Solution Concepts For Cooperative Games Research Articles

Gately values of cooperative games

AbstractWe investigate Gately’s solution concept for cooperative games with transferable utilities. Gately’s conception introduced a bargaining solution that minimises the maximal quantified “propensity to disrupt” the negotiation process of the players over the allocation of the generated collective payoffs. We show that Gately’s solution concept is well-defined for a broad class of games and that it can be interpreted as a compromise solution. We also consider a generalisation based on a parameter-based quantification of the propensity to disrupt. We provide an axiomatic characterisation of the original Gately value as well as these generalised Gately values. Furthermore, we investigate the relationship of these Gately values with the Core and the Nucleolus and show that Gately’s solution is in the Core for all regular 3-player games, but is fundamentally different from the Nucleolus. We identify exact conditions under which these Gately values are Core imputations for arbitrary regular cooperative games. Finally, we investigate the relationship of the Gately value with the Shapley value.

Optimization implementation of solution concepts for cooperative games with stochastic payoffs

Optimization implementation of solution concepts for cooperative games with stochastic payoffs

The Prekernel of Cooperative Games with Alpha-Excess

In this paper, we introduce a new approach to measure the dissatisfaction for coalitions of players in cooperative transferable utility games. This is done by considering affine (and convex) combinations of the classical excess and the proportional excess. Based on this so-called alpha-excess, we defi ne new solution concepts for cooperative games, such as the alpha-prenucleolus and the alpha-prekernel. The classical prenucleolus and prekernel are a special case. We characterize the alpha-prekernel by strong stability and the alpha-balanced surplus property. Also, we show that the payoff vector generated by the alpha-prenucleolus belongs to the alpha-prekernel.

The Egalitarian Shapley value: a generalization based on coalition sizes

In designing solution concepts for cooperative games with transferable utilities, consolidation of marginalism and egalitarianism has been widely studied. The $$\alpha $$ -Egalitarian Shapley value is one such solution that combines the Shapley value and the Equal Division rule, the two most popular extreme instances of marginalism and egalitarianism respectively. This value gives the planner the flexibility to choose the level of marginality for the players by varying the convexity parameter $$\alpha $$ . In this paper, we define the Generalized Egalitarian Shapley value that gives the planner more flexibility in choosing the level of marginality based on the coalition size. We then provide two characterizations of the Generalized Egalitarian Shapley value.

On the Values of Bayesian Cooperative Games with Sidepayments

In this paper we explore the relationship between several value-like solution concepts for cooperative games with incomplete information and utility transfers in the form of sidepayments. In our model, state-contingent contracts are required to be incentive compatible, and thus utility might not be not fully transferable (as it would be in the complete information case). When we restrict our attention to games with orthogonal coalitions (i.e., which do not involve strategic externalities), our first main result states that Myerson’s [Cooperative games with incomplete information. Int. J. Game Theory. (1984), 13, 69-96] generalization of the Shapley NTU value and Salamanca’s [A generalization of the Harsanyi NTU value to games with incomplete information. (2016), HAL 01579898] extension of the Harsanyi NTU value are interim utility equivalent. If we allow for arbitrary informational and strategic externalities, our second main result establishes that the ex-ante evaluation of Myerson’s solution equals Kalai and Kalai’s [Cooperation in strategic games revisited. Q. J. Econ. (2013) 128, 917-966] cooperative-competitive value in two-player games with verifiable types.

COALITION FORMATION AND BARGAINING PROTOCOLS: A REVIEW OF THE LITERATURE

AbstractThis paper offers a review of the vast literature regarding bargaining and coalition formation. This topic has been generally described as the attempt to provide strategic foundations to cooperative solution concepts. It can therefore be seen as the linking ring between the non‐cooperative and the cooperative game‐theoretic approach to coalition formation. Its central role in economic theory and its relatively long history that goes back to the Nash program have fostered a large academic production, including surveys. Nonetheless, this paper will focus on an aspect that is often neglected in the dedicated surveys: the specificities of the bargaining protocols leading to different outcomes. Although generally downgraded to the rank of details, the differences in bargaining protocols, even when minor, can cause significant changes in fundamental aspects such as the possibility to reach full cooperation, the distribution of final pay‐offs and the time taken to reach an agreement. Focused on externalities‐free games, therefore on bargaining protocols sustaining solution concepts for cooperative games in characteristic function form, the paper aims at providing a brief but exhaustive review of the topic that could result in a very useful tool for any researcher approaching the subject of coalitional bargaining.

