The Pre-ANSC及其相關結果

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(1)國立屏東大學應用數學系碩士班碩士論文 Department of Applied Mathematics National Pingtung University Master’s Thesis. The Pre-ANSC 及其相關結果 The Pre-ANSC and related results. 指導教授：廖于賢博士 Advisor: Dr. LIAO-YU HSIEN 研究生：吳家賢 Student: CHIA-HSIEN WU. 中. 華. 民. 國. 107. June 2018. 年. 6. 月.

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(3) Abstract By considering the marginal contributions for the players on grand coalitions, we introduce a different extension of the equal allocation of nonseparable costs (EANSC) due to Ransmeier(1942). Inspired by the works of Hart and Mas-Colell (1989),Moulin (1985) and Wu and Liao (2013), a specific reduction and related consistency are adopted to characterize this extended solution. Chia-Hsien. Keywords: The EANSC, marginal contribution, reduction, consistency. 中文摘要考慮最大團體的邊際貢獻，我們對於 EANSC 提不一樣擴張的解，藉由 Hart 和 Mas-Colell 和 Moulin 這三個人的研究工作，我們提出不一樣的退化組，還有相關的一致性來刻畫這個擴張的解。. 關鍵字：The EANSC，邊際貢獻，退化局，一致性. Ⅰ.

(4) Contents. Abstract ........................................................................ Ⅰ Contents ....................................................................... Ⅱ Introduction...................................................................... 2 A modification of the EANSC ......................................... 3 The Pre-ANSC and related axiomatiza-tions ................. 5 Some comparisions ...................................................... 10 References .................................................................... 12. Ⅱ.

(5) The Pre-ANSC and related results∗ Chia-Hsien Wu† April 26, 2018. Abstract. By considering the marginal contributions for the players on grand coalitions, we introduce a different extension of the equal allocation of nonseparable costs (EANSC) due to Ransmeier [12]. Inspired by the works of Hart, et al. [4], Moulin [9] and Wu, et al. [16], a specific reduction and related consistency are adopted to characterize this extended solution. Keywords: The EANSC, marginal contribution, reduction, consistency. AMS classification codes: 91A, 91B. ∗. The advisor: Yu-Hsien Liao Department of Applied Mathematics, National Pingtung University, Pingtung 900, Taiwan. email: [email protected] †. 1.

(6) 1. Introduction. Consistency, initially proposed by Harsanyi [3] under the name of bilateral equilibrium, is a important property of solutions. If a solution violates consistency property, then a subgroup of agents might not respect the original compromise but revise the payoff distribution within the subgroup. Based on the notions of reduced game, this property has been discussed in different classes of problems always. Many solutions have been characterized by applying some reductions and related consistency properties, such as the core, the EANSC, the kernel, the nucleolus, the prekernel, the prenucleolus, the Shapley value, and so on. Several reductions and related results could be founded in Davis, et al. [2], Hart, et al. [4], Hwang, et al. [6], Moulin [9], Peleg [10, 11], Sobolev [14], Tadenuma [15], and so on. The equal allocation of nonseparable costs (EANSC, Ransmeier [12]) is a well-known solution concept in cooperative game theory. Moulin [9] introduced a specific reduction and related consistency to characterize the EANSC. Later, Hwang [5] extended the EANSC to non-transferableutility (NTU) games, and proposed several results of the EANSC in the frameworks of NTU game and TU game respectively. Hwang, et al. [7] characterized the EANSC by applying the reduction due to Moulin [9], and provided a dynamic approach for the EANSC by applying excess function. Inspired by the dynamic result due to Maschler, et al. [8], Chou, et al. [1] defined alternative reduction to provide a dynamics approach for the EANSC. Wu, et al. [16] adopted the weighted functions to propose the weighted EANSC (WEANSC). Here we build on the previous results. Two main results of this note are as follows. • First, we propose the Pre-ANSC based on the marginal contributions for the players on grand coalitions. Further, we also study the reduction and related consistency introduced by Moulin [9]. • Inspired by Hart, et al. [4], Moulin [9] and Wu, et al. [16], we characterize the Pre-ANSC by means of related properties of consistency and two-person standardness. Finally, some comparisons are also proposed. 2.

