Impossibility Theorem Research Articles

Aggregating credences into beliefs: agenda conditions for impossibility results

AbstractHybrid belief aggregation addresses aggregation of individual probabilistic beliefs into collective binary beliefs. In line with the development of judgment aggregation theory, our research delves into the identification of precise agenda conditions associated with some key impossibility theorems in the context of hybrid belief aggregation. We determine the necessary and sufficient level of logical interconnection between the propositions in an agenda for some key impossibilities to arise. Specifically, we prove three characterization theorems about hybrid belief aggregation: (i) Precisely the path-connected and pair-negatable agendas lead to the ‘oligarchy result’—only oligarchic rules satisfy universal domain, proposition-wise independence, respect for unanimity, and deductive closure of collective beliefs. (ii) Precisely the negation-connected agendas lead to the ‘triviality result’—only unanimity rules satisfy those conditions as well as anonymity. (iii) Precisely the blocked agendas lead to the ‘non-existence result’—no rules satisfy those conditions as well as completeness and consistency of collective beliefs. Furthermore, we compare these novel findings with existing agenda-theoretic characterization theorems in the domains of judgment aggregation and belief binarization.

Escaping Arrow’s theorem: the Advantage-Standard model

AbstractThere is an extensive literature in social choice theory studying the consequences of weakening the assumptions of Arrow’s Impossibility Theorem. Much of this literature suggests that there is no escape from Arrow-style impossibility theorems, while remaining in an ordinal preference setting, unless one drastically violates the Independence of Irrelevant Alternatives (IIA). In this paper, we present a more positive outlook. We propose a model of comparing candidates in elections, which we call the Advantage-Standard (AS) model. The requirement that a collective choice rule (CCR) be representable by the AS model captures a key insight of IIA but is weaker than IIA; yet it is stronger than what is known in the literature as weak IIA (two profiles alike on x, y cannot have opposite strict social preferences on x and y). In addition to motivating violations of IIA, the AS model makes intelligible violations of another Arrovian assumption: the negative transitivity of the strict social preference relation P. While previous literature shows that only weakening IIA to weak IIA or only weakening negative transitivity of P to acyclicity still leads to impossibility theorems, we show that jointly weakening IIA to AS representability and weakening negative transitivity of P leads to no such impossibility theorems. Indeed, we show that several appealing CCRs are AS representable, including even transitive CCRs.

From Condorcet’s paradox to Arrow: yet another simple proof of the impossibility theorem

From Condorcet’s paradox to Arrow: yet another simple proof of the impossibility theorem

An impossibility theorem for Base Rate Tracking and Equalized Odds

An impossibility theorem for Base Rate Tracking and Equalized Odds

A Human-in-the-Loop Fairness-Aware Model Selection Framework for Complex Fairness Objective Landscapes

Fairness-aware Machine Learning (FairML) applications are often characterized by complex social objectives and legal requirements, frequently involving multiple, potentially conflicting notions of fairness. Despite the well-known Impossibility Theorem of Fairness and extensive theoretical research on the statistical and socio-technical trade-offs between fairness metrics, many FairML tools still optimize or constrain for a single fairness objective. However, this one-sided optimization can inadvertently lead to violations of other relevant notions of fairness. In this socio-technical and empirical study, we frame fairness as a Many-Objective (MaO) problem by treating fairness metrics as conflicting objectives in a multi-objective (MO) sense. We introduce ManyFairHPO, a human-in-the-loop, fairness-aware model selection framework that enables practitioners to effectively navigate complex and nuanced fairness objective landscapes. ManyFairHPO aids in the identification, evaluation, and balancing of fairness metric conflicts and their related social consequences, leading to more informed and socially responsible model-selection decisions. Through a comprehensive empirical evaluation and a case study on the Law School Admissions problem, we demonstrate the effectiveness of ManyFairHPO in balancing multiple fairness objectives, mitigating risks such as self-fulfilling prophecies, and providing interpretable insights to guide stakeholders in making fairness-aware modeling decisions.

Conditional normative reasoning as a fragment of HOL

We report on the mechanisation of (preference-based) conditional normative reasoning. Our focus is on Åqvist's system E for conditional obligation, and its extensions. Our mechanisation is achieved via a shallow semantical embedding in Isabelle/HOL. We consider two possible uses of the framework. The first one is as a tool for meta-reasoning about the considered logic. We employ it for the automated verification of deontic correspondences (broadly conceived) and related matters, analogous to what has been previously achieved for the modal logic cube. The equivalence is automatically verified in one direction, leading from the property to the axiom. The second use is as a tool for assessing ethical arguments. We provide a computer encoding of a well-known paradox (or impossibility theorem) in population ethics, Parfit's repugnant conclusion. While some have proposed overcoming the impossibility theorem by abandoning the presupposed transitivity of ‘better than’, our formalisation unveils a less extreme approach, suggesting among other things the option of weakening transitivity suitably rather than discarding it entirely. Whether the presented encoding increases or decreases the attractiveness and persuasiveness of the repugnant conclusion is a question we would like to pass on to philosophy and ethics.

