# Solution in game theory

An outcome (or set of outcomes) that satisfies the optimality principle accepted in the given model. One distinguishes between the following types of solutions: 1) a Nash solution (see Non-cooperative game) and, in particular, a saddle point in game theory of the pay-off function in a two-person zero-sum game; 2) a Neumann–Morgenstern solution, which is a set of sharings no two of which dominate the other, whereby for each sharing not belonging to this set there is a sharing from this set dominating it (see Domination); and 3) a Nash solution in an arbitration scheme.

#### References

[1] | J. von Neumann, O. Morgenstern, "Theory of games and economic behavior" , Princeton Univ. Press (1947) |

[2] | G. Owen, "Game theory" , Acad. Press (1982) |

#### Comments

The Neumann–Morgenstern solution is also referred to as the Pareto solution. Still another solution concept is called the Stackelberg solution. If one player in a game enforces his strategy onto the other player(s) by announcing it before the other player(s) made his (their) decision(s), then the Stackelberg solution is the appropriate concept. It is also related to the theory of incentives in economy. See [a1], [a2].

#### References

[a1] | T. Basar, G.J. Olsder, "Dynamic noncooperative game theory" , Acad. Press (1989) |

[a2] | Y.C. Ho, P.B. Luh, G.J. Olsder, "A control-theoretical view on incentives" Automatica , 18 : 2 (1982) pp. 167–179 |

**How to Cite This Entry:**

Solution in game theory. A.N. Lyapunov (originator),

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Solution_in_game_theory&oldid=15577