Nonempty Core Research Articles

Indivisible goods, equilibrium and convex economies

We study the existence of competitive equilibria in economies with indivisible goods. The first result provides a novel sufficient condition for an economy with indivisible goods to have competitive equilibrium. This condition states that the cooperative game associated with the economy has a non-empty core. Crucially, this sufficient condition is also necessary when the economy is convex. This characterization implies that every two-player convex economy has at least one competitive equilibrium.

Two-stage network data envelopment analysis production games

<abstract> <p>DEA (Data Envelopment Analysis) production games combine DEA theory with cooperation games and assess the benefits to production organizations with single-stage structure. However, in practical production problems, the production organizations are always with network structures. The structure of the production organization not only affects its own benefits, but also relates to the cooperation among organizations. Therefore, it is necessary to study DEA production games with network structures. In this paper, we consider the production organizations with two-stage processes, wherein the organizations are assumed to possess available resources and own technologies. The technology level of each organization is reflected by the observed units based on the network DEA (NDEA) production possibility set. Suppose that the organizations can cooperate through the ways of pooling the initial resources and (or) sharing the technology in each production process. According to the different cooperation styles of each stage in the alliance, seven types of cooperation among organizations are considered. The models of maximizing the revenues of coalitions, namely the NDEA production games, are established corresponding to the seven types, by which the maximal revenue for each coalition can be calculated. We prove that two-stage DEA production games have the super-additive property, and can be expressed as linear programming games. Hence, they are equivalent to the linear production games, and they are totally balanced. Therefore, the proposed cooperative games have a non-empty core, and hence have nucleolus, and the Owen set belongs to the core. In addition, based on the basic conceptions of the nucleolus and the Owen set, the revenue can be allocated among organizations in the alliance. Finally, a numerical example and an empirical application to 17 bank branches of the China Construction Bank in the Anhui Province are presented to illustrate the applicability of the proposed approach, and the relationship between the cooperative manners and the revenue allocation is reflected in analytical results.</p> </abstract>

The coalitional aggregation of wind energy: A stochastic duality approach

The inherent uncertainty of wind power poses a major challenge to modern electricity grid. This paper mainly investigates a scenario where a crowd of wind power producers form a coalition by cooperatively offering their aggregated power output, aiming to mitigate these risks created by the inherent uncertainty of wind power. The main results are as follows. Firstly, a stochastic two-stage programming problem is proposed to formulate the capacity management with respect to a wind power producers’ coalitional game. Secondly, we give a constructive proof to illustrate that a wind power producers’ coalitional game has a nonempty core using the theory of cooperative game. Meanwhile, we also show that the form of a coalition always reduces their expected costs. Thirdly, we propose a stochastic duality approach to design a fair cost allocation scheme for players of a wind power producers’ coalition. Finally, we consider a concave function structure, indicating that there exists economies of scale, and extend this proposed duality theory to the wind power producers’ coalitional game under a concave capacity investment cost structure. Furthermore, we find that a wind power producers’ coalition is balanced under the concave capacity investment cost function. Our results provide an incentive for collaboration strategies in wind power setting, and promote sustainable development of wind power.

Peer-to-Peer Sharing of Energy Storage Systems Under Net Metering and Time-of-Use Pricing

Sharing economy has become a socio-economic trend in the transportation and housing sectors. It develops business models leveraging underutilized resources. Like those sectors, power grid is also becoming smarter with many flexible resources, and researchers are investigating the impact of sharing resources here as well that can help to reduce cost and extract value. In this work, we investigate sharing of energy storage devices among individual households in a cooperative fashion. Coalitional game theory is used to model the scenario where the utility company imposes time-of-use (ToU) price and net metering (NM) billing mechanism. The resulting game has a non-empty core and we can develop a cost allocation mechanism with easy to compute analytical formula. Allocation is fair and cost-effective for every household. We design the price for the peer-to-peer (P2P) network and an algorithm for sharing that keeps the grand coalition always stable. Thus sharing electricity of storage devices among consumers can be effective in this set-up. Our mechanism is implemented in a community of 80 households in Texas using real data of load demand and solar irradiance and the results show significant cost savings for our method.

