[учебник] Введение в теорию игр (с приложениями к экономике). Васин, Морозов (2003) (1186146), страница 44
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Íî èãðîê 1, èñïîëüçóÿ ñòðàòåãèþ β = (7/12, 0, 0), îáåñïå÷èâàåò ñåáå âûèãðûø 1/2. Ñëåäîâàòåëüíî, óêàçàííûå ñòðàòåãèè îïòèìàëüíû è çíà÷åíèå èãðû v = 1/2.14.3.  ïðîöåññå Áðàóíà ht = ((ik , jk ), k = 1, ..., t) è p1 (j|ht ) =|{k|jk = j, 1 ≤ k ≤ t}|/t → 0 ïðè t → ∞, åñëè ñòðàòåãèÿ j èãðîêà 2 âòðàåêòîðèè ((it , jt ), t = 1, 2, ...) ïðèìåíÿëàñü êîíå÷íîå ÷èñëî ðàç.251 22.Ðåøåíèå óïðàæíåíèé15.1. Âîçüìåì äâå íåïåðåñåêàþùèåñÿ êîàëèöèè K è T . Ïóñòü P K −ìíîæåñòâî ñìåøàííûõ ñòðàòåãèé pK êîàëèöèè K. Íåòðóäíî âèäåòü, ÷òîP K∪T ⊃ P K × P T . Îòñþäàv(K ∪ T ) =minmaxpK∪T ∈P K∪T sA\(K∪T ) ∈S A\(K∪T )≥ max maxminpK ∈P K pT ∈P T sA\(K∪T ) ∈S A\(K∪T )[uK (pK , pT , sA\(K∪T ) )++uT (pT , pK , sA\(K∪T ) )] ≥ max maxpK ∈P K pT ∈P T+uK∪T (pK∪T , sA\(K∪T ) ) ≥uK (pK , sA\K )+minsA\K ∈S A\KuT (pT , sA\T ) = v(K) + v(T ).minsA\T ∈S A\TÄîêàæåì ðàâåíñòâî (15.1).v(K) = maxminpK ∈P K sA\K ∈S A\K= maxminpK ∈P K sA\K ∈S A\K= v(A) − min[v(A) − uA\K (pK , sA\K )] =maxpK ∈P K sA\K ∈S A\K= v(A) −maxpA\K ∈P A\KuK (pK , sA\K ) =uA\K (pK , sA\K ) =min uA\K (pA\K , sK ) = v(A) − v(A\K).sK ∈S K15.2.
v(2) = 1, v(13) = 9, v(3) = 4, v(12) = 6.15.3. c = 1/500, b1 = −2/5, b2 = −3/5, b3 = 0, v 0 (12) = v 0 (13) =3/5, v 0 (23) = 7/10.P a15.4. Çàìåòèì, ÷òî ìíîæåñòâî C 0 íå ïóñòî, à ôóíêöèÿy îãðàíè÷åa∈APíà íà íåì ñíèçó âåëè÷èíîév(a). Îòñþäà ñëåäóåò, ÷òî çàäà÷à ëèíåéa∈Aíîãîz. ÏóñòüP aïðîãðàììèðîâàíèÿ â (15.2) èìååò îïòèìàëüíîåP ðåøåíèå0az ≤ v(A). Âîçüìåì òàêîé âåêòîð h ∈ C , ÷òîh > v(A). Òîãäàa∈Aa∈Aâûïóêëàÿ êîìáèíàöèÿ λz + (1 − λ)h, ãäå λ ∈ (0, 1] îïðåäåëÿåòñÿ èç óðàâíåíèÿXXλz a + (1 − λ)ha = v(A),a∈Aa∈Aïðèíàäëåæèò ÿäðó C. Îáðàòíî, äîïóñòèì, ÷òî ÿäðî C íå ïóñòî. Òîãäà(15.2) ñëåäóåò èç âêëþ÷åíèÿ ìíîæåñòâ C ⊂ C 0 .252 22.Ðåøåíèå óïðàæíåíèé15.5.