Decision with belief functions and generalized independence: Two impossibility theorems

Dempster-Shafer theory of evidence is a framework that is expressive enough to represent both ignorance and probabilistic information. However, decision models based on belief functions proposed in the literature face limitations in a sequential context: they either abandon the principle of dynamic consistency, restrict the combination of lotteries, or relax the requirement for transitive and complete comparisons. This work formally establishes that these requirements are indeed incompatible when any form of compensation is considered. It then demonstrates that these requirements can be satisfied in non-compensatory frameworks by introducting and characterizing a dynamically consistent rule based on first-order dominance.

Social choice models and topological dynamics

In this paper, we propose a topological social choice model in an equivariant setting. We generalize the Chichilnisky–Heal impossibility theorem to the realm of proper actions of almost-connected locally compact groups. We also improve one result stated in Juárez-Anguiano (Topol Appl 279:107246, 2020) about the preservation of equivariant ANR’s by symmetric powers.

Dynamically rational judgment aggregation

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. 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.

Exploring the constraints on artificial general intelligence: A game-theoretic model of human vs machine interaction

The potential emergence of artificial general intelligence (AGI) systems has sparked intense debate among researchers, policymakers, and the public due to their potential to surpass human intelligence in all domains. This note argues that for an AI to be considered “general”, it should achieve superhuman performance not only in zero-sum games but also in general-sum games, where winning or losing is not clearly defined. In this note, I propose a game-theoretic framework that captures the strategic interactions between a representative human agent and a potential superhuman machine agent. Four assumptions underpin this framework: Superhuman Machine, Machine Strategy, Rationality, and Strategic Unpredictability. The main result is an impossibility theorem, establishing that these assumptions are inconsistent when taken together, but relaxing any one of them results in a consistent set of assumptions. This note contributes to a better understanding of the theoretical context that can shape the development of superhuman AI.

A general impossibility theorem on Pareto efficiency and Bayesian incentive compatibility

A general impossibility theorem on Pareto efficiency and Bayesian incentive compatibility

ARROW’S THEOREM, ULTRAFILTERS, AND REVERSE MATHEMATICS

Abstract This paper initiates the reverse mathematics of social choice theory, studying Arrow’s impossibility theorem and related results including Fishburn’s possibility theorem and the Kirman–Sondermann theorem within the framework of reverse mathematics. We formalise fundamental notions of social choice theory in second-order arithmetic, yielding a definition of countable society which is tractable in ${\mathsf {RCA}}_0$ . We then show that the Kirman–Sondermann analysis of social welfare functions can be carried out in ${\mathsf {RCA}}_0$ . This approach yields a proof of Arrow’s theorem in ${\mathsf {RCA}}_0$ , and thus in $\mathrm {PRA}$ , since Arrow’s theorem can be formalised as a $\Pi ^0_1$ sentence. Finally we show that Fishburn’s possibility theorem for countable societies is equivalent to ${\mathsf {ACA}}_0$ over ${\mathsf {RCA}}_0$ .

Which set of agents plays a key role? An impossibility in transforming binary relations

When provided with the performance ranking of multiple sets of agents as input, which set of agents is expected to play the key role? To address this question, we introduce a new rule (transformation rule) that maps a performance ranking over sets of agents into a contributing ranking over sets of agents. Preference extension (PE) and social ranking problem (SRP) represent two special cases. We prove an impossibility theorem: in a sufficiently rich environment, there is no transformation rule that satisfies ceteris paribus weak dominance, self-reflection, and triple-acyclicity. The impossibility is novel in that it is degenerated in PE/SRP models.

The advantages of the corporate form—an impossibility theorem on persons and things

Abstract The literature tends to view the firm either as a person or as a thing. Due to this dichotomy, it struggles with the proposition that the corporate form brings about efficiency gains that cannot be accomplished by other types of profit-seeking firms. This study supplies game-theoretic proof for this proposition. It identifies a capital-intensive business that requires external financing and is vulnerable to a holdup by a supplier. It shows that only the corporate form can organise the business while other types of firms fail. This advantage of the corporate form over other types of firms is due to its Janus-faced nature as both person, that is endowed with legal capacity, and thing, such as a transferable bundle of assets. A long-held tradition of strictly separating persons and things fails to account for this advantage.

Three Unique Virtues of Approval Voting

Approval Voting offers advantages over other voting systems for single-winner elections. This manuscript analyzes three unique virtues of Approval Voting. First, the procedure does not violate the independence of irrelevant alternatives criterion for rational choice. Second, it prevents manipulation of outcomes through agenda setting. Third, it avoids intransitive majority preference cycles like Condorcet paradoxes and so escapes Arrow’s Impossibility Theorem constraints. As a result of these virtues, which are generally not shared by its best known competitors, Approval voting emerges as a strong option for realizing majoritarian democratic will.

