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½ ! "#" $!% 9 =+ 3& : '"+?%+& G J!+ !" #$%" @ ! % ?/ N ?& !/ '% !-'% !+"!, !%-% Γ = N, {Xi }i∈N , {fi (x)}i∈N . (1.1) K'% , -% !+ "'+ %+ + N = {1, 2, . . . , N }& (!% N > 1L -'3 4 N +& % M%'"" , ' + ?/& +%! +/ !!%/ $'%3 !+%* xi ∈ Xi ⊆ n k ⊆ R $ + R , k 1& 4'% , '%% 4(%! k.%% %+'+ '%3 !+!%,% ). %!(% % ! !+& #%%! ! "+"/! " "'(%% 4 k '%3 !+. !%, ( %& 4 +% + +'% !?+& !'! " 4+%'%% %+'+3 3*L + %4,!!% ! + 4%! " i x = (x1 , . . . , xN ) ∈ X = Xi ⊆ Rn (n = i∈N ni ); i∈N -% !+% X %'%% )?" +> fi(x)& ( % ?%+/0" (% !+ ). ?+" i. (i ∈ N) L e'%% (xz i ) = (x1 , . . . , xi−1 , zi , xi+1 , . . . , xN ) f = (f1 , . . . , fN ) e e #$% 1 (x , f ) = ((x1 , . . . , xeN ), (f1 (xe ), . . . , fN (xe ))) ∈ X × RN 4+%! " + % (1.1)& % max fi (xe xi ) = fi (xe ) (i ∈ N); xi ∈Xi (1.2) -% xe 4+%! " + % (1.1) B B B N #$% & 1 (xB , f B ) = (xB 1 , . . . , xN ), (f1 (x ), . . . , fN (x )) ∈ X × R 4+%!. " + % (1.1)& % B max fi (xxB i ) = fi (x ) (i ∈ N); x∈X (1.3) '%% xB 4+%! " + % (1.1) & %'#%" ($ $ )$ )' + !% 56 !% & ! '%! % , . +& 4+'"0 ' !! -% !+ 56 !%! "& (! ( ! /! 4 (%! (% !+ !+"% !+ N% -% % !+. +%! " + %4,!!% ++%>+" %'-%" " '%, + %' !+ 4+ !& !,/ '%-+" , #!3 '% ?%4+"& ! / %& '' '!& ! +4'%3 !+/! N + 2 4+'!%%3 7'% (!!,& (! + % +% "'+% % ! 1 ' N -%. !+ + {1, 2, . . . , N } 4( (%%4 N BM% +0%3 i. 4+'!%% (i ∈ N) 4 %!3 $4'3 * %-! +%% '? 4( (%%4 qi 1 #! -'3 4 + % -%! !+!, !+ + (% !+% %,>%& (% α > 0& ,>%& (% β& ! % !, +% %+% !+ α qi β (i = 1, . . . , N ). (2.1) 1+% %+% !+ + $* ! 4(%! !! )!& (! 4+' !+%% 0 ! -' 4 4+'!%%3 )4(% (% K'% , '" !! (!%& (! + % #! 0 ! '+ G!, %+ %+% !+ 4 $* + !& (! + '% ! !+%! %3 ! $%& ' !+ % #%!#%* C!! ! ' %! !, ' !!( + %& !%& ! !%!& %4+ ! -+>%3 " % ?%& !+ !+ + 4%% % %%% 4%% %'%%3 +%( α @++% % 7%- + '% 9 4+' !+ i. (i ∈ N) %'/! " %3 4+ ! . (% !+ +0%3 '? qi ! !, %' !+% + +'% cqi + d& 4'% , c d !. +%! !+% %'% %%%% !"% 4'%- $ %%% 4'%- ! "! "& %& 4!! 4! (& 4 ,"& !4?/ '+"& !" 2 %' %0%3& 4%& !+& ?%43 !* : % + 4+ ! ! !++%! " ?% '?& !/ !-% (!% %3 4+ "0%3 ! 0% (% !+ q̄ = q1 +q2 +. . .+qN !+>% '- !+ G(!%& (! ?% p !+ %3 4+ ! ! %'-%"& %& %' !+"% %% + +'% p(q̄) = a − bq̄, (2.2) '% a = const > 0 2 (," ?% !+& !"3 -!%,3 #))?%! # !( ! b > 0 4+%!& , ;'%!< ?% !% + '- %'? '? 1%'-& (! ?% %'%"%! " !