Ïóñòü λ − çàäàííîå â óñëîâèè ïðèâåäåííîå ñáàëàíñèðîâàííîå ïîêðûòèå. Òîãäà λT ∪L = 0, ïîñêîëüêó â ïðîòèâíîì ñëó÷àå âåêòîðû χ(T ), χ(L)è χ(T ∪ L) áûëè áû ëèíåéíî çàâèñèìûìè. ÈìååìλT χ(T ) + λL χ(L) = (λT − λL )χ(T ) + λL (χ(T ) + χ(L)) == µT χ(T ) + µT ∪L χ(T ∪ L).ÏîýòîìóXµK χ(K) =K6=AXλK χ(K) = χ(A),K6=Aà ñèñòåìà âåêòîðîâ{χ(K) | µK > 0} = {χ(K) | λK > 0} ∪ {χ(T ∪ L)}\{χ(L)}ëèíåéíî íåçàâèñèìà. Òàêèì îáðàçîì, âåêòîð µ − ïðèâåäåííîå ñáàëàíñèðîâàííîå ïîêðûòèå. Äàëåå,λT v(T ) + λL v(L) = (λT − λL )v(T ) + λL (v(T ) + v(L)) ≤≤ (λT − λL )v(T ) + λL v(T ∪ L) = µT v(T ) + µT ∪L v(T ∪ L).Îòñþäà âûòåêàåò íåðàâåíñòâî (15.4).15.6.
Ïðîåêöèÿ ÿäðà C íà ïëîñêîñòü (y 1 , y 2 ) èìååò âèä{(y 1 , y 2 ) | v(12) ≤ y 1 + y 2 ≤ v(123) − v(3),v(1) ≤ y 1 ≤ v(123) − v(23), v(2) ≤ y 2 ≤ v(123) − v(13)} == {(y 1 , y 2 ) | 800 ≤ y 1 + y 2 ≤ 1000, 200 ≤ y 1 ≤ 350, 300 ≤ y 2 ≤ 500}.Âåðøèíû ìíîæåñòâà C :y(1) = (300, 500, 200), y(2) = (350, 450, 200), y(3) = (350, 500, 150).15.7. Íåîáõîäèìîñòü. Âîçüìåì äåëåæ y(0) èç ÿäðà C. Îáîçíà÷èì ÷åðåçy(k), k = 1, ..., |A| − 1, äåëåæè, ïîëó÷åííûå èç y(0) öèêëè÷åñêèì ñäâèãîìíà k êîìïîíåíò âïðàâî. Òîãäà äåëåæ|A|−1z=Xy(k)/|A| = (v|A| /|A|, ..., v|A| /|A|)k=0ïðèíàäëåæèò ÿäðó è, ñëåäîâàòåëüíî,Xa∈Kza =|K|v|A|≥ v|K| ∀ K ⊂ A ⇒ (15.5).|A|253 22.Ðåøåíèå óïðàæíåíèéÄîñòàòî÷íîñòü. Ïóñòü âûïîëíåíû íåðàâåíñòâà (15.5).
Òîãäà óêàçàííûéâåêòîð z ïðèíàäëåæèò ÿäðó C.P15.8. Ïðîâåðèì ðàâåíñòâî a∈A ϕa ==X X (|K| − 1)! (|A| − |K|)!(v(K) − v(K\{a})) = v(A).|A|!a∈A K:a∈K(22.6)Âîçüìåì êîàëèöèþ K 6= A è ïîäñ÷èòàåì â ïîñëåäíåé äâîéíîé ñóììåKKêîýôôèöèåíò cK = cK+ + c− ïðè v(K). Îí âêëþ÷àåò ñóììó c+ ïîëîæèòåëüíûõ ñëàãàåìûõ, âñòðå÷àþùèõñÿ ïðè âêëàäàõ â êîàëèöèþ K è ñóììócK− îòðèöàòåëüíûõ ñëàãàåìûõ, âñòðå÷àþùèõñÿ ïðè âêëàäàõ â êîàëèöèþK ∪ {a}, ãäå a ∈/ K. Ïîñêîëüêó êàæäûé èç èãðîêîâ êîàëèöèè K èìååòñâîé âêëàä â K,cK+ = |K|(|K| − 1)! (|A| − |K|)!|K|= C|A| .|A|!Àíàëîãè÷íî,cK− = −(|A| − |K|)|K|! (|A| − |K| − 1)!|K|= −C|A| .|A|!|A|Îòñþäà cK = 0. Åñëè K = A, òî cA+ = C|A| = 1 è ðàâåíñòâî (22.6) äîêàçàíî.Îñòàëîñü ïðîâåðèòü óñëîâèå ϕa ≥ v(a), ∀ a ∈ A.
