А.А. Васин, В.В. Морозов - Введение в теорию игр с приложениями к экономике (1184512), страница 53
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Åñë� v(123) = 997, ò� ϕ = (349, 474, 174) � ϕ2 + ϕ3 < v(23) = 650. 15.10. � ñèììåòðè÷íî� èãð� âñ� êîìïîíåíò� âåêòîð� Øåïë� ðàâí� ìåæä� ñîáîé. 16.1. Í� ó÷òåí� çàòðàò� í� õðàíåíè� ïðîäóêöèè. Ïóñòü, íàïðèìåð, ïðåäïðèÿòè� äîëæí� ïîñòàâëÿò� åæåäíåâí� 5 èçäåëèé. Çàòðàò� í� õðàíåíè� âîçíèêàþò, åñë� ÷åðå� äåí� èñïîëüçóåòñ� òåõíîëîãèÿ, äàþùà� 10 èçäåëèé.
⎧ 0, 0 ≤ p < 1, [0, 1], p = 1,16.2. S a (p) = 1, 1 < p ≤ 2, ⎩ p/2, p > . 17.1. Ôóíêöè� ñïðîñ� D(p) = 3/p2 � ôóíêöè� ïðåäëîæåíè� S a (p) (ðèñ. 16.5) ïåðåñåêàþòñ� � òî÷ê� p̃ = 1. Ïîýòîì� ôóíêöè� ïðèáûë� èìåå� âè� (p3/p − (3/p2 − 1), 1 ≤ p ≤ 3/2,� W (p) = pD(p) − C(D(p)) = 3/p − 3/(2p2 ), p > 3/2. Å� ìàêñèìó� í� ïîëóèíòåðâàë� [1, ∞) äîñòèãàåòñ� ïð� p∗ = 1.
17.2. Ôóíêöè� pD(p) ÿâëÿåòñ� íåóáûâàþùå� í� îòðåçê� [p1 , p2 ], ïîñêîëüê� å� ïðîèçâîäíà� D(p) + pḊ(p) = D(p)(1 − e(D(p))) ≥ 0. Ïîýòîì� ôóíêöè� ñïðîñ� D(p) − ìåäëåíí� óáûâàþùàÿ. Ôóíêöè� ïðèáûë� W (p) = pD(p) − C(D(p)) âîçðàñòàå� � îïòèìàëüíà� ñòðàòåãè� ìîíîïîëè� í� îòðåçê� [p1 , p2 ] ðàâí� p∗ = p2 .
19.1. Èìåå� D(p) = K/pα , D−1 (V ) = K 1/α /V 1/α , Ḋ(D−1 (V )) = −αV 1+1/α /K 1/α .Ïóñò� v − ñèòóàöè� ðàâíîâåñèÿ. Ï� óòâåðæäåíè� 19.1 v a > 0 ∀ a ∈ A.Îòñþä� ï� ëåìì� 19.1 âûïîëíåí� íåðàâåíñòâ�u0va (v) ≥ 0 ∀ a ∈ A èë� (ñì. ôîðìóë� 19.5) � 1/α � 1+1/α � K 1/α /v b − c − K 1/α v a / α v b ≥ 0 ∀ a ∈ A. b∈Ab∈A Ñêëàäûâà� ýò� íåðàâåíñòâà, ïîëó÷è� � 1/α � 1/αK (mα − 1)/ α v b − cm ≥ 0, b∈A 255 22. Ðåøåíè� óïðàæíåíè�÷ò� íåâîçìîæí� ïð� 0 < α ≤ 1/m.
19.2. Âûïèøå� âòîðó� ÷àñòíó� ïðîèçâîäíó� ôóíêöè� âûèãðûø� au (v) ï� ïåðåìåííî� v a : � � 2+1/α � � 001/α ba uva va (v) = K− 2( v ) + (1 + 1/α)v / α v b . b∈Ab∈A Îòñþä� âèäíî, ÷ò� ïð� α ≥ 1 ôóíêöè� ua (v) âîãíóòà, � ïð�1/m < α < 1 îí� èìåå� åäèíñòâåííó� òî÷ê� ìàêñèìóì� ï� ïåðåìåííî� v aí� ïîëóïðÿìî� [0, +∞) ( ïð� ôèêñèðîâàííû� ïåðåìåííû� v b , b ∈ A\{a}).Îòñþä� ñëåäóåò, ÷ò� íåîáõîäèìû� óñëîâè� äë� ñèòóàöè� ðàâíîâåñè� v,ñôîðìóëèðîâàííû� � ëåìì� 19.1, ÿâëÿþòñ� òàêæ� � äîñòàòî÷íûì� óñë�âèÿìè.1. Åñë� V ≥ K/(cα m), ò� def p̃ = c, ṽ a = K/(cα m), p∗ = αcm/(mα − 1), v a ≡ v ∗ = K/(m(p∗ )α ).