& (! ++% !, %'-%% C! 4(%!& (! -'3 4 4+'!%%3 '%! + %& (! 4+'! ( i. (i ∈ N) #! !+"%! p(q̄)qi = (a − bq̄)qi = a − b qk qi , k∈N % $+( 4 +(%! 4'%-%* '%! πi (qi , . . . , qN ) = a − b qk qi − (cqi + d). (2.3) k∈N 1%'%! " !-%& (!& %'%"" +3 M% 4+' !+& +' !+ -'3 ).4+'!%" %!%! " ;?,%< +%'%% + %!+ !%!(% " '%, !% +4'%3 !+" %' !+"%! 3 % ?. / N ?A N, {Qi = [α; β]}i∈N , {πi (q1 , . . . , qN )÷(2.3)}i∈N . (2.4) K'% ,& + $*& N = {1, 2, . . . , N } 2 -% !+ "'+ %+ +& Qi = [α; β] 2 -% !+ !!%3 i (i ∈ N) G!? q = (q1, . . . , qN ) ∈ Q = Q1 × Q2 × . . . × QN & )?" +> i. πi(q) = πi(q1, . . . , qN ) %'%% + $* )$* ! a > c" (2.4) q B = B )" q B = α (i ∈ N)" # = (q1B , q2B , . . . , qN i πiB = πi (q B ) = [a − N bα]α − (cα + d) = [a − c]α − bN α2 − d. 4 ! % , ! + G!?" ++% " 7%- + % $9* %'%"%! " !%3 4 N %+% !+A B B πi (qqi ) πi (q ) ∀q∈Q '%& & = $* %+% !+ $ * /! +' (qqiB ) (i ∈ N), (2.5) (q1 , q2 , . . . , qi−1 , qiB , qi+1 , . . . , qN ) B B ⎧ + q + . . . + q a − b q q1 − (cq1B ⎪ 2 N 1 ⎪ ⎨ a − b q1 + q2B + . . . + qN q2B − (cq2B . . . . . . . . . ⎪ ⎪ ⎩ B B − (cq B qN a − b q1 + q2 + . . . + qN N + d) + d) . . + d) B B B + . . . + qB a − b q + q q1 − (cq1B + d), 1 2 N B B a − b q1 + q2B + . . . + qN q2B − (cq2B + d), . . . . . . . B . B . . . B B − (cq B + d) a − b q1 + q2 + . . . + qN qN N !/! " +%'+ + % qi ∈ Qi (i ∈ N) O% 4%!!,& (! !?%3 ++%B " 7%- + $9* '%! qB = (α, α, . . . , α)& #! ++% % 7%- +> πi (i ∈ N) !+"! πiB = πi (q B ) = [a − N bα]α − (cα + d) = [a − c]α − bN α2 − d. =+ 3& : '"+?%+& G J!+ %3 !+!%,& , 0" !+ !+%3 '?& -'3 i.3 (i ∈ N) !% +%(+%! , + % !, ( !+ $9* 1%%3'% !%%, ++% / :#> + % $9* )$* & ! a > c" (2.4) e q e = (q1e , q2e , . . . , qN ), i ∈ N ⎧ a−c ⎪ α, ⎪ α, ⎪ ⎪ b(N + 1) ⎪ ⎪ ⎨ a−c a−c , α < < β, qie = b(N + 1) b(N + 1) ⎪ ⎪ ⎪ ⎪ a−c ⎪ ⎪ ⎩ β, β. b(N + 1) $ (i ∈ N) (2.6) + ⎧ ⎪ (a − c)α − bN α2 − d, ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ (a − c)2 e e πi = πi (q ) = − d, ⎪ (N + 1)2 b ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ (a − c)β − bN β 2 − d, % % % a−c α, b(N + 1) a−c < β, α< b(N + 1) a−c β. b(N + 1) 4 ! % , ! + G!?" ++% " :#> + $9* %'%"%! " !%3 . +% !