Äåéñòâèòåëüíî, èçñâîéñòâà ñóïåðàääèòèâíîñòè õàðàêòåðèñòè÷åñêîé ôóíêöèè è ðàâåíñòâà(15.6)ϕa =X (|K| − 1)! (|A| − |K|)!(v(K) − v(K\{a})) ≥|A|!K:a∈K≥X (|K| − 1)! (|A| − |K|)!v(a) = v(a).|A|!K:a∈K15.9.1111ϕ1 = (v(123) − v(23)) + (v(12) − v(2)) + (v(13) − v(3)) + v(1),36631111ϕ2 = (v(123) − v(13)) + (v(12) − v(1)) + (v(23) − v(3)) + v(2),3663254 22.Ðåøåíèå óïðàæíåíèé1111ϕ3 = (v(123) − v(12)) + (v(13) − v(1)) + (v(23) − v(2)) + v(3).3663 èãðå "äæàç-îðêåñòð"âåêòîð Øåïëè ϕ = (350, 475, 175). Åñëè v(123) =997, òî ϕ = (349, 474, 174) è ϕ2 + ϕ3 < v(23) = 650.15.10.  ñèììåòðè÷íîé èãðå âñå êîìïîíåíòû âåêòîðà Øåïëè ðàâíûìåæäó ñîáîé.16.1.
Íå ó÷òåíû çàòðàòû íà õðàíåíèå ïðîäóêöèè. Ïóñòü, íàïðèìåð,ïðåäïðèÿòèå äîëæíî ïîñòàâëÿòü åæåäíåâíî 5 èçäåëèé. Çàòðàòû íà õðàíåíèå âîçíèêàþò, åñëè ÷åðåç äåíü èñïîëüçóåòñÿ òåõíîëîãèÿ, äàþùàÿ 10èçäåëèé.0,0 ≤ p < 1,[0, 1], p = 1,16.2. S a (p) =1,1 < p ≤ 2,p/2, p > .17.1. Ôóíêöèÿ ñïðîñà D(p) = 3/p2 è ôóíêöèÿ ïðåäëîæåíèÿ S a (p) (ðèñ.16.5) ïåðåñåêàþòñÿ â òî÷êå p̃ = 1. Ïîýòîìó ôóíêöèÿ ïðèáûëè èìååò âèä(p3/p − (3/p2 − 1), 1 ≤ p ≤ 3/2,pW (p) = pD(p) − C(D(p)) =3/p − 3/(2p2 ),p > 3/2.Åå ìàêñèìóì íà ïîëóèíòåðâàëå [1, ∞) äîñòèãàåòñÿ ïðè p∗ = 1.17.2. Ôóíêöèÿ pD(p) ÿâëÿåòñÿ íåóáûâàþùåé íà îòðåçêå [p1 , p2 ], ïîñêîëüêó åå ïðîèçâîäíàÿ D(p) + pḊ(p) = D(p)(1 − e(D(p))) ≥ 0.
Ïîýòîìó ôóíêöèÿ ñïðîñà D(p) − ìåäëåííî óáûâàþùàÿ. Ôóíêöèÿ ïðèáûëèW (p) = pD(p) − C(D(p)) âîçðàñòàåò è îïòèìàëüíàÿ ñòðàòåãèÿ ìîíîïîëèèíà îòðåçêå [p1 , p2 ] ðàâíà p∗ = p2 .19.1. ÈìååìD(p) = K/pα , D−1 (V ) = K 1/α /V 1/α , Ḋ(D−1 (V )) = −αV 1+1/α /K 1/α .Ïóñòü v − ñèòóàöèÿ ðàâíîâåñèÿ. Ïî óòâåðæäåíèþ 19.1 v a > 0 ∀ a ∈ A.Îòñþäà ïî ëåììå 19.1 âûïîëíåíû íåðàâåíñòâàu0va (v) ≥ 0 ∀ a ∈ A èëè (ñì.
