# Cooperative games

Cooperative games are those games in which coalitions can be formed. As a distribution of the payments can be agreed, they are also known as coalition games.

Game theory is a mathematical tool with which you can analyze strategic rational decision-making problems. That is, where the decision of the other agents affects mine and vice versa.

Parallel to the development of non-cooperative game theory, cooperative game theory began to take shape. The first contributions came from John Nash, Howard Raiffa, followed by Lloyd Shapley, David Gale, Martin Shubik, and Robert Aumann.

## Central concepts in cooperative game theory

In cooperative game theory players are allowed to form coalitions to distribute a certain amount of something, which can be food, money, power, costs, etc. Therefore, there are incentives for players to work together, with a view to obtaining the maximum benefit.

The analysis of cooperative games focuses on the concepts of solutions to the different types of games. In addition to verifying that the coalition is stable. That is, that no member is dissatisfied and wants to withdraw from it.

## Types of cooperative games

The fundamental problem in cooperative games is how the total payout for the game is shared among the players. There the theory is divided into two: coalitional games with transferable payouts (UT) and games without transferable payouts (UNT).

### Cooperative games with transferable payments

The most popular types of coalitional games with transferable payouts are super additive games, convex games, bankruptcy games, market games, voting games, auction games, cost games, flow games, etc.

Example: Three player auction game (luxury car market)

Player 1 owns a luxury car and there are two other players who want to buy it. Player 2 values it more than the owner and Player 3 values it more than Player 2.

This auction can be modeled as a coalitional game UT where v = p1, v = v = v = 0, v = p2, v = p3, v = p3

That is, the following scenarios can occur:

- Only player one is in the auction. The value is what its owner gives it and is not sold.
- In the auction there are players 2 and 3. Then, the value is zero, because they cannot buy the car only among themselves,
- Players 1 and 2 are in the auction. The value is the one given by player 2 and sold at that value.
- Players 1 and 3 are in the auction. The value is the one given by player 3 and sold at that value.
- Players 1, 2 and 3 are in the auction. The value is the one given by player 3 and it is sold at that value (which is higher than the value given by player 2).

### Cooperative games with non-transferable payments

The most popular types of coalition games with non-transferable payouts are market games, voting games, auction games, matching games, optimization games, etc.

Example: banker game

There are 3 players, who by themselves can obtain nothing. Player 1, with the help of Player 2, can get $ 100. Player 1 can give back to Player 2 by giving him money, but the money sent is lost or stolen with probability 0.75. Player 3 is the banker, so Player 1 can rest assured that his transactions are safely sent to Player 2 by using Player 3 as an intermediary.

The problem is determining how much Player 1 should pay Player 2 for his help in obtaining the $ 100, and how much Player 3 (intermediary banker) should pay for helping player 2 make transactions less expensive. allowed to make transfers between players.

This game has "infinite solutions" (as long as it is a space and not a point). The solutions involve collaboration between player 1 and 2, on the condition that something is paid to the intermediary.

## Cooperative game theory application

The main solution concepts in cooperative game theory (the Shapley kernel and value) have implicit moral judgments such as justice, fairness, and the social optimum. The economic and social applications are numerous, the concepts offered by cooperative game theory have been implemented in situations such as:

- Cost distribution.
- Evaluation of investment projects.
- Assignment of taxes and subsidies.
- Distribution of power in political and military affairs.
- Development of public service supply models.