Two-step coalition values for multichoice games

We introduce and compare several coalition values for multichoice games. Albizuri defined coalition structures and an extension of the Owen coalition value for multichoice games using the average marginal contribution of a player over a set of orderings of the player’s representatives. Following an approach used for cooperative games, we introduce a set of nested or two-step coalition values on multichoice games which measure the value of each coalition and then divide this among the players in the coalition using either a Shapley or Banzhaf value at each step. We show that when a Shapley value is used in both steps, the resulting coalition value coincides with that of Albizuri. We axiomatize the three new coalition values and show that each set of axioms, including that of Albizuri, is independent. Further we show how the multilinear extension can be used to compute the coalition values. We conclude with a brief discussion about the applicability of the different values.     

Semivalues as power indices

A restricted notion of semivalue as a power index, i.e. as a value on the lattice of simple games, is axiomatically introduced by using the symmetry, positivity and dummy player standard properties together with the transfer property. The main theorem, that parallels the existing statement for semivalues on general cooperative games, provides a combinatorial definition of each semivalue on simple games in terms of weighting coefficients, and shows the crucial role of the transfer property in this class of games. A similar characterization is also given that refers to unanimity coefficients, which describe the action of the semivalue on unanimity games. We then combine the notion of induced semivalue on lower cardinalities with regularity and obtain a series of characteristic properties of regular semivalues on simple games, that concern null and nonnull players, subgames, quotients, and weighted majority games.     

Reconstructing a simple game from a uniparametric family of allocations

Several relationships between simple games and a particular type of solutions for cooperative games are studied in this paper. These solutions belong to the set of semivalues and they are related to a unique parameter that explicitly provides their weighting coefficients. Through the allocations offered by this family of solutions, so-called binomial semivalues, and also from their respective potentials, some characteristics of the simple games can be recovered. The paper analyzes the capacity of binomial semivalues to summarize the structure of simple games, and, moreover, a property of separation among simple games is given.     

Partnership formation and binomial semivalues

Partnership formation in cooperative games is studied, and binomial semivalues are used to measure the effects of such a type of coalition arising from an agreement between (a group of) players. The joint effect on the set of involved players is also compared with that of the alternative alliance formation. The simple game case is especially considered, and the application to a real life example illustrates the use of coalitional values closely related to the binomial semivalues when dealing with partnership formation and coalitional bargaining simultaneously.     

Compensations in the Shapley value and the compensation solutions for graph games

We consider an alternative expression of the Shapley value that reveals a system of compensations: each player receives an equal share of the worth of each coalition he belongs to, and has to compensate an equal share of the worth of any coalition he does not belong to. We give a representation in terms of formation of the grand coalition according to an ordering of the players and define the corresponding compensation vector. Then, we generalize this idea to cooperative games with a communication graph in order to construct new allocation rules called the compensation solutions. Firstly, we consider cooperative games with arbitrary graphs and construct rooted spanning trees (see Demange, J Political Econ 112:754–778, 2004) instead of orderings of the players by using the classical algorithms DFS and BFS. If the graph is complete, we show that the compensation solutions associated with DFS and BFS coincide with the Shapley value and the equal surplus division respectively. Secondly, we consider cooperative games with a forest (cycle-free graph) and all its rooted spanning trees. The compensation solution is characterized by component efficiency and relative fairness. The latter axiom takes into account the relative position of a player with respect to his component in the communication graph.     

On ordinal equivalence of the Shapley and Banzhaf values for cooperative games

In this paper I consider the ordinal equivalence of the Shapley and Banzhaf values for TU cooperative games, i.e., cooperative games for which the preorderings on the set of players induced by these two values coincide. To this end I consider several solution concepts within semivalues and introduce three subclasses of games which are called, respectively, weakly complete, semicoherent and coherent cooperative games. A characterization theorem in terms of the ordinal equivalence of some semivalues is given for each of these three classes of cooperative games. In particular, the Shapley and Banzhaf values as well as the segment of semivalues they limit are ordinally equivalent for weakly complete, semicoherent and coherent cooperative games.     

