Totally balanced combinatorial optimization games

Combinatorial optimization games deal with cooperative games for which the value of every subset of players is obtained by solving a combinatorial optimization problem on the resources collectively owned by this subset. A solution of the game is in the core if no subset of players is able to gain advantage by breaking away from this collective decision of all players. The game is totally balanced if and only if the core is non-empty for every induced subgame of it.?We study the total balancedness of several combinatorial optimization games in this paper. For a class of the partition game [5], we have a complete characterization for the total balancedness. For the packing and covering games [3], we completely clarify the relationship between the related primal/dual linear programs for the corresponding games to be totally balanced. Our work opens up the question of fully characterizing the combinatorial structures of totally balanced packing and covering games, for which we present some interesting examples: the totally balanced matching, vertex cover, and minimum coloring games. Received: November 5, 1998 / Accepted: September 8, 1999?Published online February 23, 2000     

On the balancedness of multiple machine sequencing games

This paper takes a game theoretical approach to sequencing situations with m parallel and identical machines. We show that in a cooperative environment cooperative m-sequencing games, which involve n players, give rise to m-machine games, which involve m players. Here, n corresponds to the number of jobs in an m-sequencing situation, and m corresponds to the number of machines in the same m-sequencing situation. We prove that an m-sequencing game is balanced if and only if the corresponding m-machine game is balanced. Furthermore, it is shown that m-sequencing games are balanced if m∈{1,2}. Finally, if m⩾3, balancedness is established for two special classes of m-sequencing games. Furthermore, we consider a special class of m-sequencing situations in a noncooperative setting and show that a transfer payments scheme exists that is both incentive compatible and budget balanced.     

Totally balanced games arising from controlled programming problems

A cooperative game in characteristic-function form is obtained by allowing a number of individuals to esercise partial control over the constraints of a (generally nonlinear) mathematical programming problem, either directly or through committee voting. Conditions are imposed on the functions defining the programming problem and the control system which suffice to make the game totally balanced. This assures a nonempty core and hence a stable allocation of the full value of the programming problem among the controlling palyers. In the linear case the core is closely related to the solutions of the dual problem. Applications are made to a variety of economic models, including the transferable utility trading economies of Shapley and Shubik and a multishipper one-commodity transshipment model with convex cost functions and concave revenue functions. Dropping the assumption of transferable utility leads to a class of controlled multiobjective or ‘Pareto programming’ problems, which again yield totally balanced games.     

A fundamental study for partially defined cooperative games

The payoff of each coalition has been assumed to be known precisely in the conventional cooperative games. However, we may come across situations where some coalitional values remain unknown. This paper treats cooperative games whose coalitional values are not known completely. In the cooperative games it is assumed that some of coalitional values are known precisely but others remain unknown. Some complete games associated with such incomplete games are proposed. Solution concepts are studied in a special case where only values of the grand coalition and singleton coalitions are known. Through the investigations of solutions of complete games associated with the given incomplete game, we show a focal point solution suggested commonly from different viewpoints.     

Communication and Cooperation in Public Network Situations

This paper focuses on sharing the costs and revenues of maintaining a public network communication structure. Revenues are assumed to be bilateral and communication links are publicly available but costly. It is assumed that agents are located at the vertices of an undirected graph in which the edges represent all possible communication links. We take the approach from cooperative game theory and focus on the corresponding network game in coalitional form which relates any coalition of agents to its highest possible net benefit, i.e., the net benefit corresponding to an optimal operative network. Although finding an optimal network in general is a difficult problem, it is shown that corresponding network games are (totally) balanced. In the proof of this result a specific relaxation, duality and techniques of linear production games with committee control play a role. Sufficient conditions for convexity of network games are derived. Possible extensions of the model and its results are discussed. The research of Jeroen Suijs has been made possible by a fellowship of the Royal Netherlands Academy of Arts and Sciences.     

