Scarf's, Shapley's, and Shubik's Applications of the “Core” to General Equilibrium

The concept of ‘the core’ originates in cooperative game theory and its introduction to economics in the 1960s as a basis for proofs of existence of general equilibrium is one of the earliest attempts to use game theory to address big questions in economics. Discovery of the core was met with enthusiasm among the community of economic theorists at the time. However, use of the core eventually waned and the concept faded into the backdrop of economic theory. This paper makes use of unpublished correspondence between Herbert Scarf, Lloyd Shapley, and Martin Shubik, as well as other archival and secondary resources, to provide an account of the development of the core and the trajectory of this concept, including those who developed it, after its initial appearance. It is found that the core’s eventual decline is explained by the combined effect of the slowing general equilibrium research program in the 1970s, the increasing prominence of non-cooperative game theory, and subtle issues with the concept that shaped Scarf and Shubik’s research programs after the 1960s.

Game theory provides a powerful mathematical framework for modeling and analyzing systems with multiple decision makers, referred to as players, with possibly conflicting objectives. A game studied in game theory consists of a set of players, a set of strategies (or moves) available to the players, and their payoffs (or utilities) for each combination of their strategies. Depending on whether the players can sign enforceable binding agreements, game theory consists of two branches: noncooperative game theory and cooperative game theory. Noncooperative game theory provides concepts and tools to study the behaviors of the players when they make their decisions independently. Cooperative game theory, on the other hand, assumes that it is possible for the players to sign enforceable binding agreements and provides concepts describing basic principles these binding agreements should follow. Both noncooperative game theory and cooperative game theory have been widely used in many disciplines, such as economics, political science, social science, as well as biology and computer science, among others. They have also received considerable attention in supply chain management literature in recent years. In this chapter, we provide a concise introduction to some of the key concepts and results that are most relevant in our context. We refer to Osborne (2003) and Myerson (1997) for both noncooperative game theory and cooperative game theory, Fudenberg and Tirole (1991) and Başar and Olsder (1999) for noncooperative game theory, Vives (2000) on oligopoly pricing from the perspective of noncooperative game theory, and Peleg and Sudhölter (2007) for cooperative game theory, respectively.

Within economics, game theory occupied a rather isolated niche in the 1960s and 1970s. It was pursued by people who were known specifically as game theorists and who did almost nothing but game theory, while other economists had little idea what game theory was. Game theory is now a standard tool in economics. Contributions to game theory are made by economists across the spectrum of fields and interests, and economists regularly combine work in game theory with work in other areas. Students learn the basic techniques of game theory in the first-year graduate theory core. Excitement over game theory in economics has given way to an easy familiarity. This essay first examines this transition, arguing that the initial excitement surrounding game theory has dissipated not because game theory has retreated from its initial bridgehead, but because it has extended its reach throughout economics. Next, it discusses some key challenges for game theory, including the continuing problem of dealing with multiple equilibria, the need to make game theory useful in applications, and the need to better integrate noncooperative and cooperative game theory. Finally it considers the current status and future prospects of game theory.

Cooperative games and cooperative organizations

Game theory is a branch of mathematics aimed at the modeling and understanding of resource conflict problems. Essentially, the theory splits into two branches: noncooperative and cooperative game theory. The distinction between the two is whether or not the players in the game can make joint decisions regarding the choice of strategy. Noncooperative game theory is closely connected to minimax optimization and typically results in the study of various equilibria, most notably the Nash equilibrium. Cooperative game theory examines how strictly rational (selfish) actors can benefit from voluntary cooperation by reaching bargaining agreements. Another distinction is between static and dynamic game theory, where the latter can be viewed as a combination of game theory and optimal control. In general, the theory provides a structured approach to many important problems arising in signal processing and communications, notably resource allocation and robust transceiver optimization. Recent applications also occur in other emerging fields, such as cognitive radio, spectrum sharing, and in multihop-sensor and adhoc networks.

Game theory is a branch of mathematics aimed at the modeling and understanding of resource conflict problems. Essentially, the theory splits into two branches: noncooperative and cooperative game theory. The distinction between the two is whether or not the players in the game can make joint decisions regarding the choice of strategy. Noncooperative game theory is closely connected to minimax optimization and typically results in the study of various equilibria, most notably the Nash equilibrium. Cooperative game theory examines how strictly rational (selfish) actors can benefit from voluntary cooperation by reaching bargaining agreements. Another distinction is between static and dynamic game theory, where the latter can be viewed as a combination of game theory and optimal control. In general, the theory provides a structured approach to many important problems arising in signal processing and communications, notably resource allocation and robust transceiver optimization. Recent applications also occur in other emerging fields, such as cognitive radio, spectrum sharing, and in multihop-sensor and adhoc networks.

This essay examines game theory’s transformation to becoming the major tool in economics, arguing that the initial excitement surrounding game theory has dissipated not because game theory has retreated from its initial bridgehead, but because it has extended its reach throughout economics. Next, it discusses some key challenges for game theory, including the continuing problem of dealing with multiple equilibria, the need to make game theory useful in applications, and the need to better integrate noncooperative and cooperative game theory. Finally it considers the current status and future prospects of game theory.

