Shapley-Shubik model

Shapley-Shubik model. The Shapley-Shubik model is a well-known concept in Game theory, developed by Lloyd Shapley and Martin Shubik in the 1950s, building on the work of John von Neumann and Oskar Morgenstern. This model is closely related to the Nash equilibrium concept, introduced by John Nash, and has been influential in the development of Mechanism design theory by Leonid Hurwicz, Eric Maskin, and Roger Myerson. The Shapley-Shubik model has been applied in various fields, including Economics, Politics, and Sociology, with notable contributions from Kenneth Arrow, James Buchanan, and Gordon Tullock.

Introduction

The Shapley-Shubik model is a Cooperative game theory model that aims to describe the behavior of players in a Coalition formation process. This model is based on the idea that players have different levels of Power (social and political), which is measured by their ability to influence the outcome of the game. The model is closely related to the work of Robert Aumann and Michael Maschler on Bargaining theory and has been applied in various fields, including International relations and Conflict resolution, with notable contributions from Thomas Schelling and Anatol Rapoport. The Shapley-Shubik model has also been used to study the behavior of Political parties and Interest groups in Democratic systems, with insights from Anthony Downs and Mancur Olson. Furthermore, the model has been applied in the context of European Union decision-making, with research by Robert Axelrod and Robert Keohane.

Theory

The Shapley-Shubik model is based on the concept of a Weighted voting system, where each player has a certain number of votes, and the outcome of the game is determined by the coalition that forms. The model assumes that players are rationally motivated and will form coalitions that maximize their expected payoff. The Shapley-Shubik model is closely related to the Shapley value, which is a solution concept in Cooperative game theory that assigns a value to each player in a game based on their contribution to the coalition. This concept has been influential in the development of Auction theory by William Vickrey and Roger B. Myerson. The model has also been applied in the context of Network theory, with research by Mark Granovetter and Barry Wellman. Additionally, the Shapley-Shubik model has been used to study the behavior of Firms and Markets in Industrial organization, with insights from Joseph Schumpeter and George Stigler.

Application

The Shapley-Shubik model has been applied in various fields, including Politics, Economics, and Sociology. In Politics, the model has been used to study the behavior of Political parties and Interest groups in Democratic systems. For example, the model has been applied to the study of United States Congress decision-making, with research by David Mayhew and Keith Krehbiel. In Economics, the model has been used to study the behavior of Firms and Markets in Industrial organization, with insights from Oliver Williamson and Herbert Simon. The model has also been applied in the context of International trade, with research by Paul Krugman and Jagdish Bhagwati. Furthermore, the Shapley-Shubik model has been used to study the behavior of Social networks and Communities, with notable contributions from James Coleman and Peter Blau.

Limitations

The Shapley-Shubik model has several limitations, including the assumption of rational behavior and the difficulty of measuring Power (social and political). The model also assumes that players have complete information about the game, which may not always be the case in real-world situations. Additionally, the model does not account for Externalities and Uncertainty, which can affect the outcome of the game. Despite these limitations, the Shapley-Shubik model remains a widely used and influential concept in Game theory and has been applied in various fields, including Public choice theory by James Buchanan and Gordon Tullock, and Mechanism design theory by Leonid Hurwicz, Eric Maskin, and Roger Myerson. The model has also been criticized by Herbert Simon and Amos Tversky for its assumption of rational behavior, and has been extended by Robert Aumann and Sergiu Hart to account for Bounded rationality.

Mathematical_Formulation

The Shapley-Shubik model can be formulated mathematically using the concept of a Weighted voting system. Let N be the set of players, and let w_i be the weight of player i. The model assumes that the outcome of the game is determined by the coalition that forms, and that players are rationally motivated. The Shapley-Shubik model can be represented by the following equation: φ_i = S|, where φ_i is the Shapley value of player i, v(S) is the value of coalition S, and S| is the size of coalition S. This equation has been used to study the behavior of Firms and Markets in Industrial organization, with insights from Joseph Schumpeter and George Stigler. The model has also been applied in the context of Network theory, with research by Mark Granovetter and Barry Wellman. Additionally, the Shapley-Shubik model has been used to study the behavior of Social networks and Communities, with notable contributions from James Coleman and Peter Blau.

Category:Game theory