A convex representation of totally balanced games

We analyze the least increment function, a convex function of n variables associated to an n-person cooperative game. Another convex representation of cooperative games, the indirect function, has previously been studied. At every point the least increment function is greater than or equal to the indirect function, and both functions coincide in the case of convex games, but an example shows that they do not necessarily coincide if the game is totally balanced but not convex. We prove that the least increment function of a game contains all the information of the game if and only if the game is totally balanced. We also give necessary and sufficient conditions for a function to be the least increment function of a game as well as an expression for the core of a game in terms of its least increment function.     

A general framework for cooperation under uncertainty

In this paper, we introduce a general framework for situations with decision making under uncertainty and cooperation possibilities. This framework is based upon a two stage stochastic programming approach. We show that under relatively mild assumptions the associated cooperative games are totally balanced. Finally, we consider several example situations.     

Constrained core solutions for totally positive games with ordered players

In many applications of cooperative game theory to economic allocation problems, such as river-, polluted river- and sequencing games, the game is totally positive (i.e., all dividends are nonnegative), and there is some ordering on the set of the players. A totally positive game has a nonempty core. In this paper we introduce constrained core solutions for totally positive games with ordered players which assign to every such a game a subset of the core. These solutions are based on the distribution of dividends taking into account the hierarchical ordering of the players. The Harsanyi constrained core of a totally positive game with ordered players is a subset of the core of the game and contains the Shapley value. For special orderings it coincides with the core or the Shapley value. The selectope constrained core is defined for acyclic orderings and yields a subset of the Harsanyi constrained core. We provide a characterization for both solutions.     

Balancedness conditions for exact games

We provide two new characterizations of exact games. First, a game is exact if and only if it is exactly balanced; and second, a game is exact if and only if it is totally balanced and overbalanced. The condition of exact balancedness is identical to the one of balancedness, except that one of the balancing weights may be negative, while for overbalancedness one of the balancing weights is required to be non-positive and no weight is put on the grand coalition. Exact balancedness and overbalancedness are both easy to formulate conditions with a natural game-theoretic interpretation and are shown to be useful in applications. Using exact balancedness we show that exact games are convex for the grand coalition and we provide an alternative proof that the classes of convex and totally exact games coincide. We provide an example of a game that is totally balanced and convex for the grand coalition, but not exact. Finally we relate classes of balanced, totally balanced, convex for the grand coalition, exact, totally exact, and convex games to one another.     

Convex decomposition of games and axiomatizations of the core and the D-core

This paper provides an axiomatic framework to compare the D-core (the set of undominated imputations) and the core of a cooperative game with transferable utility. Theorem 1 states that the D-core is the only solution satisfying projection consistency, reasonableness (from above), (*)-antimonotonicity, and modularity. Theorem 2 characterizes the core replacing (*)-antimonotonicity by antimonotonicity. Moreover, these axioms also characterize the core on the domain of convex games, totally balanced games, balanced games, and superadditive games.      

On the equivalence between some local and global Chinese postman and traveling salesman graphs

A connected graph G=(V,E), a vertex in V and a non-negative weight function defined on E can be used to induce Chinese postman and traveling salesman (cooperative) games. A graph G=(V,E) is said to be locally (respectively, globally) Chinese postman balanced (respectively, totally balanced, submodular) if for at least one vertex (respectively, for all vertices) in V and any non-negative weight function defined on E, the corresponding Chinese postman game is balanced (respectively, totally balanced, submodular). Local and global traveling salesman balanced (respectively, totally balanced, submodular) graphs are similarly defined.In this paper, we study the equivalence between local and global Chinese postman balanced (respectively, totally balanced, submodular) graphs, and between local and global traveling salesman submodular graphs.     

A polynomial time algorithm for computing the nucleolus for a class of disjunctive games with a permission structure

Recently, applications of cooperative game theory to economic allocation problems have gained popularity. In many such allocation problems there is some hierarchical ordering of the players. In this paper we consider a class of games with a permission structure describing situations in which players in a cooperative TU-game are hierarchically ordered in the sense that there are players that need permission from other players before they are allowed to cooperate. The corresponding restricted game takes account of the limited cooperation possibilities by assigning to every coalition the worth of its largest feasible subset. In this paper we provide a polynomial time algorithm for computing the nucleolus of the restricted games corresponding to a class of games with a permission structure which economic applications include auction games, dual airport games, dual polluted river games and information market games.     

