Classical Shapley Value Research Articles

A value for cooperative games on simplicial complexes with a filtration

The classical Shapley value for cooperative games determines a payoff vector considering that the formation of the grand coalition is made by incorporating players one by one. Later, this method was generalized for games with restricted cooperation by several known mathematical structures: partitions, graphs, convex geometries, antimatroids, matroids or simplicial complexes. In this paper we consider games over simplicial complex with an extra information about the relationships of the agents, a filtration of the complex. Filtrations are very known simplicial structures in the social and neuronal networks. We propose a Shapley value for these situations and an axiomatization.

Cooperation Preference Aware Shapley Value: Modeling, Algorithms and Applications

The Shapley value is a cornerstone in cooperative game theory and has been widely applied in networking, data science, etc. The classical Shapley value assumes that each player has an equal preference to cooperate with each other. Since the cooperation preference is an important factor of a variety of networking applications, we first generalize the classical Shapley value to allow general degree of the cooperation preference. In particular, we develop mathematical models to solicit two types of cooperation preferences, i.e., (1) group-wise preferences and (2) pair-wise preferences, and extend the classical Shapley value to capture this feature. Our second contribution is tackling the intrinsic computational challenge because even for the classical Shapley value, it is computationally expensive to evaluate. We design computationally efficient randomized algorithms with theoretical guarantees to fully cover the computational space of our generalized Shapley value. We also extend our models and algorithms to divide payoffs for multiple coalitions with dynamic preferences. We demonstrate the versatility of our framework by applying it to divide the revenue among ISPs in deploying new Internet architectures, as well as to divide the reward among workers in crowdsourcing systems.

Optimal scheduling strategy and benefit allocation of multiple virtual power plants based on general nash bargaining theory

The benefits of small-scale renewable energy resources are restricted by a feed-in-tariff strategy that works with distribution grids only. Virtual Power Plant (VPP) aggregates diverse distributed energy resources on the demand side for efficient control, which can improve their revenue and encourage the growth of renewable energy. Based on the general Nash bargaining theory, this paper proposes a multi-VPP energy-sharing mechanism. This approach enables VPPs to increase their revenue by engaging in peer-to-peer energy transactions. First, a general Nash bargaining-based model is proposed for multi-VPPs energy sharing considering peer-to-peer energy transactions. The model is then decomposed into two subproblems: minimizing the total cost of the VPP energy-sharing alliance, and the cost allocation of peer-to-peer energy transactions. Considering the privacy protection of VPPs, the first subproblem is solved by a distributed algorithm based on the alternating direction multiplier method. In the second subproblem, due to the introduction of a bargaining factor that is positively related to each VPP’s contribution to energy sharing alliance, the cost allocation better reflects their contribution. The numerical test demonstrates the proposed energy-sharing mechanism, with little computational complexity, delivers a cost distribution result that is comparable to the classical Shapley value while promoting the development of renewable energy resources.

Fairness of prosumers’ incentives in residential demand response: A practical decentralized optimization approach

This paper presents a new decentralized optimization-based approach for an aggregator involved in residential demand response programs. The proposed approach relies on an existing two-phase optimization framework, whereby prosumers optimize first their self-generation and household consumption, which are subsequently coordinated by the aggregator via incentives through the solution of a mixed-integer linear program. As a distinctive aspect, the notion of fairness in the distribution of incentives among prosumers is incorporated through new nonlinear constraints based on the Shapley Value while respecting standard demand response requirements. A mathematical approximation, which is proven to be exact under certain practical assumptions, is proposed to accommodate the new constraints into the linear model. Furthermore, a novel solution methodology, which combines the divide-and-conquer principle and Dantzig–Wolfe decomposition, is devised to tackle the resulting problem in a distributed fashion. For validation purposes, the behavior of the linear approximations is examined by comparing the results with those attained with their nonlinear counterparts. Moreover, scenarios for prosumers featuring different flexibility levels are generated to assess the practicality of the assumptions adopted. Simulations also show the scalability of the proposed tool by considering cases with up to 10,000 prosumers with computing times within an hour if distributed computing is implemented. Furthermore, as compared to classical Shapley Value approaches, a similar distribution of incentives is attained by the new solution methodology with relative less and acceptable computational effort. Finally, the impact of the new constraints on Peak-to-Average Ratio is quantified, obtaining an increment of up to 13.2% with respect to cases disregarding fairness.

