А.А. Васин, В.В. Морозов - Введение в теорию игр с приложениями к экономике (1184512), страница 55
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Ï� ïðåäûäóùå� òåîðåì� ñóùåñòâóå� ñèìïëåê� y 1 (k) · · · y n+1 (k) ðàçáèåíè� πk , äë� êîòîðîã� ν(y i (k)) = xi , i = 1, ..., n + 1.Çàìåòèì, ÷ò� äë� êàæäîã� âåêòîð� y i (k) ï� îïðåäåëåíè� îòîáðàæåíè� ν jt = i � ïîýòîì� y i (k) ∈ Ci , i = 1, ..., n + 1. Ïîñêîëüê� K − êîìïàêò, áå� ïîòåð� îáùíîñò� ìîæí� ñ÷èòàòü, ÷ò� ïîñëåäîâàòåëüíîñò� {y i (k)}ñõîäÿòñÿ. Í� lim δ(πk ) = 0.
Ïîýòîì� ïð� k → ∞ äèàìåò� ñèìïëåêñ� k→∞ y 1 (k) · · · y n+1 (k) ñòðåìèòñ� � íóëþ. Ñëåäîâàòåëüíî, lim y i (k) = y ∗ , i = k→∞1, ..., n + 1. Ïîñêîëüê� ïð� âñå� i y i (k) ∈ Ci , � ìíîæåñòâ� Ci çàìêíóòû, n+1Tèìåå� y ∗ ∈ Ci . i=1 265 Ïðèëîæåíè�Óòî÷íè� ôîðìóëèðîâê� òåîðåì� 9.1. Ïóñò� Z − âûïóêëû� êîìïàê� � E m � y 1 · · · y n+1 − ñîäåðæàùèéñ� � íå� ñèìïëåê� ìàêñèìàëüíî� ðàçìåðíîñòè.
Òîãä� ÷èñë� n ≤ m íàçûâàåòñ� ðàçìåðíîñòü� ìíîæåñòâ� Z. Âûïóêëû� êîìïàê� Z ìàêñèìàëüíî� ðàçìåðíîñò� m èìåå� âíóòðåííþ� òî÷ê� (íàïðèìåð, áàðèöåíò� ñèìïëåêñ� y 1 · · · y m+1 ). Îáðàòíî, åñë� Z èìåå� âíóòðåííþ� òî÷êó, ò� åã� ðàçìåðíîñò� ìàêñèìàëüíà. Ïðåäïîëîæèì, ÷ò� Z èìåå� ðàçìåðíîñò� n < m � ñîäåðæè� íóëü. Òîãä� ìîæí� ñ÷èòàòü, ÷ò� y n+1 = 0. Ïð� ýòî� Z ÿâëÿåòñ� ïîäìíîæåñòâî� åâêëèäîâ� ïðîñòðàíñòâ� E n � áàçèñî� y 1 , ..., y n . È� ñêàçàííîã� âûòåêàåò, ÷ò� � òåîðåì� 9.1 áå� ïîòåð� îáùíîñò� ìîæí� ïðåäïîëîæèòü, ÷ò� âûïóêëû� êîìïàê� Z èìåå� ìàêñèìàëüíó� ðàçìåðíîñò� m.
Äîêàçàòåëüñòâ� òåîðåì� 9.1 (Áðàóýðà). Ïóñò� K = x1 · · · xm+1 − m-ìåðíû� ñèìïëåê� � E m . Ï� òåîðåì� Ï.2 (ñì. Ï3) íàéäåòñ� ãîìåîìîðôèç� τ : Z → K. Îòîáðàæåíè� τ ◦ f ◦ τ −1 ïåðåâîäè� òî÷ê� y ∈ K � òî÷ê� τ (f (τ −1 (y))) ∈ K � ÿâëÿåòñ� íåïðåðûâíûì. Åñë� y 0 ∈ K − íåïîäâèæíà� òî÷ê� îòîáðàæåíè� τ ◦ f ◦ τ −1 , ò� x0 = τ −1 (y 0 ) − íåïîäâèæíà� òî÷ê� îòîáðàæåíè� f : Z → Z. Ïîýòîì� áå� ïîòåð� îáùíîñò� ìîæí� ñ÷èòàòü, ÷ò� Z = K. Ïîëîæè� Ci = {x ∈ K | λi (f (x)) ≤ λi (x)}, i = 1, ..., m + 1. Ïîñêîëüê� îòîáðàæåíè� f � ôóíêöè� λi (x) íåïðåðûâí� í� K , ìíîæåñòâ� Ci çàìêíóòû. Ïîêàæåì, ÷ò� ìíîæåñòâ� Ci óäîâëåòâîðÿþ� óñëîâè� ïðåäûäóùå� òåîðåìû.
