Contributions to Game Theory and Management, 2020, Volume 13, Pages 218–243
Mechanism for shifting Nash equilibrium trajectories to cooperative Pareto solutions in dynamic bimatrix games
Nikolay A. Krasovskiia, Alexander M. Tarasyevba
a Krasovskii Institute of Mathematics and Mechanics UrB RAS,
S. Kovalevskoy Str., 16, Yekaterinburg, 620990, Russia
b Ural Federal University named after the first President of Russia B.N. Yeltsin, Mira Str., 19, Yekaterinburg, 620002, Russia
In the paper,
constructions of the generalized method of characteristics
are applied for calculating
the generalized minimax (viscosity) solutions
of Hamilton-Jacobi equations
in dynamic bimatrix games.
The structure of the game presumes
interactions of two players in the framework of the evolutionary game model.
Stochastic contacts between players
occur according to the dynamic process,
which can be interpreted
as a system of Kolmogorov's differential equations
instead of probability parameters.
It is assumed
that control parameters are not fixed
and can be constructed
by the feedback principle.
Two types of payoff functions
short-term payoffs are determined
in the current moments of time,
and long-term payoffs
are determined as limit functionals
on the infinite time horizon.
The notion of dynamic Nash equilibrium
in the class of controlled feedbacks
is considered for the long-term payoffs.
In the framework of constructions
of dynamic equilibrium,
the solutions are designed on the basis
of maximization of guaranteed payoffs.
Such guaranteeing strategies
are built in the framework
of the theory of minimax (viscosity) solutions
of Hamilton-Jacobi equations.
The analytical formulas are obtained
for the value functions
in the cases of different orientations
for the “zigzags” (broken lines) of acceptable situations
in the static game.
The equilibrium trajectories
generated by the minimax solutions
shift the system in the direction of cooperative Pareto points. The proposed approach
provides new qualitative properties
of the equilibrium trajectories
in the dynamic bimatrix games
which guarantee better results of payoffs
for both players
than static Nash equilibria.
As an example,
interactions of two firms
on the market of innovative electronic devices
are examined within the proposed approach
for treating dynamic bimatrix games.
dynamic bimatrix games,
minimax solutions of Hamilton-Jacobi equations,
dynamic Nash equilibrium trajectories,
shift to Pareto maximum.
|Russian Foundation for Basic Research
|The first author, Nikolay A. Krasovskii, is supported by the Russian Foundation for Basic Research (Project No. 18-01-00221a).
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Nikolay A. Krasovskii, Alexander M. Tarasyev, “Mechanism for shifting Nash equilibrium trajectories to cooperative Pareto solutions in dynamic bimatrix games”, Contributions to Game Theory and Management, 13 (2020), 218–243
Citation in format AMSBIB
\by Nikolay~A.~Krasovskii, Alexander~M.~Tarasyev
\paper Mechanism for shifting Nash equilibrium trajectories to cooperative Pareto solutions in dynamic bimatrix games
\jour Contributions to Game Theory and Management
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