# 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

• What happens if they all rank $a$ highest? – Giskard Mar 28 '18 at 22:16
• Can you specify which four properties you're considering? – Herr K. Mar 28 '18 at 22:58
• @HerrK. I've now added these four properties to my question – user98937 Mar 29 '18 at 0:25
• @denesp according to the setup, only agent 1 ranks $a$ highest; other agents highest rankings are as shown in the table. No additional information was said about agents 1,2, and 3's rankings for their 2nd to 4th preferences. – user98937 Mar 29 '18 at 0:27
• I think this kind of answers your question. These properties have to hold over universal domain. – Giskard Mar 29 '18 at 4:30

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.

• Is there a typo in your Agent 1, Profile 2? Why are there 5 rankings, with $a$ appearing twice? – user98937 Mar 29 '18 at 12:31
• @user98937 You are right; I corrected the typo. – Michael Greinecker Mar 29 '18 at 12:32