Axiomatic Characterization Research Articles

The priority value for cooperative games with a priority structure

We study cooperative games with a priority structure modeled by a poset on the agent set. We introduce the Priority value, which splits the Harsanyi dividend of each coalition among the set of its members over which no other coalition member has priority. This allocation shares many desirable properties with the classical Shapley value: it is efficient, additive and satisfies the null agent axiom. We provide two axiomatic characterizations of the Priority value which invoke both classical axioms and new axioms describing the effects of the priority structure on the payoff allocation. Finally, in the special case where agents are ranked by level, a link between the Priority value, the weighted Shapley values and the Owen-type values can be drawn.

Axiomatic Characterizations of a Proportional Influence Measure for Sequential Projects with Imperfect Reliability

We define and axiomatically characterize a new proportional influence measure for sequential projects with imperfect reliability. We consider a model in which a finite set of players aims to complete a project, consisting of a finite number of tasks, which can only be carried out by certain specific players. Moreover, we assume the players to be imperfectly reliable, i.e., players are not guaranteed to carry out a task successfully. To determine which players are most important for the completion of a project, we use a proportional influence measure. This paper provides two characterizations of this influence measure. The most prominent property in the first characterization is task decomposability. This property describes the relationship between the influence measure of a project and the measures of influence one would obtain if one divides the tasks of the project over multiple independent smaller projects. Invariance under replacement is the most prominent property of the second characterization. If, in a certain task group, a specific player is replaced by a new player who was not in the original player set, this property states that this should have no effect on the allocated measure of influence of any other original player.

Assessment of the influence of features on a classification problem: An application to COVID-19 patients

This paper deals with an important subject in classification problems addressed by machine learning techniques: the evaluation of the influence of each of the features on the classification of individuals. Specifically, a measure of that influence is introduced using the Shapley value of cooperative games. In addition, an axiomatic characterisation of the proposed measure is provided based on properties of efficiency and balanced contributions. Furthermore, some experiments have been designed in order to validate the appropriate performance of such measure. Finally, the methodology introduced is applied to a sample of COVID-19 patients to study the influence of certain demographic or risk factors on various events of interest related to the evolution of the disease.

Axiomatic foundations of the core for games in effectiveness form

We provide an axiomatic characterization of the core of games in effectiveness form. We point out that the core, whenever it applies to appropriate classes of these games, coincides with a wide variety of prominent stability concepts in social choice and game theory, such as the Condorcet winner, the Nash equilibrium, pairwise stability, and stable matchings, among others. Our characterization of the core invokes the axioms of weak nonemptiness, coalitional unanimity, and Maskin monotonicity together with a principle of independence of irrelevant states, and uses in its proof a holdover property echoing the conventional ancestor property. Taking special cases of this general characterization of the core, we derive new characterizations of the previously mentioned stability concepts.

Scenario-based risk evaluation

Risk measures such as expected shortfall (ES) and value-at-risk (VaR) have been prominent in banking regulation and financial risk management. Motivated by practical considerations in the assessment and management of risks, including tractability, scenario relevance and robustness, we consider theoretical properties of scenario-based risk evaluation. We establish axiomatic characterisations of scenario-based risk measures that are comonotonic-additive or coherent, and we obtain a novel ES-based representation result. We propose several novel scenario-based risk measures, including various versions of Max-ES and Max-VaR, and study their properties. The theory is illustrated with financial data examples.

Axiomatic characterizations of [formula omitted]-valued rough sets using a single axiom

Fuzzy rough approximation operators are the underlying concepts in fuzzy rough set theory. There are at least two approaches to develop these primary concepts, i.e., the constructive approach and the axiomatic approach. Single axiomatic characterizations of fuzzy rough approximation operators have got tons of attention. In this paper, considering L being a GL-quantale, we will develop the theory of L-valued rough sets with an L-set as the basic universe of defining L-valued rough approximation operators. Adopting the idea of single axiomatic characterizations of fuzzy rough sets, we will present the axiomatic characterizations of L-valued upper and lower rough approximation operators on an L-set with respect to reflexive, symmetric, transitive L-valued relations on an L-set as well as their compositions. Choosing an L-set as the universe will break the rules that adopting Zadeh’s fuzzy sets as the universe. By these results, we will further provide a new framework of axiomatic research of fuzzy rough set theory.

