By L P J. Groenewegen
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Extra info for Characterization of Optimal Strategies in Dynamic Games
3, there exists an n>O with the property (i) d * «f" ••• ,f r ), (g" •.. •. ,fr,a" ••. ,a r ) for arbitrary discrete strategies IS Let now F, G : n -+ n ===> (g" ••. e. (i=1 , ••• ,r) The latter relation implies (iv) (i=1, ••• r ) . (w),s) = d(f~(w),G(w»}. ~ ~ Then, the correspondence G~ : n ~ by A. 5 and A. e. e. By definition of gi (i=1, ••• ,r), the discrete strategy IS (g1,···,gr,Q1,···,Qr) satisfies d*«f 1 ,···,f ),(g1,···,g» r r = Jl~i~r max :> JSEOF(w) max = Jl:>i:>r max d(fi(·),gi(·» d(f.
Hence, the function Vi : E .... 1. +g) 1. d~@K (Kd) is upper semicontinuous on E. A similar argumentation shows that the function -1. V. : E .... (K) -1. -g) 1. d~@K is lower semicontinuous on E. As the equality Vi = Yi holds, Vi and + 2 Yi Jc g d\l are upper and lower continuous functions, and are therefore continuous. Define Vi: E .... lR by V. (K) 1. := fcxs u. 1. d~@K (Kd:) • 1 V. 1. + -1. , is a continuous function 27 on r and satisfies s fn-c d~ g S (Ker) g This shows that Ui can be uniformly approximated by continuous 0 functions and is therefore itself a continuous function.
9, we obtain a continuous, convex- and compact-valued correspondence rr i : E .... C(E i ). Let the correspondence rr*: E .... ) i=l ~ ~ ~ = ~ max v. (K, L! (K) ~ ~ ~ (KEE) • ~ Then, because v. 4 shows that rr* is a ~ (nonempty) compact-valued, upper semi continuous correspondence. Moreover, since v. is quasiconcave in the second variable, ~ rr* is convex-valued. 1 applies to rr* and delivers a strategy K E E with K E rr*(K) . By definition of rr. t. K for every player i and satisfies for abitrary admissible strategies L.