ôîðìóëó 19.5) X 1/α X 1+1/α K 1/α /vb− c − K 1/α v a / αvb≥ 0 ∀ a ∈ A.b∈Ab∈AÑêëàäûâàÿ ýòè íåðàâåíñòâà, ïîëó÷èì X 1/α 1/αK (mα − 1)/ αvb− cm ≥ 0,b∈A255 22.Ðåøåíèå óïðàæíåíèé÷òî íåâîçìîæíî ïðè 0 < α ≤ 1/m.19.2. Âûïèøåì âòîðóþ ÷àñòíóþ ïðîèçâîäíóþ ôóíêöèè âûèãðûøàau (v) ïî ïåðåìåííîé v a : X 2+1/α X001/αbavb.uva va (v) = K− 2(v ) + (1 + 1/α)v / αb∈Ab∈AÎòñþäà âèäíî, ÷òî ïðè α ≥ 1 ôóíêöèÿ ua (v) âîãíóòà, à ïðè1/m < α < 1 îíà èìååò åäèíñòâåííóþ òî÷êó ìàêñèìóìà ïî ïåðåìåííîé v aíà ïîëóïðÿìîé [0, +∞) ( ïðè ôèêñèðîâàííûõ ïåðåìåííûõ v b , b ∈ A\{a}).Îòñþäà ñëåäóåò, ÷òî íåîáõîäèìûå óñëîâèÿ äëÿ ñèòóàöèè ðàâíîâåñèÿ v,ñôîðìóëèðîâàííûå â ëåììå 19.1, ÿâëÿþòñÿ òàêæå è äîñòàòî÷íûìè óñëîâèÿìè.1. Åñëè V ≥ K/(cα m), òîdefp̃ = c, ṽ a = K/(cα m), p∗ = αcm/(mα − 1), v a ≡ v ∗ = K/(m(p∗ )α ).Åñëè K/(cα m) > V > v ∗ , òîp̃ = K/(mV α ), ṽ a ≡ V, p∗ = αcm/(mα − 1), v a ≡ v ∗ .Åñëè V ≤ v ∗ , òî p̃ = p∗ = K/(mV α ), ṽ a = v a ≡ V.mP2.
Ïîëîæèì c̃ = c/K, tl =V a , l = 0, 1, ..., m − 1, tm = 0.a=l+1Ïóñòü íàéäåòñÿ öåëîå k ∈ {1, ..., m}, óäîâëåòâîðÿþùåå óñëîâèþtk < 1/c̃ ≤ tk−1 . Òîãäà èç óðàâíåíèÿ 19.6 00 íàõîäèìpv ∗ (k) = k − 1 − 2c̃ktk + (k − 1)2 + 4c̃ktk /(2c̃k 2 ).Ïîñêîëüêó c̃tk < 1 (≤ c̃tk−1 , òî v ∗ (k) < (k − 1 − 2c̃ktk + k + 1)/(2c̃k 2 ) ≤ V k .V a,a > k,Ïîýòîìó v : v a =− ñèòóàöèÿ ðàâíîâåñèÿ. Ñîîòâåòñòâó∗v (k), a ≤ k.þùàÿ åé öåíà ðàâíàpp∗ = K/(tk + kv ∗ (k)) = 2c̃kK/(k − 1 + (k − 1)2 + 4c̃ktk ) > c = p̃.Ïóñòü 1/c̃ ≥ t0 . Òîãäà v a = V a ∀ a ∈ A è p∗ = p̃ = K/t0 .kP3.
