Inconsistency indices for pairwise comparison matrices: a numerical study

Evaluating the level of inconsistency of pairwise comparisons is often a crucial step in multi criteria decision analysis. Several inconsistency indices have been proposed in the literature to estimate the deviation of expert’s judgments from a situation of full consistency. This paper surveys and analyzes ten indices from the numerical point of view. Specifically, we investigate degrees of agreement between them to check how similar they are. Results show a wide range of behaviors, ranging from very strong to very weak degrees of agreement.     

Extension of the LINMAP for multiattribute decision making under Atanassov’s intuitionistic fuzzy environment

The aim of this article is further extending the linear programming techniques for multidimensional analysis of preference (LINMAP) to develop a new methodology for solving multiattribute decision making (MADM) problems under Atanassov’s intuitionistic fuzzy (IF) environments. The LINMAP only can deal with MADM problems in crisp environments. However, fuzziness is inherent in decision data and decision making processes. In this methodology, Atanassov’s IF sets are used to describe fuzziness in decision information and decision making processes by means of an Atanassov’s IF decision matrix. A Euclidean distance is proposed to measure the difference between Atanassov’s IF sets. Consistency and inconsistency indices are defined on the basis of preferences between alternatives given by the decision maker. Each alternative is assessed on the basis of its distance to an Atanassov’s IF positive ideal solution (IFPIS) which is unknown a prior. The Atanassov’s IFPIS and the weights of attributes are then estimated using a new linear programming model based upon the consistency and inconsistency indices defined. Finally, the distance of each alternative to the Atanassov’s IFPIS can be calculated to determine the ranking order of all alternatives. A numerical example is examined to demonstrate the implementation process of this methodology. Also it has been proved that the methodology proposed in this article can deal with MADM problems under not only Atanassov’s IF environments but also both fuzzy and crisp environments.     

Analysis of pairwise comparison matrices: an empirical research

Pairwise comparison (PC) matrices are used in multi-attribute decision problems (MADM) in order to express the preferences of the decision maker. Our research focused on testing various characteristics of PC matrices. In a controlled experiment with university students (N=227) we have obtained 454 PC matrices. The cases have been divided into 18 subgroups according to the key factors to be analyzed. Our team conducted experiments with matrices of different size given from different types of MADM problems. Additionally, the matrix elements have been obtained by different questioning procedures differing in the order of the questions. Results are organized to answer five research questions. Three of them are directly connected to the inconsistency of a PC matrix. Various types of inconsistency indices have been applied. We have found that the type of the problem and the size of the matrix had impact on the inconsistency of the PC matrix. However, we have not found any impact of the questioning order. Incomplete PC matrices played an important role in our research. The decision makers behavioral consistency was as well analyzed in case of incomplete matrices using indicators measuring the deviation from the final order of alternatives and from the final score vector.     

Interval weight generation approaches for reciprocal relations

In this paper, we propose methods to derive interval weight vectors from reciprocal relations for reflecting the inconsistency when decision makers provide preferences over alternatives (or criteria). Several goal programming models are established to minimize the inconsistency based on multiplicative and additive consistency, respectively. Especially, if we obtain a crisp weight vector from a reciprocal relation, then it is consistent. Then, we extend the proposed methods to incomplete reciprocal relations and interval reciprocal relations and develop the corresponding models to derive interval weight vectors. Several examples are also given to compare the developed methods with the existing ones.     

Power indices and the veil of ignorance

We provide an axiomatic foundation of the expected utility preferences over lotteries on roles in simple superadditive games represented by the two main power indices, the Shapley-Shubik index and the Banzhaf index, when they are interpreted as von Neumann-Morgenstern utility functions. Our axioms admit meaningful interpretations in the setting proposed by Roth in terms of different attitudes toward risk involving roles in collective decision procedures under the veil of ignorance. In particular, an illuminating interpretation of `efficiency', up to now missing in this set up, as well as of the corresponding axiom for the Banzhaf index, is provided. November 7, 2002 RID="*" ID="*"  We want to thank M. Maschler, J. M. Zarzuelo and two referees for their comments. This research has been supported by the IVIE, and by the DGES of the Spanish Ministerio de Educación y Cultura, under project PB96-0247. The first author also acknowledges the financial support from the Ramón y Cajal Program initiated by the Spanish MCyT.     