Closed-loop supply chain coalitional cooperation and coordination under differentiated carbon tax regulation

This paper presents a systematic two-stage analysis to improve a three-echelon closed-loop supply chain cooperation and coordination under differentiated carbon tax regulation, which enriches non-cooperative and cooperative game theory applications in remanufacturing circular economies. Firstly, we utilize non-cooperative games to explore the optimal operations of new and remanufactured products across five coalitional models. Then, we adopt cooperative games to achieve profit coordination. Specifically, Shapley values are employed for allocating a grand coalition profit in a centralized setting. In order to alleviate carbon taxpayers' fairness concerns, we proposed a novel revised Shapley value algorithm incorporating differentiated carbon tax costs to improve their utility. The results show that regulators can reduce emissions by adjusting carbon tax rates on different products more effectively. The optimal environmental and economic performance can also be achieved by forming a grand coalition. Shapley value algorithms achieve the supply chain system Pareto improvement. Moreover, carbon taxpayers' (i.e., manufacturer and remanufacturer) utilities in our revised Shapley are improved by 20.15% and 77.71%, respectively, compared with the classical Shapley value under certain conditions. These results benefit regulators' carbon tax policy formulation and industrial managers’ remanufacturing practices.

Cost Sharing in Insurance Communities: A Hybrid Approach Based on Multiple-Choice Objective Programming and Cooperative Games

At present, utilizing the insurance community is a common method to deal with investment risks along the Belt and Road; however, there is no clear method or mechanism to deal with the decision-making optimization and cost allocation of the insurance community participants. We propose a hybrid approach to solve this problem. First, we construct an underwriting decision optimization model for the insurance community using the multi-choice goal programming method, which generates the cost characteristic function based on a cooperative alliance. Second, we use the cooperative game method combined with the modified Shapley value method to take risk factors into consideration, which allows us to optimize the cost allocation among members of the insurance community. Finally, our simulation analysis results show that the multi-choice goal programming method can optimize the insurance community’s underwriting decisions. Specifically, the total underwriting cost is lower than the sum of the underwriting costs under the insurance company’s single-action strategy, and the total underwriting scale is as large as possible. Compared with the classical Shapley value method, the modified Shapley value method can better reflect differences in the underwriting risks of different regions, encouraging governments to take measures to reduce underwriting risks. To conclude, we propose some suggestions based on our research findings. The possible contributions of this paper are as follows: our research provides a hybrid optimization method based on multiple-choice objective programming and cooperative games to solve the cost allocation problem facing the insurance community, and it has some reference value for improving the cost-sharing system of the insurance community.

The priority value for cooperative games with a priority structure

We study cooperative games with a priority structure modeled by a poset on the agent set. We introduce the Priority value, which splits the Harsanyi dividend of each coalition among the set of its members over which no other coalition member has priority. This allocation shares many desirable properties with the classical Shapley value: it is efficient, additive and satisfies the null agent axiom. We provide two axiomatic characterizations of the Priority value which invoke both classical axioms and new axioms describing the effects of the priority structure on the payoff allocation. Finally, in the special case where agents are ranked by level, a link between the Priority value, the weighted Shapley values and the Owen-type values can be drawn.

Hodge decomposition and the Shapley value of a cooperative game

We show that a cooperative game may be decomposed into a sum of component games, one for each player, using the combinatorial Hodge decomposition on a graph. This decomposition is shown to satisfy certain efficiency, null-player, symmetry, and linearity properties. Consequently, we obtain a new characterization of the classical Shapley value as the value of the grand coalition in each player's component game. We also relate this decomposition to a least-squares problem involving inessential games (in a similar spirit to previous work on least-squares and minimum-norm solution concepts) and to the graph Laplacian. Finally, we generalize this approach to games with weights and/or constraints on coalition formation.

A concise axiomatization of a Shapley-type value for stochastic coalition processes

The classical Shapley value is the average marginal contribution of a player, taken over all possible ways to form the grand coalition \(N\) when one starts from the empty coalition and adds players one by one. In a previous paper, the authors have introduced an allocation scheme for a general coalition formation model where the evolution of the coalition of active players is ruled by a Markov chain and need not finish with the grand coalition. This note provides an axiomatization which is only slightly weaker than the original one but allows a much more transparent proof. Moreover, the logical independence of the axioms is exhibited.

