Введение в теорию игр (сторонняя методичка) (1184510), страница 47
Текст из файла (страница 47)
Ïîñêîëüêó K − êîìïàêò,áåç ïîòåðè îáùíîñòè ìîæíî ñ÷èòàòü, ÷òî ïîñëåäîâàòåëüíîñòè {y i (k)}ñõîäÿòñÿ. Íî lim δ(πk ) = 0. Ïîýòîìó ïðè k → ∞ äèàìåòð ñèìïëåêñàk→∞y 1 (k) · · · y n+1 (k) ñòðåìèòñÿ ê íóëþ. Ñëåäîâàòåëüíî, lim y i (k) = y ∗ , i =k→∞1, ..., n + 1. Ïîñêîëüêó ïðè âñåõ i y i (k) ∈ Ci , à ìíîæåñòâà Ci çàìêíóòû,n+1Tèìååì y ∗ ∈Ci .i=1265ÏðèëîæåíèåÓòî÷íèì ôîðìóëèðîâêó òåîðåìû 9.1. Ïóñòü Z − âûïóêëûé êîìïàêòâ E m è y 1 · · · y n+1 − ñîäåðæàùèéñÿ â íåì ñèìïëåêñ ìàêñèìàëüíîé ðàçìåðíîñòè.
Òîãäà ÷èñëî n ≤ m íàçûâàåòñÿ ðàçìåðíîñòüþ ìíîæåñòâà Z.Âûïóêëûé êîìïàêò Z ìàêñèìàëüíîé ðàçìåðíîñòè m èìååò âíóòðåííþþòî÷êó (íàïðèìåð, áàðèöåíòð ñèìïëåêñà y 1 · · · y m+1 ). Îáðàòíî, åñëè Z èìååò âíóòðåííþþ òî÷êó, òî åãî ðàçìåðíîñòü ìàêñèìàëüíà. Ïðåäïîëîæèì,÷òî Z èìååò ðàçìåðíîñòü n < m è ñîäåðæèò íóëü. Òîãäà ìîæíî ñ÷èòàòü,÷òî y n+1 = 0. Ïðè ýòîì Z ÿâëÿåòñÿ ïîäìíîæåñòâîì åâêëèäîâà ïðîñòðàíñòâà E n ñ áàçèñîì y 1 , ..., y n .Èç ñêàçàííîãî âûòåêàåò, ÷òî â òåîðåìå 9.1 áåç ïîòåðè îáùíîñòè ìîæíîïðåäïîëîæèòü, ÷òî âûïóêëûé êîìïàêò Z èìååò ìàêñèìàëüíóþ ðàçìåðíîñòü m.Äîêàçàòåëüñòâî òåîðåìû 9.1 (Áðàóýðà).
Ïóñòü K = x1 · · · xm+1 −m-ìåðíûé ñèìïëåêñ â E m . Ïî òåîðåìå Ï.2 (ñì. Ï3) íàéäåòñÿ ãîìåîìîðôèçì τ : Z → K. Îòîáðàæåíèå τ ◦ f ◦ τ −1 ïåðåâîäèò òî÷êó y ∈ K â òî÷êóτ (f (τ −1 (y))) ∈ K è ÿâëÿåòñÿ íåïðåðûâíûì. Åñëè y 0 ∈ K − íåïîäâèæíàÿòî÷êà îòîáðàæåíèÿ τ ◦ f ◦ τ −1 , òî x0 = τ −1 (y 0 ) − íåïîäâèæíàÿ òî÷êàîòîáðàæåíèÿ f : Z → Z. Ïîýòîìó áåç ïîòåðè îáùíîñòè ìîæíî ñ÷èòàòü,÷òî Z = K.ÏîëîæèìCi = {x ∈ K | λi (f (x)) ≤ λi (x)}, i = 1, ..., m + 1.Ïîñêîëüêó îòîáðàæåíèå f è ôóíêöèè λi (x) íåïðåðûâíû íà K , ìíîæåñòâà Ci çàìêíóòû.
