Complete Preorder Research Articles

On the computation of median linear orders, of median complete preorders and of median weak orders

Given a finite set X and a collection Π, called a profile, of binary relations defined on X (which can be linear orders, complete preorders, any relations, and so on), a relation R is said to be median if it minimizes the total number of disagreements with respect to Π. In the context of voting theory, X can be considered as a set of candidates and the relations of Π as the preferences of voters, while a median relation can be adopted as the collective preference with respect to the voters’ opinions. If the relations of Π are tournaments (which includes linear orders), then there always exists a median complete preorder (i.e. a median complete and transitive relation) which is in fact a linear order. Moreover, if there is no tie when aggregating the tournaments of Π, then all the median complete preorders are linear orders. We show the same when the median is assumed to be a weak order (a weak order being the asymmetric part of a complete preorder). We then deduce from this that the computation of a median complete preorder or of a median weak order of a profile Π of m linear orders is NP-hard for any even m greater than or equal to 4 or for odd m large enough with respect to |X| (about |X|2). We then sharpen these complexity results when coping with other kinds of profiles Π for odd values of m. In particular, when the relations of Π and the median relation are complete preorders, we obtain the same results for the NP-hardness of Kemeny’s problem.

A Short Note on the Higher Level Version of the Krull–Baer Theorem

AbstractKlep and Velušček generalized the Krull–Baer theorem for higher level preorderings to the non-commutative setting. A n-real valuation v on a skew field D induces a group homomorphism . A section of is a crucial ingredient of the construction of a complete preordering on the base field D such that its projection on the residue skew field kv equals the given level 1 ordering on kv. In the article we give a proof of the existence of the section of , which was left as an open problem by Klep and Velušček, and thus complete the generalization of the Krull–Baer theorem for preorderings.

A comprehensive framework for project selection problem under uncertainty and real-world constraints

This paper proposes a comprehensive framework for project selection problem under uncertainty and subject to real-world constraints, like segmentation, logical, and budget constraints. The framework consists of two main phases. In the first phase, the candidate projects are ranked considering the uncertainty, through a Monte Carlo simulation linked to a multi-criteria approach. In the second phase, the overall complete preorder of the projects in different iterations is first determined and then used in another Monte Carlo simulation linked to an integer programming module in order to effectively drive the final portfolio selection while satisfying the budget, segmentation and other logical constraints. The proposed framework is implemented in a case study to show its usefulness and applicability in practice. Finally, a comparison is carried out between the proposed approach and its deterministic counterpart and the corresponding results are discussed.

On complete fuzzy preorders and their characterizations

In the context of crisp or classical relations, one may find several alternative characterizations of the concept of a total preorder. In this contribution, we first discuss the way of translating those characterizations to the framework of fuzzy relations. Those new properties depend on t-norms. We focus on two important families of t-norms, namely those that do not admit zero divisors and those that are rotation invariant. For these families, we study whether or not the properties shown for fuzzy relations lead to characterizations of complete fuzzy preorders. Special attention is paid to the minimum operator, which shows the best behaviour in preserving most of the characterizations known for crisp relations.

Complete solution of the integrability problem for homothetic demand functions

For any Walrasian demand function, the Strong Axiom implies (and is implied by) rationalizability by a complete preorder. However, these equivalent conditions do not ensure the existence of a continuous utility function or complete preorder giving raise to the primitive demand. We here propose a self-contained proof of a related fact: if the demand is homothetic and continuous, the Strong Axiom characterizes the existence of a continuous and homogeneous of degree one subjective utility function (or a continuous and homothetic complete preorder) representing the demand. Our contruction depends upon standard tools and overturns the need for ad-hoc axioms that were used in prior published literature on the topic.

The Computational Complexity of Rationalizing Behavior

We study the computational complexity of rationalizing choice behavior. We do so by analyzing two polar cases, and a number of intermediate ones. In our most structured case, that is where choice behavior is defined in universal choice domains and satisfies the “weak axiom of revealed preference,” finding the complete preorder rationalizing choice behavior is a simple matter. In the polar case, where no restriction whatsoever is imposed, either on choice behavior or on choice domain, finding a collection of complete preorders that rationalizes behavior turns out to be intractable. We also show that the task of finding the rationalizing complete preorders is equivalent to a graph problem. This allows the search for existing algorithms in the graph theory literature, for the rationalization of choice.

GMPL Logic of Kinds of Preferences

针对缺乏多类型偏好共存的偏好逻辑系统的现状,MPL(logic of many kinds of preference)被构造为一种能够表示和推理四类型偏好的偏好逻辑,但是MPL的语义基于全前序偏好结构,因而不能表示不完全偏好.为此,提出了偏好逻辑GMPL(a generalized edition of MPL).此外,通过常见逻辑偏好的GMPL重写表明GMPL较强的表达能力和实际应用前景,并提出一种将GMPL的SAT问题归结为命题逻辑的SAT问题的方法.;Because of the absence of a whole logic to represent and reason various kinds of preferences, MPL (logic of many kinds of preference) is presently constructed to fill the gap. But, the semantics of MPL is based on the complete pre-order, so incomplete preferences cannot be expressed in it. In this paper, GMPL (a generalized edition of MPL) is introduced to supply the gap. In addition, the expressive power of GMPL is showed by rewriting several familiar logical preferences. Moreover, a decision procedure is introduced to reduce SAT problem of GMPL into that of propositional logic.