Solutions for Games with General Coalitional Structure and Choice Sets

In this paper we introduce the concept of quasi-building set that may underlie the coalitional structure of a cooperative game with restricted communication between the players. Each feasible coalition, including the set of all players, contains a nonempty subset called the choice set of the coalition. Only players that are in the choice set of a coalition are able to join to feasible subcoalitions to form the coalition and to obtain a marginal contribution. We demonstrate that all restricted communication systems that have been studied in the literature take the form of a quasi-building set for an appropriate set system and choice set. Every quasi-building set determines a nonempty collection of maximal strictly nested sets and each such set induces a rooted tree satisfying that every node of the tree is a player that is in the choice set of the feasible coalition that consists of himself and all his successors in the tree. Each tree corresponds to a marginal vector of the underlying game at which each player gets as payoff his marginal contribution when he joins his successors in the tree. As solution concept of a quasi-building set game we propose the average marginal vector (AMV) value, being the average of the marginal vectors that correspond to the trees induced by all maximal strictly nested sets of the quasi-building set. Properties of this solution are also studied. To establish core stability we introduce appropriate convexity conditions of the game with respect to the underlying quasi-building set. For some specifcations of quasi-building sets, the AMV-value coincides with solutions known in the literature, for example, for building set games the solution coincides with the gravity center solution and the Shapley value recently defined for this class. For graph games it therefore differs from the well-known Myerson value. For a full communication system the solution coincides with the classical Shapley value.

Improved Weighted Shapley Value Model for the Fourth Party Logistics Supply Chain Coalition

How to make the individual get the reasonable and practical profit among the fourth party logistics supply chain coalition system is still a question for further study. Considering the characteristics of the fourth party logistics supply chain coalition, this paper combines Shapley Value with Distribution according to Contribution, two methods in the application, and then adjusts the profit allocated to each member reasonably based on the actual coalition situation named improved weighted Shapley Value model. In this paper, we first analyze the fourth party logistics supply chain coalition profit allocation models, the classical Shapley value method. Then, we analyze the weight of individual enterprise in the coalition by the analytic hierarchy process. To each enterprise, the weight is determined by the investment risks, information divulging risks, and failure risks. Finally, the numerical study shows that the profit allocation method improved weighted Shapley value model is relatively rational and practical. Thus, the proposed combined model is a useful profit allocation mechanism for the fourth party logistics supply chain coalition that the contribution and risks are fully considered.

Values for Markovian coalition processes

Time series of coalitions (so-called scenarios) are studied that describe processes of coalition formation where several players may enter or leave the current coalition at any point in (discrete) time and convergence to the grand coalition is not necessarily prescribed. Transitions from one coalition to the next are assumed to be random and to yield a Markov chain. Three examples of such processes (the Shapley-Weber process, the Metropolis process, and an example of a voting situation) and their properties are presented. A main contribution includes notions of value for such series, \emph{i.e.}, schemes for the evaluation of the expected contribution of a player to the coalition process relative to a given cooperative game. Particular processes permit to recover the classical Shapley value. This methodology's power is illustrated with well-known examples from exchange economies due to Shafer (1980) and Scafuri and Yannelis (1984), where the classical Shapley value leads to counterintuitive allocations. The Markovian process value avoids these drawbacks and provides plausible results.

The Shapley value for bicooperative games

The aim of the present paper is to study a one-point solution concept for bicooperative games. For these games introduced by Bilbao (Cooperative Games on Combinatorial Structures, 2000) , we define a one-point solution called the Shapley value, since this value can be interpreted in a similar way to the classical Shapley value for cooperative games. The main result of the paper is an axiomatic characterization of this value.

Profit allocation scheme among partners in virtual enterprises based on Shapley values with fuzzy payoffs

Virtual enterprises (VE) can be viewed as a cooperative alliance of member companies. An important part of the operation of VE is how to allocate the profit among partners in uncertain environment. This case can be viewed as a fuzzy cooperation game. In this paper, we make a study of the Shapley values with the fuzzy payoff functions from the viewpoint that the payoffs of each coalition are often only imprecisely or ambiguously known to the players. Axioms of the classical Shapley value are extended to the fuzzy Shapley values. From the viewpoint that the allocation for each partner should be a crisp value rather a fuzzy membership function at the end of cooperation, we proposed a crisp allocation scheme based on fuzzy Shapley values.

The Shapley value for games on matroids: The dynamic model

According to the work of Faigle [3] a static Shapley value for games on matroids has been introduced in Bilbao, Driessen, Jimenez-Losada and Lebron [1]. In this paper we present a dynamic Shapley value by using a dynamic model which is based on a recursive sequence of static models. In this new model for games on matroids, our main result is that there exists a unique value satisfying analogous axioms to the classical Shapley value. Moreover, we obtain a recursive formula to calculate this dynamic Shapley value. Finally, we prove that its components are probabilistic values.