(7) 2. A modification of the EANSC. Let U be a non-empty and finite set of players. A coalition is a nonempty subset of U . A coalitional game with transferable utility (TU game) is a pair (N, v) where N is a coalition and v is a mapping such that v : 2N −→ IR and v(∅) = 0. Denote the class of all TU games by G. A solution on G is a function ψ which associates with each game (N, v) ∈ G an element ψ(N, v) of IRN . A solution ψ satisfies efficiency P (EFF) if ψi (N, v) = v(N ) for all (N, v) ∈ G. i∈N. Definition 1 The equal allocation of nonseparable costs (EANSC, Ransmeier [12]), denoted by β, is the solution on G which associates with (N, v) ∈ G and each player i ∈ N the value X 1 βi (N, v) = βi (N, v) + · v(N ) − βk (N, v) , (1) |N | k∈N where βi (N, v) = v(N ) − v(N \ {i}) is the marginal contribution for player i in N . Let w : U → R+ be a positive function, then w is called a weight function. Given (N, v) ∈ G and a weight function w, we define |S|w = P i∈S w(i) for all S ⊆ N . Based on the weight function, Wu, et al. [16] proposed a weighted extension of the EANSC as follows. Definition 2 Let w be a weight function. The weighted EANSC (WEANSC, Wu and Liao [16]), β w , is the solution on G which associates with (N, v) ∈ G and all players i ∈ N the value X w(i) βiw (N, v) = βi (N, v) + · v(N ) − βk (N, v) . (2) |N |w k∈N By the definition of the WEANSC, all players receive their marginal contributions in N firstly, and further allocate the remaining utilities by applying weights proportionally. The WEANSC could be applied in many fields such as economics, political sciences, accounting and so on. In economic models, these weights could be treated as parameters to modify the differences among different players. 3.

(8) Inspired by the notions of the EANSC and the WEANSC, we would like to propose a different extension of the EANSC as follows. P Definition 3 Let G∗ = {(N, v) ∈ G| k∈N βk (N, v) 6= 0}. The PreANSC, η p , is the solution on G∗ which associates with (N, v) ∈ G∗ and all players i ∈ N the value X βi (N, v) ηip (N, v) = βi (N, v) + P · v(N ) − βk (N, v) . βk (N, v) k∈N. (3). k∈N. By the definition of the Pre-ANSC, all players receive their marginal contributions in N firstly, and further allocate the remaining utilities by applying marginal contributions proportionally.. 4.

(9) 3. The Pre-ANSC and related axiomatizations. In this section, we firstly show that the Pre-ANSC satisfies some axioms, and further provide an axiomatization of the Pre-ANSC. Lemma 1 The Pre-ANSC satisfies EFF on G∗ . Proof. Let (N, v) ∈ G∗ . By Definition 3, P p ηi (N, v) i∈N h i P P = βi (N, v) + Pβiβ(N,v) · v(N ) − β (N, v) k k (N,v) i∈N k∈N k∈N P P P Pβi (N,v) β (N, v) · v(N ) − βi (N, v) + = k βk (N,v) i∈N. =. P. i∈N P. βi (N, v) +. i∈N. k∈N. k∈N. βi (N,v). i∈N. P. βk (N,v). P · v(N ) − βk (N, v) k∈N. k∈N. = v(N ). Hence, the Pre-ANSC satisfies EFF on G∗ . Moulin [9] introduced a notion of reduced game and related consistency as follows. Definition 4 Given a solution ψ, (N, v) ∈ G and S ⊆ N . The Mreduced game (S, vS,ψ ) with respect to ψ and S is defined by for all T ⊆ S,   0 , if T = ∅, P vS,ψ (T ) = ψi (N, v) , otherwise.  v(T ∪ (N \ S)) − i∈N \S. For the M-reduced game, there is a corresponding consistency as follows. A solution ψ satisfies M-consistency (MCon) if ψi (S, vS,ψ ) = ψi (N, v) for all (N, v) ∈ G, for all S ⊆ N and for all i ∈ S. Moulin [9] characterized the EANSC by means of the M-consistency property. Wu, et al. [16] also characterized the WEANSC by means of P the M-consistency property. It is easy to check that k∈S βk (N, v) = 0 for some (N, v) ∈ G and for some S ⊆ N , i.e., η p (S, vS,ψ ) doesn’t exist for some (N, v) ∈ G and for some S ⊆ N . Therefore, we consider 5.

