Set Of Feasible Coalitions Research Articles

Robust equilibria in tournaments

We study robust equilibria in tournaments, where agents endowed with power form coalitions, and the coalition formed with the highest power prevails. We introduce the No-Threat Equilibrium (NTE), a stable partition where if a coalition deviates, then a new coalition could counter by forming an even stronger coalition. The NTE exists for any power function and preferences if and only if the set of feasible coalitions is a ‘Helly’ family.In contrast, the core is a partition in which no group of agents can profitably deviate by forming a feasible coalition, assuming that other agents do not react to such a deviation. The core is not empty for any power function and preferences if and only if the set of feasible coalitions has a ‘hierarchical-structure’.The paper also adapts and characterizes other stability concepts to tournaments, including the α-core, β-core, farsighted core and bargaining set.

Properties of Solutions for Games on Union-Closed Systems

A situation in which a finite set of players can obtain certain payoffs by cooperation can be described by a cooperative game with transferable utility, or simply a TU-game. A solution for TU-games assigns a set of payoff distributions to every TU-game. In the literature, various models of games with restricted cooperation can be found where, instead of allowing all subsets of the player set N to form, it is assumed that the set of feasible coalitions is a subset of the power set of N. In this paper, we consider games on a union-closed system where the set of feasible coalitions is closed under the union, i.e., for any two feasible coalitions also, their union is feasible. Properties of solutions (the core, the nucleolus, and the prekernel) are discussed for games on a union-closed system.

Cooperative games on intersection closed systems and the Shapley value

A situation in which a finite set of agents can obtain certain payoffs by cooperation can be described by a cooperative game with transferable utility, or simply a TU-game. In the literature, various models of games with restricted cooperation can be found, in which only certain subsets of the agent set are allowed to form. In this article, we consider such sets of feasible coalitions that are closed under intersection, i.e., for any two feasible coalitions, their intersection is also feasible. Such set systems, called intersection closed systems, are a generalization of the convex geometries. We use the concept of closure operator for intersection closed systems and we define the restricted TU-game taking into account the limited possibilities of cooperation determined by the intersection closed system. Next, we study the properties of this restricted TU-game. Finally, we introduce and axiomatically characterize a family of allocation rules for TU-games on intersection closed systems, which contains a natural extension of the Shapley value.

On a class of vertices of the core

It is known that for supermodular TU-games, the vertices of the core are the marginal vectors, and this result remains true for games where the set of feasible coalitions is a distributive lattice. Such games are induced by a hierarchy (partial order) on players. We propose a larger class of vertices for games on distributive lattices, called min–max vertices, obtained by minimizing or maximizing in a given order the coordinates of a core element. We give a simple formula which does not need to solve an optimization problem to compute these vertices, valid for connected hierarchies and for the general case under some restrictions. We find under which conditions two different orders induce the same vertex for every game, and show that there exist balanced games whose core has vertices which are not min–max vertices if and only if n>4.

A cooperative game-theoretic approach to the social ridesharing problem

In this work, we adopt a cooperative game theoretic approach in order to tackle the social ridesharing (SR) problem, where a set of commuters, connected through a social network, form coalitions and arrange one-time rides at short notice. In particular, we address two fundamental aspects of this problem. First, we focus on the optimisation problem of forming the travellers' coalitions that minimise the travel cost of the overall system. To this end, we model the formation problem as a Graph-Constrained Coalition Formation (GCCF) one, where the set of feasible coalitions is restricted by a graph (i.e., the social network). Our approach allows users to specify both spatial and temporal preferences for the trips. Second, we tackle the payment allocation aspect of SR, by proposing the first approach that computes kernel-stable payments for systems with thousands of agents. We conduct a systematic empirical evaluation that uses real-world datasets (i.e., GeoLife and Twitter). We are able to compute optimal solutions for medium-sized systems (i.e., with 100 agents), and high quality solutions for very large systems (i.e., up to 2000 agents). Our results show that our approach improves the social welfare (i.e., reduces travel costs) by up to 36.22% with respect to the scenario with no ridesharing. Finally, our payment allocation method computes kernel-stable payments for 2000 agents in less than an hour—while the state of the art is able to compute payments only for up to 100 agents, and does so 84 times slower than our approach.

