25 research outputs found
Approximate solutions of continuous-time stochastic games
The paper is concerned with a zero-sum continuous-time stochastic
differential game with a dynamics controlled by a Markov process and a terminal
payoff. The value function of the original game is estimated using the value
function of a model game. The dynamics of the model game differs from the
original one. The general result applied to differential games yields the
approximation of value function of differential game by the solution of
countable system of ODEs.Comment: 23 page
Universal Nash Equilibrium Strategies for Differential Games
The paper is concerned with a two-player nonzero-sum differential game in the
case when players are informed about the current position. We consider the game
in control with guide strategies first proposed by Krasovskii and Subbotin. The
construction of universal strategies is given both for the case of continuous
and discontinuous value functions. The existence of a discontinuous value
function is established. The continuous value function does not exist in the
general case. In addition, we show the example of smooth value function not
being a solution of the system of Hamilton--Jacobi equation.Comment: 23 page
Approximate public-signal correlated equilibria for nonzero-sum differential games
We construct an approximate public-signal correlated equilibrium for a
nonzero-sum differential game in the class of stochastic strategies with
memory. The construction is based on a solution of an auxiliary nonzero-sum
continuous-time stochastic game. This class of games includes stochastic
differential games and continuous-time Markov games. Moreover, we study the
limit of approximate equilibrium outcomes in the case when the auxiliary
stochastic games tend to the original deterministic one. We show that it lies
in the convex hull of the set of equilibrium values provided by deterministic
punishment strategies.Comment: 35 page
Viability analysis of the first-order mean field games
The paper is concerned with the dependence of the solution of the
deterministic mean field game on the initial distribution of players. The main
object of study is the mapping which assigns to the initial time and the
initial distribution of players the set of expected rewards of the
representative player corresponding to solutions of mean field game. This
mapping can be regarded as a value multifunction. We obtain the sufficient
condition for a multifunction to be a value multifunction. It states that if a
multifunction is viable with respect to the dynamics generated by the original
mean field game, then it is a value multifunction. Furthermore, the
infinitesimal variant of this condition is derived.Comment: 35 page