of our local conditions. (shrink)

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 su¢ - cient 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 constrasted 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. (shrink)

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 Rubinstein’s group identi…cation problem. (shrink)

All existing impossibility theorems on judgment aggregation require individual and collective judgment sets to be consistent and complete (in some recent results with completeness relaxed to deductive closure), arguably a demanding 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 function that is "neutral" between the acceptance and rejection of each (...) proposition is dictatorial. We relate this theorem to the literature. (shrink)

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: taste for Mozart changes by listening to it, sympathy for another player changes by 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 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. The …ndings help model selection, for both theoretical purposes (e.g. in game and decision theory) and empirical purposes (e.g. testing hypotheses or estimating parameters of change). (shrink)

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 generalizes 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. (shrink)

How can di¤erent individuals’probability assignments to some events be aggregated into a collective probability assignment? Although there are several classic results on this problem, they all assume that the ‘agenda’ of relevant events forms a -algebra, an overly demanding assumption for many practical applications. We drop this assumption and explore probabilistic opinion pooling on general agendas. Our main theorems characterize linear pooling and neutral pooling for large classes of agendas, with standard results as special cases.

If a group as a whole 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 di¤erent priors and di¤erent information. Group beliefs are shown to take a simple multiplicative form if people’s information is independent, and a more complex form if information can overlap 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 su¢ ces. (shrink)

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 systematicity that applies to most standard agendas: Every judgment aggregation function (with rational inputs and outputs) satisfying a condition called unbiasedness is dictatorial (or e¤ectively dictatorial if we remove one of the agenda conditions). Our agenda conditions are tight. Applied (...) illustratively to (strict) preference aggregation represented in our model, the result implies that every unbiased social welfare function with universal domain is e¤ectively dictatorial. Keywords: judgment aggregation, logic, impossibility, May’s neutrality.. (shrink)

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. (shrink)

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. (shrink)

According to standard rational choice theory, as commonly used in political science and economics, an agent’s fundamental preferences are exogenously …xed, and any preference change over decision options is due to Bayesian information learning. Although elegant and parsimonious, this model fails to account for preference change driven by experiences or psychological changes distinct from information learning. We develop a model of non-informational preference change. Alternatives are modelled as points in some multidimensional space, only some of whose dimensions play a role (...) in shaping the agent’s preferences. Any change in these ‘motivationally salient’ dimensions can change the agent’s preferences. How it does so is described by a new representation theorem. Our model not only captures a wide range of frequently observed phenomena, but also generalizes some standard representations of preferences in political science and economics. (shrink)

We show that, when a group takes independent majority votes on interconnected propositions, the outcome is consistent once the pro…le of individual judgment sets respects appropriate structural conditions. We introduce several such conditions on pro…les, based on ordering the propositions or ordering the individuals, and we clarify the relations between these conditions. By restricting the conditions to appropriate subagendas, we obtain local conditions that are less demanding but still guarantee consistent majority judgments. By applying the conditions to agendas representing preference (...) aggregation problems, we show parallels of some conditions to existing social-choice-theoretic conditions, speci…cally to order restriction and intermediateness, restricted to triples of alternatives in the case of our local conditions. (shrink)

Behaviourism is the view that preferences, beliefs, and other mental states in social-scientific theories are nothing but constructs re-describing people's behavioural dispositions. Mentalism is the view that they capture real phenomena, no less existent than the unobservable entities and properties in the natural sciences, such as electrons and electromagnetic fields. While behaviourism has long gone out of fashion in psychology and linguistics, it remains influential in economics, especially in `revealed preference' theory. We aim to (i) clear up some common confusions (...) about the two views, (ii) situate the debate in a historical context, and (iii) defend a mentalist approach to economics. Setting aside normative concerns about behaviourism, we show that mentalism is in line with best scientific practice even if economics is treated as a purely positive science of economic behaviour. We distinguish mentalism from, and reject, the radical neuroeconomic view that behaviour should be explained in terms of people's brain processes, as distinct from their mental states. (shrink)

Bayesian epistemology tells us with great precision how we should move from prior to posterior beliefs in light of new evidence or information, but says little about where our prior beliefs come from. It offers few resources to describe some prior beliefs as rational or well-justified, and others as irrational or unreasonable. A different strand of epistemology takes the central epistemological question to be not how to change one’s beliefs in light of new evidence, but what reasons justify a given (...) set of beliefs in the first place. We offer an account of rational belief formation that closes some of the gap between Bayesianism and its reason-based alternative, formalizing the idea that an agent can have reasons for his or her (prior) beliefs, in addition to evidence or information in the ordinary Bayesian sense. Our analysis of reasons for belief is part of a larger programme of research on the role of reasons in rational agency (Dietrich and List, Nous, 2012a, in press; Int J Game Theory, 2012b, in press). (shrink)