Optimal deterrence of cooperation

We introduce axiomatically a new solution concept for cooperative games with transferable utility inspired by the core. While core solution concepts have investigated the sustainability of cooperation among players, our solution concept, called contraction core, focuses on the deterrence of cooperation. The main interest of the contraction core is to provide a monetary measure of the robustness of cooperation into the grand coalition. We motivate this concept by providing optimal fine imposed by competition authorities for the dismantling of cartels in oligopolistic markets. We characterize the contraction core on the set of balanced cooperative games with transferable utility by four axioms: the two classic axioms of non-emptiness and individual rationality, a superadditivity principle and a new axiom of consistency.

Optimizing Multiple-Pollutant Waste Load Allocation in Rivers: An Interval Parameter Game Theoretic Model

In this paper, a new methodology is developed for optimal multiple-pollutant waste load allocation (MPWLA) in rivers considering the main existing uncertainties. An interval optimization method is used to solve the MPWLA problem. Different possible scenarios for treatment of pollution loads are defined and corresponding treatment costs are taken into account in an interval parameter optimization model. A QUAL2Kw-based water quality simulation model is developed and calibrated to estimate the concentration of the water quality variables along the river. Two non-cooperative and cooperative multiple-pollutant scenario-based models are proposed for determining waste load allocation policies in rivers. Finally, a new fuzzy interval solution concept for cooperative games, namely, Fuzzy Boundary Interval Variable Least Core (FIVLC), is developed for reallocating the total fuzzy benefit obtained from discharge permit trading among waste load dischargers. The results of applying the proposed methodology to the Zarjub River in Iran illustrate its effectiveness and applicability in multiple-pollutant waste load allocation in rivers.

Cross-Evaluation Cost Allocation for Vehicle Routing Games

In this paper, we study a cost allocation problem that arises in goods distribution by vehicles. This problem can be formulated as a vehicle-routing game (VRG), where the total distribution cost must be divided among all customers visited by the vehicles. We apply our recently developed solution concept for cooperative games, cross-evaluation value, to the cost allocation. In the solution concept, the allocation of the total cost of a VRG among its players is based on self-evaluation and peer-evaluations of the cost contribution of each player to the grand coalition, subject to the core constraints when the core of the game is not empty. Since the vehicle-routing game may have an empty core, an extended core concept is also proposed for the game, and the cross-evaluation value is then applied to its cost allocation. Numerical examples are presented and discussed.

Competitive outcomes and the inner core of NTU market games

We consider the inner core as a solution concept for cooperative games with non-transferable utility (NTU) and its relationship to payoffs of competitive equilibria of markets that are induced by NTU games. An NTU game is an NTU market game if there exists a market such that the set of utility allocations a coalition can achieve in the market coincides with the set of utility allocations the coalition can achieve in the game. In this paper, we introduce a new construction of a market based on a closed subset of the inner core which satisfies a strict positive separability. We show that the constructed market represents the NTU game and, further, has the given closed set as the set of payoff vectors of competitive equilibria. It turns out that this market is not uniquely determined, and thus, we obtain a class of markets. Our results generalize those relating to competitive outcomes of NTU market games in the literature.

A proportional value for cooperative games with a coalition structure

We introduce a solution concept for cooperative games with transferable utility and a coalition structure that is proportional for two-player games. Our value is obtained from generalizing a proportional value for cooperative games with transferable utility (Ortmann 2000) in a way that parallels the extension of the Shapley value to the Owen value. We provide two characterizations of our solution concept, one that employs a property that can be seen as the proportional analog to Myerson’s balanced contribution property; and a second one that relies on a consistency property.

Games with General Coalitional Structure

Abstract: This paper introduces a new solution concept for cooperative games with general coalitional structure in which only certain sets of players, including the set of all players, are able to form feasible coalitions. The solution concept takes into account the marginal contribution of players. This marginal contribution can be a joint contribution of several players and is equally divided among those players. Any set system representing a coalitional structure induces a collection of coalitional trees, whose nodes may consist of subsets of players. As solution we take the average of the marginal contribution vectors that correspond to all coalitional trees. The solution is ecient and several other properties are studied and some special cases are considered.

Sharing costs in Swedish road ownership associations

Usually transport systems, and roads in particular, are viewed as public goods. However, this is not always the case. In Sweden a large part of the road system is privately owned. Most of these privately owned roads are rural roads used by farmers and summer cottage owners, or used for forest transport. These roads are mainly provided by ownership associations.An important difference between public roads and these privately owned roads is that all investments- and maintenance decisions are made by the users themselves, who also have to pay the costs, whereas the usual case is that the owners/providers of a road-system are different agents than the users. Here the question is not how to charge the roads but how to split the costs of the roads among the users in an efficient and “fair” way.The motivation of this paper is the practical problem of how such an ownership association can divide the costs for the road network among the members in an efficient and “fair” way. The problem is treated from a game theoretical point of view, making use of the Shapley value. This means that the problem is associated with a game – a mathematical representation of the conflict situation. The Shapley value is a very important solution concept for cooperative games, like the game in this case. For games corresponding to this specific type of problems, it is shown that the Shapley value has excellent properties, such as beeing an element of the core, and beeing very easy to compute.