Representing interval orders by weighted bases: Some complexity results

This paper is centered on the notion of interval order as a model for preferences. We introduce a family of representation languages for such orders, parameterized by a scale and an aggregation function. We show how interval orders can be represented by elements of those languages, called weighted bases. We identify the complexity of the main decision problems to be considered for exploiting such representations of interval orders (including the comparison problems and the non-dominance problem). We also show that our representation of interval orders based on weighted bases encompasses the penalty-based representation of complete preorders as a specific case.

Deriving preference order of post-mining land-uses through MLSA framework: application of an outranking technique

This study intends to take advantage of a previously developed framework for mined land suitability analysis (MLSA) consisted of economical, social, technical and mine site factors to achieve a partial and also a complete pre-order of feasible post-mining land-uses. Analysis by an outranking multi-attribute decision-making (MADM) technique, called PROMETHEE (preference ranking organization method for enrichment evaluation), was taken into consideration because of its clear advantages on the field of MLSA as compared with MADM ranking techniques. Application of the proposed approach on a mined land can be completed through some successive steps. First, performance of the MLSA attributes is scored locally by each individual decision maker (DM). Then the assigned performance scores are normalized and the deviation amplitudes of non-dominated alternatives are calculated. Weights of the attributes are calculated by another MADM technique namely, analytical hierarchy process (AHP) in a separate procedure. Using the Gaussian preference function beside the weights, the preference indexes of the land-use alternatives are obtained. Calculation of the outgoing and entering flows of the alternatives and one by one comparison of these values will lead to partial pre-order of them and calculation of the net flows, will lead to a ranked preference for each land-use. At the final step, utilizing the PROMETHEE group decision support system which incorporates judgments of all the DMs, a consensual ranking can be derived. In this paper, preference order of post-mining land-uses for a hypothetical mined land has been derived according to judgments of one DM to reveal applicability of the proposed approach.

NP-hardness results for the aggregation of linear orders into median orders

Given a collection Π of individual preferences defined on a same finite set of candidates, we consider the problem of aggregating them into a collective preference minimizing the number of disagreements with respect to Π and verifying some structural properties. We study the complexity of this problem when the individual preferences belong to any set containing linear orders and when the collective preference must verify different properties, for instance transitivity. We show that the considered aggregation problems are NP-hard for different types of collective preferences (including linear orders, acyclic relations, complete preorders, interval orders, semiorders, quasi-orders or weak orders), if the number of individual preferences is sufficiently large.

Selection among ranked projects under segmentation, policy and logical constraints

The paper proposes a method for project selection under a specific decision situation, where a final selection is guided by two aspects: (1) satisfaction of certain segmentation, policy and/or logical constraints, and (2) assurance that the individual evaluation of the projects is respected to the maximum degree. This approach is somewhat different than the usual portfolio optimization, where combinations of projects are compared without special concern on respecting the project’s ranking. The entire process is implemented in two phases: the projects are first ranked, usually through a multicriteria approach. The obtained complete preorder of the projects is then used in an integer programming module in order to effectively drive the final selection that satisfies the segmentation and/or logical constraints. The innovative part of the proposed approach is the way it overcomes the well-known bias towards low cost projects which is caused by the knapsack formulation commonly used in the integer programming phase. Actually this is the main source of divergence between the final selection and the initial complete preorder of the projects. The proposed method improves an agreement between the final selection of projects obtained from the integer programming model and the ranking obtained from the multicriteria approach.

Hierarchies achievable in simple games

A previous work by Friedman et al. (Theory and Decision, 61:305–318, 2006) introduces the concept of a hierarchy of a simple voting game and characterizes which hierarchies, induced by the desirability relation, are achievable in linear games. In this paper, we consider the problem of determining all hierarchies, conserving the ordinal equivalence between the Shapley–Shubik and the Penrose–Banzhaf–Coleman power indices, achievable in simple games. It is proved that only four hierarchies are non-achievable in simple games. Moreover, it is also proved that all achievable hierarchies are already obtainable in the class of weakly linear games. Our results prove that given an arbitrary complete pre-ordering defined on a finite set with more than five elements, it is possible to construct a simple game such that the pre-ordering induced by the Shapley–Shubik and the Penrose–Banzhaf–Coleman power indices coincides with the given pre-ordering.

