Which of Arrow's four desirable properties' is violated in this scenario? So the scenario is as follows: there are 3 agents and 4 alternatives, $a,b,c,$ and $d$. Society's ranking of the 4 alternatives is such that 


  the highest-ranked alternative is agent 1's highest-ranked
  alternative,
  
  The alternative that agent 2 ranks highest among the three remaining alternatives is ranked second highest by society 
  
  The alternative that agent 3 ranks highest among the two remaining alternatives is ranked third highest by society. 
  
  Society ranks the 4th alternative lowest.


So my interpretation of the ranking would look something like this:



Which of Arrow's four desirable properties is violated?

I'm guessing that the "Unanimity" preference would be violated, because agent 2 prefers $b$ over $c$, but this is not reflected in society's ranking?

The four desired properties I am referring to are:

1/ Complete and transitive preferences

2/ Respect unanimity

3/ Non-dictatorial outcomes

4/ Independence of irrelevant alternatives
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.