The Component Fairness Solution for Cycle- Free Graph Games

In this paper we study cooperative games with limited cooperation possibilities, representedby an undirected cycle-free communication graph. Players in the game can cooperate if andonly if they are connected in the graph, i.e. they can communicate with one another. Weintroduce a new single-valued solution concept, the component fairness solution. Our solution is characterized by component efficiency and component fairness. The interpretationof component fairness is that deleting a link between two players yields for both resultingcomponents the same average change in payoff, where the average is taken over the players in the component. Component fairness replaces the axiom of fairness characterizing the Myerson value, where the players whose link is deleted face the same loss in payoff. Thecomponent fairness solution is always in the core of the restricted game in case the gameis superadditive and can be easily computed as the average of n specific marginal vectors,where n is the number of players. We also show that the component fairness solution canbe generated by a specific distribution of the Harsanyi-dividends.operations research and management science;

The Average Tree Solution for Cooperative Games with Communication Structure

We study cooperative games with communication structure, represented by an undirectedgraph. Players in the game are able to cooperate only if they can form a network in the graph. A single-valued solution, the average tree solution, is proposed for this class ofgames. Given the graph structure we define a collection of spanning trees, where eachspanning tree specifies a particular way by which players communicate and determines a payoff vector of marginal contributions of all the players. The average tree solution is defined to be the average of all these payoff vectors. It is shown that if a game has acomplete communication structure, then the proposed solution coincides with the Shapleyvalue, and that if the game has a cycle-free communication structure, it is the solutionproposed by Herings, van der Laan and Talman (2008). We introduce the notion of linkconvexity, under which the game is shown to have a non-empty core and the average tree solution lies in the core. In general, link-convexity is weaker than convexity. For games with a cycle-free communication structure, link-convexity is even weaker than super-additivity.operations research and management science;

Measuring the Power of Nodes in Digraphs

Many economic and social situations can be represented by a digraph. Both axiomatic and iterative methods to determine the strength or power of all the nodes in a digraph have been proposed in the literature. We propose a new method, where the power of a node is determined by both the number of its successors, as in axiomatic methods, and the powers of its successors, as in iterative methods. Contrary to other iterative methods, we obtain a full ranking of the nodes for any digraph. The new power function, called the positional power function, can either be determined as the unique solution to a system of equations, or as the limit point of an iterative process. The solution is also explicitly characterized. This characterization enables us to derive a number of interesting properties of the positional power function. Next we consider a number of extensions, like the positional weakness function and the position function.mathematical economics and econometrics ;

Socially Structured Games and Their Applications

In this paper we generalize the concept of a non-transferable utility game by introducing the concept of a socially structured game. A socially structured game is given by a set of players, a possibly empty collection of internal organizations on any subset of players, for any internal organization a set of attainable payoffs and a function on the collection of all internal organizations measuring the power of every player within the internal organization. Any socially structured game induces a non-transferable utility game. In the derived non-transferable utility game, all information concerning the dependence of attainable payoffs on the internal organization gets lost. We show this information to be useful for studying non-emptiness and refinements of the core.For a socially structured game we generalize the concept of π-balancedness to social stability and show that a socially stable game has a non-empty socially stable core. In order to derive this result, we formulate a new intersection theorem that generalizes the KKM-Shapley intersection theorem. The socially stable core is a subset of the core of the game. We give an example of a socially structured game that satisfies social stability, whose induced non-transferable utility game therefore has a non-empty core, but does not satisfy π-balanced for any choice of πThe usefulness of the new concept is illustrated by some applications and examples. In particular we define a socially structured game, whose unique element of the socially stable core corresponds to the Cournot-Nash equilibrium of a Cournot duopoly. This places the paper in the Nash research program, looking for a unifying approach to cooperative and non-cooperative behavior in which each theory helps to justify and clarify the other.microeconomics ;

