Ordinal Invariance Research Articles

An Axiomatic Characterization of the Leximin Choice Rule

This study discusses the characterization of the lexicographic maximin (leximin) choice rule using an axiom related to the Lorenz criterion, which states that a utility vector that Lorentz dominates a solution vector should not be Pareto dominated by any feasible vector, named Pareto Undominatedness of Lorenz-superior Distribution (PULSD). PULSD in itself does not imply that a solution vector of the choice rule is Lorenz undominated by any utility vectors in the feasible set. Therefore, we may say that the PULSD’s requirement for inequality aversion is weak. Indeed, PULSD is consistent with the utilitarian choice rule, which is viewed as indifferent to equity. We show that the leximin choice rule is characterized by PULSD, combined with the Suppes-Sen optimality and the common ordinal invariance axiom.

A non-cooperative approach to the ordinal Shapley–Shubik rule

In bargaining problems, a rule satisfies ordinal invariance if it does not depend on order-preserving transformations of the agents’ utilities. In this paper, a non-cooperative game for three agents, based on bilateral offers, is presented. The ordinal Shapley–Shubik rule arises in subgame perfect equilibrium as the agents have more time to reach an agreement.

Screening-Off and Causal Incompleteness: A No-Go Theorem

We begin by considering two principles, each having the form causal completeness ergo screening-off. The first concerns a common cause of two or more effects; the second describes an intermediate link in a causal chain. They are logically independent of each other, each is independent of Reichenbach's principle of the common cause, and each is a consequence of the causal Markov condition. Simple examples show that causal incompleteness means that screening-off may fail to obtain. We derive a stronger result: in a rather general setting, if the composite cause C1 & C2 & … & Cn screens-off one event from another, then each of the n component causes C1, C2, …, Cn must fail to screen-off. The idea that a cause may be ordinally invariant in its impact on different effects is defined; it plays an important role in establishing this no-go theorem. Along the way, we describe how composite and component causes can all screen-off when ordinal invariance fails. We argue that this theorem is relevant to assessing the plausibility of the two screening-off principles. The discovery of incomplete causes that screen-off is not evidence that causal completeness must engender screening-off. Formal and epistemic analogies between screening-off and determinism are discussed. 1 Introduction 2 Influence and Non-degeneracy Conditions 3 A No-Go Theorem for Two Dichotomous Causes and its Limitations 4 A More General No-Go Theorem: Allowing Several Causes, Possibly Non-dichotomous 5 Examples Illustrating Corollary 3 and Theorem 5a 6 Determinism and Screening-Off: Disanalogy and Analogy Appendix

Nash bargaining in ordinal environments

We analyze the implications of Nash’s (Econometrica 18:155–162, 1950) axioms in ordinal bargaining environments; there, the scale invariance axiom needs to be strenghtened to take into account all order-preserving transformations of the agents’ utilities. This axiom, called ordinal invariance, is a very demanding one. For two-agents, it is violated by every strongly individually rational bargaining rule. In general, no ordinally invariant bargaining rule satisfies the other three axioms of Nash. Parallel to Roth (J Econ Theory 16:247–251, 1977), we introduce a weaker independence of irrelevant alternatives (IIA) axiom that we argue is better suited for ordinally invariant bargaining rules. We show that the three-agent Shapley–Shubik bargaining rule uniquely satisfies ordinal invariance, Pareto optimality, symmetry, and this weaker IIA axiom. We also analyze the implications of other independence axioms.

The ordinal egalitarian bargaining solution for finite choice sets

Rubinstein et al. (Econometrica 60:1171–1186, 1992) introduced the Ordinal Nash Bargaining Solution. They prove that Pareto optimality, ordinal invariance, ordinal symmetry, and IIA characterize this solution. A feature of their work is that attention is restricted to a domain of social choice problems with an infinite set of basic allocations. We introduce an alternative approach to solving finite social choice problems using a new notion called the Ordinal Egalitarian (OE) bargaining solution. This suggests the middle ranked allocation (or a lottery over the two middle ranked allocations) of the Pareto set as an outcome. We show that the OE solution is characterized by weak credible optimality, ordinal symmetry and independence of redundant alternatives. We conclude by arguing that what allows us to make progress on this problem is that with finite choice sets, the counting metric is a natural and fully ordinal way to measure gains and losses to agents seeking to solve bargaining problems.

Rawlsian maximin, Dutch books, and non-additive expected utility

The utilitarian and Rawlsian principles for choosing between income distributions in society are usually considered as alternatives and to some degree opposite approaches to the problem of social ranking. However, if the utilitarian principle is considered in the broader perspective of non-additive probabilities, contradictions disappear and the Rawlsian maximin principle appears as a special case of generalized expected utility maximization. In this paper we consider a set of axioms for a preference relation defined on a set of n-vectors, describing the distribution of welfare among a population of n individuals, which gives a utility representation as expected utility with respect to a suitable non-additive measure as in the rank-dependent utility models. The specific approach taken in the present paper consists in (1) relating the notion of Dutch books, which has proved useful in other applications, to the problem of ranking income distribution, and (2) using ordinal invariance to obtain a characterization in the form of a so-called k-Rawlsian social utility function, which reduces to the standard Rawlsian function by way of the Gärdenfors principle.