Ïîêàæåì, ÷òî ìíîæåñòâà Ci óäîâëåòâîðÿþò óñëîâèþïðåäûäóùåé òåîðåìû. Ïóñòü x ∈ xj1 · · · xjs . Òîãäàj1jsλj (x) = 0 ∀j ∈/ {x , ..., x } ⇒sXλjr (x) = 1 ≥r=1sXλjr (f (x)).r=1Ñëåäîâàòåëüíî, íàéäåòñÿ íîìåð r, ïðè êîòîðîì λjr (x) ≥ λjr (f (x)) èx ∈ Cjr .
Ïî ïðåäûäóùåé òåîðåìå íàéäåòñÿ∗x ∈m+1\Ci ⇒ λi (x∗ ) ≥ λi (f (x∗ )), i = 1, ..., m + 1.i=1Íîm+1Xi=1∗λi (x ) =m+1Xλi (f (x∗ )) = 1.i=1266ÏðèëîæåíèåÏîýòîìóλi (x∗ ) = λi (f (x∗ )), i = 1, ..., m + 1 ⇒ x∗ = f (x∗ ).Äîêàçàòåëüñòâî òåîðåìû 12.1 (Êàêóòàíè). Êàê è ïðè äîêàçà-òåëüñòâå òåîðåìû Áðàóýðà, áåç ïîòåðè îáùíîñòè ìîæíî ñ÷èòàòü, ÷òî âûïóêëûé êîìïàêò Z èìååò ìàêñèìàëüíóþ ðàçìåðíîñòü m.Ïóñòü K = x1 · · · xm+1 − m-ìåðíûé ñèìïëåêñ â E m . Ïî òåîðåìå Ï.2(ñì.
Ï3) íàéäåòñÿ ãîìåîìîðôèçì ϕ : K → Z. Ðàññìîòðèì ïîñëåäîâàòåëüíîñòü {πk } ñèìïëèöèàëüíûõ ðàçáèåíèé ñèìïëåêñà K, äëÿ êîòîðîélim δ(πk ) = 0. Äëÿ ïðîèçêîëüíîãî k îïðåäåëèì ñëåäóþùåå íåïðåðûâíîåk→∞îòîáðàæåíèå Φk : K → Z. Äëÿ ëþáîé âåðøèíû x èç {πk } âûáåðåì òî÷êóΦk (x) ∈ Φ(ϕ(x)) è çàòåì ïðîäîëæèì îòîáðàæåíèå ëèíåéíî íà êàæäûé èçñèìïëåêñîâ, âõîäÿùèõ â {πk }, ò.å. åñëèx=m+1Xjλj x ,j=1m+1Xλj = 1, λj ≥ 0, j = 1, ..., m + 1,j=1− òî÷êà ñèìïëåêñà x1 · · · xm+1 ∈ {πk }, òî ïîëàãàåìkΦ (x) =m+1XΦk (xj ).j=1Î÷åâèäíî, ÷òî ïîñëåäíåå ðàâåíñòâî îïðåäåëÿåò íåïðåðûâíîå îòáðàæåíèå Φk : K → Z .
Òîãäà f k = ϕ−1 Φk : K → K ÿâëÿåòñÿ íåïðåðûâíûìîòîáðàæåíèåì è èìååò íåïîäâèæíóþ òî÷êó xk . Ïóñòü xk ïðèíàäëåæèòñèìïëåêñó x1k · · · xm+1k ∈ {πk }, ò.å.kx =m+1Xj=1λkj xjk ,m+1Xλkj = 1, λkj ≥ 0, j = 1, ..., m + 1.j=1Íå óìàëÿÿ îáùíîñòè, ìîæíî ñ÷èòàòü, ÷òî ïîñëåäîâàòåëüíîñòè{xk }, {xjk },{λkj },{Φk (xjk )}, j = 1, ..., m + 1, ñõîäÿòñÿ ïðè k → ∞. Ïîñêîëüêó äèàìåòðû ñèìïëåêñîâ x1k · · · xm+1k ñòðåìÿòñÿ ê íóëþ, ïîñëåäîâàòåëüíîñòè {xk } è {xjk } äîëæíû èìåòü îáùèé ïðåäåë x∗ ∈ K. Ïóñòülim λkj = λ∗j , lim Φk (xjk ) = η j , j = 1, ..., m + 1.k→∞k→∞267Èìååìϕ(xk ) = Φk (xk ) =m+1Xλkj Φk (xjk )(Π.2)j=1è lim ϕ(xjk ) = ϕ(x∗ ), lim ϕ(xk ) = ϕ(x∗ ) â ñèëó íåïðåðûâíîñòè îòáðàæåk→∞k→∞íèÿ ϕ. Ïåðåõîäÿ â ðàâåíñòâå (Π.2) ê ïðåäåëó ïðè k → ∞, ïîëó÷èìϕ(x∗ ) =m+1Xj=1λ∗j η j ,m+1Xλ∗j = 1, λ∗j ≥ 0, j = 1, ..., m + 1.j=1Èç çàìêíóòîñòè îòîáðàæåíèÿ Φ âûòåêàåò, ÷òîη j ∈ Φ(ϕ(x∗ )), j = 1, ..., m + 1.
