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 bargaining set for monotonic simple games based on external and internal stability

A new bargaining set based on notions of both internal and external stability is developed in the context of endogenous coalition formation. It allows one to make an explicit distinction between within-group and outside-group deviation options. This type of distinction is not present in current bargaining sets. For the class of monotonic proper simple games, the outcomes in the bargaining set are characterized. Furthermore, it is shown that the bargaining set of any homogeneous weighted majority game contains an outcome for which the underlying coalition structure consists of a minimal winning coalition and its complement.     

Accessibility and stability of the coalition structure core

This article shows that, for any transferable utility game in coalitional form with a nonempty coalition structure core, the number of steps required to switch from a payoff configuration out of the coalition structure core to a payoff configuration in the coalition structure core is less than or equal to $(n^2+4n)/4$ , where $n$ is the cardinality of the player set. This number improves the upper bounds found so far. We also provide a sufficient condition for the stability of the coalition structure core, i.e. a condition which ensures the accessibility of the coalition structure core in one step. On the class of simple games, this sufficient condition is also necessary and has a meaningful interpretation.     

Coalitional games and contracts based on individual deviations

We study what coalitions form and how the members of each coalition split the coalition value in coalitional games in which only individual deviations are allowed. In this context we employ three stability notions: individual, contractual, and compensational stability. These notions differ in terms of the underlying contractual assumptions. We characterize the coalitional games in which individually stable outcomes exist by means of the top-partition property. Furthermore, we show that any coalition structure of maximum social worth is both contractually and compensationally stable.     

The aspiration approach to predicting coalition formation and payoff distribution in sidepayment games

This paper presents the aspiration approach to coalition formation and payoff distribution in games with sidepayments. The approach is based on the idea that players set prices for their participation within coalitions. The solution space which is appropriate for price-setting players is different from that of the usual solution concepts and is called the space of aspirations. Solution concepts defined on the space of aspirations correspond to notions of how players bargain over their prices. Once the players choose a vector of prices, the coalitions which can afford to pay these prices are the coalitions which are predicted to form in the game.     

Shapley mappings and the cumulative value for n-person games with fuzzy coalitions

In this paper we prove existence and uniqueness of the so-called Shapley mapping, which is a solution concept for a class of n-person games with fuzzy coalitions whose elements are defined by the specific structure of their characteristic functions. The Shapley mapping, when it exists, associates to each fuzzy coalition in the game an allocation of the coalitional worth satisfying the efficiency, the symmetry, and the null-player conditions. It determines a “cumulative value” that is the “sum” of all coalitional allocations for whose computation we provide an explicit formula.     

Size of subsets of groups and Haar null sets

This is a study of several notions of size of subsets of groups. The first part (sections 3–5) concerns a purely algebraic setting with the underlying group discrete. The various notions of size considered there are similar to each other in that each of them assesses the size of a set using a family of measures and translations of the set; they differ in the type of measures used and the type of translations allowed. The way these various notions relate to each other is tightly and, perhaps, unexpectedly connected with the algebraic structure of the group. An important role is played by amenable, ICC (infinite conjugacy class), and FC (finite conjugacy class) groups.The second part of the paper (section 6), which was the original motivation for the present work, deals with a well-studied notion of smallness of subsets of Polish, not necessarily locally compact, groups – Haar null sets. It contains applications of the results from the first part in solving some problems posed by Christensen and by Mycielski. These applications are the first results detecting connections between properties of Haar null sets and algebraic properties (amenability, FC) of the underlying group.Received: October 2003 Revision: January 2004 Accepted: January 2004     

Constructive equivalence relations on computable probability measures

A central object of study in the field of algorithmic randomness are notions of randomness for sequences, i.e., infinite sequences of zeros and ones. These notions are usually defined with respect to the uniform measure on the set of all sequences, but extend canonically to other computable probability measures. This way each notion of randomness induces an equivalence relation on the computable probability measures where two measures are equivalent if they have the same set of random sequences.In what follows, we study the equivalence relations induced by Martin-Löf randomness, computable randomness, Schnorr randomness and Kurtz randomness, together with the relations of equivalence and consistency from probability theory. We show that all these relations coincide when restricted to the class of computable strongly positive generalized Bernoulli measures. For the case of arbitrary computable measures, we obtain a complete and somewhat surprising picture of the implications between these relations that hold in general.     

The Nakamura Theorem for coalition structures of quota games

This paper considers a model of society $S$ with a finite number of individuals,n, a finite set off alternatives, Ω effective coalitions that must contain ana priori given numberq of individuals. Its purpose is to extend the Nakamura Theorem (1979) to the quota games where individuals are allowed to form groups of sizeq which are smaller than the grand coalition. Our main result determines the upper bound on the number of alternatives which would guarantee, for a given e andq, the existence of a stable coalition structure for any profile of complete transitive preference relations. Our notion of stability, $S$ -equilibrium, introduced by Greenberg-Weber (1993), combines bothfree entry andfree mobility and represents the natural extension of the core to improper or non-superadditive games where coalition structures, and not only the grand coalition, are allowed to form.     

Endolocality meets homomorphism-homogeneity: a new approach in the study of relational algebras

We study the problem of characterizing all relations that can be defined from the fundamental relations of a given relational structure using positive existential formulæ. The notion of κ-endolocality is introduced in order to measure the complexity of relational structures with respect to this task. The hierarchy of κ-endolocal structures is thoroughly analysed in algebraic and model-theoretic ways. Interesting cross-connections with homomorphism-homogeneous relational structures are revealed. The interrelations between endolocal relational structures and several model-theoretic notions are collected in the Main Theorem. This Main Theorem is demonstrated to be a useful tool for studying relational algebras and, in particular, weak Krasner algebras. For example, a short proof of F. Börners characterization of weak Krasner clones on a countable set is given.     