There is a surprising disconnect between formal rational choice theory and philosophical work on reasons. The one is silent on the role of reasons in rational choices, the other rarely engages with the formal models of decision problems used by social scientists. To bridge this gap, we propose a new, reason-based theory of rational choice. At its core is an account of preference formation, according to which an agent’s preferences are determined by his or her motivating reasons, together with a (...) ‘weighing relation’ between different combinations of reasons. By explaining how someone’s preferences may vary with changes in his or her motivating reasons, our theory illuminates the relationship between deliberation about reasons and rational choices. Although primarily positive, the theory can also help us think about how those preferences and choices ought to respond to normative reasons. (shrink)

We investigate judgment aggregation by assuming that some formulas of the agenda are singled out as premisses, and the Independence condition (formula-wise aggregation) holds for them, though perhaps not for others. Whether premiss-based aggregation thus de…ned is non-degenerate depends on how premisses are logically connected, both among themselves and with other formulas. We identify necessary and su¢ cient conditions for dictatorship or oligarchy on the premisses, and investigate when these results extend to the whole agenda. Our theorems recover or strengthen (...) several existing ones and are formulated for in…nite populations, an innovation of this paper. JEL identi…cation numbers: D70, D71. (shrink)

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 (such as not to affirm both a proposition and its negation) and requirements dictated by the environment (such as budgetary constraints). We present some general impossibility results on constrained judgment aggregation; they are immediate corollaries of known results on (unconstrained) judgment aggregation. (shrink)

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). (shrink)

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 justified depends on the notion of probability considered and (ii) none of the notions renders both premises simultaneously justified. Under the perhaps most interesting notions, the independence assumption should be weakened.

How can the propositional attitudes of several individuals be aggregated into overall collective propositional attitudes? Although there are large bodies of work on the aggregation of various special kinds of propositional attitudes, such as preferences, judgments, probabilities and utilities, the aggregation of propositional attitudes is seldom studied in full generality. In this paper, we seek to contribute to filling this gap in the literature. We sketch the ingredients of a general theory of propositional attitude aggregation and prove two new theorems. (...) Our first theorem simultaneously characterizes some prominent aggregation rules in the cases of probability, judgment and preference aggregation, including linear opinion pooling and Arrovian dictatorships. Our second theorem abstracts even further from the specific kinds of attitudes in question and describes the properties of a large class of aggregation rules applicable to a variety of belief-like attitudes. Our approach integrates some previously disconnected areas of investigation. (shrink)

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 nonmanipulability and strategy-proofness. Comparing two frequently discussed aggregation rules, we show that “conclusion-based voting” is less vulnerable to manipulation than “premisebased 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. (shrink)

In this paper, we present a simple axiomatic justification for indifference before opening, avoiding any expectation reasoning, which is often considered problematic in infinite cases. Although the two-envelope paradox assumes an expectation-maximizing agent, we show that analogous paradoxes arise for agents using difierent decision principles such as maximin and maximax, and that our justification for indifierence before opening applies here too.

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 usually, 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. (shrink)

Under the independence and competence assumptions of Condorcet’s classical jury model, the probability of a correct majority decision converges to certainty as the jury size increases, a seemingly unrealistic result. Using Bayesian networks, we argue that the model’s independence assumption requires that the state of the world (guilty or not guilty) is the latest common cause of all jurors’ votes. But often – arguably in all courtroom cases and in many expert panels – the latest such common cause is a (...) shared ‘body of evidence’ observed by the jurors. In the corresponding Bayesian network, the votes are direct descendants not of the state of the world, but of the body of evidence, which in turn is a direct descendant of the state of the world. We develop a model of jury decisions based on this Bayesian network. Our model permits the possibility of misleading evidence, even for a maximally competent observer, which cannot easily be accommodated in the classical model. We prove that (i) the probability of a correct majority verdict converges to the probability that the body of evidence is not misleading, a value typically below 1; (ii) depending on the required threshold of ‘no reasonable doubt’, it may be impossible, even in an arbitrarily large jury, to establish guilt of a defendant ‘beyond any reasonable doubt’. (shrink)

There is a surprising disconnect between formal rational choice theory and philosophical work on reasons. The one is silent on the role of reasons in rational choices, the other rarely engages with the formal models of decision problems used by social scientists. To bridge this gap, we propose a new, reason-based theory of rational choice. At its core is an account of preference formation, according to which an agent's preferences are determined by his or her motivating reasons, together with a (...) 'weighing relation' between different combinations of reasons. By explaining how someone's preferences may vary with changes in his or her motivating reasons, our theory illuminates the relationship between deliberation about reasons and rational choices. Although primarily positive, the theory can also help us think about how those preferences and choices ought to respond to normative reasons. (shrink)

Rational choice theory analyzes how an agent can rationally act, given his or her preferences, but says little about where those preferences come from. Instead, preferences are usually assumed to be …xed and exogenously given. We introduce a framework for conceptualizing preference formation and preference change. In our model, an agent’s preferences are based on certain ‘motivationally salient’properties of the alternatives over which the preferences are held. Preferences may change as new properties of the alternatives become salient or previously salient (...) ones cease to be so. We suggest that our approach captures endogenous preferences in various contexts, and helps to illuminate the distinction between formal and substantive concepts of rationality, as well as the role of perception in rational choice. (shrink)