Axiomatic characterizations of the symmetric coalitional binomial semivalues

The symmetric coalitional binomial semivalues extend the notion of binomial semivalue to games with a coalition structure, in such a way that they generalize the symmetric coalitional Banzhaf value. By considering the property of balanced contributions within unions, two axiomatic characterizations for each one of these values are provided.     

Semivalues: weighting coefficients and allocations on unanimity games

Each semivalue, as a solution concept defined on cooperative games with a finite set of players, is univocally determined by weighting coefficients that apply to players’ marginal contributions. Taking into account that a semivalue induces semivalues on lower cardinalities, we prove that its weighting coefficients can be reconstructed from the last weighting coefficients of its induced semivalues. Moreover, we provide the conditions of a sequence of numbers in order to be the family of the last coefficients of any induced semivalues. As a consequence of this fact, we give two characterizations of each semivalue defined on cooperative games with a finite set of players: one, among all semivalues; another, among all solution concepts on cooperative games.     

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.     

Axiomatic characterizations of generalized values

In the framework of cooperative game theory, the concept of generalized value, which is an extension of that of value, has been recently proposed to measure the overall influence of coalitions in games. Axiomatizations of two classes of generalized values, namely probabilistic generalized values and generalized semivalues, which extend probabilistic values and semivalues, respectively, are first proposed. The axioms we utilize are based on natural extensions of axioms involved in the axiomatizations of values. In the second half of the paper, special instances of generalized semivalues are also axiomatized.     

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.     

Multilinear extensions and values for multichoice games

We define multilinear extensions for multichoice games and relate them to probabilistic values and semivalues. We apply multilinear extensions to show that the Banzhaf value for a compound multichoice game is not the product of the Banzhaf values of the component games, in contrast to the behavior in simple games. Following Owen (Manag Sci 18:64–79, 1972), we integrate the multilinear extension over a simplex to construct a version of the Shapley value for multichoice games. We compare this new Shapley value to other extensions of the Shapley value to multichoice games. We also show how the probabilistic value (resp. semivalue, Banzhaf value, Shapley value) of a multichoice game is equal to the probabilistic value (resp. semivalue, Banzhaf value, Shapley value) of an appropriately defined TU decomposition game. Finally, we explain how semivalues, probabilistic values, the Banzhaf value, and this Shapley value may be viewed as the probability that a player makes a difference to the outcome of a simple multichoice game.     

Sharing Nonconvex Costs

The paper analyzes cooperative games with side payments. Each player faces a possibly non-convex optimization problem, interpreted as production planning, constrained by his resources or technology. Coalitions can aggregate (or pool) members' contributions. We discuss instances where such aggregation eliminates or reduces the lack of convexity. Core solutions are computed or approximated via dual programs associated to the grand coalition.     

Accessibility measures to nodes of directed graphs using solutions for generalized cooperative games

The aim of this paper consists of constructing accessibility measures to the nodes of directed graphs using methods of Game Theory. Since digraphs without a predefined game are considered, the main part of the paper is devoted to establish conditions on cooperative games so that they can be used to measure accessibility. Games that satisfy desirable properties are called test games. Each ranking on the nodes is then obtained according to a pair formed by a test game and a solution defined on cooperative games whose utilities are given on ordered coalitions. The solutions proposed here are extensions of the wide family of semivalues to games in generalized characteristic function form.     

The interval Shapley value: an axiomatization

The Shapley value, one of the most widespread concepts in operations Research applications of cooperative game theory, was defined and axiomatically characterized in different game-theoretic models. Recently much research work has been done in order to extend OR models and methods, in particular cooperative game theory, for situations with interval data. This paper focuses on the Shapley value for cooperative games where the set of players is finite and the coalition values are compact intervals of real numbers. The interval Shapley value is characterized with the aid of the properties of additivity, efficiency, symmetry and dummy player, which are straightforward generalizations of the corresponding properties in the classical cooperative game theory.     