(10) the resilient consistency as follows. A solution ψ satisfies resilientconsistency (RCon) if (S, vS,ψ ) and ψ(S, vS,ψ ) exist for some (N, v) ∈ G and for some S ⊆ N , it holds that ψi (S, vS,ψ ) = ψi (N, v) for all i ∈ S. Lemma 2 The Pre-ANSC η p satisfies RCon on G∗ . Proof. Let (N, v) ∈ G∗ and S ⊆ N . Assume that (S, vS,ηp ) and η p (S, vS,ηp ) exist. If S = {i} for some i ∈ N , then by EFF of η p , ηip (S, vS,ηp ) = vS,ηp (S) = v(N ) −. X. ηkp (N, v) = ηip (N, v).. k6=i. Assume that |N | ≥ 2 and |S| ≥ 2. For all i ∈ S,. = = = =. β (S, vS,ηp ) i vS,ηp (S) − vS,ηp (S \ {i}) P p P p v(N ) − ηi (N, v) − v(N \ {i}) + ηi (N, v) i∈N \S i∈N \S v(N ) − v(N \ {i}) βi (N, v).. By equations (3), (4) and Definition 4, for all i ∈ S, ηip (S, vS,ηp ) = βi (S, vS,ηp ) +. βi (S,vS,ηp ) P βi (S,vS,ηp ). k∈S. = βi (N, v) +. Pβi (N,v) βk (N,v). P βk (S, vS,ηp ) · vS,ηp (S) − P k∈S βk (N, v) · vS,ηp (S) − k∈S. k∈S. ( by equation (4) ) P P p = βi (N, v) + Pβiβ(N,v) · v(N ) − η (N, v) − β (N, v) k k (N,v) k k∈N \S. k∈S. k∈S. ( by Definition 4 ) P p P = βi (N, v) + Pβiβ(N,v) · ηk (N, v) − βk (N, v) k (N,v) k∈S. k∈S. k∈S. ( by EFF of η p ) P. = βi (N, v) +. Pβi (N,v) βk (N,v) k∈S. ·. h. βk (N,v). k∈S. P. βt (N,v). i P · v(N ) − βt (N, v). t∈N. t∈N. ( by equation (3) ) P = βi (N, v) + Pβiβ(N,v) · v(N ) − β (N, v) t t (N,v) t∈N. t∈N. =. ηip (N, v). 6. (4).

(11) Hence, η p satisfies RCon on G∗ . Inspired by Hart, et al. [4] and Wu, et al. [16], we characterize the Pre-ANSC by applying the properties of resilient-consistency and modified standard for two-person games. Let ψ be a solution on G∗ . • Revised standard for two-person games (RST): For all (N, v) ∈ G∗ with |N | ≤ 2, ψ(N, v) = η p (N, v). Lemma 3 Let ψ be an solution on G∗ . If ψ satisfies RST and RCon, then it also satisfies EFF on G∗ . Proof. Suppose ψ satisfies RST and RCon. Let (N, v) ∈ G∗ . If |N | ≤ 2, it is trivial that ψ satisfies EFF by RST of ψ. Assume that |N | > 2. Since (N, v) ∈ G∗ , there exists i ∈ N such that βi (N, v) 6= 0. Thus, ({i}, v{i},ψ ) and ψ({i}, v{i},ψ ) exist. By the definition of v{i},ψ , X ψk (N, v). (5) ψi ({i}, v{i},ψ ) = v{i},ψ ({i}) = v(N ) − k6=i. By RCon of ψ, ψi ({i}, v{i},ψ ) = ψi (N, v).. (6). By equations (5) and (6), v(N ) =. X. ψk (N, v).. k∈N. Hence, ψ satisfies EFF on G∗ . Theorem 1 A solution ψ on G∗ satisfies RST and RCon if and only if ψ = η p on G∗ . Proof. By Lemma 2, η p satisfies RCon on G∗ . Clearly, η p satisfies RST on G∗ . To prove uniqueness, suppose ψ satisfies RST and RCon. By Lemma 3, ψ satisfies EFF. Let (N, v) ∈ G∗ . If |N | ≤ 2, it is trivial that ψ(N, v) = η p (N, v) by RST. Assume that |N | > 2. Let i ∈ N . Assume that there. 7.

