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## The cooperative game theory foundations of network bargaining games,”

Venue: | in International Colloquium on Automata, Languages and Programming, |

Citations: | 10 - 0 self |

### Citations

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(Show Context)
Citation Context ...l as possible to different players. As the nucleolus defined below is a more widely accepted solution concept and achieves complete impartiality, we do not give detailed information about the lexicographic kernel. We simply note that it has been studied in [22, 32, 49], and the result of [22] in addition to Lemma 3 allows us to compute the lexicographic kernel for any general bipartite (or even non-bipartite) bargaining game. 4 Given a graph G(V,E) and a function f : V 7→ Z≥0, an f -factor is a subset F ⊆ E of edges such that each vertex v has exactly f(v) edges incident on it in F . See West [48] for a discussion and for a polynomial-time algorithm to find an f -factor. The approach can be extended to the case where f(v) values are upper bounds on the degrees, and we are interested in finding the maximum-weight solution. 5 In fact, kernel and pre-kernel coincide in our game because ν({i}) = 0 for any i ∈ N—indeed, the two closely-related solution concepts coincide for any zero-monotonic TU-game [34], and our game is one of this class. 5 2.2 A unique outcome None of the solution concepts proposed above are unique. For any given game instance, and any of the solution concepts above, the... |

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Citation Context ...ng game. The work of Chakraborty et al. [7] as well as that of Chakraborty and Kearns [6] considers a related problem, in which there is no capacity constraints on the vertices but agents have non-linear utilities. They explore the existence and computability of the Nash bargaining solution as well as the proportional bargaining solution in their setting. Much recent literature has focused on the computability of various solution concepts in the economics literature. In the non-cooperative game theoretic setting, the complexity of Nash and approximate Nash equilibria has a rich recent history [47, 30, 29, 12, 13, 9, 10, 5, 14]. In cooperative game-theoretic settings, the core of a game defined by a combinatorial optimization problem is fundamentally related to the integrality gap of a natural linear program, as observed in numerous prior work [4, 42, 26, 31, 25]. The computability of the nucleolus has also been studied for some special games [16, 27, 45, 20, 21, 19, 24]. Much of our work leverages existing results in the cooperative game theory literature; these results will be cited as they are used. 2 Preliminaries In the network bargaining game, there is a setN of n agents. For bipartite graphs, the setN is part... |

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3 |
The lexicographic kernel of a cooperative game,
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- 1981
(Show Context)
Citation Context ...ive, it turns out to be similar to the notion of balanced solutions in certain networked settings. A further refinement of this definition is that of lexicographic kernel which is, roughly speaking, a subset of the pre-kernel that lexicographically maximizes the vector of all sij values. In some sense, this definition tries to be as impartial as possible to different players. As the nucleolus defined below is a more widely accepted solution concept and achieves complete impartiality, we do not give detailed information about the lexicographic kernel. We simply note that it has been studied in [22, 32, 49], and the result of [22] in addition to Lemma 3 allows us to compute the lexicographic kernel for any general bipartite (or even non-bipartite) bargaining game. 4 Given a graph G(V,E) and a function f : V 7→ Z≥0, an f -factor is a subset F ⊆ E of edges such that each vertex v has exactly f(v) edges incident on it in F . See West [48] for a discussion and for a polynomial-time algorithm to find an f -factor. The approach can be extended to the case where f(v) values are upper bounds on the degrees, and we are interested in finding the maximum-weight solution. 5 In fact, kernel and pre-kernel co... |

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