We analyze the problem of computing the Banzhaf and Shapley power indices for graph restricted voting games, defined in a particular class of graphs, that we called line-clique. A line-clique graph is a model of a uni-dimensional political space in which voters with the same bliss point are the connected vertices of a clique and then other arcs connect nodes of consecutive cliques. The interest to this model comes from its correspondence to the spatial voting game: a model that has been proposed and used by political analysts to understand nations’ behavior and the political outcome of the bargaining process within the EU Council. Broadly speaking, the computation of a power index of a graph restricted game is strongly #P-complete, as it includes the enumeration of all winning coalitions. Nevertheless, we show that in this special class of graph coalitions can be enumerated by dynamic programming, resulting in a pseudo-polynomial algorithm and proving that the problem only weakly #P-complete. After implementing our new algorithms and finding that they are very fast in practice, we analyze the voting behavior in the EU Council, as for this application previous research compiled a large data set concerning nations’ political positions and political outcomes. We will test whether voting power has an effect on the political outcome, more precisely, whether nations that are favored by their weight and position can influence the political outcome to their advantages. Using linear regressions, we will see that unrestricted power indices are not capable of any predictive property, but graph restricted indices are. The statistic evidence shows that the combination of voting weight and network position is a source of power that affects the political outcome to the advantage of a country.

我们分析了在特定类别的图表中定义的图受限投票游戏的Banzhaf和Shapley功效指数的计算问题，我们将其称为线形。线状斜线图是一维政治空间的模型，其中具有相同幸福点的选民是集团的连接顶点，然后其他弧线连接连续的集团的节点。该模型的兴趣来自其与空间投票博弈的对应关系：该模型已被政治分析家提出并用于理解各国的行为以及欧盟理事会内部议价过程的政治结果。广义地说，图受限游戏的功效指数的计算强烈地是＃P-完全的，因为它包括所有获胜联盟的枚举。尽管如此，我们表明，在这种特殊的图联盟中，可以通过动态规划来枚举联盟，从而得出伪多项式算法，并证明该问题仅能弱地完成＃P-完全。在实施了我们的新算法并发现它们在实践中非常快之后，我们在欧盟理事会中分析了投票行为，为此，针对先前的研究，该研究针对国家的政治立场和政治成果汇编了大数据集。我们将测试投票权是否对政治结果产生影响，更确切地说，是受权重和地位所偏爱的国家是否会影响政治结果以发挥其优势。使用线性回归，我们将看到无限制的幂指数不具有任何预测属性，而图的限制指数则具有。