Shapley value

Shapley value. The Shapley value is a concept developed by Lloyd Shapley in 1953, in the field of cooperative game theory, which is a subfield of mathematics and economics, closely related to the work of John von Neumann and Oskar Morgenstern. This concept is used to fairly distribute the total value created by a coalition of players, taking into account their respective contributions, as studied by Kenneth Arrow and Gerard Debreu. The Shapley value has been influential in various fields, including game theory, economics, and political science, with notable applications in the work of Robert Aumann and Thomas Schelling.

Introduction

The Shapley value is a solution concept in cooperative game theory, which is a branch of mathematics that studies the behavior of players in cooperative situations, as developed by John Nash and Reinhard Selten. It is used to allocate the total value created by a coalition of players, taking into account their respective contributions, as analyzed by Leonid Hurwicz and Eric Maskin. The Shapley value is named after Lloyd Shapley, who introduced it in his 1953 paper, "A Value for n-Person Games," which was influenced by the work of Von Neumann and Morgenstern on game theory. The concept has been widely used in various fields, including economics, political science, and computer science, with applications in the work of Andrew Carnegie and Bill Gates. Notable researchers, such as Amartya Sen and Joseph Stiglitz, have also applied the Shapley value in their studies on social choice theory and information economics.

Definition and Formula

The Shapley value is defined as a function that assigns a value to each player in a coalition, based on their marginal contribution to the coalition, as formalized by Shapley and Shubik. The formula for the Shapley value is based on the concept of marginal contribution, which is the difference between the value of the coalition with and without a particular player, as discussed by Milton Friedman and Gary Becker. The Shapley value is calculated using the following formula: φi = ∑(S⊆N\{i}) (|S|! \* (n-|S|-1)! / n!) \* (v(S∪{i}) - v(S)), where φi is the Shapley value of player i, S is a subset of the set of players N, |S| is the number of players in S, n is the total number of players, and v(S) is the value of the coalition S, as used by Robert J. Aumann and Thomas C. Schelling. This formula has been applied in various contexts, including auction theory and mechanism design, with contributions from William Vickrey and Roger Myerson.

Properties and Axioms

The Shapley value has several properties and axioms that make it a desirable solution concept, as discussed by Kenneth Arrow and Gerard Debreu. One of the key properties is efficiency, which means that the sum of the Shapley values of all players equals the total value created by the coalition, as analyzed by Leonid Hurwicz and Eric Maskin. Another property is symmetry, which means that the Shapley value is invariant to the ordering of the players, as studied by John Harsanyi and Reinhard Selten. The Shapley value also satisfies the axiom of additivity, which means that the Shapley value of a player in a coalition is the sum of their Shapley values in the sub-coalitions, as formalized by Lloyd Shapley and Martin Shubik. These properties and axioms have been influential in the development of cooperative game theory, with contributions from Robert Aumann and Thomas Schelling.

Calculation and Examples

The calculation of the Shapley value can be complex, especially for large coalitions, as discussed by Andrew Carnegie and Bill Gates. However, there are several examples that illustrate the calculation of the Shapley value, as analyzed by Amartya Sen and Joseph Stiglitz. One example is the Gloves Game, where two players, a buyer and a seller, form a coalition to trade gloves, as studied by William Vickrey and Roger Myerson. Another example is the Airport Game, where three players, representing different airlines, form a coalition to share the cost of building an airport, as used by Robert J. Aumann and Thomas C. Schelling. These examples demonstrate how the Shapley value can be used to allocate the total value created by a coalition, taking into account the respective contributions of each player, as applied by Leonid Hurwicz and Eric Maskin.

Applications and Interpretations

The Shapley value has been applied in various fields, including economics, political science, and computer science, with notable contributions from John Nash and Reinhard Selten. One of the key applications is in the allocation of costs and benefits in cooperative games, as studied by Lloyd Shapley and Martin Shubik. The Shapley value has also been used in auction theory and mechanism design, with applications in the work of William Vickrey and Roger Myerson. In addition, the Shapley value has been used to study voting systems and political coalitions, as analyzed by Kenneth Arrow and Gerard Debreu. The Shapley value has also been interpreted as a measure of a player's power or influence in a coalition, as discussed by Robert Aumann and Thomas Schelling. Overall, the Shapley value is a powerful tool for analyzing and designing cooperative systems, with contributions from Andrew Carnegie and Bill Gates. Category:Game theory