Core 與 Dcore 的等價條件

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(1)國立屏東教育大學應用數學系碩士班碩士論文 Department of Applied Mathematics National Pintung University of Education Master’s Thesis. Core 與 Dcore 的等價條件 Alternative coincidences between the core and the dominance core. 指導教授：廖于賢博士 Advisor:Dr. Yu-Hsien Liao 研究生:黃子誼 Student: Zih-Yi Huang. 中華民國 102 年 5 月 May ,2013.

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(3) 致謝很感謝這兩年系上老師的指導，尤其是在論文方面認真指導我的廖于賢老師，不厭其煩的將其所知道的學識傳授給我們。其次感謝詹勳國教授願意撥空審閱論文，並在口試時給予論文修正的寶貴意見。最後感謝應用數學系在碩士學習的期間，給予許多的幫助，不論是學術上或是論文指導，讓我能順利的拿到碩士學位. 子誼僅至於國立屏東教育大學應用數學系民國 102 年 5 月.

(4) 摘要在這篇論文當中，我們首先研讀一些先前有關於 CORE 與 DCORE 的結果，與先前不同的理論，我們有找到 CORE 跟 DCORE 另外一個等價關係關鍵字：core、dcore、賽局、等價條件.

(5) Abstract In this paper we first study some pre-existing results of the core and the dominance core. Different from pre-existing results, we further investigate alternative coincidences between the core and the dominance core. Keywords : core、dcore、game theory、coincidences.

(6) Contents Abstract. 1. 1 Introduction. 2. 2 Preliminaries. 3. 3 Dominance core and alternative coincidences. 4. References. 8.

(7) Alternative coincidences between the core and the dominance core∗ Zih-Yi Huang† April 24, 2013 Abstract. In this paper, we first study some pre-existing results of the core and the dominance core. Different from these pre-existing results, we further investigate alternative coincidences between the core and the dominance core. Keywords: The core, the dominance core, coincidences. AMS classification codes: 91A. ∗. The advisor: Yu-Hsien Liao Department of Applied Mathematics, National Pingtung University of Education, Pingtung 900, Taiwan. email: [email protected] †. 1.

(8) 1. Introduction. In order to investigate allocation rules, many solution concepts and related results have been proposed in game theory, such the equal allocation of nonseparable costs (EANSC) (Ransmeier, 1942), the Shapley value (1953) and so on. The core is, perhaps, the most intuitive solution concept in game theory. In general, there are two main representations of the core in literature: 1. The core is a set of the payoff vectors satisfying efficiency and coalitional rationality. The efficiency asserts that all utilities produced by all players should be allocated by all players completely. The coalitional rationality asserts that the sum of the payoffs assigned to all players of a coalition should be greater than the utility produced by this coalition. 2. For all proper balanced games, the core coincides with the set of all undominated imputations. Some more related results of the core could be founded in Bondareva (1963), Shapley (1967), Peleg (1985,1986,1989), Tadenuma (1992), Serrano and Volij (1998), Voorneveld and Nouweland (1998), Hwang and Sudh¨olter (2001) and so on. Different from previous results, we first study the domination among all payoff vectors of imputation sets and the dominance core. Further, we investigate alternative coincidences between the core and the dominance core.. 2.

(9) 2. Preliminaries. 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 with v(∅) = 0. Denote the class of all TU games by G. Given (N, v) ∈ G, x ∈ RN and P S ⊆ N . We denote xS ∈ RS to be the restriction of x to S, and x(S) = i∈S xi . A payoff vector of (N, v) is a vector x = (xi )i∈N ∈ RN . Then a payoff vector x of (N, v) ∈ G is • efficient (EFF) if x(N ) = v(N ). . . • individually rational (IR) if for all i ∈ N , xi ≥ v {i} . • coalitionally rational (CR) if for all S ⊆ N , x(S) ≥ v(S). Moreover, x is an imputation of (N, v) if it is EFF and IR. The set of feasible payoff vectors of (N, v) is denoted by X ∗ (N, v) = {x ∈ RN | x(N ) ≤ v(N )}, whereas X(N, v) = { x ∈ RN | x is EFF } is the set of preimputations of (N, v) and the set of imputations of (N, v) is denoted by I(N, v). A solution on G is a function σ which associates with each game (N, v) ∈ G an element σ(N, v) which associates with each (N, v) ∈ G a subset σ(N, v) of X ∗ (N, v). The core of a TU game (N, v) is as follows. Definition 1 • The core C(N, v) of (N, v) ∈ G consists of all x ∈ X(N, v) which satisfy CR, i.e., C(N, v) = { x ∈ X(N, v) | x is EFF and CR }. • A TU game (N, v) is balanced if for all maps λ : 2N → R+ satisfying for all i ∈ N , X λ(S) = 1, S∈2N. i∈S. it holds that. P. λ(S) · v(S) ≤ v(N ). Let Gb denote the set of all. S∈2N. balanced games. Remark 1 Let (N, v) ∈ G. It is shown that the core C(N, v) is nonempty if and only if (N, v) is balanced. 3.