Ïóñò� x ∈ xj1 · · · xjs . Òîãä� j1 js λj (x) = 0 ∀j ∈/ {x , ..., x } ⇒ sXλjr (x) = 1 ≥ r=1 sXλjr (f (x)). r=1 Ñëåäîâàòåëüíî, íàéäåòñ� íîìå� r, ïð� êîòîðî� λjr (x) ≥ λjr (f (x)) � x ∈ Cjr . Ï� ïðåäûäóùå� òåîðåì� íàéäåòñ� ∗ x ∈ m+1\Ci ⇒ λi (x∗ ) ≥ λi (f (x∗ )), i = 1, ..., m + 1. i=1 Í� m+1Xi=1 ∗λi (x ) = m+1Xλi (f (x∗ )) = 1. i=1 266Ïðèëîæåíè�Ïîýòîì� λi (x∗ ) = λi (f (x∗ )), i = 1, ..., m + 1 ⇒ x∗ = f (x∗ ).Äîêàçàòåëüñòâ� òåîðåì� 12.1 (Êàêóòàíè). Êà� � ïð� äîêàçàòåëüñòâ� òåîðåì� Áðàóýðà, áå� ïîòåð� îáùíîñò� ìîæí� ñ÷èòàòü, ÷ò� âûïóêëû� êîìïàê� Z èìåå� ìàêñèìàëüíó� ðàçìåðíîñò� m.
Ïóñò� K = x1 · · · xm+1 − m-ìåðíû� ñèìïëåê� � E m . Ï� òåîðåì� Ï.2 (ñì. Ï3) íàéäåòñ� ãîìåîìîðôèç� ϕ : K → Z. Ðàññìîòðè� ïîñëåäîâàòåëüíîñò� {πk } ñèìïëèöèàëüíû� ðàçáèåíè� ñèìïëåêñ� K, äë� êîòîðî� lim δ(πk ) = 0. Äë� ïðîèçêîëüíîã� k îïðåäåëè� ñëåäóþùå� íåïðåðûâíî� k→∞îòîáðàæåíè� Φk : K → Z. Äë� ëþáî� âåðøèí� x è� {πk } âûáåðå� òî÷ê� Φk (x) ∈ Φ(ϕ(x)) � çàòå� ïðîäîëæè� îòîáðàæåíè� ëèíåéí� í� êàæäû� è� ñèìïëåêñîâ, âõîäÿùè� � {πk }, ò.å. åñë� x = m+1Xjλj x ,j=1 m+1Xλj = 1, λj ≥ 0, j = 1, ..., m + 1, j=1 − òî÷ê� ñèìïëåêñ� x1 · · · xm+1 ∈ {πk }, ò� ïîëàãàå� kΦ (x) = m+1XΦk (xj ).
j=1 Î÷åâèäíî, ÷ò� ïîñëåäíå� ðàâåíñòâ� îïðåäåëÿå� íåïðåðûâíî� îòáðàæåíè� Φk : K → Z . Òîãä� f k = ϕ−1 Φk : K → K ÿâëÿåòñ� íåïðåðûâíû� îòîáðàæåíèå� � èìåå� íåïîäâèæíó� òî÷ê� xk . Ïóñò� xk ïðèíàäëåæè� ñèìïëåêñ� x1k · · · xm+1k ∈ {πk }, ò.å. k x = m+1Xj=1 λkj xjk ,m+1Xλkj = 1, λkj ≥ 0, j = 1, ..., m + 1. j=1 Í� óìàëÿ� îáùíîñòè, ìîæí� ñ÷èòàòü, ÷ò� ïîñëåäîâàòåëüíîñò� {xk }, {xjk },{λkj },{Φk (xjk )}, j = 1, ..., m + 1, ñõîäÿòñ� ïð� k → ∞.
Ïîñêîëüê� äèàìåòð� ñèìïëåêñî� x1k · · · xm+1k ñòðåìÿòñ� � íóëþ, ïîñëåäîâàòåëüíîñò� {xk } � {xjk } äîëæí� èìåò� îáùè� ïðåäå� x∗ ∈ K. Ïóñò� lim λkj = λ∗ j , lim Φk (xjk ) = η j , j = 1, ..., m + 1. k→∞ k→∞ 267 Èìåå�ϕ(x k ) = Φk (x k ) = m+1Xλkj Φk (xjk ) (Π.2) j=1 � lim ϕ(xjk ) = ϕ(x∗ ), lim ϕ(xk ) = ϕ(x∗ ) � ñèë� íåïðåðûâíîñò� îòáðàæåk→∞k→∞íè� ϕ.
Ïåðåõîä� � ðàâåíñòâ� (Π.2) � ïðåäåë� ïð� k → ∞, ïîëó÷è� ϕ(x∗ ) = m+1Xj=1 λ∗j η j ,m+1Xλ∗ j = 1, λ∗ j ≥ 0, j = 1, ..., m + 1.j=1È� çàìêíóòîñò� îòîáðàæåíè� Φ âûòåêàåò, ÷ò�η j ∈ Φ(ϕ(x∗ )), j = 1, ..., m + 1. Ïîñêîëüê� ìíîæåñòâ� Φ(ϕ(x∗ )) âûïóêëî,òî÷ê� ϕ(x∗ ) åì� ïðèíàäëåæè� � ÿâëÿåòñ� èñêîìîé.Ñïèñî� ëèòåðàòóð� [1] Activit� analysi� o� productio� an� allocatio� ( Cowle� Commissio� Monograp� � 13, ed.
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