+ $ %'%%% * e )]q − (cq + d)}, π1 (q e ) = max π1 (q e q1 ) = max {[a − b(q1 + q2e + . . . + qN 1 1 q1 ∈[α;β] q1 ∈[α;β] q2 ∈[α;β] q2 ∈[α;β] e )]q − (cq + d)}, π2 (q e ) = max π2 (q e q2 ) = max {[a − b(q1e + q2 + . . . + qN 2 2 . . . . . . . . . . . . . . . . . . . . πN (q e ) = max πN (q e qN ) = max {[a − b(q1e + q2e + . . . + qN )]qN − (cqN + d)}. qN ∈[α;β] (2.7) qN ∈[α;β] $D*& ! -% +>%& '" / i ∈ N (%%4 (qeqi) 4(% !?" qe& + !3 e !!%" i. qi 4%% qi " -' i ∈ N )? πi(qeqi) %%%3 qi ' !%! " +. % '+ !%+3A ∂πi (q e qi ) e e e )−c = a − 2bqi − b(q1e + . . . + qi−1 + qi+1 + . . . + qN = 0, qi =qie ∂qi qi =qie ∂ 2 πi (q e qi ) = −2b < 0. ∂qi2 qi =q e i (2.8) !% +% 4 $* %%! % !& , #))?%! # !( ! b > 0& 4 %+ +% !+ -' i ∈ N (% !% N %3 +%3A ⎧ ⎪ e = a − c, ⎪ 2q1e + q2e + q3e + . . . + qN ⎪ ⎪ b ⎪ ⎪ ⎨ e e = a − c, q1 + 2q2e + q3e + . . . + qN b ⎪ ⎪ . . . . . . . . . ⎪ ⎪ ⎪ ⎪ q e + q e + q e + . . . + 2q e = a − c , ⎩ 1 2 3 N b @++% % 7%- + '% %>%% !3 '%! !?" e q = e (q1e , q2e , . . . , qN ) = a−c a−c a−c , ,..., . (N + 1)b (N + 1)b (N + 1)b 1 +% +" α < b(Na −+c1) < β 3'%% qie (i ∈ N) ' !+"/! ,% 4(%% )?3 πi(qeqi) !%4% [α; β]& %'+!%,& "+"/! " ++% :#> !!%" + % $9* -% b(Na −+c1) α& !& + +! %+% !+ 4 $*& !%4% [α; β] -'" 4 )?3 πi (q e qi ) (i ∈ N) ! +%! G%'+!%,& + $D* ' !/! " e qi = α (i ∈ N) (% ' b(Na −+c1) β& )? πi(qeqi) + % i ∈ N '! -% ! +4 !/0 [α; β] G!+%! !+%& +% !+ $D* +"/! " qie = β (i ∈ N) %%& M%'"" + % ! +%'% ("& +'& (! ++% % :#> !!% + $9* %'%"/! " )3 $F* 1%%3'% !%/ +>%3 πi(qe) + 3'%3 !? $F* (i ∈ N) 1' !++ $F* + )? +> $*& '" -' i. (i ∈ N) ( ++% % :#> +> + % $9* %& b(Na −+c1) α ' πie = πi (α, α, . . . , α) = [a − bN α] α − (cα + d) = (a − c)α − bN α2 − d. b(Na −+c1) β& ! ++% 3 :#> +> i. πie '%! πie = πi (β, β, . . . , β) = [a − bN β] β − (cβ + d) = (a − c)β − bN β 2 − d. :%?& + (% α < b(Na −+c1) < β !+! πie a−c a−c a−c a−c a−c a−c , ,..., = a − bN − c +d = (N + 1)b (N + 1)b (N + 1)b (N + 1)b (N + 1)b (N + 1)b = πi = (a − c)2 N (a − c)2 (a − c)2 − · −d= − d. (N + 1)b (N + 1)b N + 1 (N + 1)2 b 4& +> i. + !? ++% " :#> %%! +' ⎧ a−c ⎪ (a − c)α − bN α2 − d, % α, ⎪ ⎪ b(N + 1) ⎪ πie e = πi (q ) = ⎪ ⎪ ⎨ (a − c)2 − d, ⎪ (N + 1)2 b ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ (a − c)β − bN β 2 − d, % % a−c < β, b(N + 1) a−c β. b(N + 1) α< + $ $ , # $*( '$-. $ / # 0-( #! )% +%'% +%% +>%3& !% (! & + ,4++. > , !?%3 ++% " 7%-& !% +>& !% -'/! + !? ++% " :#> " #! "!,.! ! ! (" =+ 3& : '"+?%+& G J!+ a−c α ++% % :#> !!