Åñë� K/(cα m) > V > v ∗ , ò� p̃ = K/(mV α ), ṽ a ≡ V, p∗ = αcm/(mα − 1), v a ≡ v ∗ . Åñë� V ≤ v ∗ , ò� p̃ = p∗ = K/(mV α ), ṽ a = v a ≡ V. mP2. Ïîëîæè� c̃ = c/K, tl = V a , l = 0, 1, ..., m − 1, tm = 0. a=l+1 Ïóñò� íàéäåòñ� öåëî� k ∈ {1, ..., m}, óäîâëåòâîðÿþùå� óñëîâè� tk < 1/c̃ ≤ tk−1 . Òîãä� è� óðàâíåíè� 19.6 0� íàõîäè� pv ∗ (k) = k − 1 − 2c̃ktk + (k − 1)2 + 4c̃ktk /(2c̃k 2 ).
Ïîñêîëüê� ct˜ k < 1 (≤ ct˜ k−1 , ò� v ∗ (k) < (k − 1 − 2c̃ktk + k + 1)/(2c̃k 2 ) ≤ V k . V a , a > k, Ïîýòîì� v : v a = − ñèòóàöè� ðàâíîâåñèÿ. Ñîîòâåòñòâó∗v (k), a ≤ k. þùà� å� öåí� ðàâí� pp∗ = K/(tk + kv ∗ (k)) = 2c̃kK/(k − 1 + (k − 1)2 + 4c̃ktk ) > c = p.˜Ïóñò� 1/c̃ ≥ t0 . Òîãä� v a = V a ∀ a ∈ A � p∗ = p̃ = K/t0 .
kP3. Ïîëîæè� η = K(k − 1)/cb . Òîãä� ñèòóàöè� ðàâíîâåñè� v èìåå� âè� b=1� η − η 2 ca /K, a = 1, ..., k, v a = 0,a = k + 1, ..., m, 256 22. ãä� k = max{l | lPÐåøåíè� óïðàæíåíè�cb > (l − 1)cl }. Ñîîòâåòñòâóþùà� å� öåí� ðàâí� b=1 p∗ = kPcb /(k − 1) > p̃ = c1 . b=1 19.3.
1. Îáúåì� Ṽp� , ãä� p� < p, ïðèîáðåòàþ� ñíà÷àë� ïîòðåáèòåë� � ðåçåðâíî� öåíî� r ≥ p. Äë� ïîêóïê� òîâàð� ï� i öåí� p ÷èñë� òàêè� hïîòðåáèòåëå� ñòàíå� ðàâíû� max 0, D(p) − max Vp� . 0p <p 2. Ïóñò� P (s) = {pi } − óïîðÿäî÷åííî� ìíîæåñòâ� öå� p1 < p2 < ... < pk < p ≤ pk+1 < ....