Egalitarianism in ordinal bargaining: the Shapley–Shubik rule

A bargaining rule is ordinally invariant if its solutions are independent of which utility functions are chosen to represent the agents' preferences. For two agents, only dictatorial bargaining rules satisfy this property (Shapley, L., La Décision: Agrégation et Dynamique des Ordres de Préférence, Editions du CNRS (1969) 251). For three agents, we construct a “normalized subclass” of problems through which an infinite variety of such rules can be defined. We then analyze the implications of various properties on these rules. We show that a class of monotone path rules uniquely satisfy ordinal invariance, Pareto optimality, and “monotonicity” and that the Shapley–Shubik rule is the only symmetric member of this class. We also show that the only ordinal rules to satisfy a stronger monotonicity property are the dictatorial ones.

Orderings of monomial ideals

We study the set of monomial ideals in a polynomial ring as an ordered set, with the ordering given by reverse inclusion. We give a short proof of the fact that every antichain of monomial ideals is finite. Then we investigate ordinal invariants for the complexity of this ordered set. In particular, we give an interpretation of the height function in terms of the Hilbert-Samuel polynomial, and we compute bounds on the maximal order type. 2000Mathematics Subject Classification. Primary 06A06; Secondary 13D40, 13P10

Ordinal invariance in multicoalitional bargaining

A multicoalitional bargaining problem is a non-transferable utility game and for each coalition, a bargaining rule. We look for ordinally invariant solutions to such problems and discover a subrule of Bennett's (1997, Games Econ. Behav. 19, 151–179) that satisfies the property. On a subclass of problems that is closely related to standard bargaining problems and allocation problems with majority decision-making, the two rules coincide. Therefore, Bennett solutions to such problems are immune to misrepresentation of cardinal utility information. We also show that Shapley–Shubik solution to any bargaining problem is the limit of a sequence of unique Bennett solutions to associated multicoalitional problems.

Qualitative decision theory with preference relations and comparative uncertainty: An axiomatic approach

This paper investigates a purely qualitative approach to decision making under uncertainty. Since the pioneering work of Savage, most models of decision under uncertainty rely on a numerical representation where utility and uncertainty are commensurate. Giving up this tradition, we relax this assumption and introduce an axiom of ordinal invariance requiring that the Decision Maker's preference between two acts only depends on the relative position of their consequences for each state. Within this qualitative framework, we determine the only possible form of the corresponding decision rule. Then assuming the transitivity of the strict preference, the underlying partial confidence relations are those at work in non-monotonic inference and thus satisfy one of the main properties of possibility theory. The satisfaction of additional postulates of unanimity and anonymity enforces the use of a necessity measure, unique up to a monotonic transformation, for encoding the relative likelihood of events.

Qualitative decision theory

This paper investigates to what extent a purely symbolic approach to decision making under uncertainty is possible, in the scope of artificial intelligence. Contrary to classical approaches to decision theory, we try to rank acts without resorting to any numerical representation of utility or uncertainty, and without using any scale on which both uncertainty and preference could be mapped. Our approach is a variant of Savage's where the setting is finite, and the strict preference on acts is a partial order. It is shown that although many axioms of Savage theory are preserved and despite the intuitive appeal of the ordinal method for constructing a preference over acts, the approach is inconsistent with a probabilistic representation of uncertainty. The latter leads to the kind of paradoxes encountered in the theory of voting. It is shown that the assumption of ordinal invariance enforces a qualitative decision procedure that presupposes a comparative possibility representation of uncertainty, originally due to Lewis, and usual in nonmonotonic reasoning. Our axiomatic investigation thus provides decision-theoretic foundations to the preferential inference of Lehmann and colleagues. However, the obtained decision rules are sometimes either not very decisive or may lead to overconfident decisions, although their basic principles look sound. This paper points out some limitations of purely ordinal approaches to Savage-like decision making under uncertainty, in perfect analogy with similar difficulties in voting theory.

Universal Bounds and Monotonicity of Ordinal Invariants

The tightness of a topological space provides a universal bound for all ordinal invariants determined by weak topologies generated by collections of subsets. All ordinal invariants are monotonic decreasing for pseudo-open mappings.

Universal bounds and monotonicity of ordinal invariants

Universal bounds and monotonicity of ordinal invariants

Ordinal invariants in topology

Ordinal invariants in topology

On open mappings, closed mappings, and ordinal invariants

In this brief note new characterizations of open mappings and closed mappings are presented which involve the compact order of the domain. The paper answers some of the questions which resulted from the study of the ordinal monotonicity of pseudo-open mappings and thus constitutes an extension of that work. Another characterization of open mappings is introduced.

Cardinal functions for 𝑘-spaces

In this paper, four cardinal functions are defined on the class of k-spaces. Some of the relationships between these cardinal functions are studied. Characterizations of various k-spaces are presented in terms of the existence of these cardinal functions. A bound for the ordinal invariant κ \kappa of Arhangelskii and Franklin is established in terms of the tightness of the space. Examples are presented which exhibit the interaction between these cardinal invariants and the ordinal invariants of Arhangelskii and Franklin.

Ordinal invariants in topology I. on two questions of Arhangel'skiǐ and Franklin

Group invariants such as homotopy groups, and cardinal invariants such as weights and density character are very often used in the problem of classifying topological spaces. Some interesting and useful ordinal invariants introduced and investigated in the recent past have proved equally significant in this direction. The k-order and sequential order are two of them. In this paper, two natural open questions concerning them are answered fully. The main result of the paper is the theorem which assetts, for each ordinal α, the existence of a nice k-space of k-order α.

Modules and rings in the Cayley algebra

We derive all integral Z-modules and rings (in the Cayley algebra) which contain a module with the norm-form y 0 2 + … + y 7 2. The norm-forms of these modules comprise eight classes of quadratic forms each of which is uniquely characterized by the values of certain of its ordinal invariants.