Sequencing games

Sequencing situations with linear cost functions are considered. A division rule for the cost savings is introduced and characterized axiomatically. Cooperative game theory is applied to analyse these situations and expressions for division rules induced by solution concepts from cooperative game theory are derived.     

Stability of plane parallel shear flow in a rotating layer heated from below

The discussion of stability of plane parallel shear flow in an infinite rotating layer heated from below requires a mathematical analysis of this problem in dependence on four parameters. These are the Reynolds- and Rayleigh-number, controlling the strength of the shear flow and the heating power, respectively, the Prandtl-number, which measures the relative influence of viscosity and thermal conductivity, and the rotation rate of the layer. After discussing some physical background, possible applications and laboratory experiments two major problems are addressed: i) To find out the cases where unconditional (global) stability up to criticality takes place. In these situations theory makes the clearest predictions and coincidence between experiments and mathematical theory can be expected. ii) To prove that the (monotonic) energy-stability limit is assumed by 2-dimensional (with respect to the spatial variables) perturbations. The solution of this variational problem shows that in certain situations the critical perturbations are 2-dimensional. In these situations, at least, the stability problem is completely solved. Received May 21, 1996 / Accepted December 17, 1996     

Multi-issue allocation situations

This paper introduces a new class of transferable utility games, called multi-issue allocation games. These games arise from various allocation situations and are based on the concepts underlying the bankruptcy model, as introduced by O'Neill [Math. Social Sci. 2 (1982) 345]. In this model, a perfectly divisible good (estate) has to be divided amongst a given set of agents, each of whom has some claim on the estate. Contrary to the standard bankruptcy model, the current model deals with situations in which the agents' claims are multi-dimensional, where the dimensions correspond to various issues.It is shown that the class of multi-issue allocation games coincides with the class of (non-negative) exact games. The run-to-the-bank rule is introduced as a solution for multi-issue allocation situations and turns out to be Shapley value of the corresponding game. Finally, this run-to-the-bank rule is characterised by means of a consistency property in the spirit of O'Neill.     

Sequencing interval situations and related games

Uncertainty accompanies almost every situation in real world and it influences our decisions. In sequencing situations it may affect parameters used to determine an optimal order in the queue, and consequently the decision of whether (or not) to rearrange the queue by sharing the realized cost savings. This paper extends the analysis of one-machine sequencing situations and their related cooperative games to a setting with interval data, i.e. when the agents’ costs per unit of time and/or processing time in the system lie in intervals of real numbers obtained by forecasting their values. The question addressed here is: How to determine an optimal order (if the initial order in the queue is not so) and which approach should be used to motivate the agents to adopt the optimal order? This question is an important one that deserves attention both in theory and practice.     

Divergence of choices despite similarity of characteristics: An application of catastrophe theory

In this paper we use catastrophe theory to analyse situations in which agents with similar characteristics and objectives and facing identical or similar environments make choices which are considerably different. We first provide two simple analytical examples of this phenomenon and then set up a general framework to which we apply the classification theorem of catastrophe theory.     

一个肿瘤生长自由边界问题解的渐近性态

该文研究根据Byrne和Chaplain的思想建立的一个描述抑制物作用下无坏死核肿瘤生长的数学模型, 这个模型是一个非线性反应扩散方程组的自由边界问题. 作者运用反应扩散方程理论中的上下解方法结合自由边界问题的迭代技巧, 研究了解的渐近性态, 在营养物消耗函数f、抑制物消耗函数g和肿瘤细胞繁衍函数S的一些一般条件下,证明当常数c₁,c₂(肿瘤细胞分裂速率和营养物、抑制物扩散速率的比值)都非常小时,在一定的初边值条件下肿瘤趋于消失,在另外一些初边值条件下肿瘤半径趋于一个常数,进而时变解将趋于一个稳态解.     

Fuzzy approaches for multiple objective linear fractional optimization

The problem of finding a solution to a multiple objective linear fractional program arises in several real world situations.In this paper we advocate that fuzzy sets theory provides a basis for solving this problem with sufficient consistency and rigorousness.After representing imprecise aspirations of the decision maker by structured linguistic variables or converting the original problem via approximations or change of variables into a multiple objective linear program, techniques of fuzzy linear programming may be used to reach a satisfactory solution.It is shown that under reasonable restrictions, this solution is efficient (Pareto optimal) for the original problem. Numerical examples are also included for illustration.     

模糊支付合作博弈的可信性准核仁

合作博弈是处理局中人之间协同行为的数学理论。有诸如核心、稳定集、沙普利值、准核仁和核仁等不同的解概念。在很多情形,除了借助专家经验和主观直觉,没有恰当的方式来确定支付函数,由此产生了具有模糊支付的合作博弈模型。准核仁是一种重要的解概念,在模糊支付合作博弈中如何恰当定义准核仁是个重要的问题。本文在可信性理论的框架下研究了这个问题,定义了两类可信性准核仁概念并证明了它们的存在性和唯一性,同时研究了可信性核心、可信性核仁与它们之间的关系。     

Sequencing Games with Slack Due Windows and Group Technology Considerations

We study sequencing situations in which the customers are initially sequenced to be served by a single server. We consider both slack due windows and group technology simultaneously. We introduce two division rules to divide among the customers the cost saving from resequencing the customers to follow the optimal sequence and characterize the rules axiomatically. Applying cooperative game theory to analyze the sequencing games corresponding to the sequencing situations, we use the theory’s solution concepts to solve the games.     

