Background
The Shapley value, named after Nobel laureate Lloyd Shapley, is a key concept in cooperative game theory used to determine a fair allocation of resources or payoffs among players in a cooperative game. It provides a systematic way to distribute the total gains of a coalition among its participants based on their individual contributions.
Historical Context
The Shapley value was introduced by Lloyd Shapley in 1953 as a solution concept in cooperative game theory. Shapley’s work built on earlier mathematical frameworks and aimed to address the equitable distribution of outcomes when parties cooperate and form coalitions.
Definitions and Concepts
The Shapley value for a player in a cooperative game is calculated by:
- Determining the marginal contribution of each player by evaluating the difference between the payoff of any coalition that includes the player and the payoff of the coalition without the player.
- Calculating the weighted average of these marginal contributions. The weights are the probabilities that each coalition can form, given by the combinatorial structure of the coalitions.
Mathematically, if \( v \) is the characteristic function representing the total payoff of a coalition and \( S \) is any subset of players, the Shapley value \( \phi_i(v) \) for player \( i \) is given by: \[ \phi_i(v) = \sum_{S \subseteq N \setminus {i}} \frac{|S|!(|N|-|S|-1)!}{|N|!} (v(S \cup {i}) - v(S)) \] where \( N \) is the set of all players, and \( |S| \) denotes the size of the coalition \( S \).
Major Analytical Frameworks
Classical Economics
Classical economics has limited direct overlap with game theory concepts such as the Shapley value, which are more commonly used in the analysis of cooperative behaviors within competitive environments.
Neoclassical Economics
Neoclassical economics occasionally uses game theory and the Shapley value to analyze market structures and the behavior of firms in industries characterized by oligopoly and other cooperative scenarios.
Keynesian Economics
Keynesian economics focuses on macroeconomic variables but may use the Shapley value indirectly in discussions of cooperative behaviors in public policy or international economic frameworks.
Marxian Economics
Marxian economics typically does not engage directly with game theory concepts like the Shapley value, focusing more on class struggle and the dynamics of capital and labor.
Institutional Economics
Institutional economics may leverage ideas in cooperative game theory, including the Shapley value, to explore institutional structures and the behavior of organizations and groups in economic contexts.
Behavioral Economics
Behavioral economics could apply the Shapley value to understand how real human behaviors deviate from the fully rational decision-making processes in cooperative game settings.
Post-Keynesian Economics
Post-Keynesian economics might use game theory concepts, including the Shapley value, to examine asymmetric information, bargaining power, and other scenarios where cooperative agreements significantly impact economic outcomes.
Austrian Economics
Austrian economics is less likely to use the Shapley value, given its focus on individual actions and subjective values over cooperative game-theoretical scenarios.
Development Economics
In development economics, the Shapley value can be instrumental in understanding how to distribute aid, resources, or investments fairly among cooperative groups or countries.
Monetarism
Monetarism typically focuses on the role of money supply and would not conventionally leverage the Shapley value directly.
Comparative Analysis
By comparing the applications across different economic frameworks, it is evident that the Shapley value is most applicable in fields that deal with cooperative behaviors, shared resources, and coalition formations. This includes institutional economics, development economics, and certain applications in behavioral and neoclassical economics.
Case Studies
- Application in Environmental Economics: Using the Shapley value to allocate costs fairly among countries for mitigating climate change.
- Telecommunications Industry Analysis: Distributing profits among companies forming a shared network infrastructure.
Suggested Books for Further Studies
- “A Course in Game Theory” by Osborne and Rubinstein.
- “Game Theory for Applied Economists” by Robert Gibbons.
- “Microeconomic Theory” by Andreu Mas-Colell, Michael D. Whinston, and Jerry R. Green.
Related Terms with Definitions
- Cooperative Game Theory: A branch of game theory that studies how cooperative groups interact and how they share incomes or benefits.
- Characteristic Function: A function in cooperative game theory that assigns a value to each coalition, representing the total payoff that can be achieved by the coalition.
- Marginal Contribution: The