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Кілька рівноваг: яку вибрати?
Є два агенти i=1,2i=1,2i=1,2 . Розглянемо наступну програму s.t. V1(x0):=maxu∫∞0e−ρtF1(x(t),u(t),v(t))dtV2(x0):=maxv∫∞0e−ρtF2(x(t),u(t),v(t))dtx˙(t)=f(x(t),u(t),v(t))x(0)=x0V1(x0):=maxu∫0∞e−ρtF1(x(t),u(t),v(t))dtV2(x0):=maxv∫0∞e−ρtF2(x(t),u(t),v(t))dts.t. x˙(t)=f(x(t),u(t),v(t))x(0)=x0\begin{align} &V_1(x_0) := \max_u \int^\infty_0 e^{-\rho t}F_1(x(t),u(t),v(t))dt\\ &V_2(x_0) := \max_v \int^\infty_0 e^{-\rho t}F_2(x(t),u(t),v(t))dt\\ s.t.~&\dot x(t)=f(x(t),u(t),v(t))\\ &x(0) = x_0 \end{align}ρ>0ρ>0\rho > 0Vi(⋅)Vi(⋅)V_i(\cdot)Fi(⋅)Fi(⋅)F_i(\cdot)x∈X=[0,2]x∈X=[0,2]x\in X = [0,2]u∈U=[0,1]u∈U=[0,1]u\in U=[0,1]v∈V=[0,1]v∈V=[0,1]v\in V=[0,1]f(⋅)f(⋅)f(\cdot)ρV1(x)=maxu[F(x,u,v∗)+V′1(x)f(x,u,v∗)],∀t∈[0,∞)ρV2(x)=maxv[F(x,u∗,v)+V′2(x)f(x,u∗,v)],∀t∈[0,∞)ρV1(x)=maxu[F(x,u,v∗)+V1′(x)f(x,u,v∗)],∀t∈[0,∞)ρV2(x)=maxv[F(x,u∗,v)+V2′(x)f(x,u∗,v)],∀t∈[0,∞)\begin{align} \rho V_1(x)=\max_u [F(x,u,v^*) + V_1'(x)f(x,u,v^*)],\quad \forall t\in[0,\infty)\\ \rho V_2(x)=\max_v [F(x,u^*,v) + V_2'(x)f(x,u^*,v)],\quad \forall t\in[0,\infty)\\ …