Lecture
Non-cooperative game — a term of game theory. A non-cooperative game is a mathematical model of the interaction of several parties (players), during which they cannot form coalitions and coordinate their actions.
Both zero-sum and non-zero-sum games can be cooperative or non-cooperative. Therefore, non-cooperative games can be divided into non-zero-sum games and zero-sum games.
A non-cooperative game in normal form is a triple 
, where 
— is the set of participants of the game (parties, players); Si — is the set of strategies of participant 
; 
— is the payoff function of participant i
, defined on the set of situations 
and mapping it into the set of real numbers.
A non-cooperative game in normal form assumes the following order of play.
1. Players simultaneously and independently of each other choose their strategies from the sets Si . The vector of strategies 
of all players constitutes a situation in the game.
2. Each player receives a payoff, determined by the value of the function 
, after which interaction between them ceases.
The normal form of the game describes the static interaction of players, without providing for the possibility of sequential moves, accumulation of information about the opponent's actions, and repeated interaction. The extensive form of the game is used to model these aspects.
A non-cooperative game in extensive form with a set of players is represented using a directed tree (game tree) as follows.
The vertices of the tree represent states (positions) in which the game may find itself, the edges — the moves that players may use. It is assumed that at most one player can make a move at each position. Three kinds of positions in the game are distinguished:
The initial and intermediate positions form the set of non-terminal positions.
For each vertex of the tree v
, corresponding to a non-terminal position, a player i is defined, who makes a move in it, and a set of moves of this player Sv . Each move 
corresponds to an edge leaving vertex v.
To account for the imperfect information available to players, non-terminal vertices may be combined into information sets.
For each vertex v, corresponding to a terminal position, the payoff functions of all players are defined 
.
The game assumes the following order of play:
1. The game starts from the initial position.
2. At any non-terminal position v the player who has the right to move in it chooses a move , as a result of which the game passes to the next position, into which the edge corresponding to move s leads. If this position is non-terminal, step 2 is repeated.
3. If the game reaches a terminal position v, then all players receive payoffs Hi(v), and the game ends.
The main principle of strategy optimality for non-cooperative games in normal form is the Nash equilibrium, based on the impossibility of participants deviating from their chosen strategies. To date, a family of principles based on the Nash equilibrium has been developed, called Nash equilibrium refinements, the most commonly used of which are:
Less universal principles, used in certain classes of non-cooperative games, are the following:
For non-cooperative games in extensive form, optimality principles based on the Nash equilibrium are also used, but taking into account the specifics of the dynamic interaction of players. The main ones include:
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