Discrete and stochastic coalitional storage games

Problem definition: Energy storage has become an indispensable part of power distribution systems, necessitating prudent investment decisions. We analyze an energy storage facility location problem and compare the benefits of centralized storage (adjacent to a central energy generation site) versus distributed storage (localized at demand sites). This problem encompasses optimizing storage capacities across all locations, with the objective of minimizing the total storage investment and energy generation costs. Methodology/results: We employ a stylized model that captures essential features of an energy distribution system, including convex costs, stochastic demand, storage efficiency, and line losses. Using dynamic programming, we optimize storage operations and derive value function properties that are key to analyzing the storage investment decisions. We discern fundamental differences between centralization/localization decisions at the capacity investment stage and the centralization/localization decisions at the storage operations level. Operationally, centrally stored energy offers more flexibility, which is consistent with the conventional understanding of inventory pooling. However, we find that localized storage often emerges as the preferred option at the investment stage under various circumstances. Managerial implications: Storage investment should first be made at the demand locations with positive minimum demand regardless of the level of demand variability. Subsequent storage investment should consider the tradeoffs between centralized versus localized investment. Operationally, the relative magnitudes of storage and line losses drive different optimal storage policies. Despite the differences, these policies are guided by common principles such as pooling inventory and balancing local storage levels. Supplemental Material: The online appendices are available at https://doi.org/10.1287/msom.2020.0652 .

One of the most significant solution concepts in cooperative grey game theory is the grey Shapley value. This value is a fascinating one among the models and methods of operations research, and has been the subject of extensive study by other researchers. The objective of this study is to characterize and redefine this value in cooperative games where coalition values are grey numbers. In this study, the grey Shapley value is characterized by the following axioms: G-gain loss, G-null player, and G-differential marginality. Finally, this study concludes with an investigation of some applications involving production costs. This study is based on an investigation of the costs incurred when milk producers collaborate.

One of the most difficult and major problems of game theory consists in understanding how players may choose to organize themselves in a game in order to get joint-maximizing profits. The problem raises in fact two different questions: which organizations are likely to emerge from a game (first problem) and how are members of these organizations going to share its gains (second problem)? Hence, if these organizations are coalitions of players, a solution to the game shall specify those coalitions that are likely to form (ex post stable coalitions and/or coalitions that could form during the process at an intermediary step) and the way players, in each coalition, shall share common utility they get through their coalition. In Theory of Games and Economic Behavior (TGEB) Von Neumann and Morgenstern proposed developing a theory of n-person games in the framework of the characteristic function model in order to solve the second problem. Later, Shapley (1953), Aumann and Maschler (1964), and many other authors have responded to the question by proposing various solution concepts: respectively, the Shapley value, the bargaining set, the core, the kernel, the nucleolus, etc. (Although the core is a major solution concept in cooperative games, we will not tackle it here, except in comparing it to bargaining sets, because we want to focus our attention on coalition structure. We will only give a brief definition and speak of the “core of a coalition structure.” We refer to Herve Moulin’s chapter in this book for a presentation of the core.)In cooperative game theory it seems hard to disentangle the two problems: the way players share the worth of the coalition depends on which coalitions are likely to emerge from the game. It is this link between the two problems that research on coalition structure games tries to elucidate, seeking a better understanding of how a given coalition structure affects each player’s individual payoff. Once this question is answered (and there are many ways to answer it), one can face the first problem (how players choose to organize themselves) from different viewpoints. Either one links explicitly the two problems considering that the “sharing” game within each coalition and the game between coalitions that determines stable coalitions should havecommon features,1 or one separates them. A way of avoiding arbitrary or ad hoc aspects of a specific choice of a solution concept may consist in introducing first a non-cooperative step into the game, which describes the whole set of possible links among players. Starting from a graph-theoretic extension of the Shapley value, Aumann and Myerson (1988) propose for instance justifying the existence of a cooperation structure in a cooperative game by a non-cooperative “linking” game, the issue of which is a subgame perfect Nash equilibrium.

This article outlines the differences between cooperative and non-cooperative game theory. It introduces some of the main concepts of cooperative game theory as they apply to strategic management research.

Nucleolus is one of the fundamental solution concepts in cooperative game theory. There has been considerable progress in locating the nucleolus in the last three years. The paper motivates through examples how the recent algorithms work efficiently for certain structured class of coperative games. Though the data of a cooperative game grows exponentially in size with the number of players, assignment games, and balanced connected games, grow only polynomially in size, on the number of players. The algorithm for assignment games is based on an efficient graph theoretic algorithm which counts the longest paths to each vertex and trimming of cycles to quickly arrive at the lexicographic geometric centre. Connected games are solved by the technique of feasible direction, initiated in the assignment case. The sellers market corner of the core for assignment games has its counterpart, the lexmin vertex in balanced connected games. Nucleolus has also been characterized via a set of anxioms based on subgame consistency. This is exploited for standard tree games to arrive at an efficient and intuitively explainable algorithm. Improvements on the pivoting manipulations to locate coalitions with constant excess are possible and the paper initially discusses such an algorithm at the beginning.KeywordsCooperative GameGrand CoalitionTree GameCooperative Game TheoryEdge CostThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

