$x$ to $y$ then so should society.

<span style="font-variant:small-caps;">Universal Domain Property</span>No matter what kind of wacky preferences people may have, so long as they are well-behaved, ${\cal R}$ has to be able to deal with them. In other words there are no restrictions on the profiles of preferences, i.e. on the elements in ${\cal P}^m$.

<span style="font-variant:small-caps;">Independence of Irrelevant Alternatives</span>Whether or not society prefers $x$ to $y$ does not depend on what people think of any other [*irrelevant alternative*]{} $z$. This can be formally stated by saying that if there are two profiles of individual preferences $(\succeq_1,\dotsc,\succeq_m)$ and $(\succeq'_1,\dotsc,\succeq'_m)$ such that $$x \succeq_i y\quad \text{if and only if}\quad x \succeq'_i y,$$ then $$x\ {\cal R}(\succeq_1, \ldots , \succeq_m)\ y\quad \text{if and only if}\quad
x\ {\cal R}(\succeq'_1, \ldots , \succeq'_m)\ y.$$

<span style="font-variant:small-caps;">Nondictatorship</span>An agent $a_i$ is said to be [*dictatorial*]{} if, for all $x,y \in X$, whenever $a_i$ prefers $x$ to $y$ society prefers $x$ to $y$, i.e. ${\cal R}$ is the projection on the $i$-th component. The social decision rule ${\cal R}$ is said to be [*nondictatorial*]{} if it is not a projection map.

Arrow [@Arrow] proved that such a function does not exist. Therefore, in order to overcome this 