Cooperative voltage management by demand resources and fair payoff allocation for distribution systems

This paper proposes a socially acceptable framework for the realization of efficient voltage management by cooperation among consumers in electric distribution systems where the frequent voltage variation problem is caused by large amounts of photovoltaic (PV) generation systems. In the proposed framework, the profit maximization problem of the coalition under the constraints to suppress the voltage variations caused by PVs is formulated as a linear programming problem, and the payoff allocation among consumers is provided using a solution concept from cooperative game theory called the core. The proposed method efficiently computes the payoff allocation in the nonempty core to accomplish the cooperative voltage management system. Unlike most related works, this paper employs cooperative game theory to explicitly address incentives for cooperation with the consumers. Although the computation for finding an allocation in the core generally requires a significant amount of time, this paper demonstrates that the allocation can be efficiently calculated by using the dual optimal solution of a formulated linear programming problem. The proposed method is validated by computational experiments on a large-scale distribution system model in which consumers with demand resources (including PVs and controllable loads) are connected.

Nucleolus based cost allocation methods for a class of constrained lane covering games

In truckload transportation procurement networks, shippers purchasing the services of truckload carriers seek to form multi-company tours which consist of regularly scheduled shipments with minimal empty truck movements with the aim of getting better rates from the carriers in return. Identifying the minimum cost collaborative solution and sharing the costs in a fair manner are two interrelated and critical tasks for the success of such networks. The restrictions on the tours which can be formed make both of these tasks very challenging. In this paper, we take a cooperative game theory approach and study the resulting constrained lane covering game. We first show the set of conditions which must be satisfied for the game to have a non-empty core. We propose nucleolus based cost allocation methods. We also develop column and row generation methods for solving these mathematical models. We compare our proposed cost allocation models with alternative cost allocation methods and each other using a set of core stability metrics through computational experiments on randomly generated instances from the literature.

On the core of m$m$‐attribute games

We study a special class of cooperative games with transferable utility (TU), called [Formula: see text]‐ attribute games. Every player in an [Formula: see text]‐attribute game is endowed with a vector of [Formula: see text] attributes that can be combined in an additive fashion; that is, if players form a coalition, the attribute vector of this coalition is obtained by adding the attributes of its members. Another fundamental feature of [Formula: see text]‐attribute games is that their characteristic function is defined by a continuous attribute function [Formula: see text]—the value of a coalition depends only on evaluation of [Formula: see text] on the attribute vector possessed by the coalition, and not on the identity of coalition members. This class of games encompasses many well‐known examples, such as queueing games and economic lot‐sizing games. We believe that by studying attribute function [Formula: see text] and its properties, instead of specific examples of games, we are able to develop a common platform for studying different situations and obtain more general results with wider applicability. In this paper, we first show the relationship between nonemptiness of the core and identification of attribute prices that can be used to calculate core allocations. We then derive necessary and sufficient conditions under which every [Formula: see text]‐attribute game embedded in attribute function [Formula: see text] has a nonempty core, and a set of necessary and sufficient conditions that [Formula: see text] should satisfy for the embedded game to be convex. We also develop several sufficient conditions for nonemptiness of the core of [Formula: see text]‐attribute games, which are easier to check, and show how to find a core allocation when these conditions hold. Finally, we establish natural connections between TU games and [Formula: see text]‐attribute games.

Gain measurement and payoff allocation for the internal resource sharing based on DEA approach

As an effective strategy for improving resource utilization and increasing profits, resource sharing exists not only between independent systems but also between sub-systems in the same system. Many researchers study the external cooperation among independent systems through building Data Envelopment Analysis (DEA) games. However, they do not consider the internal resource sharing between sub-systems and think players possess the same risk attitudes for gain and loss when allocating potential gains, which may be inconsistent with practice. To fill these gaps and answer the question: “For the system containing multiple independent DMUs, how to measure and allocate the potential gains derived from the internal resource sharing”, we construct an internal resource-sharing DEA game based on the DEA approach and propose a novel payoff allocation method considering players’ bounded rationality. Our proposed game is super-additive, monotone, not necessarily convex, and has a non-empty core. By the proposed payoff method, we obtain a unique, efficient, stable, and fair payoff allocation. Finally, to validate our method, we conduct a numerical experiment with inland transportation systems and compare the novel payoff allocation method with the core, Shapley value, and nucleolus.

