Many bodies around the world make their decisions through voting systems in which voters have several options and the collective result also has several options. Many of these voting systems are anonymous, i.e., all voters have an identical role in voting. Anonymous simple voting games, a binary vote for voters and a binary collective decision, can be represented by an easy weighted game, i.e., by means of a quota and an identical weight for the voters. Widely used voting systems of this type are the majority and the unanimity decision rules. In this article, we analyze the case in which voters have two or more voting options and the collective result of the vote has also two or more options. We prove that anonymity implies being representable through a weighted game if and only if the voting options for voters are binary. As a consequence of this result, several significant enumerations are obtained.

世界各地的许多机构都是通过投票系统做出决定的，投票系统中的选民有多种选择，而集体选举结果也有多种选择。这些投票系统中有许多是匿名的，即所有选民在投票中都具有相同的作用。匿名简单投票游戏，对投票者的二进制投票和二进制集体决定，可以通过简单的加权游戏来表示，即通过配额和对投票者相同的权重。多数和一致决定规则是这种类型的广泛使用的投票系统。在本文中，我们分析了选民具有两个或多个投票选项，而投票的集体结果也具有两个或多个选项的情况。我们证明了匿名意味着通过加权游戏是可表示的当且仅当选民的投票选项是二进制的。