(12) exists j ∈ N \ {i} such that βi (N, v) + βj (N, v) 6= 0. Without loss of generality, we can assume that βi (N, v) 6= 0. Let S = {i, j}. Then ψi (N, v) − ηip (N, v) = ψi (S, vS,ψ ) − ηip (S, vS,ηp ) (by RCon of ψ, η p ) = ηip (S, vS,ψ ) − ηiph(S, vS,ηp ) (by RST of ψ, η p ) i =. (7). βi (N,v) · vS,ψ (S) + vS,ψ ({i}) − vS,ψ ({j}) βi (N,v)+βj (N,v) i h βi (N,v) − βi (N,v)+βj (N,v) · vS,ηp (S) + vS,ηp ({i}) − vS,ηp ({j}) .. By the definitions of vS,ψ and vS,ηp , h i vS,ψ ({i}) − vS,ψ ({j}) = v(N \ {j}) − v(N \ {i}) = vS,ηp ({i}) − vS,ηp ({j}).. (8). By equations (7), (8), the definition of vS,ψ and EFF of ψ and η p , = =. ψi (N, v) − ηip (N, h v) βi (N,v) βi (N,v)+βj (N,v) βi (N,v) βi (N,v)+βj (N,v). i · vS,ψ (S) − vS,ηp (S) h i p p · ψi (N, v) + ψj (N, v) − ηi (N, v) − ηj (N, v) .. That is, =. βj (N,v) βi (N,v)+βj (N,v) βi (N,v) βi (N,v)+βj (N,v). h · ψi (N, v) − ηip (N, v) h i · ψj (N, v) − ηjp (N, v) .. By EFF of ψ and η p , 0 = v(N ) − v(N ) P = ψj (N, v) − ηjp (N, v) j∈N P βj (N,v) = ψi (N, v) − ηip (N, v) βi (N,v) j∈N P. =. βj (N,v). j∈N. βi (N,v). · ψi (N, v) − ηip (N, v) .. Hence, ψi (N, v) = ηip (N, v). Assume that βi (N, v) + βj (N, v) = 0 for all j ∈ N \ {i}. Since (N, v) ∈ G∗ , βj (N, v) + βt (N, v) 6= 0 for all j, t ∈ N \ {i}. Similar to above process, ψj (N, v) = ηjp (N, v) for all j ∈ N \ {i}. So, P ψi (N, v) = v(N ) − ψj (N, v) (By EFF of ψ) j∈N \{i} P = v(N ) − ηjp (N, v) =. j∈N \{i} p ηi (N, v).. 8. (By EFF of η p ).

(13) The proof is completed. The following examples are to show that each of the axioms used in Theorem 1 is logically independent of the remaining axioms. Example 1 Define a solution ψ to be that for all (N, v) ∈ G∗ and for all i ∈ N , ψi (N, v) = 0. Clearly, ψ satisfies RCon, but it violates RST. Example 2 Define a solution ψ to be that for all (N, v) ∈ G∗ and for all i ∈ N , ( , if |N | ≤ 2, ηip (N, v) ψi (N, v) = p ηi (N, v) − ε , otherwise. where ε ∈ R \ {0}. Clearly, ψ satisfies RST, but it violates RCon.. 9.

