Weighted voting games are simple games that can be represented by a collection of integer weights for each player so that a coalition wins if the sum of the player weights matches or exceeds a given quota. It is known that a simple game can be expressed as the intersection or the union of weighted voting games. The dimension (codimension) of a simple game is the minimum number of weighted voting games such that their intersection (union) is the given game. In this work, we analyze some subclasses of weighted voting games and their closure under intersection or union. We introduce generalized notions of dimension and codimension regarding some subclasses of weighted voting games. In particular, we show that not all simple games can be expressed as intersection (union) of pure weighted voting games (those in which dummy players are not allowed) and we provide a characterization of such simple games. Finally, we experimentally study the generalized dimension (codimension) for some subclasses defined by establishing restrictions on the representations of weighted voting games.

加权投票游戏是简单的游戏，可以用每个玩家的整数权重集合来表示，这样如果玩家权重的总和匹配或超过给定的配额，联盟就会获胜。众所周知，一个简单的博弈可以表示为加权投票博弈的交集或并集。简单博弈的维度（codimension）是加权投票博弈的最小数量，使得它们的交集（并集）是给定的博弈。在这项工作中，我们分析了加权投票博弈的一些子类及其在交集或并集下的闭包。我们介绍了关于加权投票游戏的某些子类的维度和余维度的广义概念。特别是，我们表明并非所有简单游戏都可以表示为纯加权投票游戏（不允许虚拟玩家的游戏）的交集（并集），并且我们提供了此类简单游戏的特征。最后，我们通过实验研究了一些子类的广义维度（余维度），这些子类是通过对加权投票游戏的表示形式建立限制来定义的。