The Transition from a Drèze Equilibrium to a Walrasian Equilibrium

In this paper a continuous time price and quantity adjustment process is considered for an economy facing price rigidities. In the short run prices are assumed to be completely fixed and the markets are cleared by quantity adjustments until a fixed price equilibrium is reached where every market is typically characterized by either supply rationing or demand rationing. Next prices are assumed to move upwards in case of demand rationing on a market and downwards when supply rationing occurs. Markets are kept in equilibrium by infinitesimal quantity adjustments such that at every moment in time a fixed price equilibrium results. Using only standard assumptions on the primitive concepts of the economy it is shown that the price and quantity adjustment process indeed converges to a fixed price equilibrium for the initially given prices. Moreover, in the long run, when prices are allowed to change, the process is shown to reach a Walrasian equilibrium. A simplicial algorithm is developed to approximate the price and quantity adjustment process arbitrarily close. It is shown that the path of price systems and rationing schemes generated by the algorithm converges to the path of the adjustment process.Adjustment processes; price rigidities; simplicial algorithms

Cooperative Games in Graph Structure

By a cooperative game in coalitional structure or shortly coalitional game we mean the standard cooperative non-transferable utility game described by a set of payoffs for each coalition that is a nonempty subset of the grand coalition of all players. It is well-known that balancedness is a sufficient condition for the nonemptiness of the core of such a cooperative non-transferable utility game. For this result any information on the internal organization of the coalition is neglected.In this paper we generalize the concept of coalitional games and allow for organizational structure within coalitions. For a subset of players any arbitrarily given structural relation represented by a graph is allowed for. We then consider non-transferable utility games in which a possibly empty set of payoff vectors is assigned to any graph on every subset of players. Such a game will be called a cooperative game in graph structure or shortly graph game. A payoff vector lies in the core of the game if there is no graph on a group of players which can make all of its members better off.We define the balanced-core of a graph game as a refinement of the core. To do so, for each graph a power vector is determined that depends on the relative positions of the players within the graph. A collection of graphs will be called balanced if to any graph in the collection a positive weight can be assigned such that the weighted power vectors sum up to the vector of ones. A payoff vector lies in the balanced-core if it lies in the core and the payoff vector is an element of payoff sets of all graphs in some balanced collection of graphs. We prove that any balanced graph game has a nonempty balanced-core and therefore a nonempty core. We conclude by some examples showing the usefulness of the concepts of graph games and balanced-core. In particular these examples show a close relationship between solutions to noncooperative games and balanced-core elements of a well-defined graph game. This places the paper in the Nash research program, looking for a unifying theory in which each approach helps to justify and clarify the other.microeconomics ;

Socially structured games

We generalize the concept of a cooperative non-transferable utility game by introducing a socially structured game. In a socially structured game every coalition of players can organize themselves according to one or more internal organizations to generate payoffs. Each admissible internal organization on a coalition yields a set of payoffs attainable by the members of this coalition. The strengths of the players within an internal organization depend on the structure of the internal organization and are represented by an exogenously given power vector. More powerful players have the power to take away payoffs of the less powerful players as long as those latter players are not able to guarantee their payoffs by forming a different internal organization within some coalition in which they have more power.we introduce the socially stable core as a solution concept that contains those payoffs that are both stable in an economic sense, i.e., belong to the core of the underlying cooperative game, and stable in a social sense, i.e., payoffs are sustained by a collection of internal organizations of coalitions for which power is distributed over all players in a balanced way. The socially stable core is a subset and therefore a refinement of the core. We show by means of examples that in many cases the socially stable core is a very small subset of the core.we will state conditions for which the socially stable core is non-empty. In order to derive this result, we formulate a new intersection theorem that generalizes the kkms intersection theorem. We also discuss the relationship between social stability and the wellknown concept of balancedness for ntu-games, a sufficient condition for non-emptiness of the core. In particular we give an example of a socially structured game that satisfies social stability and therefore has a non-empty core, but whose induced ntu-game does not satisfy balancedness in the general sense of billera