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    An anytime approximation method for the inverse Shapley value problem

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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 players 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 players 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. Copyright © 2008, International Foundation for Autonomous Agents and Multiagent Systems (www.ifaarnas.org). All rights reserved

    Bookmaker Consensus and Agreement for the UEFA Champions League 2008/09

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    Bookmakers odds are an easily available source of ``prospective" information that is thus often employed for forecasting the outcome of sports events. To investigate the statistical properties of bookmakers odds from a variety of bookmakers for a number of different potential outcomes of a sports event, a class of mixed-effects models is explored, providing information about both consensus and (dis)agreement across bookmakers. In an empirical study for the UEFA Champions League, the most prestigious football club competition in Europe, model selection yields a simple and intuitive model with team-specific means for capturing consensus and team-specific standard deviations reflecting agreement across bookmakers. The resulting consensus forecast performs well in practice, exhibiting high correlation with the actual tournament outcome. Furthermore, the teams' agreement can be shown to be strongly correlated with the predicted consensus and can thus be incorporated in a more parsimonious model for agreement while preserving the same consensus fit.Series: Research Report Series / Department of Statistics and Mathematic

    Non-centralized Control for Flow-based Distribution Networks: A Game-theoretical Insight

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    This paper solves a data-driven control problem for a flow-based distribution network with two objectives: a resource allocation and a fair distribution of costs. These objectives represent both cooperation and competition directions. It is proposed a solution that combines either a centralized or distributed cooperative game approach using the Shapley value to determine a proper partitioning of the system and a fair communication cost distribution. On the other hand, a decentralized noncooperative game approach computing the Nash equilibrium is used to achieve the control objective of the resource allocation under a non-complete information topology. Furthermore, an invariant-set property is presented and the closed-loop system stability is analyzed for the non cooperative game approach. Another contribution regarding the cooperative game approach is an alternative way to compute the Shapley value for the proposed specific characteristic function. Unlike the classical cooperative-games approach, which has a limited application due to the combinatorial explosion issues, the alternative method allows calculating the Shapley value in polynomial time and hence can be applied to large-scale problems.Generalitat de Catalunya FI 2014Ministerio de Ciencia y Educación DPI2016-76493-C3-3-RMinisterio de Ciencia y Educación DPI2008-05818Proyecto europeo FP7-ICT DYMASO
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