The Banzhaf power index for games admits several extensions if the players have more than two ordered voting options. In this paper we prove that the most intuitive and recognized extension of the index fails to preserve the desirability relation for games with more than three ordered input levels of approval, a failure that undermines the index to be a good measure of power. This leads us to think of an alternative to the Banzhaf index for several input levels of approval. We propose a candidate for which it is proved that: (1) coincides with the Banzhaf index for simple games, (2) it is proportional to its known extension for three levels of approval, and (3) preserves the desirability relation regardless of the number of input levels of approval. This new index is based on measuring the total capacity the player has to alter the outcome. In addition, it can be expressed through a very appropriate mathematical formulation that greatly facilitates its computation. Defining extensions of well-established notions in a wider context requires a careful analysis. Different extensions can provide complementary nuances and, when this occurs, none of them can be considered to be ‘the’ extension. As shown in this paper, this situation applies when trying to extend the Banzhaf power index from simple games to the broader context of games with several ordered input levels of approval.

如果玩家具有两个以上的订购投票选项，则游戏的Banzhaf力量指数将允许若干扩展。在本文中，我们证明了索引的最直观和公认的扩展无法保留具有三个以上有序输入批准级别的游戏的可取性关系，而这种失败破坏了索引，是衡量权力的一种好方法。这使我们考虑了Banzhaf指数的替代方案，以用于多个输入批准级别。我们提出了一个候选人，证明了这一点：（1）与简单游戏的Banzhaf指数一致；（2）与它在三个批准级别上的已知扩展成正比；（3）保留期望关系，无论批准输入级别的数量。这个新指标是基于衡量玩家改变结果的总能力。另外，它可以通过非常方便的数学公式来表示，极大地方便了它的计算。在更广泛的上下文中定义公认概念的扩展需要仔细分析。不同的扩展名可以提供互补的细微差别，并且在发生这种情况时，都不会将其视为“扩展名”。如本文所示，这种情况适用于尝试将Banzhaf幂指数从简单游戏扩展到具有多个有序输入批准级别的较广泛游戏的情况。它们中没有一个可以被认为是“ the”扩展。如本文所示，这种情况适用于尝试将Banzhaf幂指数从简单游戏扩展到具有多个有序输入批准级别的较广泛游戏的情况。它们中没有一个可以被认为是“ the”扩展。如本文所示，这种情况适用于尝试将Banzhaf幂指数从简单游戏扩展到具有多个有序输入批准级别的较广泛游戏的情况。