(10) 3. Dominance core and alternative coincidences. In this section, we study the dominance core and related results. Further, we investigate alternative coincidences between the core and the dominance core. Let (N, v) ∈ G, x, y ∈ I(N, v) and S ⊆ N . y dominates x via S, denoted by y domS x, if y(S) ≤ v(S) and yi > xi for all i ∈ S. We say that y dominates x if there exists S ⊆ N such that y domS x. Definition 2 The dominance core DC(N, v) of (N, v) ∈ Γ consists of all x ∈ I(N, v) for which there exists no y ∈ I(N, v) such that y dominates x. A game (N, v) ∈ G is said to be a proper game if for all S, T ⊆ N with S ∩ T 6= ∅, v(S ∪ T ) ≥ v(S) + v(T ). Remark 2 It is shown that C(N, v) = DC(N, v) for all balanced proper game (N, v). Different from proposed result related to Remark 2, we investigate alternative coincidences between the core and the dominance core. Lemma 1 For all (N, v) ∈ G, C(N, v) ⊆ DC(N, v). Proof. Assume, on the contrary, that there exists (N, v) ∈ G and x ∈ C(N, v) such that x ∈ / DC(N, v). Hence, there exists y ∈ I(N, v) and S ⊆ N such that y domS x. Then, v(S) ≥ y(S) > x(S) ≥ v(S), which clearly gives a contradiction. Hence, x ∈ DC(N, v). A game (N, v) is called zero-normalized if for all i ∈ N , v({i}) = 0. Let (N, v) ∈ G be zero-normalized and x ∈ I(N, v), it is easy to see that xi ≥ 0 for all i ∈ N . The normalization of (N, v), denoted by (N, v0 ), is defined by X v0 (S) = v(S) − v {i} . i∈S. for all S ⊆ N . Subsequently, we investigate some relations for several payoff vectors between a game and its normalization. These relations are very useful in the proofs of our main results. 4.

(11) Lemma 2 Let (N, v) ∈ G, (N, v0 ) be the normalization of (N, v) and x be a payoff vector of (N, v). Define a vector y ∈ RN by yi = xi − v({i}) for all i ∈ N . Then we have that 1. x ∈ X(N, v) if and only if y ∈ X(N, v0 ) 2. x ∈ I(N, v) if and only if y ∈ I(N, v0 ) 3. x ∈ C(N, v) if and only if y ∈ C(N, v0 ) 4. x ∈ DC(N, v) if and only if y ∈ DC(N, v0 ) Proof. Let (N, v) ∈ G, (N, v0 ) be the normalization of (N, v) and x be a payoff vector of (N, v). Define a vector y ∈ RN by yi = xi − v({i}) for all i ∈ N . To prove (1), P yi = v0 (N ) y ∈ X(N, v0 ) ⇔ i∈N h i P P ⇔ xi − v {i} = v(N ) − v {i} i∈N i∈N P ⇔ xi = v(N ) i∈N. ⇔ x ∈ X(N, v). To prove (2), . y ∈ I(N, v0 ) ⇔ y ∈ X(N, v0 ), yi ≥ v0 {i} for all i ∈ N ⇔ x ∈ X(N, v), yi ≥ v0 {i} for all i ∈ N ⇔ x ∈ X(N, v), xi − v {i} ≥ v {i} − v {i} for all i ∈ N ⇔ x ∈ X(N, v), xi ≥ v {i} for all i ∈ N ⇔ x ∈ I(N, v). To prove (3), y ∈ C(N, v0 ) ⇔ y ∈ X(N, v0 ), P y(S) ≥ v0 (S) ⇔ x ∈ X(N, v), yi ≥ v0 (S) for all S ⊆ N i∈S h i P P xi − v {i} ≥ v(S) − v {i} for all S ⊆ N ⇔ x ∈ X(N, v), i∈S i∈S P ⇔ x ∈ X(N, v), xi ≥ v(S) for all S ⊆ N i∈S. ⇔ x ∈ X(N, v), x(S) ≥ v(S) for all S ⊆ N ⇔ x ∈ C(N, v).. 5.

