TY - JOUR
T1 - Approximating the Value of Zero-Sum Differential Games with Linear Payoffs and Dynamics
AU - Kuipers, Jeroen
AU - Schoenmakers, Gijs
AU - Staňková, Kateřina
PY - 2023
Y1 - 2023
N2 - We consider two-player zero-sum differential games of fixed duration, where the running payoff and the dynamics are both linear in the controls of the players. Such games have a value, which is determined by the unique viscosity solution of a Hamilton–Jacobi-type partial differential equation. Approximation schemes for computing the viscosity solution of Hamilton–Jacobi-type partial differential equations have been proposed that are valid in a more general setting, and such schemes can of course be applied to the problem at hand. However, such approximation schemes have a heavy computational burden. We introduce a discretized and probabilistic version of the differential game, which is straightforward to solve by backward induction, and prove that the solution of the discrete game converges to the viscosity solution of the partial differential equation, as the discretization becomes finer. The method removes part of the computational burden of existing approximation schemes.
AB - We consider two-player zero-sum differential games of fixed duration, where the running payoff and the dynamics are both linear in the controls of the players. Such games have a value, which is determined by the unique viscosity solution of a Hamilton–Jacobi-type partial differential equation. Approximation schemes for computing the viscosity solution of Hamilton–Jacobi-type partial differential equations have been proposed that are valid in a more general setting, and such schemes can of course be applied to the problem at hand. However, such approximation schemes have a heavy computational burden. We introduce a discretized and probabilistic version of the differential game, which is straightforward to solve by backward induction, and prove that the solution of the discrete game converges to the viscosity solution of the partial differential equation, as the discretization becomes finer. The method removes part of the computational burden of existing approximation schemes.
KW - Differential games
KW - Stochastic games
KW - Viscosity solutions
UR - http://www.scopus.com/inward/record.url?scp=85160860705&partnerID=8YFLogxK
U2 - 10.1007/s10957-023-02236-x
DO - 10.1007/s10957-023-02236-x
M3 - Article
AN - SCOPUS:85160860705
SN - 0022-3239
VL - 198
SP - 332
EP - 346
JO - Journal of Optimization Theory and Applications
JF - Journal of Optimization Theory and Applications
IS - 1
ER -