Ïîñêîëüê� D(p1 ) ïîêóïàòåëå� òîâàð� ï� öåí� p1 ðàâíîìåðí� ðàñïðåäåëåí� � î÷åðåäè, ÷èñë� ïîêóïàòåëå� ï� öåí� p1 , èìåþùè� ðåçåðâíó� öåí� r ≥ p, ñîñòàâè� âåëè÷èí� Ṽp1 D(p)/D(p1 ), � àíàëîãè÷íî� è� ÷èñë� � r ∈ [p1 , p) ðàâí� Ṽp1 (D(p1 ) − D(p))/D(p1 ). Ïîýòîì� ïîñë� ïðîäàæ� òîâàð� ï� öåí� p1 ÷èñë� ïîêóïàòåëå� ï� öåí� p2 , èìåþùè� ðåçåðâíó� öåí� r ≥ p, ñòàíå� ðàâíû� D(p)(1 − Ṽp1 /D(p1 )), ò.å. îí� óìåíüøèòñ� ïðîïîðöèîíàëüí� êîýôôèöèåíò� 1 − Ṽp1 /D(p1 ). Àíàëîãè÷íî� óìåíüøåíè� ïðîèçîéäå� � � ïîêóïàòåëÿìè, èìåþùèì� ðåçåðâíó� öåí� r ≥ p2 . Ïðîäîëæà� ðàññóæäåíèÿ, ïðèäå� � âûâîäó, ÷ò� ïîñë� ïîêóïê� òîâàð� ï� öåí� p2 ÷èñë� ïîêóïàòåëå� ï� öåí� p3 , èìåþùè� ðåçåðâíó� öåí� r ≥ p, ñîñòàâè� âåëè÷èí� !� � V˜p1 � V˜p2 V˜p1 V˜p2 � D(p) 1 − 1 − = D(p) 1 − − ṼD(p1 )D(p1 ) D(p2 )D(p2 )(1 − D(pp1 1 ) )� ò.ä.
3. Ïóñò� P (s) = {pi } − óïîðÿäî÷åííî� ìíîæåñòâ� öå� p1 < p2 < ... < pk < p ≤ pk+1 < .... Òîãä� íåòðóäí� âèäåòü, ÷ò� D(p1 , Ṽ ) = D(p1 ), D(p2 , Ṽ ) = max[min[D(p2 ), D(p1 ) − Ṽp1 ], 0], D(p3 , Ṽ ) = max[min[D(p3 ), D(p2 ) − Ṽp2 , D(p1 ) − Ṽp1 − Ṽp2 ], 0] � ò.ä. 19.4. � ñèë� ñëåäñòâè� � óòâåðæäåíè� 19.3 äë� ðàâíîâåñè� ï� Íýø� s âûïîëíåí� p(s) = p.˜ PÏóñò� íàéäåòñ� ïðîèçâîäèòåë� b, äë� êîòîðîã� b +V > S (p̃) − D(p̃) = V a − D(p̃) � cb = p.˜ Òîãä� ïð� ìàëî� ε > 0 aa:c ≤p̃Pâûïîëíåí� D(˜p + ε) − V a > 0 � ï� (19.13) îñòàòî÷íû� ñïðî� a:ca ≤˜p,a=6 bï� öåí� p̃ + ε áóäå� ïîëîæèòåëüíûì. Ñëåäîâàòåëüíî, ïðîèçâîäèòåë� b âûãîäí� îòêëîíèòüñ� � âûáðàò� öåí� sb = p̃ + ε (ïðîòèâîðå÷èå).
257 22. Ðåøåíè� óïðàæíåíè�20.1. Äë� àêöèçíîã� íàëîã� te = S −1 (D1 + D2 )(D2 − K)/(D1 + K), p̃(te ) = te + S −1 (D1 + D2 ), äë� íàëîã� í� ïðèáûë� p̃(tpr ) = S −1 (D1 + D2 ), tpr = p̃(tpr )(D2 − K)/P r, ãä� âåëè÷èí� ïðèáûë� ðàâí� p̃Z(tpr ) (D1 + D2 − S(p))dp. P r =0 20.2. Ï� óñëîâè� e(D(p)) = −p(Ḋ1 (p) + Ḋ2 (p) − K/p2 )/(D1 (p) + D2 (p) + K/p) < 1. Îòñþä� Q̇1 (p) + Q̇2 (p) = D1 (p) + D2 (p) + p(Ḋ1 (p) + Ḋ2 (p)) > 0. 20.3. Ôóíêöè� K(p) íåïðåðûâí� � í� îòðåçêàõ, ãä� îí� äèôôåðåíöè¨ 1 (p)(q(p) − p) − 2Ḋ1 (p) − Q̇2 (p) ðóåìà, å� âòîðà� ïðîèçâîäíà� K̈(p) = Díåîòðèöàòåëüíà.