Inconsistency indices for pairwise comparisons and the Pareto dominance principle

We consider the problem of quantifying inconsistency in pairwise comparisons and valued-preferences. A wide range of indices have been proposed in the literature to perform this task, and two sets of conditions have been introduced to validate such indices. We summarize some criticisms from the literature and we add more evidence to show that neither of the two systems is adequate in its current formulation. Thanks to the widely accepted concept of weak Pareto dominance, we formulate a new property. We argue that a simple regularity condition and this new property can overcome the shortcomings of the two axiomatic systems, and represents a significantly simpler framework. Finally, we claim that, if we had resorted to strict Pareto dominance, we would have needed just one axiom.     

Policy-based inconsistency management in relational databases

Though inconsistency management in databases and AI has been studied extensively for years, it does not allow the user to specify how he wants to resolve inconsistencies. In real-world applications, users may want to manage or resolve inconsistencies based not only on the data, but their own knowledge of the risks involved in decision making based on faulty data. Each user should be empowered to use reasonable policies to deal with his data and his mission needs. In this paper, we start by providing an axiomatic definition of inconsistency management policies (IMPs) that puts this power in the hands of users. Any function satisfying these axioms is an IMP. We then define three broad families of IMPs, and derive several results that show (i) how these policies relate to postulates for the revision of belief bases and to recent research in the area of consistent query answering, and (ii) how they interact with standard relational algebra operators. Finally, we present several approaches to efficiently implement an IMP-based framework.     

Using possibilistic logic for modeling qualitative decision: Answer Set Programming algorithms

A qualitative approach to decision making under uncertainty has been proposed in the setting of possibility theory, which is based on the assumption that levels of certainty and levels of priority (for expressing preferences) are commensurate. In this setting, pessimistic and optimistic decision criteria have been formally justified. This approach has been transposed into possibilistic logic in which the available knowledge is described by formulas which are more or less certainly true and the goals are described in a separate prioritized base. This paper adapts the possibilistic logic handling of qualitative decision making under uncertainty in the Answer Set Programming (ASP) setting. We show how weighted beliefs and prioritized preferences belonging to two separate knowledge bases can be handled in ASP by modeling qualitative decision making in terms of abductive logic programming where (uncertain) knowledge about the world and prioritized preferences are encoded as possibilistic definite logic programs and possibilistic literals respectively. We provide ASP-based and possibilistic ASP-based algorithms for calculating optimal decisions and utility values according to the possibilistic decision criteria. We describe a prototype implementing the algorithms proposed on top of different ASP solvers and we discuss the complexity of the different implementations.     

An action learning approach for assessing the consistency of pairwise comparison data

Pairwise comparison data are used in various contexts including the generation of weight vectors for multiple criteria decision making problems. If this data is not sufficiently consistent, then the resulting weight vector cannot be considered to be a reliable reflection of the evaluator’s opinion. Hence, it is necessary to measure its level of inconsistency. Different approaches have been proposed to measuring the level of inconsistency, but they are often based on ‘rules of thumb” and/or randomly generated matrices, and are not interpretable. In this paper we present an action learning approach for assessing the consistency of the input pairwise comparison data that offer interpretable consistency measures.     

Learning about preferences in electronic negotiations – A volume-based measurement method

To create an integrative solution in a bargaining problem, negotiators need to have information about each other’s preferences. Empirical negotiation research therefore requires methods to measure the extent to which information about preferences is available during a negotiation. We propose such a method based on Starr’s domain criterion, which was originally developed for sensitivity analysis in decision making. Our method provides indices for the amount of preference information that can be inferred both in negotiations reaching an agreement and negotiations where an agreement was not (yet) reached. To test the external validity of our proposed measures, we conduct an empirical study which shows that the proposed measures exhibit positive relationships to the success of negotiations as well as to the efficiency of outcomes that would be expected according to negotiation theory.     

