Fundamentals of simple games from a viewpoint of blockability relations

This paper shows some elementary facts on simple games with respect to blockability relations. It is verified in this paper that fundamental concepts on simple games as null players, dictators, veto players, and so on can be expressed in terms of blockability relations. More, some new concepts as “conflict-free” and so on, are introduced from the viewpoint of blockability relations into the framework of simple games.     

A new method for ranking fuzzy numbers and its application to group decision making

In this paper, a new method for comparing fuzzy numbers based on a fuzzy probabilistic preference relation is introduced. The ranking order of fuzzy numbers with the weighted confidence level is derived from the pairwise comparison matrix based on 0.5-transitivity of the fuzzy probabilistic preference relation. The main difference between the proposed method and existing ones is that the comparison result between two fuzzy numbers is expressed as a fuzzy set instead of a crisp one. As such, the ranking order of n fuzzy numbers provides more information on the uncertainty level of the comparison. Illustrated by comparative examples, the proposed method overcomes certain unreasonable (due to the violation of the inequality properties) and indiscriminative problems exhibited by some existing methods. More importantly, the proposed method is able to provide decision makers with the probability of making errors when a crisp ranking order is obtained. The proposed method is also able to provide a probability-based explanation for conflicts among the comparison results provided by some existing methods using a proper ranking order, which ensures that ties of alternatives can be broken.     

Self-consistency of decision rules for group decision making

The author treats, in this paper, a group of decision makers, where each of them already has preference on a given set of alternatives but the group as a whole does not have a decision rule to make their group decision, yet. Then, the author examines which decision rules are appropriate. As a criterion of “appropriateness” the author proposes the concepts of self-consistency and universal self-consistency of decision rules. Examining the existence of universally self-consistent decision rules in two cases: (1) decision situations with three decision makers and two alternatives, and (2) those with three decision makers and three alternatives, the author has found that all decision rules are universally self-consistent in the case (1), whereas all universally self-consistent decision rules have one and just one vetoer in the essential cases in (2). The result in the case (2) implies incompatibility of universal self-consistency with symmetry. An example of applications of the concept of self-consistency to a bankruptcy problem is also provided in this paper, where compatibility of self-consistency with symmetry in a particular decision situation is shown.     

On the use of binary decision diagrams for solving problems on simple games

Simple games are yes/no cooperative games which arise in many practical applications. Recently, we have used reduced ordered binary decision diagrams and quasi-reduced ordered binary decision diagrams (abbreviated as Robdds and Qobdds, respectively) for the representation of simple games and for the computation of some power indices. In the present paper, we continue this work. We show how further important computational problems on simple games can be solved using Qobdds, viz. the identification of some key players, the computation of the desirability relation on individuals, the test whether a simple game is proper and strong, respectively, and the computation of Qobdd-representations for the sets of all minimal winning coalitions, all shift-minimal winning coalitions and all blocking coalitions, respectively. Applications of these solutions include the computation of recent power indices based on shift-minimal winning coalitions and the test for linear separability of a directed simple game.     

An acyclic relation for comparison of bargaining powers of coalitions and its interrelationship with bargaining set

This paper proposes a method to compare bargaining power of coalitions within the framework of games in coalition form with transferable utility. The method is expressed by a relation on the set of all coalitions in a game, the relation which is defined based on the players’ bargaining power. It is shown in this paper that the newly defined relation satisfies acyclicity. Also, it is verified in this paper that the set of all individually rational payoff configurations under which all coalitions have the equal bargaining power coincides with the bargaining set. Some examples demonstrate how the newly proposed method works.     

Normalizing rank aggregation method for priority of a fuzzy preference relation and its effectiveness

The aim of this paper is to show that the normalizing rank aggregation method can not only be used to derive the priority vector for a multiplicative preference relation, but also for the additive transitive fuzzy preference relation. To do so, a simple functional equation between fuzzy preference’s element and priority weight is derived firstly, then, based on the equation, three methods are proposed to prove that the normalizing rank aggregation method is simple and effective for deriving the priority vector. Finally, a numerical example is used to illustrate the proposed methods.     

