#### DMCA

## On the Structure of Synergies in Cooperative Games

### Citations

2791 | Computational Complexity
- Papadimitriou
- 1994
(Show Context)
Citation Context ...d, while the problem of determining if a pair of agents is indifferent is coNP-hard. This is not too surprising, given that the problem of computing the Shapley value in WVGs is #P-complete (Deng and =-=Papadimitriou 1994-=-; Matsui and Matsui 2000). Theorem 2. In WVGs, the problems of determining whether a given pair of agents is synergistic, whether it is antagonistic, and whether it is indifferent are NP-hard, NP-hard... |

1659 |
Combinatorial optimization : algorithms and complexity.
- Papadimitriou, Steiglitz
- 1998
(Show Context)
Citation Context ...e elements sum to exactly k. It is also given that all the solutions are of the given size l, that is, the elements in S sum to exactly k only if |S| = l. It can be seen from the reductions given in (=-=Papadimitriou and Steiglitz 1982-=-; Papadimitriou 1994) that SUBSET-SUM-EQ is NP-complete. We argue that the problem remains NP-complete even if we assume that l ≥ dn/2e, because if l < dn/2e then we simply change k to ∑n i=1 xi − k. ... |

760 |
A value for n-person game
- Shapley
- 1953
(Show Context)
Citation Context ... 0, forming the weight vector w, and a coalition S ⊆ N has value 1 iff its total weight is at least the threshold t: if∑ i∈S wi ≥ t then v(S) = 1, else v(S) = 0. The Shapley Value. The Shapley value (=-=Shapley 1952-=-) is a method of measuring power or dividing payoff that is uniquely characterized by four important axioms (Dubey 1975). It measures the contribution of each agent to the grand coalition N by analyzi... |

168 |
On the complexity of cooperative solution concepts
- Deng, Papadimitriou
- 1994
(Show Context)
Citation Context ...re NP-hard, while the problem of determining if a pair of agents is indifferent is coNP-hard. This is not too surprising, given that the problem of computing the Shapley value in WVGs is #P-complete (=-=Deng and Papadimitriou 1994-=-; Matsui and Matsui 2000). Theorem 2. In WVGs, the problems of determining whether a given pair of agents is synergistic, whether it is antagonistic, and whether it is indifferent are NP-hard, NP-hard... |

117 |
Values of games with a priori unions
- Owen
- 1977
(Show Context)
Citation Context ...d a line of influential work building on his axiom of balanced contributions (i.e., the synergy relation is symmetric) to characterize the Shapley value or extensions thereof, such as the Owen value (=-=Owen 1977-=-); see, e.g., the work of Hart and Mas (1989), Calvo et al. (1996), and Grabisch and Roubens (1999). Myerson’s notion of synergy is slightly different from the one we employ: he considers the differen... |

110 | Multilinear extensions of games - Owen - 1972 |

100 |
Conference structures and fair allocation rules
- Myerson
- 1980
(Show Context)
Citation Context ...son (1980) modeled the structure of interactions between agents as “a series of conferences” in which agents would meet “to discuss possible cooperative plans and to sign jointly binding agreements” (=-=Myerson 1980-=-, page 169). He explicitly studied synergies from an axiomatic viewpoint by requiring that for two agents i and j, j’s contribution to i is always equal to i’s contribution to j (in a formal sense tha... |

78 | Techniques for reading fuzzy measures (III): interaction index - Murofushi, Soneda - 1993 |

64 | An axiomatic approach to the concept of interaction among players in cooperative games - Grabisch, Roubens - 1999 |

48 | Complexity of constructing solutions in the core based on synergies among coalitions - Conitzer, Sandholm |

48 | A survey of algorithms for calculating power indices of weighted majority games
- Matsui
(Show Context)
Citation Context ... of determining if a pair of agents is indifferent is coNP-hard. This is not too surprising, given that the problem of computing the Shapley value in WVGs is #P-complete (Deng and Papadimitriou 1994; =-=Matsui and Matsui 2000-=-). Theorem 2. In WVGs, the problems of determining whether a given pair of agents is synergistic, whether it is antagonistic, and whether it is indifferent are NP-hard, NP-hard, and coNP-hard, respect... |

43 |
On the Uniqueness of the Shapley Value
- Dubey
- 1975
(Show Context)
Citation Context ...i∈S wi ≥ t then v(S) = 1, else v(S) = 0. The Shapley Value. The Shapley value (Shapley 1952) is a method of measuring power or dividing payoff that is uniquely characterized by four important axioms (=-=Dubey 1975-=-). It measures the contribution of each agent to the grand coalition N by analyzing its marginal contributions to various subsets of agents. The marginal contribution of an agent i ∈ N to a coalition ... |