Aggregation of analytic hierarchy process models based on similarities in decision makers’ preferences

The analytic hierarchy process is widely used in both individual and group decision making environments. In this paper we investigate its applicability to model a specific class of decentralized decision problems where many decision makers take individual subjective decisions using locally available information. In such subjective decision making environments, it is neither possible nor appropriate to use group preference aggregation techniques to model the problem as a single group decision problem. An approach to identify homogeneous subgroups of decision makers based on similarities in preferences and to aggregate preferences within each subgroup is proposed. This approach is validated using employment preferences of 70 subjects modeled using the analytic hierarchy process.     

Discounting axioms imply risk neutrality

Although most applications of discounting occur in risky settings, the best-known axiomatic justifications are deterministic. This paper provides an axiomatic rationale for discounting in a stochastic framework. Consider a representation of time and risk preferences with a binary relation on a real vector space of vector-valued discrete-time stochastic processes on a probability space. Four axioms imply that there are unique discount factors such that preferences among stochastic processes correspond to preferences among present value random vectors. The familiar axioms are weak ordering, continuity and nontriviality. The fourth axiom, decomposition, is non-standard and key. These axioms and the converse of decomposition are assumed in previous axiomatic justifications for discounting with nonlinear intraperiod utility functions in deterministic frameworks. Thus, the results here provide the weakest known sufficient conditions for discounting in deterministic or stochastic settings. In addition to the four axioms, if there exists a von Neumann-Morgenstern utility function corresponding to the binary relation, then that function is risk neutral (i.e., affine). In this sense, discounting axioms imply risk neutrality.     

Ascending bid auctions with behaviorally consistent bidders

Decision makers whose preferences do not satisfy the independence axiom of expected utility theory, when faced with sequential decisions will act in a dynamically inconsistent manner. In order to avoid this inconsistency and maintain nonexpected utility, we suggest the idea of behavioral consistency. We implement this notion by regarding the same decision maker at different decision nodes as different agents, and then taking the Bayesian — Nash equilibrium of this game. This idea is applied to a finite ascending bid auction game. We show the condition for the existence of an equilibrium of this game, and we also characterize the equilibrium in those cases when it exists. In particular, when the utility functionals are both quasi-concave and quasi-convex, then there is an equilibrium in dominant strategies where each bidder continues to bid if and only if the prevailing price is smaller than his value. In the case of quasi-concavity it is shown that, in equilibrium, each bidder has a value such that he continues with positive probability up to it, and withdraws after that.This research was supported by the NSF under Grant No. SES87-08360. We would like to thank Professor Irving H. LaValle for his helpful suggestions.     

Two axiomatic approaches to decision making using possibility theory

This paper proposes a utility theory for decision making under uncertainty that is described by possibility theory. We show that our approach is a natural generalization of the two axiomatic systems that correspond to pessimistic and optimistic decision criteria proposed by Dubois et al. The generalization is achieved by removing axioms that are supposed to reflect attitudes toward uncertainty, namely, pessimism and optimism. In their place we adopt an axiom that imposes an order on a class of canonical lotteries that realize either in the best or in the worst prize. We prove an expected utility theorem for the generalized axiomatic system based on the newly introduced concept of binary utility.     

Generalized Choquet-like aggregation functions for handling bipolar scales

We are interested in modeling interaction between criteria in multi-criteria decision making when underlying scales are bipolar. Interacting phenomena involving behavioral bias between attractive and repulsive values are in particular considered here. We show in an example that both the Choquet integral and the cumulative prospect theory (CPT) model fail to represent these interacting phenomena. Axioms that enable the construction of the preferences of the decision maker over each attribute, and the representation of his preferences about aggregation of criteria are introduced and justified. We show there is a unique aggregation operator that fits with these axioms. It is based on the notion of bi-capacity and generalizes both the Choquet integral and the CPT model.     