On a Class of Vertices of the Core

It is known that for supermodular TU-games, the vertices of the core are the marginal vectors, and this result remains true for games where the set of feasible coalitions is a distributive lattice. Such games are induced by a hierarchy (partial order) on players. We propose a larger class of vertices for games on distributive lattices, called min-max vertices, obtained by minimizing or maximizing in a given order the coordinates of a core element. We give a simple formula which does not need to solve an optimization problem to compute these vertices, valid for connected hierarchies and for the general case under some restrictions. We find under which conditions two different orders induce the same vertex for every game, and show that there exist balanced games whose core has vertices which are not min-max vertices if and only if n > 4.

Sharing Rides with Friends: A Coalition Formation Algorithm for Ridesharing

We consider the Social Ridesharing (SR) problem, where a set of commuters, connected through a social network, arrange one-time rides at short notice. In particular, we focus on the associated optimisation problem of forming cars to minimise the travel cost of the overall system modelling such problem as a graph constrained coalition formation (GCCF) problem, where the set of feasible coalitions is restricted by a graph (i.e., the social network). Moreover, we significantly extend the state of the art algorithm for GCCF, i.e., the CFSS algorithm, to solve our GCCF model of the SR problem. Our empirical evaluation uses a real dataset for both spatial (GeoLife) and social data (Twitter), to validate the applicability of our approach in a realistic application scenario. Empirical results show that our approach computes optimal solutions for systems of medium scale (up to 100 agents) providing significant cost reductions (up to -36.22%). Moreover, we can provide approximate solutions for very large systems (i.e., up to 2000 agents) and good quality guarantees (i.e., with an approximation ratio of 1.41 in the worst case) within minutes (i.e., 100 seconds).

The Average Tree permission value for games with a permission tree

In the literature, various models of games with restricted cooperation can be found. In those models, instead of allowing for all subsets of the set of players to form, it is assumed that the set of feasible coalitions is a subset of the power set of the set of players. In this paper, we consider such sets of feasible coalitions that follow from a permission structure on the set of players, in which players need permission to cooperate with other players. We assume the permission structure to be an oriented tree. This means that there is one player at the top of the permission structure, and for every other player, there is a unique directed path from the top player to this player. We introduce a new solution for these games based on the idea of the Average Tree value for cycle-free communication graph games. We provide two axiomatizations for this new value and compare it with the conjunctive permission value.

Cooperative Games on Accessible Union Stable Systems

Agents participating in different kind of organizations, usually take different positions in some relational structure. The aim of this paper is to introduce a new framework taking into account both communication and hierachical features derived from this participation. In fact, this new set or network structure unifies and generalizes well-known models from the literature, such as communication networks and hierarchies. We introduce and analyze accessible union stable systems where union stability reflects the communication network and accessibility describes the hierarchy. Particular cases of these new structures are the sets of connected coalitions in a communication graph, antimatroids (and therefore also sets of feasible coalitions in permission structures) and augmenting systems which have numerous applications in the literature. We give special attention to th e class of cycle-free accessible union stable systems. We also consider cooperative games with restricted cooperation where the set of feasible coalitions is an accessible union stable system, and provide an axiomatization of an extension of the Shapley value to this class of games.

On the restricted cores and the bounded core of games on distributive lattices

A game with precedence constraints is a TU game with restricted cooperation, where the set of feasible coalitions is a distributive lattice, hence generated by a partial order on the set of players. Its core may be unbounded, and the bounded core, which is the union of all bounded faces of the core, proves to be a useful solution concept in the framework of games with precedence constraints. Replacing the inequalities that define the core by equations for a collection of coalitions results in a face of the core. A collection of coalitions is called normal if its resulting face is bounded. The bounded core is the union of all faces corresponding to minimal normal collections. We show that two faces corresponding to distinct normal collections may be distinct. Moreover, we prove that for superadditive games and convex games only intersecting and nested minimal collection, respectively, are necessary. Finally, it is shown that the faces corresponding to pairwise distinct nested normal collections may be pairwise distinct, and we provide a means to generate all such collections.

