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3 (N;E;d) : N =f1; 2;:::;ng : E : d = (d 1 ;d 2 ;:::;d n ) : D = P j2nd j >E f(n;e;d) = (x 1 ;x 2 ;:::;x n ) : P x i ï 0 8i2N; j2n x j =E h i (N;E;d) = E D d i 3
4 =)!! =) (f1; 2g;E; (d 1 ;d 2 ))! max(eäd 2 ; 0)! max(eäd 1 ; 0) ) EÄmax(EÄd 1 ; 0)Ämax(EÄd 2 ; 0) x 1 = 1 2 (EÄmax(EÄd 1; 0) +max(eäd 2 ; 0)) x 2 = 1 2 (E +max(eäd 1; 0)Ämax(EÄd 2 ; 0)) 4
5 f(n;e;d) =x i;j2n(i6=j) (fi;jg;x i +x j ; (d i ;d j )) (x i ;x j ) ( E >E 0 )f i (N;E;d)ïf i (N;E 0 ;d) 8i d i ïd j )f i (N;E;d)ïf j (N;E;d) f i (N;E;d) =d i Äf i (N;DÄE;d) 5
6 (N:v) N : SíN : v : 2 N!< : v(s) : S à;' X =fxjx = (x 1 ;x 2 ;:::;x n )2< N g à(n;v)íx : solution '(N;v)2X : value P i2nx i =v(n) x i ïv(fig) 8i2N I É x I x 6
7 v(s) =max(eä X i2nns d i ; 0) 8SíN K É S x (excess) e(s;x) =v(s)ä P i2sx i i j (surplus s ij (x) =maxfe(s;x)js3i; S3=jg pre-kernel K É =fx2i É js ij (x) =s ji (x) 8i;j(i6=j)g (f1; 2g;v) x 1 = 1 2 (v(f1; 2g)Äv(f1g)Äv(f2g)) +v(f1g) x 2 = 1 2 (v(f1; 2g)Äv(f1g)Äv(f2g)) +v(f2g) n n 7
8 j2n, x2i É (Nnfjg;v x ) v x (Nnfjg) =v(n)äx j v x (S) =maxfv(s);v(s[fjg)äx j g SöN v x (;) = 0 j2n;x2à(n;v) Ä! (x i ) i2nnfjg 2 à(nnfjg;v x ) x2i É ; (x i ;x j )2à(fi;jg;v x ) 8i;j(i6=j) Ä! x2à(n;v) 8
9 (N;E;d) x Ä! (N;v) x # # Ä! (S;EÄ P i2nnsx i ; (d i ) i2s ) (S;v x ) # # Ä! (fi;jg;x i +x j ; (d i ;d j )) (fi;jg;v x ) # # (x i ;x j ) Ä! (x i ;x j ) 9
10 Mimonide Mimonide Mimonide Rabad d 1 <d 2 <:::<d n ; E =d n d 1 d 2 Äd 1 nä1 d n Äd nä1 n (N;v) (N;v É ) v É (S) =v(n)äv(nns) 8SíN v É (S) =min(e; X i2s d i) 8SíN 10
11 i2s v(s)äv(snfig) i2n û i (v) û i (v) = X S:i2SíN (sä1)!(näs)! (v(s)äv(snfig)) n! s =jsj P i2nû i (v) =v(n) v(s[fig) =v(s) 8SíNnfigÄ! û i (v) = 0 (ETP) v(s[fig) =v(s[fjg) 8SíNnfi;jg Ä! û i (v) =û j (v) (N;v) (N;w) û i (v +w) =û i (v) +û i (w) 8i2N (v +w)(s) =v(s) +w(s) 8SíN 11
12 g(n;e;d) g i (N;E;d) = 1 n (min(e;d i)+ X j2n nfig g i (Nnfjg;max(0;EÄd j ); ^d j ) ^d j = (d 1 ;d 2 ;:::d jä1 ;d j+1 ;:::;d n ) # g i (Nnfjg;max(0;EÄd j ); ^d j ) = 1 nä1 (min(eäd j;d i )+ X k2n nfi;jg g i (Nnfj;kg;max(0;EÄd j Äd k ); ^d j;k ) # g i (fi;jg;e 0 ; (d i ;d j )) = 1 2 (min(e0 ;d i )+ X k2n nfi;jg g i (fjg;max(0;e 0 Äd j );d i ) = 1 2 (min(e0 ;d i ) +max(0;e 0 Äd j )) 12
13 g(f1; 2; 3g; 200; (100; 200; 300)) = ( ; ; ) 6=f(f1; 2; 3g; 200; (100; 200; 300)) = (50; 75; 75) 13
14 à(n;v)6=; 8(N;v) ã> 0;å2< N (N;w) w(s) =ãv(s) + P i2så i 8SíN à(n;w) =ãà(n:v) +å à(n;v) =f'(n;v)g 14
15 v x (Nnfjg) =v(n)äx j v x (S) = näjsjä1 jsj v(s)+ nä1 nä1 fv(s[fjg)äx jg8sön v x (;) = 0 v x (S) =v(s) v x (S) =v(snfjg)äx j 15
16 Aumann,R.J. and M. Maschler(1985): Game TheoreticAnalysisof a BankuruptcyProblem from the Talmud, Journal of Economic Theory, 36, pp Davis, M. and M. Maschler(1965): The Kernel of a Cooperative Game, Naval Res. Logist. Quart., 12,pp Driessen, T.S.H.(1988): Cooperative Games, Solutions and Applications, Kluwer Academic Publishers. Driessen, T.S.H.(1991): A Survey of Consistency Properties in Cooperative Game Theory, SIAM Review 33, pp Driessen, T.S.H. and Y. Funaki(1997): Reduced Game Properties of Egalitarian Division Rules for TU-Games, Game Theoretical Applications to Economics and Operations Research, edited by T.Parthasarathy et al., Kluwer Academic Publishers, pp Funaki, Y.(1997): The Dual Axiomatizations of Solutions of Cooperative Games, mimeo. Moulin, H. (1985): The Separability Axiom and Equal-Sharing Methods, Journal of Economic Theory 36, pp O'Neill,B.(1982): A Problem Rights Arbitration from the Talmud, Mathematical Social Science, 2, pp Peleg, B.(1986): On the Reduced Game Property and Its Converse, International Journal of Game Theory 15, pp ; Correction, International Journal of Game Theory 16 (1987), p290. Shapley, L.S.(1953): A Value for n-person Games, in Contributions to the Theory of Games II, H. Kuhn and A.W. Tucker, eds., Ann. Math. Studies 28, Princeton University Press, Princeton, NJ, pp Sobolev, A.I.(1973): The Functional Equations that Give the Payoãs of the Players in an n-person Game, in Advances in Game Theory, E. Vilkas, ed., Izdat. \Mintis" Vilnius, pp (In Russian.) Sobolev, A.I.(1975): The Characterization of Optimality Principles in Cooperative Games by Functional Equations, in Mathematical Methods in the Social Sciences 6, N.N. Vorobev, ed., Vilnius, pp (In Russian.) Young, P.T.(1994): Equity : In Theory and Practice, Princeton UniversityPress, Princeton, NJ. 16
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