Consistency Axiom Research Articles

Refined Characterizations of Approval-Based Committee Scoring Rules

In approval-based committee (ABC) elections, the goal is to select a fixed-size subset of the candidates, a so-called committee, based on the voters' approval ballots over the candidates. One of the most popular classes of ABC voting rules are ABC scoring rules, for which voters give points to each committee and the committees with maximal total points are chosen. While the set of ABC scoring rules has recently been characterized in a model where the output is a ranking of all committees, no full characterization of these rules exists in the standard model where a set of winning committees is returned. We address this issue by characterizing two important subclasses of ABC scoring rules in the standard ABC election model, thereby both extending the result for ABC ranking rules to the standard setting and refining it to subclasses. In more detail, by relying on a consistency axiom for variable electorates, we characterize (i) the prominent class of Thiele rules and (ii) a new class of ABC voting rules called ballot size weighted approval voting. Based on these theorems, we also infer characterizations of three well-known ABC voting rules, namely multi-winner approval voting, proportional approval voting, and satisfaction approval voting.

Consistent social ranking solutions

The performance of coalitions is an important measure for evaluating individuals. Sport players, researchers, and firm workers are often judged with their team performances. The social ranking solution (SRS) is a function that maps the ranking on the set of all feasible coalitions (the domain of coalitions) into the ranking of individuals. Importing the axiom of consistency from voting theory, we study consistent SRSs under the variable domains of coalitions. We suppose that there are several domains of coalitions (e.g., a set of research teams made up of only young researchers and a set of research teams including senior researchers), and the individuals are required to be evaluated consistently on each domain of coalition. Such a situation is typical because all the logically possible coalitions are often too huge to deal with. We obtain a new characterization of the lexicographic excellence solution (LES) and its dual (DLES): they are the only SRSs satisfying consistency, neutrality, weak coalitional anonymity, and complete dominance. This characterization is expected to provide a new ground for determining the impacts of individuals based on the lexicographic comparisons of their team performances.

Bayesian Contextual Choices Under Imperfect Perception of Attributes

Classic theories show that choices can be represented by a stable utility function when they satisfy consistency axioms, such as transitivity and the independence of irrelevant alternatives. Empirical choice data, however, display several contextual choice effects that violate these axioms. We study a choice model with a fixed underlying utility function and explain contextual choices with a novel type of information friction: the agent’s perception of the options is affected by attribute-specific uncertainty. Under this friction, the agent learns useful information when the agent sees more options. Therefore, the agent chooses contextually, exhibiting intransitivity, joint–separate evaluation reversal, the compromise effect, the phantom decoy effect, the attraction effect, and the similarity effect. Because the uncertainty is attribute-specific and common across alternatives, the classic axioms hold when the alternatives dominate one another in attributes. This paper was accepted by Dmitri Kuksov, marketing.

Approval voting under dichotomous preferences: A catalogue of characterizations

Approval voting allows every voter to cast a ballot of approved alternatives and chooses the alternatives with the largest number of approvals. Due to its simplicity and superior theoretical properties, it is a serious contender for use in real-world elections. We support this claim by giving eight characterizations of approval voting. All our results involve the consistency axiom, which requires choices to be consistent across different electorates. In addition, we consider strategyproofness, agreement with majority opinions, independence of cloned alternatives, and invariance under removing inferior alternatives. We prove our results by reducing them to a single base theorem, for which we give a simple and intuitive proof.

Towards continuous consistency axiom

Development of new algorithms in the area of machine learning, especially clustering, comparative studies of such algorithms as well as testing according to software engineering principles requires availability of labeled data sets. While standard benchmarks are made available, a broader range of such data sets is necessary in order to avoid the problem of overfitting. In this context, theoretical works on axiomatization of clustering algorithms, especially axioms on clustering preserving transformations are quite a cheap way to produce labeled data sets from existing ones. However, the frequently cited axiomatic system of Kleinberg:2002, as we show in this paper, is not applicable for finite dimensional Euclidean spaces, in which many algorithms like $k$-means, operate. In particular, the so-called outer-consistency axiom fails upon making small changes in datapoint positions and inner-consistency axiom is valid only for identity transformation in general settings. Hence we propose an alternative axiomatic system, in which Kleinberg's inner consistency axiom is replaced by a centric consistency axiom and outer consistency axiom is replaced by motion consistency axiom. We demonstrate that the new system is satisfiable for a hierarchical version of $k$-means with auto-adjusted $k$, hence it is not contradictory. Additionally, as $k$-means creates convex clusters only, we demonstrate that it is possible to create a version detecting concave clusters and still the axiomatic system can be satisfied. The practical application area of such an axiomatic system may be the generation of new labeled test data from existent ones for clustering algorithm testing. %We propose the gravitational consistency as a replacement which does not have this deficiency.

