Symmetry Axiom Research Articles

Alternative characterizations of the proportional solution for nonconvex bargaining problems with claims

Three alternative characterizations of the proportional solution defined on compact and comprehensive bargaining problems with claims are provided. Two new contraction-type and expansion-type axioms are used. Moreover, the single-valuedness axiom is dispensable if the classical symmetry axiom is imposed.

Values on regular games under Kirchhoff’s laws

The Shapley value is a central notion defining a rational way to share the total worth of a cooperative game among players. We address a general framework leading to applications to games with communication graphs, where the feasible coalitions form a poset whose all maximal chains have the same length. Considering a new way to define the symmetry among players, we propose an axiomatization of the Shapley value of these games. Borrowing ideas from electric networks theory, we show that our symmetry axiom and the efficiency axiom correspond to the two Kirchhoff’s laws in the circuit associated to the Hasse diagram of feasible coalitions.

Monotonicity and the Hirsch index

The Hirsch index is a number that synthesizes a researcher's output. It is the maximum number h such that the researcher has h papers with at least h citations each. Woeginger [Woeginger, G. J. (2008a). An axiomatic characterization of the Hirsch-index. Mathematical Social Sciences, 56(2), 224–232; Woeginger, G. J. (2008b). A symmetry axiom for scientific impact indices. Journal of Informetrics, 2(3), 298–303] characterizes the Hirsch index when indices are assumed to be integer-valued. In this note, the Hirsch index is characterized, when indices are allowed to be real-valued, by adding to Woeginger's monotonicity two axioms in a way related to the concept of monotonicity.

A System of Axioms for Hyperbolic Geometry

Three–dimensional hyperbolic geometry is characterized using axioms of order, incidence, dimension, continuity and, instead of an axiom of parallels, there is an axiom of “rigidity” and, rather than several axioms of congruence, there is one axiom of symmetry. It is claimed that this system of axioms is simpler than the system of independent axioms of Moore [15].

A symmetry axiom for scientific impact indices

We provide a new axiomatic characterization of the Hirsch-index. This new characterization is based on a simple and appealing symmetry axiom which essentially imposes that the number of citations and the number of publications should be treated in the same way and should be measured in the same scale.

Linking Strategic Interaction and Bargaining Theory: The Harsanyi-Schelling Debate on the Axiom of Symmetry

This paper analyzes the early contributions of John Harsanyi and Thomas C. Schelling to bargaining theory. In the 1950s, Harsanyi draws Nash's solution to two-person cooperative games from the bargaining model proposed by Zeuthen (1930), and Schelling proposes a multifaceted theory of conflict that, without dismissing the assumption of rational behavior, points out some of its paradoxical consequences. Harsanyi's and Schelling's contrasting views on the axiom of symmetry, as postulated by Nash (1950), are then presented. This debate explains why, although in the early 1960s two different approaches to link strategic interaction and bargaining theory were proposed, only Harsanyi's insights were fully developed later.

Sharing the surplus: An extension of the Shapley value for environments with externalities

Economic activities, both on the macro and micro level, often entail wide-spread externalities. This in turn leads to disputes regarding the compensation levels to the various parties affected. We propose a method of deciding upon the distribution of the gains (costs) of cooperation in the presence of externalities when forming the grand coalition is efficient. We show that any sharing rule satisfying efficiency, linearity, dummy player and a strong symmetry axioms can be obtained through an average game. Adding an additional axiom, we identify one unique rule satisfying these properties.

Nash Bargaining without Convexity

In this note I study Nash bargaining when the utility possibility set of the bargaining problem is not-convex. A simple variation of Nash's Symmetry axiom is all that is necessary to establish a set-valued version of Nash's solution in non-convex settings.

THE LOGIC OF IDEAS AND THE LOGIC OF THINGS: A REPLY TO CHAPPELL

Abstract: A continuation of the debate over the intelligibility, and plausibility, of Yolton's reading of Locke's account of perception. Here, the issue turns on the de‐reification of ideas and its implications for the putative axioms of symmetry and transitivity governing the identity of ideas. The issue is illustrated by what Locke says about confused ideas.