A chi-square method for obtaining a priority vector from multiplicative and fuzzy preference relations

Decision makers (DMs)’ preferences on decision alternatives are often characterized by multiplicative or fuzzy preference relations. This paper proposes a chi-square method (CSM) for obtaining a priority vector from multiplicative and fuzzy preference relations. The proposed CSM can be used to obtain a priority vector from either a multiplicative preference relation (i.e. a pairwise comparison matrix) or a fuzzy preference relation or a group of multiplicative preference relations or a group of fuzzy preference relations or their mixtures. Theorems and algorithm about the CSM are developed. Three numerical examples are examined to illustrate the applications of the CSM and its advantages.     

An approach to group decision making based on interval multiplicative and fuzzy preference relations by using projection

This paper presents a consensus model for group decision making with interval multiplicative and fuzzy preference relations based on two consensus criteria: (1) a consensus measure which indicates the agreement between experts’ preference relations and (2) a measure of proximity to find out how far the individual opinions are from the group opinion. These measures are calculated by using the relative projections of individual preference relations on the collective one, which are obtained by extending the relative projection of vectors. First, the weights of experts are determined by the relative projections of individual preference relations on the initial collective one. Then using the weights of experts, all individual preference relations are aggregated into a collective one. The consensus and proximity measures are calculated by using the relative projections of experts’ preference relations respectively. The consensus measure is used to guide the consensus process until the collective solution is achieved. The proximity measure is used to guide the discussion phase of consensus reaching process. In such a way, an iterative algorithm is designed to guide the experts in the consensus reaching process. Finally the expected value preference relations are defined to transform the interval collective preference relation to a crisp one and the weights of alternatives are obtained from the expected value preference relations. Two numerical examples are given to illustrate the models and approaches.     

Is It Better to Forget? Stimulus-Response, Prediction, and the Weight of Past Experience in a Fast-Paced Bargaining Task

Decision makers in dynamic environments such as air traffic control, firefighting, and call center operations adapt in real-time using outcome feedback. Understanding this adaptation is important for influencing and improving the decisions made. Recently, stimulus-response (S-R) learning models have been proposed as explanations for decision makers' adaptation. S-R models hypothesize that decision makers choose an action option based on their anticipation of its success. Decision makers learn by accumulating evidence over action options and combining that evidence with prior expectations. This study examines a standard S-R model and a simple variation of this model, in which past experience may receive an extremely low weight, as explanations for decision makers' adaptation in an evolving Internet-based bargaining environment. In Experiment 1, decision makers are taught to predict behavior in a bargaining task that follows rules that may be the opposite of, congruent to, or unrelated to a second task in which they must choose the deal terms they will offer. Both models provide a good account of the prediction task. However, only the second model, in which decision makers heavily discount all but the most recent past experience, provides a good account of subsequent behavior in the second task. To test whether Experiment 1 artificially related choice behavior and prediction, a second experiment examines both models' predictions concerning the effects of bargaining experience on subsequent prediction. In this study, decision models where long-term experience plays a dominating role do not appear to provide adequate explanations of decision makers' adaptation to their opponent's changing response behavior.     

A new data envelopment analysis method for priority determination and group decision making in the analytic hierarchy process

The DEAHP method for weight deviation and aggregation in the analytic hierarchy process (AHP) has been found flawed and sometimes produces counterintuitive priority vectors for inconsistent pairwise comparison matrices, which makes its application very restrictive. This paper proposes a new data envelopment analysis (DEA) method for priority determination in the AHP and extends it to the group AHP situation. In this new DEA methodology, two specially constructed DEA models that differ from the DEAHP model are used to derive the best local priorities from a pairwise comparison matrix or a group of pairwise comparison matrices no matter whether they are perfectly consistent or inconsistent. The new DEA method produces true weights for perfectly consistent pairwise comparison matrices and the best local priorities that are logical and consistent with decision makers (DMs)’ subjective judgments for inconsistent pairwise comparison matrices. In hierarchical structures, the new DEA method utilizes the simple additive weighting (SAW) method for aggregation of the best local priorities without the need of normalization. Numerical examples are examined throughout the paper to show the advantages of the new DEA methodology and its potential applications in both the AHP and group decision making.     

Strategic control and interests,its effects on decision outcomes

Application of the model to artificial data shows that actors with strong preferences in the center have more possibilities to realize good outcomes than other actors. On the basis of an empirical application it is shown that a Nash equilibrium does not always arise after a large number of iterations unless actors have learning capabilities or are severely restricted in their strategic behavior.