Single axiomatic characterization of a hesitant fuzzy generalization of rough approximation operators

Single axiomatic characterization of a hesitant fuzzy generalization of rough approximation operators

Revealed preference analysis and bounded rationality

Abstract The principle of revealed preference is the backbone of structural empirical work on consumer demand. It focuses on what we can learn about the processes by which economic agents make decisions, using observed choices and minimal auxiliary assumptions. Classical revealed preference methods assume that choices and preferences are stable and consistent, but many studies have found violations of these assumptions both in consumer data and experimental settings. Recent research effort extends far beyond the axiomatic characterization of neoclassical choice models. New behavioural theories explain these violations in a theoretically founded way, considering data consistency and preference recoverability for a wide class of behavioural models. This article reviews some of the themes emerging from the recent literature.

Risk aversion for losses and the Nash bargaining solution

We call a decision maker risk averse for losses if that decision maker is risk averse with respect to lotteries having alternatives below a given reference alternative in their support. A two-person bargaining solution is called invariant under risk aversion for losses if the assigned outcome does not change after correcting for risk aversion for losses with this outcome as pair of reference levels, provided that the disagreement point only changes proportionally. We present an axiomatic characterization of the Nash bargaining solution based on this condition, and we also provide a decision-theoretic characterization of the concept of risk aversion for losses.

The Efficient Proportional Myerson Values for Hypergraph Games

This study deals with a class of efficient extensions of Myerson value for games with hypergraph communication situations in which the surplus is allocated proportionally. We introduce w -fairness of surplus and provide axiomatic characterizations of the new allocation rule. Furthermore, we give an example of research fund distribution amongst researchers, compare the numerical results with several values, and realize other efficient extensions of Myerson value can be obtained depending on the different measure function w on the hypergraph.

L-Fuzzy Rough Approximation Operators Based on Co-Implication and Their (Single) Axiomatic Characterizations

For L a complete co-residuated lattice and R an L-fuzzy relation, an L-fuzzy upper approximation operator based on co-implication adjoint with L is constructed and discussed. It is proved that, when L is regular, the new approximation operator is the dual operator of the Qiao–Hu L-fuzzy lower approximation operator defined in 2018. Then, the new approximation operator is characterized by using an axiom set (in particular, by single axiom). Furthermore, the L-fuzzy upper approximation operators generated by serial, symmetric, reflexive, mediate, transitive, and Euclidean L-fuzzy relations and their compositions are characterize through axiom set (single axiom), respectively.

A general framework for the characterization of (S,N)-implications with a non-continuous negation based on completions of t-conorms

The characterization of (S,N)-implications when N is a non-continuous negation has remained one of the most significant open problems in fuzzy logic for the last decades. This paper constitutes the first progress in this topic. Namely, a general characterization of this family of fuzzy implication functions is presented, in which the central property is the existence of a completion of a binary function defined on a certain subregion of [0,1]2 to a t-conorm. In this paper, the dual problem of finding a completion of a binary function defined on a subregion of [0,1]2 to a continuous t-norm is studied and solved for the minimum and a cancellative function. These results are the basis for the novel axiomatic characterizations of (S,N)-implications in the case when N has one point of discontinuity and S is equal to the maximum t-conorm in a certain subregion of [0,1]2 or a strict t-conorm.

Weak null, necessary defender and necessary detractor players: characterizations of the Banzhaf and the Shapley bisemivalues

We focus on bicooperative games, a variation of the classic cooperative games and, in particular, on the Banzhaf and the Shapley bisemivalues defined on these games. We consider three special classes of players: weak null, necessary defender and necessary detractor players. By introducing new properties related to this kind of players, we provide new axiomatic characterizations of the Banzhaf and the Shapley bisemivalues giving, in both cases, a set of independent properties that univocally determine them. We also provide a computational procedure to calculate the allocations given by the Shapley bisemivalue in terms of the generalized multilinear extension of the game.

Compromising between the proportional and equal division values

We introduce a family of values for TU-games that offers a compromise between the proportional and equal division values. Each value, called an α-mollified value, is obtained in two steps. First, linear functions are defined that associate a real number to every TU-game. Second, the weight assigned by this function is used to weigh proportionality and equality principles in allocating the worth of the grand coalition. We provide an axiomatic characterization of this family, and show that this family contains the affine combinations of the equal division value and the equal surplus division value as the only linear values. Further, we identify the proportional division value and the affine combinations of the equal division value and the equal surplus division value as those members of this family, that satisfy projection consistency.