Possibility results for graph clustering: A novel consistency axiom

Kleinberg introduced three natural clustering properties, or axioms, and showed they cannot be simultaneously satisfied by any clustering algorithm. We present a new clustering property, Monotonic Consistency, which avoids the well-known problematic behaviour of Kleinberg’s Consistency axiom, and the impossibility result. Namely, we describe a clustering algorithm, Morse Clustering, inspired by Morse Theory in Differential Topology, which satisfies Kleinberg’s original axioms with Consistency replaced by Monotonic Consistency. Morse clustering uncovers the underlying flow structure on a set or graph and returns a partition into trees representing basins of attraction of critical vertices. We also generalise Kleinberg’s axiomatic approach to sparse graphs, showing an impossibility result for Consistency, and a possibility result for Monotonic Consistency and Morse clustering.

Consistency and the graph Banzhaf value for communication graph games

The graph Banzhaf value was introduced and axiomatically characterized by Alonso-Meijide and Fiestras-Janeiro (2006). In this paper we propose the reduced game and consistency of the graph Banzhaf value for communication situations. By establishing the relationship between the Harsanyi dividends of a coalition in a communication situation and the reduced communication situation, we provide a new axiomatization of the graph Banzhaf value by means of the axioms of consistency and standardness.

Assessing Resolvability and Consistency in OBO Foundry Ontologies: Pilot Study.

Ontologies listed in the OBO Foundry are often regarded as reliable choices to be reused but ontology interoperability of them remains unknown. This study evaluated the resolvability of URIs and consistency of axioms in the OBO Foundry library, BFO ontology, and CIDO ontology. All had nonresolvable URIs, but the OBO library and the CIDO had additional interoperability issues regarding the use of incorrect prefixes, mixing up with ontologies, and inconsistency in the use of property. These detected issues reflected the real-world common problems that were not significant from human beings' point of view but hindered the machine-processability of ontologies. The assessment performed in this study was automated and enables scale-up against more metrics over more ontologies, which remains future work.

Issues in clustering algorithm consistency in fixed dimensional spaces. Some solutions for k-means

Kleinberg introduced an axiomatic system for clustering functions. Out of three axioms, he proposed, two (scale invariance and consistency) are concerned with data transformations that should produce the same clustering under the same clustering function. The so-called consistency axiom provides the broadest range of transformations of the data set. Kleinberg claims that one of the most popular clustering algorithms, k-means does not have the property of consistency. We challenge this claim by pointing at invalid assumptions of his proof (infinite dimensionality) and show that in one dimension in Euclidean space the k-means algorithm has the consistency property. We also prove that in higher dimensional space, k-means is, in fact, inconsistent. This result is of practical importance when choosing testbeds for implementation of clustering algorithms while it tells under which circumstances clustering after consistency transformation shall return the same clusters. Two types of remedy are proposed: gravitational consistency property and dataset consistency property which both hold for k-means and hence are suitable when developing the mentioned testbeds.

Axiomatization of the counting rule for cost-sharing with possibly redundant items.

For cost allocation problems with an existing set of indivisible public resources with heterogeneous individual needs and non-rivalry access, an axiomatization is provided for the allocation rule that proportionally charges agents for a given resource with respect to their counting liability indices. The main result we obtain holds in the class of cost allocation rules that are additive in cost and simply combines a new independence property together with the well-known axioms of consistency and independence of supplementary items.

