<p>Yes, since you included the w.l.g. comments this is a proof.</p>
<p>The 'more general' property is dictatorship, proving that would need to cover all cases. Non-dictatorship says that that property is not fulfilled for any voters, hence it is enough to show with some examples that it is not.</p>

CC BY-SA 4.0
2024-02-01T12:50:22.283Z
2024-02-01T12:50:22.283Z
Giskard
<p>My question is whether using an example is enough to show that a social choice function satisfies the non-dictatorship property.</p>
<p>There are 3 alternatives (<span class="math-container">$a,b,c$</span>) and 5 individuals (<span class="math-container">$1,2,3,4,5$</span>).
Each individual needs to rank all alternatives in a strict order.
Let <span class="math-container">$P_k$</span> be the strict ranking of the <span class="math-container">$k^{th}$</span> individual.
Let <span class="math-container">$f$</span> be a social welfare function and given the rankings of the individuals it selects the society's ranking as follows:</p>
<ul>
<li><strong>First Choice of the Society:</strong> If an alternative is ranked more than 50% of the individuals, then that alternative is the first choice of the society.
If there is no such an alternative, then we remove the alternative who is ranked first by the fewest individuals and shift the rankings (e.g., if the ranking is <span class="math-container">$a - b - c$</span> and <span class="math-container">$b$</span> is removed then the new ranking is <span class="math-container">$a - c$</span>).
Repeat the same procedure with the updated rankings.</li>
<li><strong>Second Choice of the Society:</strong> Consider the original rankings.
Once the first choice of the society is determined, we remove that alternative and shift the rankings.
The second choice of the society is the remaining alternative ranked as top choice by more than 50% of the individuals among the remaining alternatives.</li>
<li><strong>Third Choice of the Society:</strong> The remaining alternative is the third choice of the society.</li>
</ul>
<p>Is it sufficient to assume that there is some dictator and show that there exist a ranking of other people that does not allow that person to be a dictator?</p>
<p>Proof.</p>
<p>Suppose there is some dictator <span class="math-container">$h$</span> whose preference orderings always chooses the society's rank.
Without losing generality, assume that <span class="math-container">$h=1$</span> and that his preferences are given <span class="math-container">$P_1 = a - b - c$</span>.
The other four individuals have preferences
<span class="math-container">$$P_j = c - b - a, \text{ for $j \in 2,3,4,5$}$$</span>
so the ranking of the society is <span class="math-container">$c - b - a$</span>.</p>
<p>Since <span class="math-container">$c - b - a \ne P_1$</span>, person 1 cannot be a dictator.
Hence this social welfare function satisfies the non-dictatorship condition.</p>
<p>q.e.d.</p>

CC BY-SA 4.0
2024-02-01T12:48:42.137Z
The question is somewhat long, next time please consider if you need to include all the information, or perhaps you could parse it. E.g.; you describe a welfare function version of ranked choice voting instead of writing "ranked choice".
Giskard
CC BY-SA 4.0
2024-02-02T02:47:58.483Z
Thank you for your comment. I didn't realize this had a name!
RobinsonWM
<social-choice><social-choice-function>
Can you prove a social choice function satisfies non-dictatorship using an example?
RobinsonWM
# Can you prove a social choice function satisfies non-dictatorship using an example?

(Asked by: username_0 on 2024-02-01)

My question is whether using an example is enough to show that a social choice function satisfies the non-dictatorship property.

There are 3 alternatives ($a,b,c$) and 5 individuals ($1,2,3,4,5$).
Each individual needs to rank all alternatives in a strict order.
Let $P\_k$ be the strict ranking of the $k^{th}$ individual.
Let $f$ be a social welfare function and given the rankings of the individuals it selects the society's ranking as follows:

* **First Choice of the Society:** If an alternative is ranked more than 50% of the individuals, then that alternative is the first choice of the society.
  If there is no such an alternative, then we remove the alternative who is ranked first by the fewest individuals and shift the rankings (e.g., if the ranking is $a - b - c$ and $b$ is removed then the new ranking is $a - c$).
  Repeat the same procedure with the updated rankings.
* **Second Choice of the Society:** Consider the original rankings.
  Once the first choice of the society is determined, we remove that alternative and shift the rankings.
  The second choice of the society is the remaining alternative ranked as top choice by more than 50% of the individuals among the remaining alternatives.
* **Third Choice of the Society:** The remaining alternative is the third choice of the society.

Is it sufficient to assume that there is some dictator and show that there exist a ranking of other people that does not allow that person to be a dictator?

Proof.

Suppose there is some dictator $h$ whose preference orderings always chooses the society's rank.
Without losing generality, assume that $h=1$ and that his preferences are given $P\_1 = a - b - c$.
The other four individuals have preferences
$$P\_j = c - b - a, \text{ for $j \in 2,3,4,5$}$$
so the ranking of the society is $c - b - a$.

Since $c - b - a \ne P\_1$, person 1 cannot be a dictator.
Hence this social welfare function satisfies the non-dictatorship condition.

q.e.d.

**Tags:** social-choice, social-choice-function
> **Comments:**
> **username_1**: The question is somewhat long, next time please consider if you need to include all the information, or perhaps you could parse it. E.g.; you describe a welfare function version of ranked choice voting instead of writing "ranked choice".  
> **username_0**: Thank you for your comment. I didn't realize this had a name!  

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## 1 answers

### Answer by username_1 on 2024-02-01. Score: 2 | Accepted answer

Yes, since you included the w.l.g. comments this is a proof.

The 'more general' property is dictatorship, proving that would need to cover all cases. Non-dictatorship says that that property is not fulfilled for any voters, hence it is enough to show with some examples that it is not.


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