Abstract
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.
Highlights
A voting game is a cooperative game in which a set of voters, or players, having different weights, must make a yes-or-no decision on a proposal
Researchers have addressed unequal coalitions probability by: (i) Restricting the family of feasible coalitions, Rodriguez-Veiga et al (2016), for example imposing that some voters always vote in opposite directions, Alonso-Meijide et al (2015) and Yakuba (2008), or assuming that voters form a-priori unions before casting their votes, Alonso-Meijide et al (2009), Amer et al (2002) and Leech and Leech (2006; ii) Assuming that some coalitions are more probable than others, for example when voters are located in Euclidean political spaces the most likely coalitions are composed of voters that are close to each other, Benati and Vittucci Marzetti (2013), Passarelli and Barr (2007) and Mielcova (2016). One of these models is relevant here: It is the spatial voting game proposed in Pajala and Widgrèn (2004), for which a large set of empirical data has been collected, Thomson et al (2012; iii) Assuming that voters are embedded in a communication graph, in which the vertices represent voters and the edges represent communication links
We have shown how to use dynamic programming to compute the number of connected coalitions in spatial voting games and we have shown that the indices of power calculated in this way are useful to understand the political games, namely, the bargaining process in EU Council
Summary
A voting game is a cooperative game in which a set of voters, or players, having different weights, must make a yes-or-no decision on a proposal. Researchers have addressed unequal coalitions probability by: (i) Restricting the family of feasible coalitions, Rodriguez-Veiga et al (2016), for example imposing that some voters always vote in opposite directions, Alonso-Meijide et al (2015) and Yakuba (2008), or assuming that voters form a-priori unions before casting their votes, Alonso-Meijide et al (2009), Amer et al (2002) and Leech and Leech (2006; ii) Assuming that some coalitions are more probable than others, for example when voters are located in Euclidean political spaces the most likely coalitions are composed of voters that are close to each other, Benati and Vittucci Marzetti (2013), Passarelli and Barr (2007) and Mielcova (2016) One of these models is relevant here: It is the spatial voting game proposed in Pajala and Widgrèn (2004), for which a large set of empirical data has been collected, Thomson et al (2012; iii) Assuming that voters are embedded in a communication graph, in which the vertices represent voters and the edges represent communication links. We will see that, when applied to decision processes, graph restricted indices are better predictors of the outcomes of the negotiation processes, perhaps because restricted indices are measuring in a more realistic way the real political forces underlying the bargaining processes
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