In political systems and large organizations, ultimate decision makers are usually just a small subset of all actors in the social system. To arrive at acceptable decisions, decision makers have to take into account the preferences of other actors in the system. Typically preferences of more interested and more powerful actors are weighted heavier than those of less interested and powerful actors. This implies that the total leverage of an actor on the decision is determined by the combination of his power (his potential) and his interest (his willingness to mobilize his power). As the exact level of an actor's leverage is difficult to estimate for the other actors in the system, an actor is able to optimize his effects on outcomes of decisions by providing strategic informatioa

In this paper, first an analytic solution is presented for the optimization of strategic leverage in collective decision making by one single actor. In this solution, the actor makes assumptions about the leverage other actors will show in decision making. Subsequently, the actor optimizes the outcomes of decisions by manipulating the distribution of his leverage over a set of issues.

The analytic solution can be theoretically interpreted by decomposing the solution into three terms, the expected external leverage of the other actors on the issue, the evaluation of the deviance of the expected from the preferred outcome of the issue, and the restrictions on the distribution of leverage over the issues. The higher the expectation of the leverages the other actors will allocate to the issue, the less an actor is inclined to allocate leverage to the issue. The higher the evaluation of the deviance, the more an actor is inclined to allocate leverage to the issue. This is restricted, however, by the required distribution of leverages over the issues. The researcher is able to manipulate these restrictions to investigate its consequences for the outcomes.

In the next step, we investigate whether we can find a Nash equilibrium if all actors optimize their leverage simultaneously. Under certain conditions, a Nash equilibrium can be found by an iterative process in which actors update their estimates oh each other's leverages on the basis of what the other actors have shown in previous iterations.     

A method of fuzzy preference relation to authorize the worldwide interoperability for microwave access license in Taiwan

This study proposes a preference relation based evaluation framework to help the National Communication Commission (NCC) in Taiwan authorize a worldwide interoperability for microwave access (WiMAX) license under a fuzzy environment where the uncertainty, subjectivity and vagueness are dealt with linguistic variables parameterized by triangular fuzzy numbers. This study applies the fuzzy multi-criteria decision making approach to determine the importance weights of evaluation criteria and consolidate the performance ratings of possible alternatives. Aggregated the evaluators’ opinions toward the criteria and alternatives, the fuzzy preference relation approach is utilized to obtain the non-dominated degree of each alternative for the decision makers to make a final decision. Simultaneously, an empirical case involving sixteen quantitative and fifteen qualitative evaluation criteria, thirteen telecommunication applicants assessed by twelve specialists from various fields of telecommunication industry in Taiwan is solicited to demonstrate the proposed approach.     

A combinatorial theory of minimal social situations

A minimal social situation is a game‐like situation in which there are two actors, each of them has two possible actions, and both evaluate the outcomes of their joint actions in terms of two categories (say, ‘success’ and ‘failure'). By fixing actors and actions and varying ‘payoffs’ the set of 256 ‘configurations’ is obtained. This set decomposes into 43 ‘structural forms’, or equivalence classes with respect to the relation of isomorphism defined on it. This main theorem and other results concerning related configurations (minimal decision situations) are derived in this paper by means of certain tools of group theory. Some extensions to larger structures are proved in the Appendix. In the introductory section after a brief explanation of the meaning given to the terms ‘structure’ and ‘isomorphism’ in mathematics (Bourbaki) it is shown how these terms can be used to formalize the concept of ‘social form’.     

Systemic risk elicitation: Using causal maps to engage stakeholders and build a comprehensive view of risks

As evidenced through both a historical and contemporary number of reported over-runs, managing projects can be a risky business. Managers are faced with the need to effectively work with a multitude of parties and deal with a wealth of interlocking uncertainties. This paper describes a modelling process developed to assist managers facing such situations. The process helps managers to develop a comprehensive appreciation of risks and gain an understanding of the impact of the interactions between these risks through explicitly engaging a wide stakeholder base using a group support system and causal mapping process. Using a real case the paper describes the modelling process and outcomes along with its implications, before reflecting on the insights, limitations and future research.     