Multiple Fund Investment Situations and Related Games

This paper deals with interactive multiple fund investment situations, in which investors can invest their capital in a number of funds. The investors, however, face some restrictions. In particular, the investment opportunities of an investor depend on the behaviour of the other investors. Moreover, the individual investment returns may differ. We consider this situation from a cooperative game theory point of view. Based on different assumptions modelling the gains of joint investment, we consider three corresponding games and analyse their properties. We propose an allocation process for the maximal total investment revenues.Ruud Hendrickx acknowledges financial support from the Netherlands Organisation for Scientific Research (NWO).     

Stability and monotonicity in newsvendor situations

This study considers a supply chain that consists of n retailers, each of them facing a newsvendor problem, and a supplier. Groups of retailers might increase their expected joint profit by joint ordering and inventory centralization. However, we assume that the retailers impose some level of stock that should be dedicated to them. In this situation, we show that the associated cooperative game has a non-empty core. Afterwards, we concentrate on a dynamic situation, where several model cost parameters and the retailers’ dedicated stock levels can change. We investigate how the profit division might be affected by these changes. We focus on four monotonicity properties. We identify several classes of games with retailers, where some of the monotonicity properties hold. Moreover, we show that pairs of cooperative games associated with newsvendor situations do not necessarily satisfy these properties in general, when changes in dedicated stock levels are in concern.     

Totally monotonic games and flow games

Equivalences between totally balanced games and flow games, and between monotonic games and pseudoflow games are well-known. This paper shows that for every totally monotonic game there exists an equivalent flow game and that for every monotonic game, there exists an equivalent flow-based secondary market game.     

The maximum and the addition of assignment games

In the framework of two-sided assignment markets, we first consider that, with several markets available, the players may choose where to trade. It is shown that the corresponding game, represented by the maximum of a finite set of assignment games, may not be balanced. Some conditions for balancedness are provided and, in that case, properties of the core are analyzed. Secondly, we consider that players may trade simultaneously in more than one market and then add up the profits. The corresponding game, represented by the sum of a finite set of assignment games, is balanced. Moreover, under some conditions, the sum of the cores of two assignment games coincides with the core of the sum game.     

Cooperative games with stochastic payoffs

This paper introduces a new class of cooperative games arising from cooperative decision making problems in a stochastic environment. Various examples of decision making problems that fall within this new class of games are provided. For a class of games with stochastic payoffs where the preferences are of a specific type, a balancedness concept is introduced. A variant of Farkas' lemma is used to prove that the core of a game within this class is non-empty if and only if the game is balanced. Further, other types of preferences are discussed. In particular, the effects the preferences have on the core of these games are considered.     

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.     

On multi-choice TU games arising from replica of economies

In this paper we derive a multi-choice TU game from r-replica of exchange economy with continuous, concave and monetary utility functions, and prove that the cores of the games converge to a subset of the set of Edgeworth equilibria of exchange economy as r approaches to infinity. We prove that the dominance core of each balanced multi-choice TU game, where each player has identical activity level r, coincides with the dominance core of its corresponding r-replica of exchange economy. We also give an extension of the concept of the cover of the game proposed by Shapley and Shubik (J Econ Theory 1: 9-25, 1969) to multi-choice TU games and derive some sufficient conditions for the nonemptyness of the core of multi-choice TU game by using the relationship among replica economies, multi-choice TU games and their covers.     

An algorithm for computing the nucleolus of disjunctive non-negative additive games with an acyclic permission structure

A cooperative game with a permission structure describes a situation in which players in a cooperative TU-game are hierarchically ordered in the sense that there are players that need permission from other players before they are allowed to cooperate. In this paper we consider non-negative additive games with an acyclic permission structure. For such a game we provide a polynomial time algorithm for computing the nucleolus of the induced restricted game. The algorithm is applied to a market situation where sellers can sell objects to buyers through a directed network of intermediaries.