Cooperative Games and the Core

Concept in game theory.

Cooperative games are a significant part of game theory, which focuses on the interactions between decision-makers. Unlike non-cooperative games, where players cannot form binding agreements, cooperative games allow for the possibility of players forming coalitions and making commitments.

Introduction to Cooperative Games

Cooperative games are scenarios where players can benefit by working together rather than competing against each other. These games are characterized by the ability of players to form binding agreements and coalitions. The main objective in these games is to achieve a mutually beneficial outcome for all players involved.

Defining the Core of a Game

The core of a game is a concept in cooperative game theory that represents the set of feasible allocations that cannot be improved upon by a subset (a coalition) of the economy's agents. An allocation is said to be in the core if there is no coalition that can improve upon it. In other words, there is no group of players who could break away and receive a larger payoff by themselves.

Understanding the Concept of the Shapley Value

The Shapley Value is a solution concept in cooperative game theory. It was named after Lloyd Shapley, who introduced it in 1953. The Shapley Value provides a way to divide the total payoff of a game among the players based on their marginal contribution to the total payoff. It is calculated by considering all possible orders in which players could join the game, and the marginal contribution each player makes in each of these orders.

The Role of Communication and Agreements in Cooperative Games

In cooperative games, communication plays a crucial role. Players can discuss their strategies, form coalitions, and make binding agreements. These agreements can significantly influence the outcome of the game. For instance, players can agree to share the payoff in a certain way, or they can agree on a particular strategy to increase the total payoff.

Examples of Cooperative Games in Real-Life Situations

Cooperative games are prevalent in real-life situations. For instance, businesses often form alliances or partnerships to increase their market share or to develop new products. Similarly, countries often form coalitions to negotiate trade agreements or to address global issues like climate change.

In conclusion, cooperative games provide a framework for understanding how individuals and groups can work together to achieve common goals. By understanding the core of a game and the Shapley Value, we can gain insights into how to distribute the benefits of cooperation fairly among all participants.