Solution Concepts for Cooperative Games with Circular Communication Structure

We study transferable utility games with limited cooperation between the agents. The focus is on communication structures where the set of agents forms a circle, so that the possibilities of cooperation are represented by the connected sets of nodes of an undirected circular graph. Agents are able to cooperate in a coalition only if they can form a network in the graph. A single-valued solution which averages marginal contributions of each player is considered. We restrict the set of permutations, which induce marginal contributions to be averaged, to the ones in which every agent is connected to the agent that precedes this agent in the permutation. Staring at a given agent, there are two permutations which satisfy this restriction, one going clockwise and one going anticlockwise along the circle. For each such permutation a marginal vector is determined that gives every player his marginal contribution when joining the preceding agents. It turns out that the average of these marginal vectors coincides with the average tree solution. We also show that the same solution is obtained if we allow an agent to join if this agent is connected to some of the agents who is preceding him in the permutation, not necessarily being the last one. In this case the number of permutations and marginal vectors is much larger, because after the initial agent each time two agents can join instead of one, but the average of the corresponding marginal vectors is the same. We further give weak forms of convexity that are necessary and sufficient conditions for the core stability of all those marginal vectors and the solution. An axiomatization of the solution on the class of circular graph games is also given.

The subcoalition-perfect core of cooperative games

The core is a set-valued solution concept for cooperative games. In situations where the characteristic function is not monotone the classical definition may not be sufficient. Hence, we propose a subset of the core that is called subcoalition-perfect core. It will be proven that the subcoalition-perfect core coincides with the set of non-negative core allocations. Furthermore, an ellipsoid algorithm is provided which may be applied in many applications to compute an element in the subcoalition-perfect core. In addition, we discuss an application where the characteristic function is not monotone and perform a computational study.

A general procedure to compute mixed modified semivalues for cooperative games with structure of coalition blocks

Semivalues are solution concepts for cooperative games that assign to each player a weighted sum of his/her marginal contributions to the coalitions, where the weights only depend on the coalition size. The Shapley value and the Banzhaf value are semivalues. Mixed modified semivalues are solutions for cooperative games when we consider a priori coalition blocks in the player set. For all these solutions, a computational procedure is offered in this paper.

The consensus value: a new solution concept for cooperative games

To generalize the standard solution for 2-person TU games into n-person cases, this paper introduces a recursive two-sided negotiation process to establish cooperation between all players. This leads to a new solution concept for cooperative games: the consensus value. An explicit comparison with the Shapley value is provided, also at the axiomatic level. Moreover, a class of possible generalizations of the consensus value is introduced and axiomatized with the Shapley value at one end and the equal surplus solution at the other. Finally, we discuss a non-cooperative mechanism which implements the consensus value.

The Equal Split-off Set for Cooperative Games

In this paper the equal split-off set is introduced as a new solution concept for cooperative games. This solution is based on egalitarian considerations and it turns out that for superadditive games the equal split-off set is a subset of the equal division core. Moreover, the proposed solution is single valued on the class of convex games and it coincides with the Dutta-Ray constrained egalitarian solution.

The Consensus Value: A New Solution Concept for Cooperative Games

By generalizing the standard solution for 2-person games into n-person cases, this paper develops a new solution concept for cooperative games: the consensus value. We characterize the consensus value as the unique function that satisfies efficiency, symmetry, the quasi dummy property and additivity. By means of the transfer property, a second characterization is provided. By defining the stand-alone reduced game, a recursive formula for the value is established. We also show that this value is the average of the Shapley value and the equal surplus solution. Furthermore, we discuss a possible generalization.

STABILITY OF THE CORE IN A CLASS OF NTU GAMES: A CHARACTERIZATION

The core and the stable set are possibly the two most crucially important solution concepts for cooperative games. The relation between the two has been investigated in the context of symmetric transferable utility games and this has been related to the notion of large core. In this paper the relation between the von-Neumann–Morgenstern stability of the core and the largeness of it is investigated in the case of non-transferable utility (NTU) games. The main findings are that under certain regularity conditions, if the core of an NTU game is large then it is a stable set and for symmetric NTU games the core is a stable set if and only if it is large.