Monotonicity in sharing the revenues from broadcasting sports leagues

We explore the implications of the principle of monotonicity in the problem of sharing the revenues from broadcasting sports leagues. We formalize different forms of this principle as several axioms for sharing rules in this setting. We show that, combined with two other basic axioms (equal treatment of equals and additivity), they provide axiomatic characterizations of focal rules for this problem, as well as families of rules compromising among them. These results highlight the normative appeal of the (focal) equal-split rule.

On the axiomatic characterization of the coalitional multinomial probabilistic values

The coalitional multinomial probabilistic values extend the notion of multinomial probabilistic value to games with a coalition structure, in such a way that they generalize the symmetric coalitional binomial semivalues and link and combine the Shapley value and the multinomial probabilistic values. By considering the property of balanced contributions within unions, a new axiomatic characterization is stated for each one of these coalitional values, provided that it is defined by a positive tendency profile, by means of a set of logically independent properties that univocally determine the value. Two applications are also shown: (a) to the Madrid Assembly in Legislature 2015–2019 and (b) to the Parliament of Andalucía in Legislature 2018–2022.

Boolean-Valued Set-Theoretic Systems: General Formalism and Basic Technique

This article is devoted to the study of the Boolean-valued universe as an algebraic system. We start with the logical backgrounds of the notion and present the formalism of extending the syntax of Boolean truth values by the use of definable symbols, internal classes, outer terms and external Boolean-valued classes. Next, we enrich the collection of Boolean-valued research tools with the technique of partial elements and the corresponding joins, mixings and ascents. Passing on to the set-theoretic signature, we prove that bounded formulas are absolute for transitive Boolean-valued subsystems. We also introduce and study intensional, predicative, cyclic and regular Boolean-valued systems, examine the maximum principle, and analyze its relationship with the ascent and mixing principles. The main applications relate to the universe over an arbitrary extensional Boolean-valued system. A close interrelation is established between such a universe and the intensional hierarchy. We prove the existence and uniqueness of the Boolean-valued universe up to a unique isomorphism and show that the conditions in the corresponding axiomatic characterization are logically independent. We also describe the structure of the universe by means of several cumulative hierarchies. Another application, based on the quantifier hierarchy of formulas, improves the transfer principle for the canonical embedding in the Boolean-valued universe.

L-fuzzy generalized neighborhood system operator-based L-fuzzy approximation operators

In this paper, we investigate the axiomatic characterizations on L-fuzzy approximation operators based on L-fuzzy generalized neighborhood system operators. In detail, we use single axioms to characterize each kind of L-fuzzy approximation operators corresponding to serial, reflexive, weak-unary and weak-transitive L-fuzzy generalized neighborhood system operators as well as any of their compositions. Moreover, we give the necessary and sufficient condition for that L-interior (L-closure) operators that is the same as L-fuzzy lower (upper) approximation operators constructed by the reflexive and weak-transitive L-fuzzy generalized neighborhood system operators. Finally, we discuss the relationship between L-fuzzy approximation operators based on L-fuzzy generalized neighborhood system operators and L-topologies.

The Folk Rule for Minimum Cost Spanning Tree Problems with Multiple Sources

In this paper, we introduce minimum cost spanning tree problems with multiple sources. This new setting is an extension of the classical model where there is a single source. We extend several definitions of the folk rule, the most prominent rule in the classical model, to this new context: first as the Shapley value of the irreducible game; second as an obligation rule; third as a partition rule and finally through a cone-wise decomposition. We prove that all the definitions provide the same cost allocation and present two axiomatic characterizations.

Costly information and random choice

Consider an agent who decides whether to employ an information structure to learn about a payoff-relevant state of the world before making a decision. Information is costly either because (1) the agent has to wait some time until the information structure becomes available and is impatient, or because (2) she has to pay a fixed and menu-independent cost to use the information structure. For each menu of options the analyst observes random choice from the menu and whether the agent acquires the information. We give an axiomatic characterization of when this random choice is consistent with either of the cases (1) or (2). We identify the information structure the agent can employ, as well as its costs. We also discuss implications of our approach for sequential sampling.