Why we use revealed preferred condition in GARP instead of directly revealed preferred? The definition of revealed preferred seems like coming out of nowhere. Why there's a string of direct revealed preferences lie between? Can we use directly revealed preferred in GARP condition ------GARP says that if $x \succsim_{D} y$, $y \nsucceq_{D} x$?
We need indirect comparisons because looking at direct comparisons is not sufficient to detect all types of inconsistencies. Suppose for instance that we observe $C(x,z)=\{x\},C(z,y)=\{z\}$ and $C(x,y)=\{y\}$. $C(x,y)=\{y\}$ implies that $y$ is "directly" revealed preferred to $x$. The reason why GARP is violated is that $C(x,z)=\{x\}$ and $C(z,y)=\{z\}$ imply that $x$ is "indirectly" revealed preferred to $y$, an inference that wouldn't be possible by looking at direct comparisons.