Ôóíêöè� K̇(p) � òî÷êà� ñâîåã� ðàçðûâ� èìåå� ïîëîæèòåëüíû� ñêà÷êè. Ñëåäîâàòåëüíî, ôóíêöè� K(p) âûïóêë� í� âñå� îòðåçê� [pD , ps ]. 21.1. Çàìåòèì, ÷ò� lim R(p) = T (qF − c)/F = qT − T c/F < R(p̂) = qT − T (1 − q)c/F. p→p̂−Åñë� qF > (1 − q)c, ò� R(p̂) > 0, R∗ = max R(p) = R(p̂), p∗ = p.ˆ0≤p≤1 Åñë� qF ≤ (1 − q)c, ò� R(p̂) ≤ 0, R∗ = p∗ = 0. 21.2.
1) È� óñëîâè� ñëåäóåò, ÷ò� F q > c. Ïîýòîì� ôóíêöè� R(p) âîçðàñòàå� í� ïîëóèíòåðâàë� [0, p̂) � ïîñòîÿíí� í� îòðåçê� [p̂, 1]. È� R(p̂) > 0 ïîëó÷àå� p∗ ∈ [p̂, 1]. 2) Èìåå� F = c/q < F = (qm + 1 − q)c/(qm). Ïîýòîì� ôóíêöè� R(p) ðàâí� íóë� í� ïîëóèíòåðâàë� [0, p̂) � óáûâàå� í� îòðåçê� [p̂, 1]. È� R(p̂) > 0 ïîëó÷àå� p∗ = p.ˆ258 Ïðèëîæåíè� Ï1. Òåîðåì� î� îòäåëÿþùå� ãèïåðïëîñêîñò� Ïóñò� A � B − äâ� âûïóêëû� íåïåðåñåêàþùèõñ� êîìïàêò� � E . Òîãä� íàéäåòñ� ãèïåðïëîñêîñò� a, x = b, ñòðîã� îòäåëÿþùà� ìíîæåñòâ� A � B, ò.å. a, x < b < a, y ∀x ∈ A, ∀y ∈ B. Òåîðåì� Ï.1. m Äîêàçàòåëüñòâî. Ðàññìîòðè� í� A × B ôóíêöè� |x − y |2 , ãä� x ∈ A, y ∈ B, � ïóñò� ïàð� (x0 , y 0 ) − òî÷ê� å� Òîãä� ìèíèìóìà.
0 0|x − y | > 0. Ïîêàæåì, ÷ò� ãèïåðïëîñêîñò� a, x = b ïðèa = y 0 − x0 , b = 12 (|y 0 |2 − |x0 |2 ) ÿâëÿåòñ� èñêîìîé. Äîêàæåì, ÷ò� a, x < b ∀x ∈ A. Ïðåäïîëîæè� ïðîòèâíîå. Òîãä� íàéäåòñ� òàêà� òî÷ê� x� ∈ A, ÷ò� a, x� ≥ b. Ðàññìîòðè� ôóíêöè� g(t) = |y 0 − (1 − t)x0 − tx0 |2 , t ∈ [0, 1]. Èìåå� g 0 (0) = 2 y 0 − x 0 , x 0 − x� = 2 y 0 − x 0 , x 0 − 2 a, x� ≤ ≤ 2 y 0 , x 0 − 2|x 0 |2 − 2b = 2 y 0 , x 0 − |x 0 |2 − |y 0 |2 = −|x 0 − y 0 |2 < 0. Ñëåäîâàòåëüíî, ïð� ïîëîæèòåëüíû� � áëèçêè� � íóë� t g(t) < g(0), ÷ò� ïðîòèâîðå÷è� îïðåäåëåíè� ïàð� (x0 , y 0 ). Âòîðî� íåðàâåíñòâ� b < a, y ∀y ∈ B äîêàçûâàåòñ� àíàëîãè÷íî.