Ïîñêîëüêó ìíîæåñòâî Φ(ϕ(x∗ )) âûïóêëî,òî÷êà ϕ(x∗ ) åìó ïðèíàäëåæèò è ÿâëÿåòñÿ èñêîìîé.Ñïèñîê ëèòåðàòóðû[1] Activity analysis of production and allocation ( Cowles CommissionMonograph 13, ed. Koopmans T.C.). − N.Y.: J.Wiley, 1951.[2] Allen B., Hellwig M. Bertrand-Edgeworth oligopoly in largemarkets. Rewiew of Economic Studies, 1986, v. 53, p. 175-204.[3] Amir R. Cournot oligopoly and the theory of supermodular games.Games and Economic Behavior, 1996, v. 15, p. 132-148.[4] Atkinson A.B., Stiglitz J.E.
Lectures on public economics. −London: McGraw-Hill, 1980.[5] Àøìàíîâ Ñ.À. Ëèíåéíîå ïðîãðàììèðîâàíèå. − Ì.: Íàóêà, 1981.[6] Àøìàíîâ Ñ.À., Òèìîõîâ À.Â. Òåîðèÿ îïòèìèçàöèè â çàäà÷àõ èóïðàæíåíèÿõ. − Ì.: Íàóêà, 1991.[7] Aumann R.J. The core of a cooperative game without sidepayments. Trans. Amer. Math. Soc., 1961, v.98, 3, p. 539-552.268Ñïèñîê ëèòåðàòóðû[8] Áåëåíüêèé Â.Ç., Âîëêîíñêèé Â.À., Èâàíêîâ Ñ.À., Ïîìàíñêèé À.Á., Øàïèðî À.Ä.
Èòåðàòèâíûå ìåòîäû â òåîðèè èãð èïðîãðàììèðîâàíèè. − Ì.: Íàóêà, 1974.[9] Áåñêîíå÷íûå àíòàãîíèñòè÷åñêèå èãðû. Ñá. ñòàòåé ïîä ðåä.Í.Í.Âîðîáüåâà. − Ì.: Ôèçìàòãèç, 1963.[10] Bertrand J. Review de théorie mathématique de la richesse sociale.Recherches sur les principes mathématique de la théorie desrichesses. Journal des Savants, 1883, p. 499-508.[11] Áëåêóýë Ä., Ãèðøèê Ì. Òåîðèÿ èãð è ñòàòèñòè÷åñêèõ ðåøåíèé.− Ì.: Èíîñòðàííàÿ ëèòåðàòóðà, 1958.[12] Áîíåíáëàñò Õ.Ô., Êàðëèí Ñ., Øåïëè Ë.Ñ.
Èãðû ñ íåïðåðûâíîéôóíêöèåé âûèãðûøà. Â ñá. [9], ñ. 337-352.[13] Áîíäàðåâà Î.Í. Íåêîòîðûå ïðèìåíåíèÿ ìåòîäîâ ëèíåéíîãî ïðîãðàììèðîâàíèÿ ê òåîðèè êîîïåðàòèâíûõ èãð. Ïðîáëåìû êèáåðíåòèêè. 1963, âûï.10, ñ. 119-140.[14] Borges T. Iterated elimination of dominated strategies in aBertrand-Edgeworth model. Rewiew of Economic Studies, 1992, v.59, p. 163-176.[15] Borel E. 1) The theory of play and integral equations with skewsymmetric kernels.