Game theory and fish wars: The case of the Northeast Atlantic mackerel fishery

In this article, we described some basic concepts from noncooperative and cooperative game theory and illustrated them by three examples using the interference channel model, namely, the power allocation game for SISO IFC, the beamforming game for MISO IFC, and the transmit covariance game for MIMO IFC. In noncooperative game theory, we restricted ourselves to discuss the NE and PoA and their interpretations in the context of our application. Extensions to other noncooperative approaches include Stackelberg equilibria and the corresponding question Who will go first? We also correlated equilibria where a certain type of common randomness can be exploited to increase the utility region. We leave the large area of coalitional game theory open.

: In RM 23, a proof was given that the nucleolus is continuous as a function of the characteristic function. This proof is not correct; the author, at least, does not know how to complete it. In the paper a correct proof for this fact is given. The proof is based on an alternative definition of the nucleolus, which is of some interest in its own right. (Author)

The reader familiar with game theory will have noticed that the models discussed in earlier chapters are treated as noncooperative games of strategy, and the reader who is totally unfamiliar with noncooperative game theory will have obtained a lengthy introduction to the subject, together with an application of the topic to oligopoly theory. Game theory has been applied to many areas of economics and used extensively in political science as well. In addition to oligopoly, it has been used in general equilibrium, public goods, voting theory, and committee decision. The role of this chapter is to make the connection explicit between game theory and oligopoly. This is done by providing a brief treatment of noncooperative game theory and then showing explicitly how several of the models of earlier chapters can be viewed as games. Section 9.1 contains a brief general discussion of games and game theory that is intended to set the subject in perspective. Section 9.2 presents a standard model of a noncooperative n -person game, and Section 9.3 connects the model of Section 9.2 directly to some of the oligopoly models of other chapters. Section 9.4 presents an important refinement of the noncooperative equilibrium, called the perfect equilibrium. Section 9.5 has some concluding remarks. Overview of game theory Essential features of many games of strategy include the following: (a) There are two or more decision makers, called players . (b) Each player wishes to maximize his own utility, called his payoff .

Commitment and weakness of will in game theory and neoclassical economics

The production environment and its related supply chain have faced with enormous changes by Industry 4.0 (I4.0) emergence. Cloud Manufacturing (CM) is one of the paradigms of I4.0. Composition of Manufacturing Cloud Service (MCS) is an approach for developing the composite services in Cloud Manufacturing System (CMS) considering the MCS providers. Game theory is considered as an effective approach for developing models according to real-world conditions for MCS providers' cooperation and competition. This research proposes a profit function and then develops a mathematical model for MCS composition using game theory and the proposed function. The MCS providers compete with each other in non-cooperative and cooperative games according to the proposed mathematical model. The result of research demonstrates that the cooperation game is more profitable than the non-cooperative game based on the formal cooperation among the MCS providers. Therefore, the payoff of players in the cooperative game is higher than the non-cooperative game. Also, the level of Quality of Service (QoS) in the cooperative game is greater than the game type of non-cooperative. Therefore, the model of cooperative game satisfies the consumers and the MCS providers mutually. The paper recommends applying the cooperation game to service composition in the CMS.

Sufficient conditions for existence of Mi1-bargaining sets.- Noncooperative game theory and social choice.- A probabilistic model of social choice.- Equilibrium points in general noncooperative games and their mixed extensions.- On the theory of optimality principles for noncooperative games.- Cooperative game theory.- A sufficient condition for the coincidence of the core of a cooperative game with its solution.- On the Shapley function for games with an infinite number of players.- Cores and generalized NM-solutions for some classes of cooperative games.- Bargaining theory.- The linear bargaining solution.- On the superlinear bargaining solution.- Stable compromises under corrupt arbitration.- Equilibrium theory.- Stability of economic equilibrium.- An algorithmic approach for searching an equilibrium in fixed budget exchange models.- Equilibrated states and theorems on the core.

Industry 4.0 is the digitalization of the manufacturing systems based on Information and Communication Technologies (ICT) for developing a manufacturing system to gain efficiency and improve productivity. Cloud Manufacturing (CM) is a paradigm of Industry 4.0. Cloud Manufacturing System (CMS) considers anything as a service. The end product is developed based on the service composition in the CMS according to consumers’ needs. Also, composite services are developed based on the interaction of MCS providers from different geographical locations. Therefore, the appropriate Manufacturing Cloud Service (MCS) composition is an important problem based on the real-world conditions in CMS. The game theory studies the mathematical model development based on interactions between MCS providers according to real-world conditions. This research develops an Equilibrial Service Composition Model in Cloud Manufacturing (ESCM) based on game theory. MCS providers and consumers get benefits mutually based on ESCM. MCS providers are players in the game. The payoff function is developed based on a profit function. Also, the game strategies are the levels of Quality of Service (QoS) based on consumers’ needs in ESCM. Firstly, the article develops a composite service based on a non-cooperative game. The Nash equilibrium point demonstrates the QoS value of composite service and the payoff value for the players. Secondly, the article develops a composite service based on a cooperative game. The players participate in coalitions to develop the composite service based on formal cooperation. The grand coalition demonstrates the QoS value of composite service and the payoff value for the players in the cooperative game. The research has compared the games’ results. The players’ payoff and the QoS value are better in the cooperative game than in the non-cooperative game. Therefore, the MCS providers and consumers are satisfied mutually in the cooperative game based on ESCM. Finally, the article has applied ESCM in a Healthcare Service to equip 24 hospitals in the best time.