A Characterization of the Average Tree Solution for Cycle-Free Graph Games

Herings et al. (2008) proposed a solution concept called the average tree solution for cycle-free graph games. We provide a characterization of the average tree solution for cycle-free graph games. The characteration underlines an important difference, in terms of symmetric treatment of agents, between the average tree solution and the Myerson value (Myerson, 1977) for cycle-free graph games.average tree solution;graph games;Myerson value;Shapley value

The Average Tree Solution for Cycle-free Graph Games

In this paper we study cooperative games with limited cooperation possibilities, represented by an undirected cycle-free communication graph. Players in the game can cooperate if and only if they are connected in the graph. We introduce a new single-valued solution concept, the average tree solution. Our solution is characterized by component efficiency and component fairness. The interpretation of component fairness is that deleting a link between two players yields for both resulting components the same average change in payoff, where the average is taken over the players in the component. The average tree solution is always in the core of the restricted game 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 average tree solution can be generated by a specific distribution of the Harsanyi dividends. © 2007 Elsevier Inc. All rights reserved

The Two-Step Average Tree Value for Graph and Hypergraph Games

We introduce the two-step average tree value for transferable utility games with restricted cooperation represented by undirected communication graphs or hypergraphs. The solution can be considered as an alternative for both the average tree solution for graph games and the average tree value for hypergraph games. Instead of averaging players' marginal contributions corresponding to all admis-sible rooted spanning trees of the underlying (hyper)graph, which determines the average tree solution or value, we consider a two-step averaging procedure, in which in the first step for each player the average of players' marginal contributions corresponding to all admissible rooted spanning trees that have this player as the root is calculated, and in the second step the average over all players of all the payoffs obtained in the first step is computed. In general these two approaches lead to different solution concepts. When each component in the underlying communication structure is cycle-free, a linear cactus with cycles, or the complete graph, the two-step average tree value coincides with the average tree value. A comparative analysis of both solution concepts is done and an axiomatization of the the two-step average tree value on the subclass of TU games with semi-cycle-free hypergraph communication structure, which is more general than that given by a cycle-free hypergraph, is obtained

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;

A Strategic Implementation of the Average Tree Solution for Cycle-Free Graph Games

In this note we provide a strategic implementation of the Average Tree solution for zero-monotonic cycle-free graph games. That is, we propose a non-cooperative mechanism of which the unique subgame perfect equilibrium payoffs correspond to the average hierarchical outcome of the game. This mechanism takes into account that a player is only able to communicate with other players (i.e., to make proposals about a division of the surplus of cooperation) when they are connected in the graph. © 2013 Elsevier Inc

Tree, web and average web value for cycle-free directed graph games

On the class of cycle-free directed graph games with transferable utility solution concepts, called web values, are introduced axiomatically, each one with respect to some specific choice of a management team of the graph. We provide their explicit formula representation and simple recursive algorithms to calculate them. Additionally the efficiency and stability of web values are studied. Web values may be considered as natural extensions of the tree and sink values as has been defined correspondingly for rooted and sink forest graph games. In case the management team consists of all sources (sinks) in the graph a kind of tree (sink) value is obtained. In general, at a web value each player receives the worth of this player together with his subordinates minus the total worths of these subordinates. It implies that every coalition of players consisting of a player with all his subordinates receives precisely its worth. We also define the average web value as the average of web values over all management teams in the graph. As application the water distribution problem of a river with multiple sources, a delta and possibly islands is considered

Tree, web and average web value for cycle-free directed graph games

On the class of cycle-free directed graph games with transferable utility solution concepts, called web values, are introduced axiomatically, each one with respect to a chosen coalition of players that is assumed to be an anti-chain in the directed graph and is considered as a management team. We provide their explicit formula representation and simple recursive algorithms to calculate them. Additionally the efficiency and stability of web values are studied. Web values may be considered as natural extensions of the tree and sink values as has been defined correspondingly for rooted and sink forest graph games. In case the management team consists of all sources (sinks) in the graph a kind of tree (sink) value is obtained. In general, at a web value each player receives the worth of this player together with his subordinates minus the total worths of these subordinates. It implies that every coalition of players consisting of a player with all his subordinates receives precisely its worth. We also define the average web value as the average of web values over all management teams in the graph. As application the water distribution problem of a river with multiple sources, a delta and possibly islands is considered