Non-cooperative game

A system

$$ \Gamma = < J, \{ S _ {i} \} _ {i \in J } ,\ \{ H _ {i} \} _ {i \in J } > , $$

where $ J $ is the set of players, $ S _ {i} $ is the set of strategies (cf. Strategy (in game theory)) of the $ i $- th player and $ H _ {i} $ is the gain function of the $ i $- th player, defined on the Cartesian product $ S = \prod _ {i \in J } S _ {i} $. A non-cooperative game is played as follows: players, who are acting individually (do not form a coalition, do not cooperate), select their strategies $ s _ {i} \in S _ {i} $, as a result of which the situation $ s = \prod _ {i \in J } s _ {i} $ appears, in which the $ i $- th player obtains the gain $ H _ {i} ( s) $. 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 $ s ^ {*} $ is called an equilibrium solution if for all $ i \in J $, $ s _ {i} \in S _ {i} $, the inequality

$$ H _ {i} ( s ^ {*} ) \geq H _ {i} ( s ^ {*} \| s _ {i} ) , $$

where $ s ^ {*} \| s _ {i} = \prod _ {j \in J \setminus i } s _ {j} ^ {*} \times s _ {i} $, 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 $ J $ and $ S _ {i} $ 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 $ s $ and $ t $ are called interchangeable if any solution $ r = \prod _ {i \in J } r _ {i} $, where $ r _ {i} = s _ {i} $ or $ r _ {i} = t _ {i} $, $ i \in J $, is also an equilibrium solution. They are called equivalent if $ H _ {i} ( s) = H _ {i} ( t) $ for all $ i \in J $. Let $ Q $ be the set of all equilibrium solutions, and let $ Q ^ \prime \subset Q $ be the set of equilibrium solutions which are Pareto optimal (cf. Arbitration scheme). A game is called Nash solvable and $ Q $ is said to be a Nash solution if all $ s \in Q $ are equivalent and interchangeable. A game is called strictly solvable if $ Q ^ \prime $ is non-empty and all $ s \in Q ^ \prime $ 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 -person game" Theory Probab. Appl. , 13 : 3 (1968) pp. 523–527 Teor. Veroyatnost. i Primenen. , 13 : 3 (1968) pp. 555–560

References

[a1]	N.N. Vorob'ev, "Game theory. Lectures for economists and system scientists" , Springer (1977) (Translated from Russian)