(12) To prove (4), y ∈ DC(N, v0 ) ⇔ y ∈ I(N, v0 ), there exists no z ∈ I(N, v0 ) such that z dominates y. ⇔ x ∈ I(N, v), there exists no w ∈ I(N, v) such that w dominates x. ⇔ x ∈ DC(N, v).. Finally, we provide alternative equivalent coincidences between the core and the dominance core. Theorem 1 Let (N, v) be a game with DC(N, v) 6= ∅. Then the core C(N, v) coincides with the dominance core DC(N, v) if and only if the normalization (N, v0 ) of (N, v) satisfies v0 (N ) ≥ v0 (S) for all S ⊆ N . Proof. By Lemma 2, it suffices to prove this theorem for zeronormalized games. Assume that (N, v) ∈ G is zero-normalized. Assume that C(N, v) = DC(N, v), DC(N, v) 6= ∅ and let x ∈ C(N, v). Since x ∈ C(N, v) and xi ≥ 0 for all i ∈ N , then for all S ⊆ N , X X xi ≥ x(S) ≥ v(S). xi ≥ v(N ) = x(N ) = i∈S. i∈N. Now assume that for all S ⊆ N , v(N ) ≥ v(S). By Lemma 1, it suffices to prove that for all x ∈ I(N, v) \ C(N, v), x ∈ / DC(N, v). Let x ∈ I(N, v) \ C(N, v). Since x ∈ I(N, v) and x ∈ / C(N, v), there exists S ⊆ N with |S| > 1 such that v(S) > x(S). Define a vector y ∈ RN by for all i ∈ N ,  v(S)−x(S)  if i ∈ S,  xi + |S| yi =   v(N )−v(S) if i ∈ / S. |N \S| By definition of y, y(N ) =. P. yi. i∈N P. P yi + yi i∈S i∈N \S i Ph P h v(N )−v(S) i xi + v(S)−x(S) + = |S| |N \S| i∈S i∈N \S h i h i P = xi + v(S) − x(S) + v(N ) − v(S) i∈S h i h i = x(S) + v(S) − x(S) + v(N ) − v(S) = v(N ).. =. 6.

(13) Hence, y ∈ X(N, v). Since for all i ∈ N , xi ≥ 0 and v(S) > x(S), we have that for all i ∈ S, yi ≥ 0. Since v(N ) ≥ v(S), we have that for all i ∈ / S, yi ≥ 0. Thus, for all i ∈ N , yi ≥ 0 = v({i}). Hence, y is individually rational in (N, v). Since y is efficient and individually rational in (N, v), y ∈ I(N, v). By definition of y, P y(S) = yi i∈S h i P v(S)−x(S) = xi + |S| i∈S h i P = xi + v(S) − x(S) i∈S h i = x(S) + v(S) − x(S) = v(S). Hence, y(S) = v(S). Clearly, by definition of y, it is easy to verify that yi > xi for all i ∈ S, i.e., y(S) > x(S). Therefore, 1. y ∈ I(N, v), 2. y(S) ≤ v(S), 3. y(S) > x(S), 4. yi > xi for all i ∈ S. By items 1, 2, 3, 4 and definition of the domination for payoff vectors, it is clear to verify that y domS x. Therefore, x ∈ / DC(N, v). Theorem 2 For all (N, v) ∈ Gb , C(N, v) = DC(N, v). Proof. By Lemma 2, it suffices to prove this theorem for zeronormalized games. Assume that (N, v) is zero-normalized. By the proof of Theorem 1, we have that C(N, v) 6= ∅ implies that for all S ⊆ N , v(S) ≤ v(N ). Since C(N, v) 6= ∅ and C(N, v) ⊆ DC(N, v), DC(N, v) 6= ∅ by Lemma 1. By Theorem 1, C(N, v) = DC(N, v). Remark 3 The result of Theorem 2 has been proposed. Here we provide alternative proof of result.. 7.

(14) References [1] Bondareva O (1963) Some applications of linear programming methods to theory of cooperative games. Problemy Kybernetiki 10:119139 (in Russian) [2] Hwang YA, Sudh¨olter P(2001) Axiomatizations of the core on the universal domain and other natural domains. International Journal of Game Theory 29:597-623 [3] Peleg B (1985) An axiomatization of the core of cooperative games without side payments. Journal of Mathematical Economics 14: 203214 [4] Peleg B (1986) On the reduced game property and its converse. International Journal of Game Theory 15:187-200 [5] Peleg B (1989) An axiomatization of the core of market games. Mathematics of Operations Research 14:448-456 [6] Ransmeier JS (1942) The Tennessee valley authority. Vanderbilt University Press, Nashville [7] Serrano R, Volij O (1998) Axiomatizations of neoclassical concepts for economies. Journal of Mathematical Economics 30:87-108 [8] Shapley LS (1953) A value for n-person game. In: Kuhn HW, Tucker AW(eds.) Contributions to the Theory of Games II, Annals of Mathematics Studies 28, Princeton University Press, Princeton, 307-317 [9] Shapley LS (1967) On balanced sets and cores. Naval Research Logistics Quarterly 14:453-460 [10] Tadenuma K (1992) Reduced games, consistency, and the core. International Journal of Game Theory 20:325-334 [11] Voorneveld M, van den Nouweland A (1998) A new axiomatization of the core of games with transferable utility.. 8.

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