The risk-neutral non-additive probability with market frictions

The fundamental theory of asset pricing has been developed under the two main assumptions that markets are frictionless and have no arbitrage opportunities. In this case the market enforces that replicable assets are valued by a linear function of their payoffs, or as the discounted expectation with respect to the so-called risk-neutral probability. Important evidence of the presence of frictions in financial markets has led to study market pricing rules in such a framework. Recently, Cerreia-Vioglio et al. (J Econ Theory 157:730–762, 2015) have extended the Fundamental Theorem of Finance by showing that, with markets frictions, requiring the put–call parity to hold, together with the mild assumption of translation invariance, is equivalent to the market pricing rule being represented as a discounted Choquet expectation with respect to a risk-neutral nonadditive probability. This paper continues this study by characterizing important properties of the (unique) risk-neutral nonadditive probability v_f associated with a Choquet pricing rule f, when it is not assumed to be subadditive. First, we show that the observed violation of the call–put parity, a condition considered by Chateauneuf et al. (Math Financ 6:323–330, 1996) similar to the put–call parity in Cerreia-Vioglio et al. (2015), is consistent with the existence of bid-ask spreads. Second, the balancedness of v_f—or equivalently the non-vacuity of its core—is characterized by an arbitrage-free condition that eliminates all the arbitrage opportunities that can be obtained by splitting payoffs in parts; moreover the (nonempty) core of v_f consists of additive probabilities below v_f whose associated (standard) expectations are all below the Choquet pricing rule f. Third, by strengthening again the previous arbitrage-free condition, we show the existence of a strictly positive risk-neutral probability below v_f, which allows to recover the standard formulation of the Fundamental Theorem of Finance for frictionless markets.

Nonempty Core of Minimum Cost Spanning Tree Games with Revenues

Nonempty Core of Minimum Cost Spanning Tree Games with Revenues

Cooperative and axiomatic approaches to the knapsack allocation problem

In the knapsack problem a group of agents want to fill a knapsack with several goods. Two issues must be considered. The first is to decide optimally what goods to select for the knapsack. This issue has been studied in many papers in the literature on Operations Research and Management Science. The second issue is to divide the total revenue among the agents. This issue has been studied in only a few papers, and this is one of them. For each knapsack problem we consider three associated cooperative games. One of them (the pessimistic game) has already been considered in the literature. The other two (realistic and optimistic games) are defined in this paper. The pessimistic and realistic games have non-empty cores but the core of the optimistic game could be empty. We then follow the axiomatic approach. We propose two rules: The first is based on the optimal solution of the knapsack problem. The second is the Shapley value of the so called optimistic game. We offer axiomatic characterizations of both rules.

Core stable coalition selection in collaborative truckload transportation procurement

We study the problem of forming a core stable coalition that minimizes system-wide cost, given a set of candidate full truckload shipment lanes, which we model as players, and a characteristic function that corresponds to solving an NP-Hard cost minimizing lane covering problem. Since not every coalition has a non-empty core in this setting, we formulate a mixed integer linear program that can identify a core stable coalition with minimal system-wide cost along with a cost allocation in the core. We propose a solution method that embeds row generation, column generation, and an upper bounding heuristic into branch-and-bound, which can be considered as a form of branch-and-cut-and-price. We evaluate the performance of our solution method through extensive numerical experiments on randomly generated problem instances.

DEA game for internal cooperation between an upper-level process and multiple lower-level processes

Cooperation is an important strategy to improve companies’ profitability. As a data-driven tool for performance evaluation, data envelopment analysis (DEA) is often used to measure the benefits from cooperation between independent decision making units (DMUs) or systems, but few involve the internal cooperation in a network structure system. For a double-level system that consists of an upper- and multiple lower-level processes, prior to cooperation, its internal processes are independent and fail to yield profit. Once the upper-level process cooperates with at least one lower-level process, the system can operate to obtain profit. At present, no research explores the internal cooperation in a double-level system from the perspective of relative performance, considering such dependency between upper- and lower-level processes. To fill this gap, we propose a double-level DEA game by integrating network DEA and cooperative game theory, and prove this new game is monotone, super-additive, and has non-empty core. We use an example of 15 supply chains including one supplier and three retailers to validate our approach and provide an example as a real-life application. Results show the Shapley value and nucleolus have the best performance in allocating the total profit in terms of fairness and stability, respectively.

