Making Hidden Priorities Explicit

Comparing and ranking information is an important topic in social and information sciences, and in particular on the web. Its objective is to measure the difference of the preferences of voters on a set of candidates and to compute a consensus ranking. Commonly, each voter provides a total order or a bucket order of all candidates, where bucket orders allow ties. In this work we consider the generalization of total and bucket orders to partial orders and compare them by the nearest neighbor and the Hausdorff Kendall tau distances. For total and bucket orders these distances can be computed in $\mathcal{O}(n \log n)$ time. We show that the computation of the nearest neighbor Kendall tau distance is NP-hard, 2-approximable and fixed-parameter tractable for a total and a partial order. The computation of the Hausdorff Kendall tau distance for a total and a partial order is shown to be coNP-hard. The rank aggregation problem is known to be NP-complete for total and bucket orders, even for four voters and solvable in $\mathcal{O}(n\log n)$ for two voters. It is NP-complete for two partial orders and the nearest neighbor Kendall tau distance. For the Hausdorff Kendall tau distance it is in $\mathbf{\Sigma_2^p}$ , but not in NP or coNP unless $\ensuremath{\mathbf{NP}} = \ensuremath{\mathbf{coNP}} $ , even for four voters.

Group decision making with preference information on alternatives is an interesting and important research topic which has been receiving more and more attention in recent years. The purpose of this paper is to investigate multiple-attribute group decision-making (MAGDM) problems with distinct uncertain preference structures. We develop some linear-programming models for dealing with the MAGDM problems, where the information about attribute weights is incomplete, and the decision makers have their preferences on alternatives. The provided preference information can be represented in the following three distinct uncertain preference structures: 1) interval utility values; 2) interval fuzzy preference relations; and 3) interval multiplicative preference relations. We first establish some linear-programming models based on decision matrix and each of the distinct uncertain preference structures and, then, develop some linear-programming models to integrate all three structures of subjective uncertain preference information provided by the decision makers and the objective information depicted in the decision matrix. Furthermore, we propose a simple and straightforward approach in ranking and selecting the given alternatives. It is worth pointing out that the developed models can also be used to deal with the situations where the three distinct uncertain preference structures are reduced to the traditional ones, i.e., utility values, fuzzy preference relations, and multiplicative preference relations. Finally, we use a practical example to illustrate in detail the calculation process of the developed approach.

In almost all current approaches to decision making under uncertainty, it is assumed that a decision problem is described by a set of states and set of outcomes, and the decision maker (DM) has a preference relation on a rather rich set of acts, which are functions from states to outcomes. The standard representation theorems of decision theory give conditions under which the preference relation can be represented by a utility function on outcomes and numerical representation of beliefs on states. For example, Savage [4] shows that if a DM’s preference order satisfies certain axioms, then the DM’s preference relation can be represented by a probability Pr on the state space and a utility function mapping outcomes to the reals such that she prefers act a to act b iff the expected utility of a (with respect to Pr) is greater than that of b. Moreover, the probability measure is unique and the utility function is unique up to affine transformations. Similar representations of preference can be given with respect to other representations of uncertainty (see, for example, [2,5]).

Robustness analysis for decision under uncertainty with rule-based preference model

On the existence of weak subgame perfect equilibria

Comparing and ranking information is an important topic in social and information sciences, and in particular on the web. Its objective is to measure the difference of the preferences of voters on a set of candidates and to compute a consensus ranking. Commonly, each voter provides a total order of all candidates. Recently, this approach was generalized to bucket orders, which allow ties. In this work we further generalize and consider total, bucket, interval and partial orders. The disagreement between two orders is measured by the nearest neighbor Spearman footrule distance, which has not been studied so far. For two bucket orders and for a total and an interval order the nearest neighbor Spearman footrule distance is shown to be computable in linear time, whereas for a total and a partial order the computation is NP-hard, 4-approximable and fixed-parameter tractable. Moreover, in contrast to the well-known efficient solution of the rank aggregation problem for total orders, we prove the NP-completeness for bucket orders and establish a 4-approximation.

Comparing genomes of different species is a crucial problem in comparative genomics. Different measures have been proposed to compare two genomes: number of common intervals, number of adjacencies, number of reversals, etc. These measures are classically used between two totally ordered genomes. However, genetic mapping techniques often give rise to different maps with some unordered genes. Starting from a partial order between genes of a genome, one method to find a total order consists in optimizing a given measure between a linear extension of this partial order and a given total order of a close and well-known genome. However, for most common measures, the problem turns out to be NP-hard. In this paper, we propose a (0,1)-linear programming approach to compute a linear extension of one genome that maximizes the number of common intervals (resp. the number of adjacencies) between this linear extension and a given total order. Next, we propose an algorithm to find linear extensions of two partial orders that maximize the number of adjacencies.KeywordsPartial OrderTotal OrderLinear ExtensionBoolean VariableReduction RuleThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We study multi-player turn-based games played on a directed graph, where the number of players and vertices can be infinite. An outcome is assigned to every play of the game. Each player has a preference relation on the set of outcomes which allows him to compare plays. We focus on the recently introduced notion of weak subgame perfect equilibrium weak SPE, a variant of the classical notion of SPE, where players who deviate can only use strategies deviating from their initial strategy in a finite number of histories. We give general conditions on the structure of the game graph and the preference relations of the players that guarantee the existence of a weak SPE, which moreover is finite-memory.

