A solidarity value forn-person transferable utility games

In this paper, we introduce axiomatically a new value for cooperative TU games satisfying the efficiency, additivity, and symmetry axioms of Shapley (1953) and some new postulate connected with the average marginal contributions of the members of coalitions which can form. Our solution is referred to as the solidarity value. The reason is that its interpretation can be based on the assumption that if a coalition, sayS, forms, then the players who contribute toS more than the average marginal contribution of a member ofS support in some sense their weaker partners inS. Sometimes, it happens that the solidarity value belongs to the core of a game while the Shapley value does not.This research was supported by the KBN Grant 664/2/91 No. 211589101.     

The nucleolus and kernel of veto-rich transferable utility games

The process of computing the nucleolus of arbitrary transferable utility games is notoriously hard. A number of papers have appeared in which the nucleolus is computed by an algorithm in which either one or a huge number of huge linear programs have to be solved. We show that on the class of veto-rich games, the nucleolus is the unique kernel element. Veto-rich games are games in which one of the players is needed by coalitions in order to obtain a non-zero payoff. We then provide a fast algorithm which does not use linear programming techniques to compute the nucleolus of these games. Furthermore, we provide a few examples of economic situations which belong to the class of veto-rich games and which are treated in the literature.     

Stable outcomes of the roommate game with transferable utility

We consider the TU version of Gale and Shapley's roommate game. We find several results that are analogous to known results for the NTU game, such as a characterization of stable outcomes by forbidden minors, a characterization of the extreme points of the core, and a median property of stable outcomes. The TU roommate game is a special case of the TU partitioning game of Kaneko and Wooders. Bondareva and Shapley's balancedness condition for the core of such games is the starting point for our forbidden minors approach. Received: April 1999/Revised version: November 2000     

Polytopes of special classes of the balanced transferable utility games

Under study are the polytopes of (0, 1)-normalized convex and 1-convex (dual simplex) n-person TU-games and monotonic big boss games.We solve the characterization problems of the extreme points of the polytopes of 1-convex games, symmetric convex games, and big boss games symmetric with respect to the coalition of powerless agents. For the remaining polytopes, some subsets of extreme points are described.     

On extensions of the core and the anticore of transferable utility games

We consider several related set extensions of the core and the anticore of games with transferable utility. An efficient allocation is undominated if it cannot be improved, in a specific way, by sidepayments changing the allocation or the game. The set of all such allocations is called the undominated set, and we show that it consists of finitely many polytopes with a core-like structure. One of these polytopes is the $L_1$ -center, consisting of all efficient allocations that minimize the sum of the absolute values of the excesses. The excess Pareto optimal set contains the allocations that are Pareto optimal in the set obtained by ordering the sums of the absolute values of the excesses of coalitions and the absolute values of the excesses of their complements. The $L_1$ -center is contained in the excess Pareto optimal set, which in turn is contained in the undominated set. For three-person games all these sets coincide. These three sets also coincide with the core for balanced games and with the anticore for antibalanced games. We study properties of these sets and provide characterizations in terms of balanced collections of coalitions. We also propose a single-valued selection from the excess Pareto optimal set, the min-prenucleolus, which is defined as the prenucleolus of the minimum of a game and its dual.     

A well-posed shooting method for transferable DAE's

Summary Consider a TPBVP for transferable nonlinear DAE's. In general the shooting equation has a singular Jacobian. A multiple shooting method which has a nonsingular Jacobian and also produces consistent initial values for the integration is presented. The estimation of the condition of the Jacobian shows the well-posedness of the method. Some illustrative examples are given     

Inference in directed evidential networks based on the transferable belief model

Inference algorithms in directed evidential networks (DEVN) obtain their efficiency by making use of the represented independencies between variables in the model. This can be done using the disjunctive rule of combination (DRC) and the generalized Bayesian theorem (GBT), both proposed by Smets [Ph. Smets, Belief functions: the disjunctive rule of combination and the generalized Bayesian theorem, International Journal of Approximate Reasoning 9 (1993) 1–35]. These rules make possible the use of conditional belief functions for reasoning in directed evidential networks, avoiding the computations of joint belief function on the product space. In this paper, new algorithms based on these two rules are proposed for the propagation of belief functions in singly and multiply directed evidential networks.     

A new value for games without transferable utilities

We define axiomatically a unique concept of value for games without transferable utilities, which is a generalization ofNash's bargaining model. Unlike other concepts, it does not coincide with theShapley value in the case of transferable utilities.     

Note on models

The first part of this note deals with the importance of the polytechnical tradition for mathematical models.¹ In the second part I discuss the question of the “semantic shift” in the mathematical usage of the word “model” (M. Epple),² that means, the passage from concrete material objects to models in the modern sense of the word – abstract entities satisfying a certain system of axioms. My aim is to provide some new information on the theme and to open a new and in my opinion essential aspect – the polytechnical tradition – to look at it. The two parts are only in a loose connection.     

Note on non-D-rings

Abstract

Recall that an integral domain R is said to be a non-D-ring if there exists a non-constant polynomial f (X) in R[X] (called a uv-polynomial) such that f (a) is a unit of R for every a in R. In this note we generalize this notion to commutative rings (that are not necessarily integral domains) as follows: for a positive integer n, we say that R is an n-non-D-ring if there exists a polynomial f of degree n in R[X] such that f (a) is a unit of R for every a in R. We then investigate the properties of this notion in di?erent contexts of commutative rings.     

Two-person cooperative uncertain differential game with transferable payoffs

Fuzzy Optimization and Decision Making - Uncertain differential game models conflicts and interests among players in the context of an uncertain dynamic system. However, cooperative behavior in...     

Note on primitive words

This note presents an example that disproves, forn=4, Weinbaum’s conjecture, that ifw is a cyclically reduced primitive word inF _n such that all the generatorsx∈X appear inw then some cyclic permutation ofw can be partitioned inton words generatingF _n :w≡uv,vu≡s ₁ s ₂…s _n , <s ₁,s ₂,…s _n >=F _n .     

A Note on CLT-Groups

In this paper, we characterize finite groups G satisfying that, for every prime power divisor p ⁿ of the order of G, there exists a subgroup H of index p ⁿ (of order p ⁿ ) in G such that H is normal or abnormal in G.     

Note on fundamental operations

A new type of generalization of fundamental calculating operations is attempted. Instead of generalizing the nature of the elements upon which operations are acting, the nature of the operations themselves is changed. By iterating the step which leads from addition to multiplication, new types of operations are generated. Some of these operations are discussed briefly.     

Note on hit-and-miss topologies

This is a continuation of [19]. We characterize first and second countability of the general hit-and-miss hyperspace topologyτ ₊ ^Δ for weakly-R ₀ base spaces. Further, metrizability ofτ ₊ ^Δ is characterized with no preliminary conditions on the base space and the generating family of closed sets and a new proof on uniformizability (i.e. complete regularity) ofτ ₊ ^Δ is given in this general setting, thus generalizing results of [3], [5] and [6].