Games with a permission structure - A survey on generalizations and applications

In the field of cooperative games with restricted cooperation, various restrictions on coalition formation are studied. The most studied restrictions are those that arise from restricted communication and hierarchies. This survey discusses several models of hierarchy restrictions and their relation with communication restrictions. In the literature, there are results on game properties, Harsanyi dividends, core stability, and various solutions that generalize existing solutions for TU-games. In this survey, we mainly focus on axiomatizations of the Shapley value in different models of games with a hierarchically structured player set, and their applications. Not only do these axiomatizations provide insight in the Shapley value for these models, but also by considering the types of axioms that characterize the Shapley value, we learn more about different network structures. A central model of games with hierarchies is that of games with a permission structure where players in a cooperative transferable utility game are part of a permission structure in the sense that there are players that need permission from other players before they are allowed to cooperate. This permission structure is represented by a directed graph. Generalizations of this model are, for example, games on antimatroids, and games with a local permission structure. Besides discussing these generalizations, we briefly discuss some applications, in particular auction games and hierarchically structured firms.     

The Shapley value for cooperative games under precedence constraints

Cooperative games are considered where only those coalitions of players are feasible that respect a given precedence structure on the set of players. Strengthening the classical symmetry axiom, we obtain three axioms that give rise to a unique Shapley value in this model. The Shapley value is seen to reflect the expected marginal contribution of a player to a feasible random coalition, which allows us to evaluate the Shapley value nondeterministically. We show that every exact algorithm for the Shapley value requires an exponential number of operations already in the classical case and that even restriction to simple games is #P-hard in general. Furthermore, we outline how the multi-choice cooperative games of Hsiao and Raghavan can be treated in our context, which leads to a Shapley value that does not depend on pre-assigned weights. Finally, the relationship between the Shapley value and the permission value of Gilles, Owen and van den Brink is discussed. Both refer to formally similar models of cooperative games but reflect complementary interpretations of the precedence constraints and thus give rise to fundamentally different solution concepts.     

Share functions for cooperative games with levels structure of cooperation

In a standard TU-game it is assumed that every subset of the player set N can form a coalition and earn its worth. One of the first models where restrictions in cooperation are considered is the one of games with coalition structure of Aumann and Drèze (1974). They assumed that the player set is partitioned into unions and that players can only cooperate within their own union. Owen (1977) introduced a value for games with coalition structure under the assumption that also the unions can cooperate among them. Winter (1989) extended this value to games with levels structure of cooperation, which consists of a game and a finite sequence of partitions defined on the player set, each of them being coarser than the previous one.     

一种资源投入不确定情形下的合作博弈形式及收益分配策略

首先,将经典合作博弈进行扩展,提出了一类模糊联盟合作博弈的通用形式,涵盖常见三种模糊联盟合作博弈,即多线性扩展博弈、比例模糊博弈与Choquet积分模糊博弈.比例模糊博弈、Choquet积分模糊博弈的Shapley值均可以作为一种特定形式下模糊联盟合作博弈的收益分配策略,但是对于多线性扩展博弈的Shapley值一直关注较少,因此利用经典Shapley值构造出多线性扩展博弈的Shapley值,以此作为一种收益分配策略.最后,通过实例分析了常见三类模糊联盟合作博弈的形式及其对应的分配策略,分析收益最大的模糊联盟合作对策形式及最优分配策略,为不确定情形下的合作问题提供了一定的收益分配依据.     

基于Choquet积分的直觉模糊联盟合作博弈的Shapley值

本文针对联盟是直觉模糊集的合作博弈Shapley值进行了研究.通过区间Choquet积分得到直觉模糊联盟合作博弈的特征函数为区间数,并研究了该博弈特征函数性质。根据拓展模糊联盟合作博弈Shapley值的计算方法,得到直觉模糊联盟合作博弈Shapley值的计算公式,该计算公式避免了区间数的减法。进一步证明了其满足经典合作博弈Shapley值的公理性。最后通过数值实例说明本文方法的合理性和有效性。