 # Nash Equilibrium

This section contains my papers devoted to the Nash Equilibrium in differential games.

Yurii V. Averboukh, Nash equilibrium for differential game and nonanticipative strategies technique // Yurii V. Averboukh, “Nash equilibrium for differential game and nonanticipative strategies technique”, Mat. Teor. Igr Pril., 4:3 (2012), 3–20.

We consider two person nonzero-sum games in the class of nonanticipative strategies. The Nash equilibrium for this case is defined. Also we give the characterization of Nash equilibrium strategies. It is shown that the Nash equilibrium solution in the class of nonanticipative strategies can be approximated by the strategies with the guide.

Ключевые слова: Nash equilibrium, nonanticipative strategies, control with model.

### On MathNet.ru (Full text is in Russian).

Yurii Averboukh, Infinitesimal characterization of Nash equilibrium for differential games with many players // Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2011, no. 2, 3–11.

We study Nash equilibrium for a differential game with many players. The condition on a multivalued map under which any value of this map is a set of Nash equilibrium payoffs is obtained. This condition is written in infinitesimal form. The sufficient condition for the given complex of continuous functions to provide a Nash equilibrium is obtained. This condition is a generalization of the method based on system of Hamilton–Jacobi equations.

Ключевые слова: Nash equilibrium, differential games, generalized derivatives.

### MatnNet.ru

Yurii Averboukh, Nash equilibrium in differential games and the construction of the programmed iteration method // Mat. Sb., 2011, Volume 202, Number 5, Pages 3–28 (Mi msb7568).

This work is devoted to the study of nonzero-sum differential games. The set of payoffs in a situation of Nash equilibrium is examined. It is shown that the set of payoffs in a situation of Nash equilibrium coincides with the set of values of consistent functions which are fixed points of the program absorption operator. A condition for functions to be consistent is given in terms of the weak invariance of the graph of the functions under a certain differential inclusion.
Bibliography: 18 titles.

Ключевые слова: differential games, Nash equilibrium, programmed iteration method.

### MathNet.ru

Yurii Averboukh, Characterization of Feedback Nash Equilibrium for Differential Games // http://www.springerlink.com/content/r4923j6786801137/, 2012.

We investigate the set of Nash equilibrium payoffs for two person differential games. The main result of the paper is the characterization of the set of Nash equilibrium payoffs in the terms of nonsmooth analysis. Also we obtain the sufficient conditions for a pair of continuous function to provide a Nash equilibrium. This result generalizes the method of system of Hamilton-Jacobi equations. See on arxiv http://arxiv.org/abs/1005.0101.

Ключевые слова: differential games, Nash equilibrium, nonsmooth analysis.

Соседние подразделы:
 The purpose of these papers is: by given value function find the differential game.... The papers devoted to continuity of stable bridges.... The transformation of the given approach problem to the approach problem "in the moment".... Supervisor: professor A.G. Chentsov Date of PhD defence: October 25, 2007 г. The text of thesis is in Russian.... Analogs of the extremal shift rule for the case of nonstable bridge.... The papers are devoted to the character of the convergence of the programmed absorption mathed. The coathor is professor A.G. Chentsov....

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20.04.13 New publication and presentation

14.02.13 Stackelberg solutions

13.08.11 English version