Arrow's impossibility theorem and the independence of irrelevant alternatives

Question

I have a question about the axiom of independence of irrelevant alternatives (IIA). According to the Wikipedia page on IIA, Arrow's formulation of IIA is as in here. I do not quite get this. In particular, what does $R_i \cap \{x,y\}^2$ mean? $R_i$ is just some permutation of objects which include $x$ and $y$ I guess. $\{x,y\}^2$ is the set $\{(x,y), (y,x)\}$ I think. But then how is the intersection above defined?

Answer 1

Let $A$ be the set of alternatives; each $R_i$ is a binary preference relation on $A$ and therefore by definition a subset of $A\times A=A^2$. Since $\{x,y\}\subseteq A$, we automatically have $\{x,y\}^2\subseteq A^2$, and $R_i\cap\{x,y\}^2$ is then just an ordinary intersection of sets.

If, for example, we have alternatives $a,b,c$, and $d$, and voter $1$ ranks them $badc$, then

$$R_1=\{\langle b,a\rangle,\langle b,d\rangle,\langle b,c\rangle,\langle a,d\rangle,\langle a,c\rangle,\langle d,c\rangle\}\;,$$

and

$$R_1\cap\{a,d\}^2=\{\langle a,d\rangle\}\;.$$