2) On games that involve chance and skillof the players. 3) On system of linear forms of skew symmetricdeterminants and the general theory of play. Econometrica, 1953,v. 21, 1, p. 97-117.[16] Borel E. Application aux jeux de hasard. Traité du calcul desprobabilités et des ses applications, Applications des jeux hasard,E.Borel et collab. − Paris: Gauthier − Villars, 1938, v. IV, fasc. 2,p. 122.[17] Brown G.W. Iterative solutions of games by fictitious play. Incollected book [1], p. 374-376.[18] Brouwer L.E.J.
On continuous vector disributions of surfaces.Amsterdam Proc., 1909, v. 11, continued in 1910, v. 12,13.269Ñïèñîê ëèòåðàòóðû[19] Walras L. Éléments d'économie politique pure. − Lausanne, 1874.[20] Wald A. Contributions to the theory of statictical estimation andtesting hypotheses. Ann. Math. Stat., 1939, v. 10, 4, p. 299-326.[21] Âàëüä À. Ñòàòèñòè÷åñêèå ðåøàþùèå ôóíêöèè.  ñá. [65], ñ. 300522.[22] Âàñèí À.À. Ìîäåëè ïðîöåññîâ ñ íåñêîëüêèìè ó÷àñòíèêàìè. −Ì.: Èçä-âî Ìîñê. óí-òà, 1983.[23] Âàñèí À.À.
Ìîäåëè äèíàìèêè êîëëåêòèâíîãî ïîâåäåíèÿ. − Ì.:Èçä-âî Ìîñê. óí-òà, 1989.[24] Âàñèí À.À., Ïàíîâà Å.È. Ñîáèðàåìîñòü íàëîãîâ è êîððóïöèÿ âíàëîãîâûõ îðãàíàõ. − Ì.: Ðîññèéñêàÿ ïðîãðàììà ýêîíîìè÷åñêèõèññëåäîâàíèé, 2000, Ñåð. "íàó÷íûå äîêëàäû", 99/10.[25] Âàñèí À.À., Àãàïîâà Î.Á. Ìàòåìàòè÷åñêàÿ ìîäåëü îïòèìàëüíîé îðãàíèçàöèè íàëîãîâîé èíñïåêöèè.  ñá.
"Ïðîãðàììíîàïïàðàòíûå ñðåäñòâà è ìàòåìàòè÷åñêîå îáåñïå÷åíèå âûñ÷èñëèòåëüíûõ ñèñòåì". − Ì.: Èçä-âî Ìîñê. óí-òà, 1993, ñ. 167-186.[26] Âàñèí À.À. , Âàñèíà Ï.À. Îïòèìèçàöèÿ íàëîãîâîé ñèñòåìû âóñëîâèÿõ óêëîíåíèÿ îò íàëîãîâ. Ðîëü îãðàíè÷åíèé íà øòðàô. −Ì.: Ðîññèéñêàÿ ïðîãðàììà ýêîíîìè÷åñêèõ èññëåäîâàíèé, 2002,Ñåð. "íàó÷íûå äîêëàäû", 01/09.[27] Âàñèí À.À.
, Âàñèíà Ï.À., Ìàðõóýíäà Ô.Õ. Íàëîãîâîå ïðèíóæäåíèå äëÿ íåîäíîðîäíûõ ôèðì./ Ïðåïðèíò 2001/025. − Ì.:Ðîññèéñêàÿ ýêîíîìè÷åñêàÿ øêîëà, 2001.[28] Âàòåëü È.À., Åðåøêî Ô.È. Ìàòåìàòèêà êîíôëèêòà è ñîòðóäíè÷åñòâà. − Ì.: Çíàíèå, 1974.[29] Âåíòöåëü Å.Ñ. Èññëåäîâàíèå îïåðàöèé. − Ì.: Ñîâåòñêîå ðàäèî,1981.[30] Ville J. Sur la théorie générale des jeux ou intervient l'abilite desjeueurs, Traité du calcul des probabilités et des ses applications,Applications des jeux hasard, E.Borel et collab.