Implementing stochastic multicriteria acceptability analysis

Stochastic multicriteria acceptability analysis (SMAA) is a family of methods for aiding multicriteria group decision making in problems with inaccurate, uncertain, or missing information. These methods are based on exploring the weight space in order to describe the preferences that make each alternative the most preferred one, or that would give a certain rank for a specific alternative. The main results of the analysis are rank acceptability indices, central weight vectors and confidence factors for different alternatives. The rank acceptability indices describe the variety of different preferences resulting in a certain rank for an alternative, the central weight vectors represent the typical preferences favouring each alternative, and the confidence factors measure whether the criteria measurements are sufficiently accurate for making an informed decision.     

Monotonicity and Consistency in Matching Markets

Objective: To obtain axiomatic characterizations of the core of one-to-one and one-to-many matching markets. Methods: The axioms recently applied to characterize the core of assignment games were adapted to the models of this paper. Results: The core of one-to-one matching markets is characterized by two different lists of axioms. The first one consists of weak unanimity, population monotonicity, and Maskin monotonicity. The second consists of weak unanimity, population monotonicity, and consistency. If we allow for weak preferences, the core is characterized by weak unanimity, population monotonicity, Maskin monotonicity, and consistency. For one-to-many matchings, the same lists as for the case of strict preferences characterize the core. Conclusions: The cores of the discrete matching markets are characterized by axioms that almost overlap with the axioms characterizing the core of the continuous matching markets. This provides an axiomatic explanation for the observations in the literature that almost parallel properties are obtained for the core of the two models. We observe that Maskin monotonicity is closely related to consistency in matching marketsThis research is financially supported by Waseda University Grant for Special Research Projects #2000A−887, 21COE-GLOPE, and Grant-in-Aid for Scientific Research #15530125, JSPS. This paper was presented at the 7th. International Meeting of the Society for Social Choice and Welfare held in Osaka, Japan. The comments of the participants are gratefully acknowledged. The author thanks Professors William Thomson, Eiichi Miyagawa and anonymous referees for their valuable comments and suggestions. Any remaining errors are independent     

Incomplete pairwise comparison and consistency optimization

This paper proposes a new method for calculating the missing elements of an incomplete matrix of pairwise comparison values for a decision problem. The matrix is completed by minimizing a measure of global inconsistency, thus obtaining a matrix which is optimal from the point of view of consistency with respect to the available judgements. The optimal values are obtained by solving a linear system and unicity of the solution is proved under general assumptions. Some other methods proposed in the literature are discussed and a numerical example is presented.     

不完全信息下的AHP优化决策方法研究

针对在紧急情况下,比如战争或灾难中利用AHP进行决策时,由于决策时间紧迫、信息掌握不完全、决策者经验限制等因素,通常会导致决策信息的不完全,提出一套完整的基于残缺判断矩阵的分析和优化的解决方案。首先,给出了残缺判断矩阵的相关定义和性质,研究了残缺判断矩阵的有效性判断的基本原理,并给出了连通图判定方法;对于无效残缺判断矩阵,通过增补最少的元素实现所有方案的互连通,从而使其成为有效残缺判断矩阵;对于有效残缺判断矩阵,提出以未知数填充残缺矩阵,构建以一致性比率最小为目标的优化决策模型;对于优化模型仍不能达到满意一致性的情形,从基本回路的不一致性分析入手,找出具有最大CR和的元素作为最不一致元素,在[1/9,9]区间上选出使得CR和最小的值作为该元素修正值,然后再构建优化模型实现最优化增补;根据以上原理,利用Matlab编程,开发了残缺判断矩阵的AHP相关决策工具软件。最后,通过算例分析验证了方法的可行性和有效性,证明了开发的软件能够满足紧急状态下决策的时效性要求,同时,通过与已有方法的对比证明了该方法更为有效。