APPLICATION OF LEARNING MATHEMATICS USING WEB BASED LEARNING METHOD

Mathematics is one branch of science that we have learned from an early age. Mathematics is the science of quantity, structure, space, and change. Mathematicians find patterns from rigorous conclusions drawn from consistent axioms and definitions, propose new conjectures, and construct truth. We often encounter a learning approach that becomes a problem in the real world as a context for learning by thinking more critically and skillfully in problem-solving and knowledge and concepts from the material to be learned. The success of learning can be seen from the results of learning. Therefore, learning methods related to technology are indispensable for practical activities with the object of study cannot be separated from theoretical learning. Learning theory makes us bored compared to the practice done directly in front of the computer with existing technology at the moment. One such technology is Web-Based Learning (WBL). Using the Web-Based Learning (WBL) method is expected to help develop its capabilities

Balanced Externalities and the Proportional Allocation of Nonseparable Contributions

In this paper, we study the implications of extending the balanced cost reduction property from queueing problems to general games. As a direct translation of the balanced cost reduction property, the axiom of balanced externalities for solutions of games, requires that the payoff of any player is equal to the total externality she inflicts on the other players with her presence. We show that this axiom and efficiency together characterize the Shapley value for 2-additive games. However, extending this axiom in a straightfoward way to general games is incompatible with efficiency. Keeping as close as possible to the idea behind balanced externalities, we weaken this axiom by requiring that every player's payoff is the same fraction of its total externality inflicted on the other players. This weakening, which we call weak balanced externalities, turns out to be compatible with efficiency. More specifically, the unique efficient solution that satisfies this weaker property is the proportional allocation of nonseparable contribution (PANSC) value, which allocates the total worth proportional to the separable costs of the players. We also provide characterizations of the PANSC value using a reduced game consistency axiom.

Dynamic decision making under ambiguity: An experimental investigation

Neoclassical economic theory assumes that whenever agents tackle dynamic decisions under ambiguity, preferences are represented by the Subjective Expected Utility (SEU) model and prior beliefs are updated according to Bayes rule, upon the arrival of partial information. Nevertheless, when one considers non-neutral ambiguity attitudes, either the axiom of dynamic consistency or of consequentialism should be relaxed. Using data from an economic experiment on dynamic choice under ambiguity, we study which of the two rationality axioms people violate, along with the question of whether this violation is part of a conscious planning strategy or not. The combination of the two, allows us to classify non-SEU subjects to three behavioural types: resolute, naïve and sophisticated. The hypothesis of Bayesian updating is rejected for more than half of the experimental population. For ambiguity non-neutral subjects, we find that the majority are sophisticated, a few are naïve and very few are resolute.

Consistent approval-based multi-winner rules

This paper is an axiomatic study of consistent approval-based committee (ABC) rules. These are multi-winner voting rules that select a committee, i.e., a fixed-size group of candidates, based on approval ballots. We introduce the class of ABC scoring rules and provide an axiomatic characterization of this class based on the consistency axiom. Building upon this result, we axiomatically characterize three important consistent multi-winner rules: Proportional Approval Voting, Multi-Winner Approval Voting and the Approval Chamberlin–Courant rule. Our results demonstrate the variety of ABC scoring rules and illustrate three different, orthogonal principles that multi-winner voting rules may represent: proportionality, diversity, and individual excellence.

Group belief

Abstract While logical formalizations of group notions of knowledge such as common and distributed knowledge have received considerable attention in the literature, most approaches being based on modal logic, group notions of belief have received much less attention. In this paper we systematically study standard notions of group belief under different assumptions about the properties of belief. In particular, we map out (lack of) preservation of belief properties against different standard definitions of group belief. It turns out that what is called group belief most often is not actually belief, i.e. does not have the properties of belief. In fact, even what is called group knowledge is sometimes not actually knowledge either. For example, under the common assumption that belief has the KD45 properties, neither common belief (does not satisfy the negative introspection axiom 5) nor distributed belief (does not satisfy the consistency axiom D) are not actually belief. There has been some confusion in the literature regarding soundness of proposed axiomatizations of logics with distributed knowledge, related to the mentioned lack of preservation. In this paper we also present detailed completeness proofs of sound and complete axiomatizations of KD45 with distributed belief, both with and without common belief.

