Game theory is a branch of mathematics that deals with the study of strategic decision making in situations where the outcomes depend on the actions of multiple individuals or players. Core game theory is a concept that refers to a set of solutions that are stable and cannot be improved upon by any group of players. In this article, we will explore how core game theory is calculated.

What Is Core Game Theory?

Core game theory is a solution concept in cooperative game theory that involves finding an allocation of resources among players such that no subset of players can improve their payoff by forming a coalition. In other words, it is a solution where no group of players has an incentive to leave and form their own group.

How Is Core Game Theory Calculated?

To calculate core game theory, we must first understand the concept of a characteristic function. A characteristic function is a mathematical representation of the value that each player assigns to each possible coalition or set of players. It maps each coalition to a real number representing the worth or value of that coalition to the player.

Once we have established the characteristic function, we can use it to determine if there exists any imputation, i.e., an allocation where every player gets at least as much as they would get by joining any other coalition. If such an imputation exists and no subset of players can improve their payoff by forming a coalition, then we can say that we have found a solution in the core.

Example:

Let us consider an example with three players – A, B, and C – who are sharing some resources amongst themselves. The worth or value that each player assigns to each possible coalition is given by the following table:

Coalition Value for Player A Value for Player B Value for Player C
{A} 4 0 0
{B} 0 3 0
{C} 0 0 5
{A,B}

In this example, the core is non-empty and consists of the imputation {4,3,5}.

In Conclusion:

Core game theory is a solution concept that involves finding an allocation of resources among players such that no subset of players can improve their payoff by forming a coalition. It is calculated using the characteristic function, which maps each coalition to a real number representing its value to each player. The core can be found by identifying an imputation where every player gets at least as much as they would get by joining any other coalition.