The Converse Consistency Principle in Bargaining

We investigate the implications of converse consistency in the context of bargaining. A solution is conversely consistent if, whenever, for some problem, a feasible alternative has the property that for all proper subgroups of the agents it involves, the solution chooses the restriction of the alternative to the subgroup for the associated reduced problem this subgroup faces, then the alternative should be the solution outcome for the problem. We present two alternative characterizations of the egalitarian solution based on converse consistency as well as either weak consistency or population monotonicity, in addition to other standard axioms of weak Pareto optimality, symmetry, and continuity. However, if we strengthen weak Pareto optimality to Pareto optimality, various impossibility results are obtained. On the other hand, the Nash solution is characterized on the basis of a weaker version of converse consistency, which applies its hypotheses only to the problem whose solution outcome is smooth. Journal of Economic Literature Classification Numbers: C71, C78, D70.

Freiling's axioms of symmetry in a general setting and some applications

We formulate C. Freiling's axioms of symmetry for general second-order structures with respect to a certain ideal of small sets contained in them and find several equivalent formulations of the principles. Then we focus on particular models, namely saturated and recursively saturated ones, and show that they are symmetric with respect to appropriate classes of small sets when their second-order part consists of definable sets. Some asymmetric models are also exhibited as well as partial asymmetric ones constructed by forcing.

Testing the assumptions of rationality, continuity and symmetry when applying discrete choice experiments in health care

In the absence of revealed preference behaviour, economists use experimental techniques to estimate welfare changes. Such an approach relies on a number of assumptions concerning individual behaviour. This paper uses a discrete choice experiment to test three of these assumptions – rationality, continuity and symmetry. The experiment was carried out with users of a rheumatology clinic in the Grampian area of Scotland and was concerned with preferences for a specialist-nurse-led clinic. Two tests of ‘rationality’ were included in the experiment. Tests were carried out to see if respondents always chose on the basis of their preferred staffing, suggesting discontinuities in the utility function. The axiom of symmetry was tested using a split sample design, with respondents divided into two groups. Each group received a different questionnaire that varied with respect to the order of the choices. Over 30% of respondents provided at least one ‘irrational’ response. Such respondents did not differ significantly in their characteristics from ‘rational’ responses, suggesting that utility estimates would not be biased if this group were excluded from the analysis. Seventeen per cent of respondents showed signs of having non-compensatory utility functions. Evidence was found to support the axiom of symmetry. Future work should explore the axioms of rationality and continuity in more detail.

Embeddings of propositional monomodal logics

The aim of this paper is to investigate the expressibility of classical propositional monomodal logics. To this end, a notion of embedding of one logic into another is introduced, which is a translation preserving theoremhood. Each translation trF is induced by a formula F(p) of one variable p; it respects boolean connectives and translates a formula of a form □A into F(trF(A)). This notion enables to measure the expressibility of a logic by a (finite or infinite) number of logics embeddable into it. This measure is calculated here for a large family of modal logics including K, K4, KB, K5, GL, GL.3, T, S4, B, S5, Grz, Grz.3, and provability logics. It is also shown that some of these logics (e.g., all normal logics containing the symmetry axiom except for the logics Triv, Ver, and the intersection of these two) are not embeddable into some others (e.g., K, K4, K5, GL, GL.3, T, S4, Grz, Grz.3, and most of provability logics).

Notions of symmetry in set theory with classes

We adapt C. Freiling's axioms of symmetry (J. Symbolic Logic 51 (1986) 190–200) to models of set theory with classes by identifying small classes with sets getting thus a sequence of principles A n , for n⩾2, of increasing strength. Several equivalents of A 2 are given. A 2 is incompatible both with the foundation axiom and the antifoundation axioms AFA ∼ considered in Aczel (Non Well Founded Sets, CSLI Lecture Notes, vol. 14, Stanford University, 1988). A hierarchy of symmetry degrees of preorderings (and of classes carrying such preorderings) is introduced and compared with A n . Models are presented in which this hierarchy is strict. The main result of the paper is that (modulo some choice principles) a class X satisfies ¬ A n iff it has symmetry degree n−2.

