Non-cooperative game
A system
![]() |
where is the set of players,
is the set of strategies (cf. Strategy (in game theory)) of the
-th player and
is the gain function of the
-th player, defined on the Cartesian product
. A non-cooperative game is played as follows: players, who are acting individually (do not form a coalition, do not cooperate), select their strategies
, as a result of which the situation
appears, in which the
-th player obtains the gain
. The main optimality principle in a non-cooperative game is the principle of realizability of the objective [1], which generates the Nash equilibrium solutions. A solution
is called an equilibrium solution if for all
,
, the inequality
![]() |
where , is valid. Thus, none of the players is interested in unilaterally disturbing the equilibrium solution previously agreed upon between them. It has been proved (Nash's theorem) that a finite non-cooperative game (the sets
and
are finite) possesses an equilibrium solution for mixed strategies. This theorem has been generalized to include infinite non-cooperative games with a finite number of players [3] and non-cooperative games with an infinite number of players (cf. Non-atomic game).
Two equilibrium solutions and
are called interchangeable if any solution
, where
or
,
, is also an equilibrium solution. They are called equivalent if
for all
. Let
be the set of all equilibrium solutions, and let
be the set of equilibrium solutions which are Pareto optimal (cf. Arbitration scheme). A game is called Nash solvable and
is said to be a Nash solution if all
are equivalent and interchangeable. A game is called strictly solvable if
is non-empty and all
are equivalent and interchangeable. Two-person zero-sum games (cf. Two-person zero-sum game) with optimal strategies are Nash solvable and strictly solvable; however, in the general case such a solvability is often impossible.
Other attempts at completing the principle of realizability of the objective were made. Thus, it was suggested [4] that the unique equilibrium solution or the maximum solution (in this last situation each player may ensure his/her own gain irrespective of the strategies chosen by the other players), the choice of which is based on the introduction of a new preference relation on the set of solutions, be considered as the solution of the non-cooperative game. In another approach the solution of a non-cooperative game is defined by a subjective prognosis of the behaviour of the players [5].
References
[1] | N.N. Vorob'ev, "The present state of the theory of games" Russian Math. Surveys , 25 : 2 (1970) pp. 77–136 Uspekhi Mat. Nauk , 25 : 2 (1970) pp. 81–140 |
[2] | J. Nash, "Noncooperative games" Ann. of Math. , 54 (1951) pp. 286–295 |
[3] | I.L. Glicksberg, "A further generalization of the Kakutani fixed point theorem, with application to Nash equilibrium points" Proc. Amer. Math. Soc. , 3 (1952) pp. 170–174 |
[4] | J.C. Harsanyi, "A general solution for finite noncooperative games based on risk-dominance" L.S. Shapley (ed.) A.W. Tucker (ed.) M. Dresher (ed.) , Advances in game theory , Princeton Univ. Press (1964) pp. 651–679 |
[5] | E.I. Vilkas, "The axiomatic definition of equilibrium points and the value of a non-coalition ![]() |
Comments
References
[a1] | N.N. Vorob'ev, "Game theory. Lectures for economists and system scientists" , Springer (1977) (Translated from Russian) |
Non-cooperative game. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Non-cooperative_game&oldid=17485