The structure of stratified systems and the structure of mobility: A first approximation to a structural theory of vertical mobility

This essay summarizes an inquiry that explores relations between the structure of stratified systems and the processes of vertical mobility. The inquiry considers economic stratification (the distribution of wealth) and is directed to determining whether the structural properties of stratification systems are sufficient to generate basic patterns in vertical mobility observed in empirical research, especially, the rank‐distance effect. In particular, the question is whether these patterns can be generated even if movement is constrained by nothing more than the size of the population over which wealth is distributed and the total amount of wealth to be distributed. Our results show that the rank‐distance effect emerges even under these minimal assumptions and, further, that rates and distances of vertical mobility are closely related to changes in these boundary parameters of a stratified system. The basic theory developed to relate structure and mobility provides results that are highly consistent with many empirical observations. It also challenges existing claims concerning the nature of the mechanisms determining the relative status immobility of most people in large scale systems. The theory implies that the way in which system structure constrains opportunity for movement is, by itself, sufficient to produce this result and others commonly observed.     

Utilization of mathematical programming for public systems: An application to effective formation of integrated regional information networks

This paper concerns a methodological reflection on the multiobjective approach to public systems which involve group decision processes. Particular attention is given to an integrated program of regional systems which include value trade-offs between multiple objectives. Our intention is to combine the judgmental processes with the optimization processes in the soft public systems. A two-layer approach is applied. At the first layer, each regional program is formulated in mathematical programming based on a utility assessment with different regional characteristics. Each subsystem independently reflects its particular concern as a single agent. The dual optimal solutions obtained for each subsystem are treated as an index, or the theoretical prices, representing the value trade-offs among the multiple objectives. At the second layer, an effective formation of interregional cooperation for compromising the conflicting regional interests is examined. Ann-person cooperative game in the characteristic function form is used to evaluate the effectiveness of the cooperation. The characteristic function for the game is derived on the incremental value of the regional benefit after the formation of a cooperation. The nucleolus and the augmented nucleolus as the solution concepts of the cooperative game are used for indicating the effectiveness of the cooperation. Finally using alternative criteria, the results in assessing the best decisions are examined comparatively.     

A method based on PSO and granular computing of linguistic information to solve group decision making problems defined in heterogeneous contexts

Group decision making is a type of decision problem in which multiple experts acting collectively, analyze problems, evaluate alternatives, and select a solution from a collection of alternatives. As the natural language is the standard representation of those concepts that humans use for communication, it seems natural that they use words (linguistic terms) instead of numerical values to provide their opinions. However, while linguistic information is readily available, it is not operational and thus it has to be made usable though expressing it in terms of information granules. To do so, Granular Computing, which has emerged as a unified and coherent framework of designing, processing, and interpretation of information granules, can be used. The aim of this paper is to present an information granulation of the linguistic information used in group decision making problems defined in heterogeneous contexts, i.e., where the experts have associated importance degrees reflecting their ability to handle the problem. The granulation of the linguistic terms is formulated as an optimization problem, solved by using the particle swarm optimization, in which a performance index is maximized by a suitable mapping of the linguistic terms on information granules formalized as sets. This performance index is expressed as a weighted aggregation of the individual consistency achieved by each expert.     

A new look at the role of players’ weights in the weighted Shapley value

everal new families of semivalues for weighted n-person transferable utility games are axiomatically constructed and discussed under increasing collections of axioms, where the weighted Shapley value arises as the resulting one member family. A more general approach to such weighted games defined in the form of two components, a weight vector λ and a classical TU-game v, is provided. The proposed axiomatizations are done both in terms of λ and v. Several new axioms related to the weight vector λ are discussed, including the so-called “amalgamating payoffs” axiom, which characterizes the value of a weighted game in terms of another game with a smaller number of players. They allow for a new look at the role of players’ weights in the context of the weighted Shapley value for the model of weighted games, giving new properties of it. Besides, another simple formula for the weighted Shapley value is found and examples illustrating some surprising behavior of it in the context of players’ weights are given. The paper contains a wide discussion of the results obtained.     

目标值不确定的协调多指标决策模型

本文建立了一类目标值不确定的多指标决策的协调模型,并对其解的概念、性质及其求解方法进行了研究,获得了一些新的结论。     

一种新广义混合比例平均算子及其在多属性群决策中的应用

给出一种新广义混合比例平均(NGHPA)算子,并提出一种基于NGHPA算子的多属性群决策方法.理论分析表明:NGHPA算子同时推广了广义有序加权比例平均(GOWPA)算子和广义加权比例平均(GWPA)算子,它不仅反映了所给数据所处位置的重要性,而且还考虑了数据本身的重要性.此外,算子具有单调性、幂等性、有界性和置换不变性.最后,通过一个算例说明所提的方法是有效的、可行的.