Majority rule for profiles of arbitrary length, with an emphasis on the consistency axiom

This paper considers voting rules for two alternatives, viz. the simple majority rule of K.O. May, the majority rule with bias of Fishburn and others, and the majority rule by difference of votes of Goodin and List, and of Llamazares. These all have been characterized axiomatically for profiles of fixed length, that is, for a fixed population. The aim of this paper is to study analogs of these results in the situation where various populations are considered and disjoint populations can be combined into one population. The effect of this shift of focus is that now the domain of the rule consists of all finite nonempty sequences of votes. Young (1974) introduced the axiom of consistency, by which two populations can be combined into one as long as they agree on the output of the voting rule. We use a version of this axiom as given by Roberts (1991). Our paper can be seen as making a strong case for this simple, and natural axiom.In the simple majority rule as well as the majority rule by difference of votes, the outcome of a tie is undecided, that is, both alternatives are in the output. In the majority rule with bias, ties are broken. In our case of profiles of variable lengths there are reasons to distinguish between strong bias (a tie is always broken) and weak bias (in certain cases a tie is not yet broken). The case of weak bias fits nicely in the context of a new characterization of simple majority rule that follows from previous results of the authors.

Tangle-tree duality in abstract separation systems

We prove a general width duality theorem for combinatorial structures with well-defined notions of cohesion and separation. These might be graphs or matroids, but can be much more general or quite different. The theorem asserts a duality between the existence of high cohesion somewhere local and a global overall tree structure.We describe cohesive substructures in a unified way in the format of tangles: as orientations of low-order separations satisfying certain consistency axioms. These axioms can be expressed without reference to the underlying structure, such as a graph or matroid, but just in terms of the poset of the separations themselves. This makes it possible to identify tangles, and apply our tangle-tree duality theorem, in very diverse settings.Our result implies all the classical duality theorems for width parameters in graph minor theory, such as path-width, tree-width, branch-width or rank-width. It yields new, tangle-type, duality theorems for tree-width and path-width. It implies the existence of width parameters dual to cohesive substructures such as k-blocks, edge-tangles, or given subsets of tangles, for which no width duality theorems were previously known.Abstract separation systems can be found also in structures quite unlike graphs and matroids. For example, our theorem can be applied to image analysis by capturing the regions of an image as tangles of separations defined as natural partitions of its set of pixels. It can be applied in big data contexts by capturing clusters as tangles. It can be applied in the social sciences, e.g. by capturing as tangles the few typical mindsets of individuals found by a survey. It could also be applied in pure mathematics, e.g. to separations of compact manifolds.

Morally Monotonic Choice in Public Good Games

Decades of robust data from public good games with positive and negative externalities challenges internal consistency axioms that comprise rational choice theory. This paper reports an extension of rational choice theory that incorporates observable moral reference points. This morally monotonic choice theory is consistent with data in the literature and has idiosyncratic features that motivate new experimental designs. We report experiments on choices in public good games with positive, negative, and mixed-sign externalities, with and without non-binding quotas on extractions or minimum contributions. Data favors choices predicted by moral monotonicity over choices predicted by: (a) conventional rational choice theory; or (b) conventional reference dependent model of loss aversion.

The Equal Surplus Division Value for Cooperative Games with a Level Structure

In this paper, we investigate the equal surplus division value for cooperative games with a level structure, which is a sequence of coalition structures becoming coarser and coarser. We propose three axiomatizations of the value. Among them, the first two use different variations of the recent population solidarity axiom, and the third one invokes a special reduced game consistency axiom. Due to the existence of a level structure, our axioms impose special restrictions on the players we focus. We show that our value can be characterized with these axioms and other variations of well-known axioms, such as efficiency, standardness, and quotient game property. Besides characterizing the value, we also connect it to the recent field of ordered tree cooperative games, wherein we find that the iterative value can be viewed as a special case of our value.

Consistency, weak fairness, and the Shapley value

The Shapley value (Shapley, 1953) has been axiomatically characterized from different points of view. van den Brink (2001) proposes a characterization by means of efficiency, fairness and the null player property. In this paper, we characterize the family of single-valued solutions obtained by relaxing fairness into weak fairness. To point out the Shapley value, we impose the additional axiom of weak self consistency and strengthen the null player property into the dummy player property. Remarkably, impossibility results emerge when replacing self consistency by a large set of consistency properties.