The Average Tree Permission Value for Games with a Permission Tree

In the literature various models of games with restricted cooperation can be found. In those models, instead of allowing for all subsets of the set of players to form, it is assumed that the set of feasible coalitions is a proper subset of the power set of the set of players. In this paper we consider such sets of feasible coalitions that follow from a permission structure on the set of players, in which players need permission to cooperate with other players. We assume the permission structure to be an oriented tree. This means that there is one player at the top of the permission structure and for every other player there is a unique directed path from the top player to this player. We introduce a new solution for these games based on the idea of the Average Tree value for cycle-free communication graph games. We provide two axiomatizations for this new value and compare it with the conjunctive permission value.

The Average Tree Permission Value for Games with a Permission Tree

This discussion paper resulted in a publication in the 'Economic Theory' , 2015, 58, 99-123. In the literature various models of games with restricted cooperation can be found. In those models, instead of allowing for all subsets of the set of players to form, it is assumed that the set of feasible coalitions is a proper subset of the power set of the set of players. In this paper we consider such sets of feasible coalitions that follow from a permission structure on the set of players, in which players need permission to cooperate with other players. We assume the permission structure to be an oriented tree. This means that there is one player at the top of the permission structure and for every other player there is a unique directed path from the top player to this player. We introduce a new solution for these games based on the idea of the Average Tree value for cycle-free communication graph games. We provide two axiomatizations for this new value and compare it with the conjunctive permission value.

The Myerson Value and Superfluous Supports in Union Stable Systems

In this paper, the set of feasible coalitions in a cooperative game is given by a union stable system. Well-known examples of such systems are communication situations and permission structures. Two games associated with a game on a union stable system are the restricted game (on the set of players in the game) and the conference game (on the set of supports of the system). We define two types of superfluous support property through these two games and provide new characterizations for the Myerson value. Finally, we analyze inheritance of properties between the restricted game and the conference game.

The Myerson value for complete coalition structures

In order to describe partial cooperation structures, this paper introduces complete coalition structures as sets of feasible coalitions. A complete coalition structure has a property that, for any coalition, if each pair of players in the coalition belongs to some feasible coalition contained in the coalition then the coalition itself is also feasible. The union stable structures, which constitute the domain of the Myerson value, are a special class of the complete coalition structures. As an allocation rule on complete coalition structures, this paper proposes an extension of the Myerson value for complete coalition structures and provides an axiomatization.

The Myerson Value and Superfluous Supports in Union Stable Systems

Cooperative games with partial cooperation cover a wider rank of real world situations than the classic model of cooperative games where every subset of a set of agents can form a coalition to execute the game. In this paper, the set of feasible coalitions which models the partial cooperation will be given by a union stable system. These systems contain, as particular cases, the communication situations and the permission structures, which are well-known both from a theoretical and applied point of view. Moreover, union stable systems are a natural framework for many other economic situations that arise in practice and which can not be modelled by these subsystems. In this paper, the goal is to make clear that there exists a close relationship between the Myerson value and the so-called conference game which player set consists of the supports of the union stable system. For that, we first analyze the relation between the restricted game and the conference game to establish later which effects a union stable system has on certain desirable properties of these games. Using the superfluous support property, defined through the conference game, new characterizations for the Myerson value are given in this context.

