Premium
Computation of nash equilibrium pairs of a stochastic differential game
Author(s) -
Yavin Y.,
Reuter G.
Publication year - 1981
Publication title -
optimal control applications and methods
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.458
H-Index - 44
eISSN - 1099-1514
pISSN - 0143-2087
DOI - 10.1002/oca.4660020303
Subject(s) - nash equilibrium , mathematical economics , epsilon equilibrium , computation , differential game , best response , differential (mechanical device) , computer science , mathematical optimization , mathematics , algorithm , physics , thermodynamics
Consider the random motion of two points M e and M p in an open and bounded domain D 0 in the plane. Each of the velocities, u = (u 1 u 2 ) T of M e and v = (v 1 , v 2 ) T of M p , are perturbed by a corresponding R 2 ‐valued Gaussian white noise. Let A and D c be two disjoint closed subsets of D 0 . Suppose that at t = 0, M e is in A and M p is anywhere in D 0 . Denote by ℰ 1 and ℰ 2 the following events: ℰ 1 = { M p intercepts M e in A before M e reaches the set D c and before either M e or M p has left D 0 }, and ℰ 2 = { M e reaches the set D c before being intercepted by M p , while M p is in A , and before either M p or M e has left D 0 }. The problem dealt with here is to find a pair of velocity strategies (u*, v*) such that, in the sense of a Nash equilibrium point, the probabilities Prob(ℰ 1 ) and Prob(ℰ 2 ) will both be maximized on a given class of velocity strategies (u, v). Sufficient conditions on (u*, v*) are derived which require the existence of a smooth solution ( V,Q ) to a pair of coupled non‐linear partial differential equations. A finite‐difference scheme for solving these equations is suggested, and two examples are treated in detail.