(14) 4. Some comparisons. Based on the viewpoints of the axiomatizations, we provide some comparisons among the EANSC, the WEANSC and the Pre-ANSC. To state the main results of this section, some more axioms are needed. A solution ψ satisfies symmetry (SYM) if for all (N, v) ∈ G and for all i, j ∈ N , v(S ∪ {i}) = v(S ∪ {j}) for all S ⊆ N \ {i, j} implies ψi (N, v) = ψj (N, v). A solution ψ satisfies covariance (COV) if for all (N, v), (N, w) ∈ G, for all α ∈ R, α > 0 and for all b ∈ RN , P ψ(N, w) = α · ψ(N, v) + b, where w(S) = α · v(S) + i∈S bi for all S ⊆ N . Remark 1 Moulin [9] showed that the EANSC is the only solution satisfying EFF, SYM, COV and MCon. It is easy to check that the WEANSC satisfies EFF, COV and MCon, but it violates SYM. In Section 3, it is shown that the Pre-ANSC satisfies EFF and RCon, but it violates MCon. Inspired by Remark 1, we would like to know that whether the PreANSC could be characterized by EFF, SYM, COV and RCon. Lemma 4 The Pre-ANSC satisfies SYM on G∗ . Proof. Let (N, v) ∈ G∗ . If |N | = 1, then the proof is trivial. Assume that |N | ≥ 2 and v(S ∪ {i}) = v(S ∪ {j}) for all i, j ∈ N and for all S ⊆ N \ {i, j}. So we have that βi (N, v) = = = = =. v(N ) − v(N \ {i}) v(N ) − v (N \ {i, j}) ∪ {j} v(N ) − v (N \ {i, j}) ∪ {i} v(N ) − v(N \ {j}) βj (N, v).. By Definition 3 and equation (9), Pβi (N,v) βk (N,v). P · v(N ) − βk (N, v) k∈N k∈N P βj (N,v) = βj (N, v) + P βk (N,v) · v(N ) − βk (N, v). ηiP (N, v) = βi (N, v) +. k∈N. k∈N. = ηjP (N, v). Hence, the Pre-ANSC satisfies symmetry on G∗ . 10. (9).

(15) Example 3 Let (N, v), (N, w) ∈ G with N = {i, j} and w(S) = v(S) + P k∈S bk for all S ⊆ N , where b = (bi , bj ) = (1, 2). Thus, w({i, j}) = v({i, j}) + 1 + 2, w({i}) = v({i}) + 1, w({j}) = v({j}) + 2 and w(∅) = v(∅) = 0. For all i ∈ N , βi (N, w) = w(N ) − w(N \ {i}) P = v(N ) + bk − v(N \ {i}) −. P. bk (10). k∈N \{i}. k∈N. = [v(N ) − v(N \ {i})] + bi = βi (N, v) + 1 Also, βj (N, w) = w(N ) − w(N \ {j}) P bk − v(N \ {j}) − = v(N ) +. P k∈N \{j}. k∈N. bk (11). = [v(N ) − v(N \ {j})] + bj = βj (N, v) + 2 By Definition 3 and equations (10), (11), ηiP (N, w) = βi (N, w) +. Pβi (N,w) βk (N,w). P · w(N ) − βk (N, w) k∈N k∈N P P = βi (N, v) + 1 + 3+βiP(N,v)+1 · v(N ) + bt − [βk (N, v) + bk ] βk (N,v) t∈N k∈N k∈N P P P = βi (N, v) + 1 + 3+βiP(N,v)+1 · v(N ) + b − βk (N, v) − bk t βk (N,v) t∈N k∈N k∈N k∈N P = βi (N, v) + 3+βiP(N,v)+1 · v(N ) − β (N, v) + 1. k βk (N,v) k∈N. k∈N. Similarly, ηjP (N, w) = βj (N, v) +. X βj (N, v) + 2 P · v(N ) − βk (N, v) + 2. 3+ βk (N, v) k∈N k∈N. In general, η P (N, w) 6= η P (N, v) + b, i.e., the Pre-ANSC violates covariance on G∗ .. 11.

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