Interaction indices for games on combinatorial structures with forbidden coalitions

The notion of interaction among a set of players has been defined on the Boolean lattice and Cartesian products of lattices. The aim of this paper is to extend this concept to combinatorial structures with forbidden coalitions. The set of feasible coalitions is supposed to fulfil some general conditions. This general representation encompasses convex geometries, antimatroids, augmenting systems and distributive lattices. Two axiomatic characterizations are obtained. They both assume that the Shapley value is already defined on the combinatorial structures. The first one is restricted to pairs of players and is based on a generalization of a recursivity axiom that uniquely specifies the interaction index from the Shapley value when all coalitions are permitted. This unique correspondence cannot be maintained when some coalitions are forbidden. From this, a weak recursivity axiom is defined. We show that this axiom together with linearity and dummy player are sufficient to specify the interaction index. The second axiomatic characterization is obtained from the linearity, dummy player and partnership axioms. An interpretation of the interaction index in the context of surplus sharing is also proposed. Finally, our interaction index is instantiated to the case of games under precedence constraints.

Games on Union Closed Systems

A situation in which a finite set of players can obtain certain payoffs by cooperation can be described by a cooperative game with transferable utility, or simply a TU-game. A solution for TU-games assigns a set of payoff distributions to every TU-game. In the literature various models of games with restricted cooperation can be found. So, instead of allowing all subsets of the player set N to form, it is assumed that the set of feasible coalitions is a subset of the power set of N. In this paper we consider such sets of feasible coalitions that are closed under union, i.e. for any two feasible coalitions also their union is feasible. Properties of solutions (the core, the nucleolus, the prekernel and the Shapley value) are given for games on union closed systems.

Axiomatizations of two types of Shapley values for games on union closed systems

A situation in which a finite set of players can obtain certain payoffs by cooperation can be described by a cooperative game with transferable utility, or simply a TU-game. A (single-valued) solution for TU-games assigns a payoff distribution to every TU-game. A well-known solution is the Shapley value. In the literature various models of games with restricted cooperation can be found. So, instead of allowing all subsets of the player set N to form, it is assumed that the set of feasible coalitions is a subset of the power set of N. In this paper, we consider such sets of feasible coalitions that are closed under union, i.e. for any two feasible coalitions also their union is feasible. We consider and axiomatize two solutions or rules for these games that generalize the Shapley value: one is obtained as the conjunctive permission value using a corresponding superior graph, the other is defined as the Shapley value of a modified game similar as the Myerson value for games with limited communication.

Strategy-proof coalition formation

We analyze coalition formation problems in which a group of agents is partitioned into coalitions and agents’ preferences only depend on the coalition to which they belong. We study rules that associate to each profile of preferences a partition of the society. We focus on strategy-proof rules on restricted domains of preferences, as the domains of additively representable or separable preferences. In such domains, the only strategy-proof and individually rational rules that satisfy either a weak version of efficiency or non-bossiness and flexibility are single-lapping rules. Single-lapping rules are characterized by severe restrictions on the set of feasible coalitions that are consistent with hierarchical organizations. These restrictions are necessary and sufficient for the existence of a unique core-stable partition. In fact, single-lapping rules always select the associated unique core-stable partition. Thus, our results highlight the relation between the non-cooperative concept of strategy-proofness and the cooperative concept of uniqueness of core-stable partitions.

Axiomatizations of Two Types of Shapley Values for Games on Union Closed Systems

A situation in which a finite set of players can obtain certain payoffs by cooperation can be described by a cooperative game with transferable utility, or simply a TU-game. A (single-valued) solution for TU-games assigns a payoff distribution to every TU-game. A well-known solution is the Shapley value. In the literature various models of games with restricted cooperation can be found. So, instead of allowing all subsets of the player set N to form, it is assumed that the set of feasible coalitions is a subset of the power set of N. In this paper we consider such sets of feasible coalitions that are closed under union, i.e. for any two feasible coalitions also their union is feasible. We consider and axiomatize two solutions or rules for these games that generalize the Shapley value: one is obtained as the conjunctive permission value using a corresponding superior graph, the other is defined as the Shapley value of a modified game similar as the Myerson rule for conference structures.