On the compositional characterization of complete fuzzy pre-orders

Complete pre-orders can be characterized in terms of the transitivity of the corresponding strict preference and indifference relations. In this paper, we investigate this characterization in a fuzzy setting. We consider two types of completeness (weak completeness and strong completeness) and decompose a fuzzy pre-order by means of an indifference generator, in particular a Frank t-norm. In the weakly complete case, we identify the strongest type of transitivity of the indifference and strict preference relations in function of the generator used for constructing them. In the strongly complete case, we lay bare a stronger type of transitivity of the strict preference relation. We conclude the paper with a rather negative result: there is no hope to obtain a compositional characterization of weakly complete fuzzy pre-orders, and hence also not of fuzzy pre-orders in general.

On the Complexity of Rationalizing Behavior

We study the complexity of rationalizing choice behavior. We do so by analyzing two polar cases, and a number of intermediate ones. In our most structured case, that is where choice behavior is defined in universal choice domains and satisfies the weak axiom of revealed preference, finding the complete preorder rationalizing choice behavior is a simple matter. In the polar case, where no restriction whatsoever is imposed, either on choice behavior or on choice domain, finding the complete preorders that rationalize behavior turns out to be intractable. We show that the task of finding the rationalizing complete preorders is equivalent to a graph problem. This allows the search for existing algorithms in the graph theory literature, for the rationalization of choice.

The Welfare Economics of Adaptive Preferences

In this paper I demonstrate that a reasonable welfare theoretic concept of progress can be made consistent with the assumption of endogenously changing preferences as long as these preference changes correspond to the pattern of adaptive preferences. The main theorem of the paper shows that under certain additional conditions adaptive preferences imply the existence of a complete pre-ordering of the consumption space in terms of improvement paths which allow endogenous preference changes. It is then shown that welfare economics of improvement paths is also possible with interpersonal influences on preferences. A conjecture is developed that results of recent empirical and experimental research into human economic behaviour corroborate the hypothesis of adaptive preferences.

Existence of homogeneous representations of interval orders on a cone in a topological vector space

Necessary and sufficient conditions are presented for the existence of a pair of positively homogeneous of degree one real functions representing an interval order Open image in new window on a real cone K in a topological vector space E (in the sense that, for every x,y∈K, xOpen image in new windowy if and only if u(x)≤v(y)), with u lower semicontinuous, v upper semicontinuous, and u and v utility functions for two complete preorders intimately connected with Open image in new window. We conclude presenting a new approach to get such kind of representations, based on the concept of a biorder.

GRAPHOID PROPERTIES OF QUALITATIVE POSSIBILISTIC INDEPENDENCE RELATIONS

Independence relations play an important role in uncertain reasoning based on Bayesian networks. In particular, they are useful in decomposing joint distributions into more elementary local ones. Recently, in a possibility theory framework, several qualitative independence relations have been proposed, where uncertainty is encoded by means of a complete pre-order between states of the world. This paper studies the well-known graphoid properties of these qualitative independences. Contrary to the probabilistic independence, several qualitative independence relations are not necessarily symmetric. Therefore, we also analyze the symmetric counterparts of graphoid properties (called reverse graphoid properties).

A stochastic spatial decision support system based on PROMETHEE

PROMETHEE is a well known multicriteria decision support method which gives insight into the preference structure of a whole set of alternatives. The retrieved preference information is used to give the partial or complete preorder of the available alternatives which makes PROMETHEE a well suited decision support tool if a ranking of available alternatives is desired. A relatively low degree of mathematical complexity makes PROMETHEE an attractive method for all non experts involved in a decision process. This paper discusses the integration of PROMETHEE in a GIS. It will be shown how the combination of a sophisticated decision support methodology with powerful spatial analysis and visualisation capabilities can be applied to evaluating decision alternatives that are made up of regular or irregular shaped zones of raster cells. Deterministic as well as stochastic PROMETHEE approaches are considered and general recommendations on the use of stochastic simulation methods are given. Moreover, a framework is presented which facilitates the application of the proposed approaches.

CHOOSING A RANKING OF ALTERNATIVES IN A TWO-STAGE GROUP DECISION PROCEDURE WITH INDIVIDUAL LINGUISTIC ASSESSMENTS

In this paper we provide a two-stage group decision-making procedure for ranking a great number of alternatives. Since individuals usually have incon-sistencies in the pairwise comparison of alternatives, we allow agents to assess alternatives one by one using linguistic labels. This information is processed by using aggregation functions, each one generating a complete preorder as a partial decision group outcome. In order to choose the final ranking of alternatives we consider an aggregation rule based on the classical Borda count, in which weights over partial outcomes are considered.

Continuous representability of homothetic preorders by means of sublinear order-preserving functions

We characterize the existence of a nonnegative, sublinear and continuous order-preserving function for a not necessarily complete preorder on a real convex cone in an arbitrary topological real vector space. As a corollary of the main result, we present necessary and sufficient conditions for the existence of such an order-preserving function for a complete preorder.