输油管线的最优设计方案

以输油管线铺设费用最省为目标,通过建立非线性规划数学模型,得到了不同情形下管线铺设的最优设计方案.综合运用多元函数极值和三角函数相关知识得出了模型的精确解.     

Solutions for some bargaining games under the Harsanyi-Selten solution theory,part II: Analysis of specific bargaining games

Part II of the paper (for Part I see Harsanyi (1982)) describes the actual solutions the Harsanyi-Selten solution theory provides for some important classes of bargaining games, such as unanimity games; trade between one seller and several potential buyers; and two-person bargaining games with incomplete information on one side or on both sides. It also discusses some concepts and theorems useful in computing the solution; and explains how our concept of risk dominance enables us to analyze game situations in terms of some intuitively very compelling probabilistic (subjective-probability) considerations disallowed by classical game theory.     

Path-following energy optimization in unilateral contact problems

Path-following (load incrementation) methods are studied in this paper for elastostatic analysis problems with unilateral contact relations in the framework of a large displacement theory by means of the parametric optimization techniques. Finite element discretization yields sparse polynomial optimization problems with equality and inequality constraints. For such sparse problems generically appearing singularities along the path of solutions are completely classified. Perturbations involving only a minimal number of parameters are shown to be sufficient to guarantee these generic situations. This clarifies stability and uniqueness questions for the solution along the examined path.     

Coalition behaviour in n‐person conflicts†

This paper presents a theory of behaviour in n‐person conflicts. The theory and its solution are predicated on the inter‐dependency of coalition formation and pay‐off determination in an n‐person conflict. The solution, therefore, involves not only pay‐off determination but also coalition formation in the game. The theory is developed using mathematical notions of fixed points together with some behavioural theories about choice behaviour and interdependency of persons in competitive situations. It predicts not only the final state of coalition formation but also transient choices and coalitions in the games. Finally, the model is tested on some experimental games of W. Riker.     

A viability approach to the skorohod problem

The theory of stochastic differential equations with reflecting boundary conditions leads to the "Skorohod" problem. Thispaper proposes a solution to this problem using techniques from viability theory and non-smooth analysis, allowing very general situations to occur.     

Validation of Computational Models Based on Multiple Heterogeneous Knowledge Sources

Theories of organizations have brought together multipleheterogeneous theories in computational models. In addition, inartificial intelligence, there has been an emphasis on the generationof knowledge-based systems that include multiple heterogeneousknowledge bases. As a result, increasingly, theory and modeldevelopers have called for the need to validate these computationalmodels. Unfortunately, there has been only limited attention givento validation of multiple knowledge source programs. The primary focus of this paper is on the identification of conflict between multiple knowledge bases. The existence of conflict is particularly critical in those situations where database evaluations are "averaged". For example, what would it mean to average the assessments of supply and demand economists, or surgeons and chemotherapists? Correlational statistics are used to identify conflict situations. In addition, a new approach, referred to as cutpoints, is developed to determine if probability distributions of multiple agents are in conflict. A case study is used to illustrate the problems of combining expertise in multiple agent systems and to demonstrate the approach.     

The restricted core of games on distributive lattices: how to share benefits in a hierarchy

Finding a solution concept is one of the central problems in cooperative game theory, and the notion of core is the most popular solution concept since it is based on some rationality condition. In many real situations, not all possible coalitions can form, so that classical TU-games cannot be used. An interesting case is when possible coalitions are defined through a partial ordering of the players (or hierarchy). Then feasible coalitions correspond to teams of players, that is, one or several players with all their subordinates. In these situations, the core in its usual formulation may be unbounded, making its use difficult in practice. We propose a new notion of core, called the restricted core, which imposes efficiency of the allocation at each level of the hierarchy, is always bounded, and answers the problem of sharing benefits in a hierarchy. We show that the core we defined has properties very close to the classical case, with respect to marginal vectors, the Weber set, and balancedness.     

An Analytical Framework to Describe the Interactions Between Individuals and a Continuum

We consider a discrete set of individual agents interacting with a continuum. Examples might be a predator facing a huge group of preys, or a few shepherd dogs driving a herd of sheep. Analytically, these situations can be described through a system of ordinary differential equations coupled with a scalar conservation law in several space dimensions. This paper provides a complete well-posedness theory for the resulting Cauchy problem. A few applications are considered in detail and numerical integrations are provided.     

A perturbation solution for long wavelength thermoacoustic propagation in dispersions

In thermoacoustic scattering the scattered field consists of a propagating acoustic wave together with a non-propagational “thermal” wave of much shorter wavelength. Although the scattered field may be obtained from a Rayleigh expansion, in cases where the particle radius is small compared with the acoustic wave length, these solutions are ill-conditioned. For this reason asymptotic or perturbation solutions are sought. In many situations the radius of the scatter is comparable to the length of the thermal wave. By exploiting the non-propagational character of the thermal field we obtain an asymptotic solution for long acoustic waves that is valid over a wide range of thermal wavelengths, on both sides of the thermal resonance condition. We show that this solution gives excellent agreement with both the full solution of the coupled Helmholtz equations and experimental measurements. This treatment provides a bridge between perturbation theory approximations in the long wavelength limit and high frequency solutions to the thermal field employing the geometric theory of diffraction.