Lecture
Cooperative game theory studies games in which groups of players — coalitions — can join their efforts. In this it differs from non-cooperative games, in which coalitions are not permitted and everyone must play for themselves.
Game theory studies conflicts, that is, situations in which a group of people needs to work out some decision concerning all of them. Non-cooperative game theory studies how players should act in order to arrive at one or another outcome, while cooperative game theory studies the question of which outcomes are attainable and the conditions for achieving these outcomes.
By definition, a cooperative game is called a pair (N,v) , where N — is the set of players, and 
— is a function: 
, from the set of all coalitions to the set of real numbers (the so-called characteristic function). It is assumed that the empty coalition earns zero, that is, 
. The characteristic function describes the amount of benefit that a given subset of players can achieve by forming a coalition. It is implied that players will decide to form a coalition depending on the size of the payoffs within the coalition.
the subset of the set of players in a cooperative game that make a non-zero contribution to some coalition is defined by the term carrier and mathematically by the formula .
where N — is the set of players in the cooperative game, v — is the characteristic function of the game.
The complement of the carrier of the game is the set of dummies or null players, that is, players who make no contribution to any of the coalitions.
. 
Simple games — a special kind of cooperative game, where all payoffs are 1 or 0, that is, coalitions either «win» or «lose». A simple game is called proper if:
.
The meaning of this: a coalition wins if and only if the complementary coalition (opposition) loses.
According to the definition of a cooperative game, the set of players N collectively possesses a certain amount of a defined good, which must be divided among the participants. The principles of this division are called solutions of the cooperative game.
A solution can be defined either for a specific game or for a class of games. Naturally, the principles of greatest importance are precisely those applicable across a wide range of cases (that is, for an extensive class of games).
A solution can be either single-valued (in which case a single distribution of payoffs is the solution for each game), or multi-valued (when several distributions can be defined for each game). Examples of single-valued solutions are the N-core (nucleolus) and the Shapley vector, examples of multi-valued ones are the C-core and the K-core.
Cooperative and non-cooperative games are related in that both study the strategic interaction of players, but do so from different points of view. Cooperative games explore the possibilities of joining forces and distributing payoffs among coalitions, while non-cooperative games explore individual strategies and equilibria. Their connection is manifested in the fact that many results of cooperative theory can be interpreted through non-cooperative models and vice versa.
Cooperative outcomes can be realized through non-cooperative mechanisms. For example, if players enter into a contract, its fulfillment can be regarded as a non-cooperative game, in which deviating from the agreement is unprofitable.
Non-cooperative games provide a basis for testing the stability of cooperative solutions. If a coalition is unstable to deviations by individual players, its solution will not be realizable.
Many concepts overlap. Thus, the Nash equilibrium can be regarded as a special case, when cooperation is impossible and decisions are made individually.
Practical connection: in economics, politics, and business, cooperative models are often formulated first (for example, profit distribution), and then tested through non-cooperative mechanisms (for example, the stability of contracts).
| Characteristic | Cooperative games | Non-cooperative games | Connection |
|---|---|---|---|
| Analysis goal | Determine which outcomes are achievable when players unite into coalitions | Study the individual strategies of players and equilibria (for example, Nash equilibrium) | Cooperative outcomes can be modeled as equilibria in non-cooperative games |
| Focus | Collective agreements, distribution of payoff | Individual decisions without binding agreements | Cooperative solutions are often tested for stability in a non-cooperative setting |
| Example | Workers unite in a union for negotiations | Each worker negotiates on their own | A cooperative outcome can be realized through non-cooperative mechanisms (for example, contracts) |
| Tools | Characteristic function, core, Shapley value | Nash equilibrium, strategy dynamics | Implementation mechanisms link both theories |
Cooperative and non-cooperative games are two complementary approaches to analyzing strategic interaction. Cooperative theory answers the question «what outcomes are possible through cooperation», while non-cooperative theory answers «how will players act without binding agreements». Their connection lies in the fact that cooperative solutions are often realized through non-cooperative mechanisms, and the stability of cooperation is tested using non-cooperative models
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