− Paris: Gauthier− Villars, 1938, v. IV, fasc. 2, p. 105-113.270Ñïèñîê ëèòåðàòóðû[31] Âèëêàñ Ý.É. Îïòèìàëüíîñòü â èãðàõ è ðåøåíèÿõ. − Ì.: Íàóêà,1990.[32] Vives X. Rationing rules and Bertrand-Edgeworth equilibria in largemarkets. Economic Letters. 1986, v. 21, p. 113-116.[33] Wolfovitz J. Minimax estimates of the mean of normal distributionwith known variance. Ann.
Math. Stat., 1950, v. 21, 2, p. 218-230.[34] Âîðîáüåâ Í.Í. Îñíîâû òåîðèè èãð. Áåñêîàëèöèîííûå èãðû. −Ì.: Íàóêà, 1984.[35] Ãåéë Ä. Òåîðèÿ ëèíåéíûõ ýêîíîìè÷åñêèõ ìîäåëåé. − Ì.:ÈË,1963.[36] Ãåðìåéåð Þ.Á. Ââåäåíèå â òåîðèþ èññëåäîâàíèÿ îïåðàöèé. −Ì.: Íàóêà, 1971.[37] Ãåðìåéåð Þ.Á. Èãðû ñ íåïðîòèâîïîëîæíûìè èíòåðåñàìè. − Ì.:Íàóêà, 1976.[38] Ãåðìåéåð Þ.Á. Îá èãðàõ äâóõ ëèö ñ ôèêñèðîâàííîé ïîñëåäîâàòåëüíîñòüþ õîäîâ. ÄÀÍ, 1971, v.
198, 5, ñ. 1001-1004.[39] Ãîðåëèê Â.À. Òåîðèÿ èãð è èññëåäîâàíèå îïåðàöèé. − Ì: Èçä-âîÌÈÍÃÏ, 1978.[40] Ãðåíü Å. Ñòàòèñòè÷åñêèå èãðû è èõ ïðèìåíåíèå. − Ì.: Ñòàòèñòèêà, 1975.[41] Äàâûäîâ Ý.Ã. Èññëåäîâàíèå îïåðàöèé. − Ì.: Âûñøàÿ øêîëà,1990.[42] Äàíöåð Ë., Ãðþíáàóì Á., Êëè Â. Òåîðåìà Õåëëè. − Ì.: Ìèð,1968.[43] Dantzig G.B. A proof of the equivalence of the programmingproblem and the problem. In collected book [1], p. 330-335.[44] Gilles D.B. Solutions to general non-zero-sum games.
Contributionsto the theory of games. IV (Kuhn H.W., Tucker A.W. eds.). Ann.Math. Studies, 40, − Princeton: Princeton Univ. Press, 1959,p. 47-86.271Ñïèñîê ëèòåðàòóðû[45] Äðåøåð Ì. Ñòðàòåãè÷åñêèå èãðû. Òåîðèÿ è ïðèëîæåíèÿ. − Ì.:Ñîâåòñêîå ðàäèî, 1964.[46] Äþáèí Ã.Í., Ñóçäàëü Â.Ã. Ââåäåíèå â ïðèêëàäíóþ òåîðèþ èãð.− Ì.: Íàóêà, 1981.[47] Kakutani S. A generalisation of Brower's fixed point theorem. DukeMath. J., 1941, v. 8, 3, p.
457-459.[48] Êàðëèí Ñ. Ìàòåìàòè÷åñêèå ìåòîäû â òåîðèè èãð, ïðîãðàììèðîâàíèè è ýêîíîìèêå. − Ì.: Ìèð, 1964.[49] Êèíè Ð.Ë., Ðàéôà Õ. Ïðèíÿòèå ðåøåíèé ïðè ìíîãèõ êðèòåðèÿõ.Ïðåäïî÷òåíèÿ è çàìåùåíèÿ. − Ì.: Ðàäèî è ñâÿçü, 1981.[50] Êîëìîãîðîâ À.Í., Ôîìèí Ñ.Â.