One of the major research streams in data envelopment analysis (DEA) is ranking decision-making units (DMUs). Utilizing a multicriteria decision-making technique, we develop a novel approach to fully rank all units. Motivated by the convex cone-based total order for multiple criteria alternatives proposed by Dehnokhalaji et al. (Nav Res Logist 61(2):155–163, 2014), we consider DMUs in DEA as multiple criteria alternatives and obtain their total ordering. Initially, some pairwise preference information is provided by the decision maker for units and the concepts of convex cones and polyhedral sets are defined in a DEA framework, correspondingly. We apply a modification of Dehnokhalaji et al. method to extract additional preference information for each pair of units and consequently obtain a full ranking (strict total ordering) of DMUs. The benefit of our approach to their method is that we apply non-radial models to overcome the instability drawback of radial models and their infeasibility occurring in DEA applications. The proposed approach is implemented for two numerical examples, and the accuracy of it is investigated through a computational test.

Deriving the priority vectors of the alternatives from preference relations is an interesting research topic for group decision making with preference information. This paper uses an example to show that the ranking or the optimal alternative could not always be derived from the existing additively consistent intuitionistic fuzzy preference relations. Thus, we provide novel additively consistent intuitionistic fuzzy preference relations and characterize them with Tanino’s normalized (T-normalized) intuitionistic fuzzy priority vectors. Then, we propose some methods to check and achieve the T-normalization, acceptably additive consistency and consensus of the intuitionistic fuzzy preference relations in group decision making using the local, individual and optimal collective intuitionistic fuzzy priority vectors, respectively. We also give some examples to show how the proposed models work and make comparisons with the existing methods to demonstrate the advantages of the proposed methods.

Method for multi-attribute group decision-making based on the compromise weights

The flow set with partial order is a mixed-integer set described by a budget on total flow and a partial order on the arcs that may carry positive flow. This set is a common substructure of resource allocation and scheduling problems with precedence constraints and robust network flow problems under demand/capacity uncertainty. We give a polyhedral analysis of the convex hull of the flow set with partial order. Unlike for the flow set without partial order, cover-type inequalities based on partial order structure are a function of a lifting sequence. We study the lifting sequences and describe structural results on the lifting coefficients for general and simpler special cases. We show that all lifting coefficients can be computed in polynomial time by solving maximum weight closure problems in general. For the special case of induced-minimal covers, we give a sequence-dependent characterization of the lifting coefficients. We prove, however, if the partial order is defined by an arborescence, then lifting is sequence-independent and all lifting coefficients can be computed in linear time. Moreover, if the partial order is defined by a path (total order), then the coefficients can be expressed explicitly. We also give a complete polyhedral description of the flow set with partial order for the polynomially-solvable total order case. We show that finding an optimal lifting order for a given induced-minimal cover and a given fractional solution is a submodular optimization problem, which is solved greedily. Finally, we present preliminary computational results with a cutting-plane algorithm based on the lifting and separation results.

Linguistic preference relations are constantly used by decision maker to express his/her preference information over alternatives in fuzzy environment where the decision maker' preference information can not be assessed precisely in a quantitative form but may be in a qualitative one. Over the last decades, many research results both theoretically and practically have been reported on linguistic preference relations, but current research mainly focuses on the situations where all the linguistic preference relations are collected from the same period, in which time is not taken into account. In this paper, we consider the decision making problems where the decision maker provides individual linguistic preference relations at different periods. We first introduce the concept of dynamic linguistic preference relation, and discuss some results related to dynamic linguistic preference relations and their aggregation technique. Then we develop an approach to multi-period decision making based on linguistic preference relations, and finally, we illustrate in detail the approach with a practical example.

The claim that, on the basis of the author's experience, IR (information retrieval) has lessons for AI (artificial intelligence) is that its methods work in those situations where information demand, rather than supply, is underspecified, and the right response is to be broadly indicative rather than narrowly assertive. The right way of being indicative is then to allow many small and individually ambiguous clues to combine and interact within whatever match between user input and file material can be found. Altogether there are many information management tasks, from finding, to sorting, to reminding, that arise in many different working contexts, that are quite crude, and that can generally be done (appropriately and not just for want of anything better), in a sufficient-to-the-day mode because they are fundamentally inexact. Using weak, multiple, redundant cues is not a novel idea in AI; but the argument here is that they may be appropriate in far more natural language cases than originally envisaged, and also in ones where, even if the representation language used is not `ordinary' natural language it shares some important properties with it, for instance in search operations on the data represented by discourse in formal languages, as in software. There is a challenge to the IR methods described in their own world, namely that of terabyte files and minimalist users. But while how far they can be pushed in that case has clearly to be investigated, this does not imply that they should not be pushed as far as they will go in the other areas mentioned, i.e. in `real' AI. (3 pages)

An intuitionistic preference relation is a powerful means to express decision makers’information of intuitionistic preference over criteria in the process of multi-criteria decision making. In this paper, we first define the concept of its consistence and give the equivalent interval fuzzy preference relation of it. Then we develop a method for estimating criteria weights from it, and then extend the method to accommodate group decision making based on them And finally, we use some numerical examples to illustrate the feasibility and validity of the developed method.