An impossibility theorem concerning positive involvement in voting

In social choice theory with ordinal preferences, a voting method satisfies the axiom of positive involvement if adding to a preference profile a voter who ranks an alternative uniquely first cannot cause that alternative to go from winning to losing. In this note, we prove a new impossibility theorem concerning this axiom: there is no ordinal voting method satisfying positive involvement that also satisfies the Condorcet winner and loser criteria, resolvability, and a common invariance property for Condorcet methods, namely that the choice of winners depends only on the ordering of majority margins by size.

Impossibility theorems for feature attribution.

Despite a sea of interpretability methods that can produce plausible explanations, the field has also empirically seen many failure cases of such methods. In light of these results, it remains unclear for practitioners how to use these methods and choose between them in a principled way. In this paper, we show that for moderately rich model classes (easily satisfied by neural networks), any feature attribution method that is complete and linear-for example, Integrated Gradients and Shapley Additive Explanations (SHAP)-can provably fail to improve on random guessing for inferring model behavior. Our results apply to common end-tasks such as characterizing local model behavior, identifying spurious features, and algorithmic recourse. One takeaway from our work is the importance of concretely defining end-tasks: Once such an end-task is defined, a simple and direct approach of repeated model evaluations can outperform many other complex feature attribution methods.

Potential rationality in collective decision-making

This study investigates Suzumura consistency as a condition for the rationality of social preferences. A preference is said to be Suzumura-consistent when all preference cycles include only indifference relations. This condition is equivalent to transitivity in the presence of completeness, but, in general, it is substantially weaker than transitivity when preference is incomplete. Notably, Suzumura consistency is especially significant for a preference because it is necessary and sufficient for the existence of an ordering (transitive and complete preference) that is compatible with the original preference. This coherency property can be regarded as a requirement for potential rationality. In this study, we examine the implications of shifting from actual rationality to potential rationality in collective decision-making. We introduce the concept of an alternative-dependent coherent collection in order to obtain a representation of a class of Suzumura-consistent collective choice rules that satisfy the axioms imposed in Arrow’s impossibility theorem. This demonstrates that the power structure to determine social choice can be alternative-dependent.

An impossibility theorem in game dynamics

The Nash equilibrium-a combination of choices by the players of a game from which no self-interested player would deviate-is the predominant solution concept in game theory. Even though every game has a Nash equilibrium, it is not known whether there are deterministic behaviors of the players who play a game repeatedly that are guaranteed to converge to a Nash equilibrium of the game from all starting points. If one assumes that the players' behavior is a discrete-time or continuous-time rule whereby the current mixed strategy profile is mapped to the next, this question becomes a problem in the theory of dynamical systems. We apply this theory, and in particular Conley index theory, to prove a general impossibility result: There exist games, for which all game dynamics fail to converge to Nash equilibria from all starting points. The games which help prove this impossibility result are degenerate, but we conjecture that the same result holds, under computational complexity assumptions, for nondegenerate games. We also prove a stronger impossibility result for the solution concept of approximate Nash equilibria: For a set of games of positive measure, no game dynamics can converge to the set of approximate Nash equilibria for a sufficiently small yet substantial approximation bound. Our results establish that, although the notions of Nash equilibrium and its computation-inspired approximations are universally applicable in all games, they are fundamentally incomplete as predictors of long-term player behavior.

아마르티아 센의 파레토적 자유주의 불가능성에 대한 공공선택학파의 반론

Kenneth’s Arrow Impossibility Theorem revealed dismal prospect to democracy, Amartya Sen’s Impossibility Theorem of Paretian Libertarianism provoked deep pessimism to Libertarianism. This paper synthesize critique by Public Choice School on Sen’s theorem comprising three points. Firstly, Sen misunderstood individuals as picking ‘social states’, which he regarded as complete description of society including all individual positions on it. Actually, however, individuals just choose particular aspect. Secondly, Sen’s impossibility basically comes from externality of individual’s preference. Public Choice scholars revealed, by designing proper game model under externality, Pareto optimality comes along with libertarian choice. Thirdly, Sen’s Impossibility Theorem is just short-term equilibrium showing contradiction between Pareto principle and Libertarian principle. Buchanan proves that paradox can be resolved through mutual coordination between individuals who are under Paretian inferior equilibrium. This paper suggests some implication as follows. Firstly, Libertarianism in Sen’s sense is far from normal meaning of liberty. Without proper conceptual modification, his theorem needs to be renamed. Secondly, If Sen’s Impossibility Theorem under U-condition can never exclude some cases in which two principles can coexist, then his theorem can be renamed as “Impossibility of Sen’s Impossibility of Paretian Libertarianism”. Thirdly, Public Choice theorists properly proved Paretian Libertarianism is possible, showing Sen exaggerated the paradox. Then, as Arrow’s theorem can be interpreted as ‘Possibility’ Theorem, considering substantial stability of actual collective choice situation, Sen’s theorem also can be re-interpreted as ‘Possibility’ Theorem of Paretian Libertarianism. Two camps needs to confront each other again towards the empirical proof on it.