% xei & %'%%% + $F*& %(! . 1 1 b(N + 1) +'/! ++% 7%- !!%" xBi = α (i ∈ N) G%'+!%,& . & '%-+" , !? ++% " :#>& (! !% -% +>& % ( & '%-+" , !? ++% " 7%-& %A a−c α. (3.1) πie = πiB b(N + 1) +% %+% !+ α < b(Na −+c1) < β& ! +> i. (i ∈ N) + !? ++% " :#> '%! %(! . 11 πie = πi (q e ) = ++% 3 7%- +> !+! (a − c)2 − d, (N + 1)2 b πiB = πi (q B ) = α[a − c − N bα] − d. 4 !, (a − c)2 − d − (a − c)α − bN α2 − d = − = 2 (N + 1) b a−c a−c (a − c)2 = bN · α − · α − . = bN α2 − (a − c)α + (N + 1)2 b (N + 1)b N (N + 1)b πie πiB 1 , ( + N > 1 a − c > 0& ! a−c a−c < . N (N + 1)b (N + 1)b + !& (! α > 0 #))?%! # !( ! b > 0& ! 4 !, πie − πiB -!%, a−c 0<α< !?!%, N (N + 1)b , a−c a−c <α< N (N + 1)b (N + 1)b + /& % α= 4& '" + % i ∈ N ⎧ ⎪ ⎪ πie > πiB , ⎪ ⎪ ⎪ ⎨ πie = πiB , ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ πie < πiB , % % % a−c . N (N + 1)b a−c a−c < β, N (N + 1)b (N + 1)b a−c a−c < β, α= N (N + 1)b (N + 1)b a−c a−c <α< < β. N (N + 1)b (N + 1)b α< (3.2) :%?& ! %'3 (3 a−c %(! . 111 1 α < β ++% 3 :#> !!%%3 i. 4+'!%" '%! (N + 1)b !+ , +4- (% !+ !+& ! % !, xei = β (i ∈ N) % , + !? ++% " :#> !+! πie = (a − c)β − bN β 2 − d. @++% % 7%- + '% K III II L I M G+%% +>%3 + !?" ++% " 7%- :#> !? ++% " B 7%- '-% 4!, !+ '! ' , 4%>%& % xi = α (i ∈ N) ++% 3 7%- +> #! '%! πiB = (a − c)α − bN α2 − d. @ ! 4 !, + +>& !/ (! ( ! $9* + !. ?" ++% " :#> 7%- & '" i (i ∈ N) !+! πie − πiB = (a − c)β − bN β 2 − d − (a − c)α − bN α2 − d = = (a − c)(β − α) − bN (β 2 − α2 ) = (β − α) · [a − c − bN (β + α)]. β > α& ! 4 +%'%3 4 ! +'%! 4 +/0%3 %33 )? a − c − bN (α + β). a−c α+β = bN a−c α+β < bN C! )?" %"%! 4 K(!& 4 !, πie − πiB + / α + β = a−c $!%4 LM * 1 4 !, πie − πiB -!%,& + (% = bN a−c α+β > '%! !?!%,3 bN c G%'+!%,& α < β (Na +− 1)b '" +>%3 i. + !? ++% " 7%- πiB + !? ++% " :#> πie %%! % ! !>%% ⎧ ⎪ % α + β < aN−bc α < β, π e > πiB , ⎪ ⎪ ⎨ i (3.3) % α + β = aN−bc α < β, πie = πiB , ⎪ ⎪ ⎪ a − c ⎩ πe < πB , % α + β > N b α < β. i i 9 =+ 3& : '"+?%+& G J!+ %%& M%'"" + % ! ("& 4 ) $*& $* $* ( % +%% ++% 7%- :#> +>%3 '" i.e (i ∈ N) + % $9* $ * ! P& 4-%3 & % +> πi + !? ++% " :#> '%! ,>%& (% +> πiB & ++% 3 7%- ! PP& !& ++% % 7%- ' !+"%! i (i ∈ N) +> ,>3& (% +> + !? ++% " :#> ! PPP 3 KLM +> + + !?" ++% " 7%- :#> '! 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