基于联盟结构的模糊合作博弈的收益分配方案

研究了具有联盟结构的企业联盟模糊情况下各局中人的收益分配问题.首先拓展了Owen联盟值在经典意义下满足的5个公理,利用Choquet积分给出了基于联盟结构的模糊合作博弈的Owen联盟值,即模糊Owen联盟值的具体形式,并证明该联盟值满足新定义的5个公理.最后用实例验证了模糊Owen联盟值方法,并对计算结果进行分析。     

On coalition formation: a game-theoretical approach

This paper deals with the question of coalition formation inn-person cooperative games. Two abstract game models of coalition formation are proposed. We then study the core and the dynamic solution of these abstract games. These models assume that there is a rule governing the allocation of payoffs to each player in each coalition structure called a payoff solution concept. The predictions of these models are characterized for the special case of games with side payments using various payoff solution concepts such as the individually rational payoffs, the core, the Shapley value and the bargaining set M₁ ⁽ⁱ⁾. Some modifications of these models are also discussed.     

The collective value: a new solution for games with coalition structures

In this study, we provide a new solution for cooperative games with coalition structures. The collective value of a player is defined as the sum of the equal division of the pure surplus obtained by his coalition from the coalitional bargaining and of his Shapley value for the internal coalition. The weighted Shapley value applied to a game played by coalitions with coalition-size weights is assigned to each coalition, reflecting the size asymmetries among coalitions. We show that the collective value matches exogenous interpretations of coalition structures and provide an axiomatic foundation of this value. A noncooperative mechanism that implements the collective value is also presented.     

Potential,value, and coalition formation

In this paper, a simple probabilistic model of coalition formation provides a unified interpretation for several extensions of the Shapley value. Weighted Shapley values, semivalues, weak (weighted or not) semivalues, and the Shapley value itself appear as variations of this model. Moreover, some notions that have been introduced in the search of alternatives to Shapley’s seminal characterization, as ‘balanced contributions’ and the ‘potential’ are reinterpreted from this point of view. Natural relationships of these conditions with some mentioned families of ‘values’ are shown. These reinterpretations strongly suggest that these conditions are more naturally interpreted in terms of coalition formation than in terms of the classical notion of ‘value.’      

Hedonic coalition formation games: A new stability notion

This paper studies hedonic coalition formation games where each player’s preferences rely only upon the members of her coalition. A new stability notion under free exit-free entry membership rights, referred to as strong Nash stability, is introduced which is stronger than both core and Nash stabilities studied earlier in the literature. Strong Nash stability has an analogue in non-cooperative games and it is the strongest stability notion appropriate to the context of hedonic coalition formation games. The weak top-choice property is introduced and shown to be sufficient for the existence of a strongly Nash stable partition. It is also shown that descending separable preferences guarantee the existence of a strongly Nash stable partition. Strong Nash stability under different membership rights is also studied.     

The Theory of Linear G-Difference Equations

We introduce the notion of difference equations defined on a structured set. The symmetry group of the structure determines the set of difference operators. All main notions in the theory of difference equations are introduced as invariants of the action of the symmetry group. Linear equations are modules over the skew group algebra, solutions are morphisms relating a given equation to other equations, symmetries of an equation are module endomorphisms, and conserved structures are invariants in the tensor algebra of the given equation.We show that the equations and their solutions can be described through representations of the isotropy group of the symmetry group of the underlying set. We relate our notion of difference equation and solutions to systems of classical difference equations and their solutions and show that out notions include these as a special case.     

Quota games with a continuum of players

In this paper the notion ofm-quota game with a continuum of players is defined and the theory of bargaining sets is generalized to this new class of games. We discuss only the bargaining setM ₀ and our results are similar to those obtained in the finite case. Our main result is that for maximal coalition structures the stable payoff functions are exactly those in which almost every non-weak player receives no more than his quota and the weak players receive zero.     

Coalition formation in the triad when two are weak and one is strong

Eighteen groups of subjects each participated in five different computer-controlled superadditive 3-person characteristic function games with sidepayments, that modeled negotiable conflicts in which two of the players are weak and one is considerably stronger. Both the degree to which the strong player was powerful and the type of communication were experimentally manipulated. The 90 game outcomes rejected any solution concept that predicts a single payoff vector for a given coalition structure, but supported the recently developed single-parameter α-power model that allows range predictions. Both the degree of power and type of communication were found to affect game outcomes and to determine the predictive power of models that make point predictions in 3-person games.     

Reducible classes of finite lattices

In this paper we study a notion of reducibility in finite lattices. An element x of a (finite) lattice L satisfying certain properties is deletable if L-x is a lattice satisfying the same properties. A class of lattices is reducible if each lattice of this class admits (at least) one deletable element (equivalently if one can go from any lattice in this class to the trivial lattice by a sequence of lattices of the class obtained by deleting one element in each step). First we characterize the deletable elements in a pseudocomplemented lattice what allows to prove that the class of pseudocomplemented lattices is reducible. Then we characterize the deletable elements in semimodular, modular and distributive lattices what allows to prove that the classes of semimodular and locally distributive lattices are reducible. In conclusion the notion of reducibility for a class of lattices is compared with some other notions like the notion of order variety.