created [*modern social choice theory*]{}, a rigorous melding of social ethics and voting theory with an economic flavor. The central aim of social choice theory is to analyze the aggregation of preferences. Assume there is a society of $n$ agents indexed by $i = 1,\dotsc,m$. Each agent has his own well-behaved preference $\succeq_i$ over some space of possibilities (or [*social outcomes*]{}) $X$, i.e. a total order on the set $X$. Let ${\cal P}$ be the set of well-behaved preferences. The element $(\succeq_1,\dotsc,\succeq_m) \in {\cal P}^m$ is the [*profile*]{} of a society. The goal is to put all of these preferences together to come up with a single [*system of social preferences*]{} or [*social rule*]{}, i.e. a total order on the set $X$, to decide matters of policy and to evaluate welfare. Namely, a [*social choice function*]{} (or [*social decision rule*]{}) $${{\mathcal{R}}}: {\cal P}^m \longrightarrow {\cal P}$$ is needed.

This social decision rule should fulfill the following properties.

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<span style="font-variant:small-caps;">Completeness and Transitivity</span>With this economists means that society can make a decision about any social outcome and can rank all social outcomes. (Obviously, this property is intrinsic in the definition of the function ${\cal R}$.)

<span style="font-variant:small-caps;">Paretianity</span>If everyone unanimously prefers 