An allocative method of hybrid electrical and thermal energy storage capacity for load shifting based on seasonal difference in district energy planning

The need for a sustainable society has grown rapidly. This trend requires new production system concepts following an era of mass customization. As one of these new concepts, “crowdsourced manufacturing” has attracted noticeable attention. In such systems, each participant shares their manufacturing resources for ecosystem co-prosperity, providing new value for the next society. To realize such a concept, it is important to (1) match resource requests and resource offers so as to achieve high efficiency, and (2) induce participants to act in a fair way. Previously, some studies showed production efficiency improvements. Nevertheless, relatively few studies have been conducted on induction mechanisms. The purpose of this study is to develop induction mechanisms for participants. Concerning induction mechanisms, we focus on two viewpoints: (a) matching stability, and (b) “strategyproofness.” These viewpoints are well-known concepts in the market design research field. We previously proposed a resource matching stability analysis method and mechanism for inducing participants to accept matching plans. Formally, a matching method is “strategyproof” when it is a dominant strategy for all participants to submit their true information. However, it is hard to satisfy this condition. Practically, it would be useful to evaluate the strength of an induction, even if the matching method is not strategyproof. In this study, we propose indices for showing the strength of induction (“strength of strategyproofness”). Subsequently, we evaluate matching methods, and show that participants will state false information to maximize their profit in a system with resource matching methods for the profit maximization of the entire system. As the resource providers, they can obtain greater profit by submitting false information regarding resource usage fees. Then, the profits of the resource requesters are unfairly impaired. Furthermore, we propose a new resource matching method, inspired from the “nucleolus” concept in cooperative game theory. The proposed method reduces the maximum dissatisfaction (i.e., profit loss) of resource requesters and resource providers, based on profit sharing. The computational results show that the proposed method induces participants to submit true information, while maintaining high production efficiency.

A New Approximation Method for the Shapley Value Applied to the WTC 9/11 Attack

This paper is a survey of some of the ways in which the representation theory of the symmetric group has been used in voting theory and game theory. In particular, we use permutation representations that arise from the action of the symmetric group on tabloids to describe, for example, a surprising relationship between the Borda count and Kemeny rule in voting. We also explain a powerful representation-theoretic approach to working with linear symmetric solution concepts in cooperative game theory. Along the way, we discuss new research questions that arise within and because of the representation-theoretic framework we are using.

According to Colman (1988), Harold Pinter’s play The Caretaker illustrates the inadequacy of the stable set, a solution concept of cooperative game theory proposed by von Neumann and Morgenstern (1944). We challenge this illustration by pointing out that it is at variance with Pinter scholars’ interpretation of the play. We argue that The Caretaker better illustrates the adequacy of the bargaining set, another solution concept introduced by Aumann and Maschler (1961; 1964).

The Shapley value is an a priori evaluation of the prospects of a player in a multi-person game. Introduced by Lloyd S. Shapley in 1953, it has become a central solution concept in cooperative game theory. The Shapley value has been applied to economic, political, and other models.

The traveling salesman game (TSG) consists of dividing the cost of a round trip among several customers. One of the most significant solution concepts in cooperative game theory is the Shapley value, which provides a fair division of the costs for a variety of games including the TSG, based on the marginal costs attributed with each customer. In this paper, we consider efficient methods for computing the Shapley value for the TSG. There exist two major variants of the TSG. In the first variant, there exists a fixed order in which the customers are serviced. We show a method for efficient computation of the Shapley value in this setting. Our result is also applicable for efficient computation of the Shapley value in ride-sharing settings, when a number of passengers would like to fairly split their ride cost. In the second variant, there is no predetermined fixed order. We show that the Shapley value cannot be efficiently computed in this setting. However, extensive simulations reveal that our approach for the first variant can serve as an excellent proxy for the second variant, outperforming the state-of-the-art methods.

Insights into the Core of the Assignment Game Via Complementarity

Storage capacity plan and transition of heterogeneous energy at regional strategic level: A financial derivative perspective

We consider dynamic cooperative games in characteristic function form in the sense that the characteristic function evolves over time in accordance with a difference or differential equation that is influenced not only by the current ("instantaneous") characteristic function but also by the solution concept used to allocate the benefits of cooperation among the players. The latter solution concept can be any one of a number of now standard solution concepts of cooperative game theory but, for demonstration purposes, we focus on the core and the Shapley value. In the process, we introduce some new mechanisms by which players may regard the evolution of cooperative game over time and analyse them with respect to the goal of attaining time consistency either in discrete or in continuous time setting. In discrete time, we illustrate the phenomena that can arise when an allocation according to a given solution concept is used to adapt the values of coalitions at successive time points. In continuous time, we introduce the notion of an "instantaneous" game and its integration over time.