Which of Arrow's four desirable properties' is violated in this scenario?
Note that an alternative being highest-ranked by an agent involves comparisons with all other alternatives, which raises doubts that your method satisfies independence of irrelevant alternatives. Indeed, it does not.  Independence off irrelevant alternatives is actually the only property that is violated.

Here are two profiles, only the preferences of the first two agents matter in them, so I will not specify the preferences of the other agents. I only assume the other agents preferences are the same in both profiles.

Agent 1, Profile $1$: $a\succ b\succ c\succ d$

Agent 2, Profile $1$: $b\succ a\succ c\succ d$

In Profile 1, the social ranking of $a$ and $b$ is $a\succ b$.

Agent 1, Profile $2$: $c\succ a\succ b\succ d$

Agent 2, Profile $2$: $b\succ a\succ c\succ d$

In Profile 2, the social ranking of $a$ and $b$ is $b\succ a$. 

Since the relative ranking of $a$ and $b$ is the same in both profiles, independence of irrelevant alternatives is violated.