Kooperative Spiele und Märkte by J. Rosenmüller (auth.)

By J. Rosenmüller (auth.)

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Es sei (xk );:1 eine Folge von Vektoren in ~m+ derart, daB xk€B und P lim u(xk ) k-+oo gilt. Mit yk = k = 1,2, ••• ~ xk",x'e gilt dann auch lim u(yk) 0(. k-+oo Notfalls nach Auswahl elner Teilfolge konvergiert yk ge~en ein - 50 ie Rm+, x ~Xe. Wegen der Stetigkeit von u ist u(i) = ex. • Ferner gilt px = lim pyk ~ lim px k ~ pa k-+'" k~oo also ist ieBp und erflillt die geforderten Bedingungen. Ware weiteres Element mit (2) und (3), so ware auch ~(i + x) € x ein Bp und u(~(i + x) ) '> ~ (u(i) + u (x) ) ~ (0(.

2. 3. 4. § 4 c T (TeV {12f, {34\ {123\, {4\ {12\, {3}, {4 \ {1}, {2l, {31, {4) 5. (12}, {13}, {23\, {4) 1, 1, 1, 1, 1 ~, 1 1 1, 1 1, 1, 1 1 1 1 ~, ~, 6. ((123), {14}, {24\ ,{3) 1 1 1 ~, ~, ~, 1 ~ Der Shapley-Wert .... Die Elemente des Cores sind ihrer Natur nach "wahrscheinliche" Ergebnisse der Verhandlungen, die durch die Gewinnzuschreibung der Spielfunktion v ausgelost werden. Der Verhandlun~smechanismus selbst, psychologische Faktoren und dgl. sind durch die Angabe einer Mengenfunktion v keineswegs bestimmt oder auch nur angedeutet; darin liegt eine Schwache , soweit konkrete Anwendungen in Betracht kommenaber aine Starke, soweit man sich mit k1ar formulierten mathematischen Modellen auseinandersetzt.

O x i,lk (18) pai (iEO), lk . i Ferner sei flir beliebiges x e B_ eine Folge x E B~ ~ P P P lk gegeben mit x + x (k "'too). also i xi E B_. A. kann x durch x lk A lk ~ lke angenommen werden, sonst ersetzt man x lke. Daher gilt: lim u i (x It.... ). lk - 54 Mithin ist ii optimal in B: und (p,X) also Gleichgewicht. 8. 1st w.... , Rm+, (~2, A) mit stetigem, konvexen, vertraglichen (~), so gibt es stets ein Gleichgewicht. A. 0 ~ u i , 1) Beweis. Sei u: ~+ ~ 0,1 irgendeine stetige, strikt konkave Funktion.

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