Computing and Combinatorics: 19th International Conference, by Susanne Albers (auth.), Ding-Zhu Du, Guochuan Zhang (eds.)

By Susanne Albers (auth.), Ding-Zhu Du, Guochuan Zhang (eds.)

This publication constitutes the refereed lawsuits of the nineteenth overseas convention on Computing and Combinatorics, COCOON 2013, held in Hangzhou, China, in June 2013. The fifty six revised complete papers offered have been rigorously reviewed and chosen from one hundred twenty submissions. there has been a co-organized workshop on discrete algorithms of which eight brief papers have been permitted and a workshop on computational social networks the place 12 papers out of 25 submissions have been accepted.

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Additional info for Computing and Combinatorics: 19th International Conference, COCOON 2013, Hangzhou, China, June 21-23, 2013. Proceedings

Example text

The power of agent i over agent j is given by sij (x) = max S⊆V : i∈S,j∈S v(S) − x(S), (1) 32 K. Georgiou et al. and the prekernel P consists of all allocations x for which sij = sji for all agents i and j. In the following first result, let stable be the projection of the collection of all stable outcomes (C, x) onto the x-space. Similarly, balance is the projection of all balanced outcomes onto the x-space. Theorem 1. Let I be an instance of GNB, and Γ (I) the corresponding cooperative game. Then C = stable, and C ∩ P ⊆ stable ∩ balance.

Similar to [1], we first identify a natural cooperative game Γ (I) associated with a given GNB instance I. Γ (I) has player set V and the value function is defined by letting v(S) = maxC⊆C(S), C feasible c∈C w(c), for all S ⊆ V , where C(S) is the set of contracts contained in the set S. We briefly introduce a few pertinent solution concepts for cooperative games, and refer the reader to [2] for a comprehensive introduction to the topic. The core C of Γ (I) consists of all allocations x ∈ ÊV+ that satisfy x(S) ≥ v(S) for all S ⊆ V , and this inequality is tight for S = V .

Thus, the claim immediately follows from Lemma 2. Theorem 9. 765. Proof. Let s be an SPE for game I. By Lemma 2, it follows that SUM(a(s)) = Ui (a(s)) i∈[k] k−1 4 = k Ui (a(s)) + i=1 k−1 4 ≥ i=1 = = Ui (a(s)) i= k−1 4 +1 k (k − 1)D + 8 i= k−1 4 (i − 1)D 2 +1 k−1 k − 1 (k − 1)D k(k − 1)D + − 4 8 4 4 D 4 k(k − 1) − k−1 4 2 + k−1 −1 4 D 4 k−1 k+1 4 2 D k−1 k+1 k−1 k(k − 1) + − 4 4 2 4 D k−1 k+1 k+2 − > k(k − 1) + 4 4 2 4 17k(k − 1)D . = 64 = Since it clearly holds that SUM(oSUM (I)) ≤ k(k − 1)D, the claim follows.

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