This is different from non-cooperative games in which there is either no possibility to forge alliances or all agreements need to be self-enforcing (e.g. through credible threats).
[1] Cooperative games are analysed by focusing on coalitions that can be formed, and the joint actions that groups can take and the resulting collective payoffs.
Formally, the coalitional game consists of a finite set of players
The function describes how much collective payoff a set of players can gain by forming a coalition.
The players must identify and agree on a common interest to establish the foundation and reasoning for cooperation.
Once the players have a clear understanding of their shared interest, they can work together to achieve it.
Players must share information about their preferences, resources, and constraints to identify opportunities for mutual gain.
By sharing information, players can better understand each other's goals and work towards achieving them together.
[citation needed] Voluntariness, equality, and mutual benefit: In cooperative games, players voluntarily come together to form coalitions and make agreements.
The players must be equal partners in the coalition, and any agreements must be mutually beneficial.
Cooperation is only sustainable if all parties feel they are receiving a fair share of the benefits.
[citation needed] Compulsory contract: In cooperative games, agreements between players are binding and mandatory.
The players must trust each other to keep their commitments, and there must be mechanisms in place to enforce the agreements.
By making agreements binding and mandatory, players can ensure that they will achieve their shared goal.
is naturally defined as In other words, we simply restrict our attention to coalitions contained in
It is sometimes assumed that a simple game is nonempty or that it does not contain an empty set.
A few relations among the above axioms have widely been recognized, such as the following (e.g., Peleg, 2002, Section 2.1[7]): More generally, a complete investigation of the relation among the four conventional axioms (monotonicity, properness, strongness, and non-weakness), finiteness, and algorithmic computability[8] has been made (Kumabe and Mihara, 2011[9]), whose results are summarized in the Table "Existence of Simple Games" below.
The restrictions that various axioms for simple games impose on their Nakamura number were also studied extensively.
The definition is very general allowing the concept to be used in a wide variety of game formats.
For simple games, there is another notion of the core, when each player is assumed to have preferences on a set
may be negative, in which case it represents a bonus for leaving the grand coalition.
Following this line of reasoning, the least-core, introduced in (Maschler, Peleg & Shapley 1979), is the intersection of all non-empty strong
The Shapley value is the unique payoff vector that is efficient, symmetric, and satisfies monotonicity.
The Shapley value of a superadditive game is individually rational, but this is not true in general.
(Maschler, Peleg & Shapley 1979) gave a more intuitive description: Starting with the least-core, record the coalitions for which the right-hand side of the inequality in the definition of
Continue decreasing the right-hand side for the remaining coalitions, until it cannot be reduced without making the set empty.
Convex cooperative games have many nice properties: Submodular and supermodular set functions are also studied in combinatorial optimization.
Many of the results in (Shapley 1971) have analogues in (Edmonds 1970), where submodular functions were first presented as generalizations of matroids.
Unfortunately, this conflicts directly with Shapley's original definition of supermodular functions as "convex".
Corporate strategic decisions can develop and create value through cooperative game theory.