‘Hydraulic’ rationing

The problem of distributing a single homogeneous divisible good among a variable set of agents, or the ‘rationing problem,’ is analyzed. Examples of rationing include bankruptcy, taxation, claims settlement, cost allocation, surplus sharing, and social choice problems. Agents are described by their personal characteristics, or types . A type may be an agent’s utility function, preference ordering, claim to an estate, financial record, etc. A rule of division that can be represented as a system of connected vessels is called hydraulic . For separable spaces of types and continuous rules, this property is equivalent to obeying the fundamental axioms of symmetry and consistency . A universal criterion is presented for deciding when a bilateral rule has a consistent extension.

On an axiomatization of the quasi-arithmetic mean values without the symmetry axiom

Kolmogoroff and Nagumo proved that the quasi-arithmetic means correspond exactly to the decomposable sequences of continuous, symmetric, strictly increasing in each variable and reflexive functions. We replace decomposability and symmetry in this characterization by a generalization of the decomposability.

Maximal symmetry and the Nash solution

The Nash Bargaining Solution is characterised by using the new axiom of Maximal Symmetry in place of Nash's Independence of Irrelevant Alternatives and Symmetry. This axiom expresses the idea that a fair arbitrator should treat symmetric alternatives in the same way, subject to the feasibility constraint. An advantage of the proposed characterisations is that they are valid on a wide set of domains, in particular domains including, or consisting of, non-convex problems.

Mew mathematical properties of the semivalues of cooperative tu games*

The Semivalues were introduced by Dubey, Neiman and Weber (1981), as the values of TU games satisfying a set of axioms, precisely:linearity, symmetry, monotomcity and projection axioms. In this paper, a potential approach is used to prove a characterization of Semivalues as the unique values satisfying a new type of consistency relative to a new implicitly defined reduced game of Hart Mas Colell type and weighted standardness for all two person games. The definition of the reduced game requires some combinatorial results on the auxiliary game of the given game. As a byproduct, we derive from these results two other combinatorial properties of the Semivalues: (i) any Semivalue is the Shapley value of the auxiliary game of the given game, and (ii) any Semivalue satisfies the fairness principle introduced by Myerson (1977) as the principle of balanced contributions

Multiple-Issue Bargaining and Axiomatic Solutions

We study two-person, multiple-issue bargaining problems and identify four procedures by which the bargaining may take place. Drawing on some logic from non-cooperative game theory, we propose axioms which relate the outcomes of the procedures. We also promote a weak monotonicity axiom on solutions, called issue-by-issue monotonicity, which is geared toward multiple-issue bargaining. Our main result concerns the relationship between a sequential bargaining procedure — with the rule that agreements are implemented only after all issues are resolved — and global bargaining (in which all issues are negotiated simultaneously). If a bargaining solution predicts the same outcome with these two procedures, then we say that it satisfiesagenda independence. We prove that a solution satisfies axioms of efficiency, symmetry, scale invariance, issue-by-issue monotonicity, and agenda independence if and only if it is the Nash solution. This result provides new intuition for Nash's independence of irrelevant alternatives axiom. Among other results, we show that a solution is invariant to all four of the procedures and satisfies efficiency and symmetry if and only if it is the utilitarian solution with equal weights. We comment on the results of other authors who address multiple-issue bargaining.

On axiomatizations of the weighted Shapley values

The family of weighted Shapley values for cooperative n-person transferable utility games is studied. We assume first that the weights of the players are given exogenously and provide two axiomatic characterizations of the corresponding weighted Shapley value. Our first characterization is based on the classical axioms determining the Shapley value with the symmetry axiom replaced by a new postulate called the ω-mutual dependence. In our second axiomatization we use among other things the strong monotonicity property of Young (1985, Int. J. Game Theory 14, 65–72). Finally, we give a new axiomatic characterization of the family of all weighted Shapley values. Journal of Economic Literature Classification Number: C71, D46.