On the complexity of cooperative solution concepts
Mathematics of Operations Research
Methods for task allocation via agent coalition formation
Artificial Intelligence
Computing Power Indices for Large Voting Games
Management Science
Marginal contribution nets: a compact representation scheme for coalitional games
Proceedings of the 6th ACM conference on Electronic commerce
Multi-attribute coalitional games
EC '06 Proceedings of the 7th ACM conference on Electronic commerce
A randomized method for the shapley value for the voting game
Proceedings of the 6th international joint conference on Autonomous agents and multiagent systems
AAAI'04 Proceedings of the 19th national conference on Artifical intelligence
Computational complexity of weighted threshold games
AAAI'07 Proceedings of the 22nd national conference on Artificial intelligence - Volume 1
False name manipulations in weighted voting games: splitting, merging and annexation
Proceedings of The 8th International Conference on Autonomous Agents and Multiagent Systems - Volume 1
Enumeration and exact design of weighted voting games
Proceedings of the 9th International Conference on Autonomous Agents and Multiagent Systems: volume 1 - Volume 1
False-name manipulations in weighted voting games
Journal of Artificial Intelligence Research
Manipulating the quota in weighted voting games
Artificial Intelligence
The inverse shapley value problem
ICALP'12 Proceedings of the 39th international colloquium conference on Automata, Languages, and Programming - Volume Part I
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Coalition formation is the process of bringing together two or more agents so as to achieve goals that individuals on their own cannot, or to achieve them more efficiently. Typically, in such situations, the agents have conflicting preferences over the set of possible joint goals. Thus, before the agents realize the benefits of cooperation, they must find a way of resolving these conflicts and reaching a consensus. In this context, cooperative game theory offers the voting game as a mechanism for agents to reach a consensus. It also offers the Shapley value as a way of measuring the influence or power a player has in determining the outcome of a voting game. Given this, the designer of a voting game wants to construct a game such that a player's Shapley value is equal to some desired value. This is called the inverse Shapley value problem. Solving this problem is necessary, for instance, to ensure fairness in the players' voting powers. However, from a computational perspective, finding a player's Shapley value for a given game is #P-complete. Consequently, the problem of verifying that a voting game does indeed yield the required powers to the agents is also #P-complete. Therefore, in order to overcome this problem we present a computationally efficient approximation algorithm for solving the inverse problem. This method is based on the technique of 'successive approximations'; it starts with some initial approximate solution and iteratively updates it such that after each iteration, the approximate gets closer to the required solution. This is an anytime algorithm and has time complexity polynomial in the number of players. We also analyze the performance of this method in terms of its approximation error and the rate of convergence of an initial solution to the required one. Specifically, we show that the former decreases after each iteration, and that the latter increases with the number of players and also with the initial approximation error.