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1.1. A VALUE FOR N-PERSON GAMES(L.S.SHAPLEY 1952) 4 Proof. by Axiom3. Definition 1.14. v and w are strategically equivalent iff w = cv + a, where c is a positive constant and a is an additive set-function on U with finite carrier. Proposition 1.15. v and w are strategically equivalent then φ i (w) = cφ i (v) + a(i) (all i ∈ U) Proof. by Axiom3 and propsition1.11 applied to the inessential game a, φ i is linear and homogeneous in v. Definition 1.16. v is constant-sum iff v(S)+v(U−S) = v(U) U). (all S ⊆ Proposition 1.17. If v is constant-sum then its <strong>value</strong> is given by the formula: ∑ φ i [v] = 2[ r n (s)v(S)] − v(N) (all i ∈ N), S⊆N,i∈S where N is any finite carrier of v. Proof. ∑ φ i [v] = r n (s)v(S) − ∑ r n (t + 1)v(T ) = = 2[ S⊆N,i∈S ∑ S⊆N,i∈S ∑ r n (s)v(S) − r n (s)v(S)] − T ⊆N,i/∈T ∑ S⊆N,i∈S ∑ S⊆N,i∈S S⊆N,i∈S r n (n − s + 1)[v(N) − v(S)] v(N). by propsition1.11 proof. Definition 1.18. the centroid of imputation set by θ: θ i = v(i) + 1 n [v(N) − ∑ v(j)]. j∈N Example1 For two-person games, three-person constant-sum games, and inessential games, we have φ = θ. 1.two-person games by theorem φ 1 = ∑ 1 inS⊆1,2 r 2(s)[v(S) − v(S − 1)] = r 2 (1)(v(1) − v(∅)) + r 2 (1)(v(1, 2) − v(2)) = 1 2 v(1) + 1 2 v(1, 2) − 1 2v(2) = v(1) +
1.1. A VALUE FOR N-PERSON GAMES(L.S.SHAPLEY 1952) 5 1 2 (v(1, 2) − v(1) − v(2)) = θ 1. 2.three-person constant-sum games by proposition1.17 φ 1 = 2[ ∑ 1∈S⊆1,2,3 r 3(s)v(S)]−v(1, 2, 3) = 2( 1 3 v(1)+ 1 6 v(1, 2) + 1 6 v(1, 3) + 1 3 v(1, 2, 3)) − v(1, 2, 3) = 2 3 v(1) + 1 3v(1, 2) + 1 3 v(1, 3) − 1 3 v(1, 2, 3) = 2 3 v(1) + 1 3 (v(1, 2, 3) − v(3)) + 1 3 (v(1, 2, 3) − v(2)) − 1 3 v(1, 2, 3) = v(1) + 1 3 (v(1, 2, 3) − v(1) − v(2) − v(3)) = θ 1. 3.inessential games φ 1 = v(1) = θ 1 Example2 Example3 Quota Games The quota games are characterized by the existence of constants w i satisfying { wi + w j = v(i, j) (all i, j ∈ N, i ≠ j) ∑ N w i = v(N). For n=3, we have φ − θ = w − θ . 2 by theorem φ 1 = 1 3 v(1) + 1 6 (v(1, 2) − v(2)) + 1 6 (v(1, 3) − v(3)) + 1 3 (v(1, 2, 3) − v(2, 3)) by definition of centroid θ 1 = 2 3 v(1) + 1 3 v(1, 2, 3) − 1 3 v(2) − 1 3 v(3) φ − θ = − 1 3 v(1) + 1 6 v(2) + 1 6 v(3) + 1 6 v(1, 2) + 1 6 v(1, 3) = − 1 3 v(1) + 1 6 v(2) + 1 6 v(3) + 1 3 w 1 − 1 6 w 2 − 1 6 w 3) = 1 2 w 1 − 1 2 (v(1) + 1 3v(1, 2, 3) − 1 3 v(1) − 1 3 v(2) − 1 3 v(3)) = w 1−θ 1 2 Example4 Quota Games All four-person constant-sum games are quota games and φ − θ = w − θ . 3 Example5 Weighted Majority Games The weighted majority games are characterized by the existence of ”weights” w i s.t. never ∑ S w i = ∑ N−S w i, and s.t. { v(S) = n − s if ∑ S w i > ∑ N−S w i v(S) = −s if ∑ S w i < ∑ N−S w i. for the game [2, 2, 2, 1, 1, 1] φ 1 = ∑ 1∈S⊆N r 6(s)(v(S) − v(S − 1)) = 1 1 6 (v(1) − v(0)) + 30 (v(1, 2) −
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