23 |
A theorem on the construction of voting paradoxes, Econometrica 21: 608–610
- McGarvey
- 1953
(Show Context)
Citation Context ...here is an edge from alternative a to alternative b if and only if a majority of the voters prefer a to b, then every complete directed graph (tournament) emerges from some collection of preferences (=-=McGarvey 1953-=-). We interpret Theorem 1 as saying that the question of whether two agents are synergistic or antagonistic is, in a sense, independent of the signs of the synergies between other pairs. In fact, we c... |

22 | Approximating power indices
- Bachrach, Markakis, et al.
- 2008
(Show Context)
Citation Context ...seful tool for the design of multiagent systems. Indeed, cooperative games and the Shapley value have been extensively studied in the AI literature (Michalak et al. 2013; Ando 2012; Aziz et al. 2009; =-=Bachrach et al. 2008-=-; Bachrach and Rosenschein 2009; Zick, Skopalik, and Elkind 2011; Bachrach, Parkes, and Rosenschein 2013). Previous work by economists has established that understanding the synergies between agents g... |

21 | Potential, value and consistency, Econometrica 57 - Hart, Mas-Colell - 1989 |

14 | R.: Power indices in spanning connectivity games
- Aziz, Lachish, et al.
- 2009
(Show Context)
Citation Context ...nsidered to be a useful tool for the design of multiagent systems. Indeed, cooperative games and the Shapley value have been extensively studied in the AI literature (Michalak et al. 2013; Ando 2012; =-=Aziz et al. 2009-=-; Bachrach et al. 2008; Bachrach and Rosenschein 2009; Zick, Skopalik, and Elkind 2011; Bachrach, Parkes, and Rosenschein 2013). Previous work by economists has established that understanding the syne... |

14 | On Coalition Formation with Sparse Synergies - Voice, Ramchurn, et al. - 2012 |

13 | The principle of balanced contributions and hierarchies of cooperation - CALVO, LASAGA, et al. - 1996 |

6 | Computational Analysis of Connectivity Games with Applications to the Investigation of Terrorist Networks
- Michalak, Rahwan, et al.
- 2013
(Show Context)
Citation Context ... reasons, the Shapley value is considered to be a useful tool for the design of multiagent systems. Indeed, cooperative games and the Shapley value have been extensively studied in the AI literature (=-=Michalak et al. 2013-=-; Ando 2012; Aziz et al. 2009; Bachrach et al. 2008; Bachrach and Rosenschein 2009; Zick, Skopalik, and Elkind 2011; Bachrach, Parkes, and Rosenschein 2013). Previous work by economists has establishe... |

5 | Computing cooperative solution concepts in coalitional skill games - Bachrach, Parkes, et al. |

5 | Anytime coalition structure generation on synergy graphs
- Bistaffa, Farinelli, et al.
- 2014
(Show Context)
Citation Context ...y the computational complexity of various cooperative solution concepts in terms of this game representation. A number of recent publications in the AI literature (Voice, Ramchurn, and Jennings 2012; =-=Bistaffa et al. 2014-=-; Vinyals et al. 2013) have used the term “synergy graph” to denote a communication graph that shows connections between agents; its connected subgraphs are the feasible coalitions. In these papers, t... |

4 | Reliability weighted voting games - Bachrach, Shah - 2013 |

2 | Computation of the shapley value of minimum cost spanning tree games: #p-hardness and polynomial cases - Ando |

2 | A hierarchical dynamic programming algorithm for optimal coalition structure generation. http://arxiv.org/abs/1310.6704
- Vinyals, Voice, et al.
- 2013
(Show Context)
Citation Context ...mplexity of various cooperative solution concepts in terms of this game representation. A number of recent publications in the AI literature (Voice, Ramchurn, and Jennings 2012; Bistaffa et al. 2014; =-=Vinyals et al. 2013-=-) have used the term “synergy graph” to denote a communication graph that shows connections between agents; its connected subgraphs are the feasible coalitions. In these papers, the feasible coalition... |

1 |
Power in threshold network flow games. Autonomous Agents and MultiAgent Systems 18(1):106–132
- Bachrach, Rosenschein
- 2009
(Show Context)
Citation Context ...ign of multiagent systems. Indeed, cooperative games and the Shapley value have been extensively studied in the AI literature (Michalak et al. 2013; Ando 2012; Aziz et al. 2009; Bachrach et al. 2008; =-=Bachrach and Rosenschein 2009-=-; Zick, Skopalik, and Elkind 2011; Bachrach, Parkes, and Rosenschein 2013). Previous work by economists has established that understanding the synergies between agents gives an important new perspecti... |

1 | International Monetary Fund: Background and Issues for Congress. CreateSpace Independent Publishing Platform - Weiss |