Ïîëîæèì η = K(k − 1)/cb . Òîãäà ñèòóàöèÿ ðàâíîâåñèÿ v èìååòâèäb=1(η − η 2 ca /K, a = 1, ..., k,va =0,a = k + 1, ..., m,256 22.ãäå k = max{l |lPÐåøåíèå óïðàæíåíèécb > (l − 1)cl }. Ñîîòâåòñòâóþùàÿ åé öåíà ðàâíàb=1p∗ =kPcb /(k − 1) > p̃ = c1 .b=119.3. 1. Îáúåìû Ṽp0 , ãäå p0 < p, ïðèîáðåòàþò ñíà÷àëà ïîòðåáèòåëèñ ðåçåðâíîé öåíîé r ≥ p. Äëÿ ïîêóïêèòîâàðà ïîi öåíå p ÷èñëî òàêèõhïîòðåáèòåëåé ñòàíåò ðàâíûì max 0, D(p) − maxVp0 .0p <p2.
Ïóñòü P (s) = {pi } − óïîðÿäî÷åííîå ìíîæåñòâî öåí p1 < p2 <... < pk < p ≤ pk+1 < .... Ïîñêîëüêó D(p1 ) ïîêóïàòåëåé òîâàðà ïî öåíåp1 ðàâíîìåðíî ðàñïðåäåëåíû â î÷åðåäè, ÷èñëî ïîêóïàòåëåé ïî öåíå p1 ,èìåþùèõ ðåçåðâíóþ öåíó r ≥ p, ñîñòàâèò âåëè÷èíó Ṽp1 D(p)/D(p1 ), ààíàëîãè÷íîå èõ ÷èñëî ñ r ∈ [p1 , p) ðàâíî Ṽp1 (D(p1 ) − D(p))/D(p1 ).
Ïîýòîìó ïîñëå ïðîäàæè òîâàðà ïî öåíå p1 ÷èñëî ïîêóïàòåëåé ïî öåíå p2 ,èìåþùèõ ðåçåðâíóþ öåíó r ≥ p, ñòàíåò ðàâíûì D(p)(1 − Ṽp1 /D(p1 )), ò.å.îíî óìåíüøèòñÿ ïðîïîðöèîíàëüíî êîýôôèöèåíòó 1 − Ṽp1 /D(p1 ). Àíàëîãè÷íîå óìåíüøåíèå ïðîèçîéäåò è ñ ïîêóïàòåëÿìè, èìåþùèìè ðåçåðâíóþöåíó r ≥ p2 . Ïðîäîëæàÿ ðàññóæäåíèÿ, ïðèäåì ê âûâîäó, ÷òî ïîñëå ïîêóïêè òîâàðà ïî öåíå p2 ÷èñëî ïîêóïàòåëåé ïî öåíå p3 , èìåþùèõ ðåçåðâíóþöåíó r ≥ p, ñîñòàâèò âåëè÷èíó!Ṽp1 Ṽp2Ṽp1Ṽp2 D(p) 1 −1−= D(p) 1 −−ṼD(p1 )D(p1 ) D(p2 )D(p )(1 − p1 )2D(p1 )è ò.ä.3. Ïóñòü P (s) = {pi } − óïîðÿäî÷åííîå ìíîæåñòâî öåí p1 < p2 < ...
<pk < p ≤ pk+1 < .... Òîãäà íåòðóäíî âèäåòü, ÷òîD(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]è ò.ä.19.4.  ñèëó ñëåäñòâèÿ ê óòâåðæäåíèþ 19.3 äëÿ ðàâíîâåñèÿ ïî Íýøós âûïîëíåíî p(s) = p̃. PÏóñòü íàéäåòñÿ ïðîèçâîäèòåëü b, äëÿ êîòîðîãîb+V > S (p̃) − D(p̃) =V a − D(p̃) è cb = p̃. Òîãäà ïðè ìàëîì ε > 0aa:c ≤p̃Pâûïîëíåíî D(p̃ + ε) −V a > 0 è ïî (19.13) îñòàòî÷íûé ñïðîña:ca ≤p̃,a6=bïî öåíå p̃ + ε áóäåò ïîëîæèòåëüíûì.
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