Characteristic Function Games

Characteristic function games are a type of cooperative game where the value of every coalition is determined by a characteristic function. A characteristic function is a mapping from every coalition to a real number that represents the value of that coalition. In other words, the characteristic function assigns a value to every possible group of players. In this lesson, we will look at the definition of characteristic function games, examples of characteristic function games, and solution concepts for these types of games.

Definition of Characteristic Function Games

A characteristic function game is a cooperative game where the value of each coalition is determined by a characteristic function. Formally, the characteristic function is defined as follows:

$$v: 2^N \rightarrow \mathbb{R}$$

Here, N represents the set of players and 2^N represents the set of all possible coalitions. The characteristic function assigns a real number to each coalition, which represents the value of that coalition. The value of the grand coalition, i.e., the coalition consisting of all players, is denoted by v(N).

Examples of Characteristic Function Games

One of the simplest examples of a characteristic function game is the public goods game. In this game, every player contributes to a public pool, and the total value of the pool is shared equally among all players. The characteristic function assigns a value of n to each coalition of size n, where n is the number of players who contribute to the public pool. Another example of a characteristic function game is the minimum cost spanning tree problem. In this game, the players are nodes of a graph and the goal is to find a minimum cost spanning tree that connects all the nodes. The value of a coalition is the cost of the cheapest spanning tree that includes all the nodes in the coalition.

Solution Concepts for Characteristic Function Games

There are several solution concepts for characteristic function games, including the Shapley value, the core, and the nucleolus. The Shapley value is a distribution that assigns a value to each player based on their marginal contribution to every possible coalition. The core is the set of payoff vectors that cannot be improved upon by any coalition of players. The nucleolus is a payoff vector that represents the most stable outcome of the game. It is defined as the set of payoff vectors that satisfy a set of axioms that capture intuitive notions of fairness and stability.

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