Retraction

Abstract Original Article Title: Existence of k-People Stable Alliance in n-Player Cooperative Games Authors: Zheng Tao,1 Yuanwei Hu,2 Ziyi Pan,2 Junyi Zhao,2 and Dongao Xing2 This article has been retracted upon request from the authors due to errors they found in the conclusions. ASTM International February 2022

Editorial: Special Section on Human-Centered Artificial Intelligence with Big Data Applications

Editorial: Special Section on Human-Centered Artificial Intelligence with Big Data Applications

Trouble comes in threes: Core stability in minimum cost connection networks

We consider a generalization of the Minimum Cost Spanning Tree (MCST) model, called the Minimum Cost Connection Network (MCCN) model, where network users have connection demands in the form of a pair of nodes they want connected directly or indirectly. For a fixed network, which satisfies all connection demands, the problem consists of allocating the total cost of the network among its users. Thereby every MCCN problem induces a cooperative cost game where the cost of every coalition of users is the cost of an efficient network satisfying the demand of the users in the coalition. Unlike the MCST-model, we show that the core of the induced cost game in the MCCN-model can be empty even when all locations are demanded. We therefore consider sufficient conditions for non-empty core. It is shown that: when the efficient network and the demand graph (i.e. the graph consisting of the direct connections between the pairs of demanded nodes) consist of the same components, the induced cost game has non-empty core (Theorem 1); and, when the demand graph consists of at most two components, the induced cost game has non-empty core (Theorem 2).

Formulating cross-sector horizontal coalition strategies for multi-product assembly systems with a common component

This study examines various coalition structures for a multiproduct assembly system with a common component. Under different forms of coalition in the channel, the common component supplier may form partial or grand coalitions with the other suppliers to achieve maximum profit and eliminate supply chain inefficiencies. The optimal pricing decisions of suppliers are characterized and possible coalition structures are proposed. Results document that coalition structures, product demand characteristics, and manufacturing costs profoundly impact optimal wholesale price decisions. In addition, component suppliers are not always worse off even when the remaining suppliers form a partial coalition. Conditions under which all suppliers form a grand coalition are also provided to introduce a fair allocation and a non-empty core. A numerical experiment is conducted to show the influence of model parameters on the profit of different coalition structures. Profit allocation between the suppliers under the multiproduct assembly system is also discussed.

Stability and values for games with coalition structures

Casajus (2008) established a link between coalition formation in hedonic games and players’ preferences over coalition structures. He showed that if a value concept for games with coalition structures (a CS-value) satisfies the splitting property and the aggregate component independence property, then the CS-value induces a hedonic game satisfying the common ranking property and yields a nonempty core. The following question was left open: What CS-values induce a hedonic game that does not satisfy the common ranking property but has a nonempty core? In this paper, we offer a set of properties that induces a hedonic game satisfying the top-coalition property, which is less demanding than the common ranking property but generates a nonempty core.

Coalitional Game Theory Based Value Sharing in Energy Communities

This paper presents a coalitional game for value sharing in energy communities (ECs). It is proved that the game is super-additive, and the grand coalition effectively increases the global payoff. It is also proved that the model is balanced and thus, it has a nonempty core. This means there always exists at least one value sharing mechanism that makes the grand coalition stable. Therefore, prosumers will always achieve lower bills if they join to form larger ECs. A counterexample is presented to demonstrate that the game is not convex and value sharing based on Shapley values does not necessarily ensure the stability of the coalition. To find a stabilizing value sharing mechanism that belongs to the core of the game, the worst-case excess minimization concept is applied. In this concept, however, size of the optimization problem increases exponentially with respect to the number of members in EC. To make the problem computationally tractable, the idea of clustering members based on their generation/load profiles and considering the same profile and share for members in the same cluster is proposed here. K-means algorithm is used for clustering prosumers’ profiles. This way, the problem would have several redundant constraints that can be removed. The redundant constraints are identified and removed via the generalized Llewellyn’s rules. Finally, value sharing in an apartment building in the southern part of Finland in the metropolitan area is studied to demonstrate effectiveness of the method.

The core and superdifferential of a fuzzy TU-cooperative game

In the paper, we consider conditions providing coincidence of the cores and superdifferentials of fuzzy cooperative games with side payments. It turned out that one of the most simple sufficient conditions consists of weak homogeneity. Moreover, by applying so-called S*-representation of a fuzzy game introduced by the author, we show that for any vwith nonempty core C(v) there exists some game u such that C(v) coincides with the superdifferential of u. By applying subdifferential calculus we describe a structure of the core forboth classic fuzzy extensions of the ordinary cooperative game (e.g., Aubin and Owen extensions) and for some new continuations, like Harsanyi extensions and generalized Airport game.