Ï2. Äîêàçàòåëüñòâ� òåîðåì� 6.1 Òåîðåì� î÷åâèäí� äë� E 0 . Ïóñò� îí� âåðí� äë� E m−1 . Ðàññìîòðè� ñåìåéñòâ� âûïóêëû� êîìïàêòî� Dα , α ∈ L è� E m , êàæäû� m+1 è� êîòîðû� èìåþ� íåïóñòî� ïåðåñå÷åíèå. Äîêàæå� ñíà÷àëà, ÷ò� ëþáî� êîíå÷íî� ïîäñåìåéñòâ� ñåìåéñòâ� Dα , α ∈ L èìåå� íåïóñòî� ïåðåñå÷åíèå. Ïðåäïîëîæè� ïðîòèâíîå. Òîãä� íàéäåòñ� ìèíèìàëüíî� öåëî� k > m + 1, äë� êîòîðîã� íàéäåòñ� òàêî� kkT−1 � ïîäñåìåéñòâ� Dαi , i = 1, ..., k, ÷ò� Dαi = ∅,Dαi =6 ∅.
Ïîëîæè� i=1 A = Dαk , B = kT−1 i=1 Dαi . Âûïóêëû� êîìïàêò� A � B í� ïåðåñåêàþòñ� � i=1 íàéäåòñ� ñòðîã� îòäåëÿþùà� è� ãèïåðïëîñêîñò� H (ñì. Ï1.) Ïóñò� C −ïåðåñå÷åíè� êàêèõ-ëèá� m ìíîæåñò� è� Dα1 , ..., Dαk−1 . Òîãä� B ⊆ C � ï� óñëîâè� C ∩ A =6 ∅. Âîçüìå� x0 ∈ C ∩ A � y 0 ∈ B.
Òîãä� îòðåçî� 259Ïðèëîæåíè�[x0 , y 0 ] ïðèíàäëåæè� C � ïåðåñåêàåòñ� � H. Ñëåäîâàòåëüíî, C ∩ H =6∅. Èòàê, ëþáû� m ìíîæåñò� è� Dα1 ∩ H, ..., Dαk−1 ∩ H èìåþ� íåïóñòî� kT−1 ïåðåñå÷åíèå. Ï� èíäóêòèâíîì� ïðåäïîëîæåíè� Dαi ∩ H = B ∩ H =6 ∅, i=1 ÷ò� ïðîòèâîðå÷è� ïîñòðîåíè� ãèïåðïëîñêîñò� H. T Çàâåðøè� äîêàçàòåëüñòâ� òåîðåìû.
Ïðåäïîëîæè� , ÷ò� Dα = ∅. Ïóñò� X = Dα� − íåêîòîðû� êîìïàê� è� ñåìåéñòâ� α∈L Dα , α ∈ L. Îïðåäåëè� íîâî� ñåìåéñòâ� Dα 0 = Dα ∩ X, α ∈ L. Òîãä� T 0� Dα = ∅ � ìíîæåñòâ� Dα = E m \Dα0 îáðàçóþ� îòêðûòî� ïîêðûòè� êîìα∈L � � ïàêò� X, è� êîòîðîã� ìîæí� âûäåëèò� êîíå÷íî� ïîäïîêðûòè� Dα1 , ..., Dαp .ppTTÎòñþä� âûòåêàåò, ÷ò� Dα0 i = Dαi ∩ Dα� = ∅. Ï� äîêàçàííîì� ëþáî� i=1 i=1 êîíå÷íîå ïîäñåìåéñòâî ñåìåéñòâ� Dα , α ∈ L èìååò íåïóñòîå ïåðåñå÷åíè� (ïðîòèâîðå÷èå).
Ï3. Ôóíêöè� Ìèíêîâñêîã� � ãîìåîìîðôèç� âûïóêëû� êîìïàêòî� Ïóñò� A − âûïóêëû� êîìïàê� � E m , ñîäåðæàùè� âíóòð� ñåá� íóë� âìåñò� � ε0 -îêðåñòíîñòü� íóë� Oε0 . Îïðåäåëè� í� E m ôóíêöè� Ìèíêîâñêîã� ìíîæåñòâ� A () xpA (x) = inf t > 0 ∈ A . t 6 0 pA (x) > 0 Ðàññìîòðè� å� ñâîéñòâà. Î÷åâèäíî, ÷ò� pA (0) = 0, � ïð� x =� íèæíÿ� ãðàí� äîñòèãàåòñÿ, ïîñêîëüê� ìíîæåñòâ� A îãðàíè÷åí� � çàìêíóòî. Êðîì� òîãî, pA (x) ≤ 1 ⇔ x ∈ A.
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