<p>Note that an alternative being highest-ranked by an agent involves comparisons with all other alternatives, which raises doubts that your method satisfies <em>independence of irrelevant alternatives</em>. Indeed, it does not.  Independence off irrelevant alternatives is actually the only property that is violated.</p>

<p>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.</p>

<p>Agent 1, Profile $1$: $a\succ b\succ c\succ d$</p>

<p>Agent 2, Profile $1$: $b\succ a\succ c\succ d$</p>

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

<p>Agent 1, Profile $2$: $c\succ a\succ b\succ d$</p>

<p>Agent 2, Profile $2$: $b\succ a\succ c\succ d$</p>

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

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

CC BY-SA 3.0
2018-03-29T12:31:07.813Z
Is there a typo in your Agent 1, Profile 2? Why are there 5 rankings, with $a$ appearing twice?
tsp216
CC BY-SA 3.0
2018-03-29T12:32:31.713Z
@user98937 You are right; I corrected the typo.
Michael Greinecker
CC BY-SA 3.0
2018-03-29T08:22:41.537Z
2018-03-29T12:32:01.030Z
2018-03-29T12:32:01.030Z
Michael Greinecker
<p>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 </p>

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

<p>So my interpretation of the ranking would look something like this:</p>

<p><a href="https://i.sstatic.net/05pRG.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/05pRG.png" alt="enter image description here"></a></p>

<p>Which of Arrow's four desirable properties is violated?</p>

<p>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?</p>

<p>The four desired properties I am referring to are:</p>

<p>1/ Complete and transitive preferences</p>

<p>2/ Respect unanimity</p>

<p>3/ Non-dictatorial outcomes</p>

<p>4/ Independence of irrelevant alternatives</p>

CC BY-SA 3.0
2018-03-28T22:16:58.627Z
What happens if they all rank $a$ highest?
Giskard
CC BY-SA 3.0
2018-03-28T22:58:25.553Z
Can you specify *which four* properties you're considering?
Herr K.
CC BY-SA 3.0
2018-03-29T00:25:57.060Z
@HerrK. I've now added these four properties to my question
tsp216
CC BY-SA 3.0
2018-03-29T00:27:10.993Z
@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.
tsp216
CC BY-SA 3.0
2018-03-29T00:36:41.467Z
@denesp I've also updated the scenario so that the ranking procedure is a little clearer.
tsp216
CC BY-SA 3.0
2018-03-29T04:30:57.143Z
I think this kind of answers your question. These properties have to hold over universal domain.
Giskard
CC BY-SA 3.0
2018-03-29T08:39:44.607Z
I think you misunderstand what unanimity means here. If *all* voters prefer alternative $x$ to $y$, so must the social ranking.
Michael Greinecker
<social-choice>
Which of Arrow's four desirable properties' is violated in this scenario?
tsp216
# Which of Arrow's four desirable properties' is violated in this scenario?

(Asked by: username_0 on 2018-03-28)

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:

[![enter image description here](https://i.sstatic.net/05pRG.png)](https://i.sstatic.net/05pRG.png)

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

**Tags:** social-choice
> **Comments:**
> **username_1**: What happens if they all rank $a$ highest?  
> **username_2**: Can you specify *which four* properties you're considering?  
> **username_0**: @HerrK. I've now added these four properties to my question  
> **username_0**: @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.  
> **username_0**: @denesp I've also updated the scenario so that the ranking procedure is a little clearer.  
> **username_1**: I think this kind of answers your question. These properties have to hold over universal domain.  
> **username_3**: I think you misunderstand what unanimity means here. If *all* voters prefer alternative $x$ to $y$, so must the social ranking.  

---

## 1 answers

### Answer by username_3 on 2018-03-29. Score: 5 | Accepted answer

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.

> **Comments:**
> **username_0**: Is there a typo in your Agent 1, Profile 2? Why are there 5 rankings, with $a$ appearing twice?  
> **username_3**: @user98937 You are right; I corrected the typo.  

---

