|Budget Amount *help
¥700,000 (Direct Cost : ¥700,000)
Fiscal Year 1997 : ¥300,000 (Direct Cost : ¥300,000)
Fiscal Year 1996 : ¥200,000 (Direct Cost : ¥200,000)
Fiscal Year 1995 : ¥200,000 (Direct Cost : ¥200,000)
Let us atate the main theorem that was proved in the present project Consider a differential game as follows :
(1) The numbers of playrs, state variables and control variables of each playr, say M,N and J_i, i=1, ..., M,are arbitrarily fixed.
(2) Let x and c_i are an N-dimensional vector of state variables and an J_<ii>-dimensional vector of control variables of playr i. Let us assume that the motion of x is determined by the system of differential equations
x^^・=F (x, c),
where c= (c_1, ...c_M). Let us denote the objective functional of playr i by
V_i=*u_i (c_i) exp [-rt] dt
where the rate of time preference r is assumed to be positive and constant.
In this differential game, we can state that :
Assume that u_<ii> is homogeneous of degree alpha>0, that F is homogeneous of degree one in x and c, and that all playr i's opponents use Markov strategies that are homogeneous of degree one. Then
(ii) Playr i's best reply is homogeneous of degree one in x.
(ii) The maximized value function V_i is homogeneous of degree alpha in x.
The proof can be found in Long and Shimomura (Journal of Economic Behavior and Organization, December 1997).
証明は本研究成果報告書第1章、あるいはLong and Shimomura(Journal of Economic Behavior and Organization,December1997)を参照されたい。