<p>I am trying to understand the proof of the world-famous <a href="https://en.wikipedia.org/wiki/Arrow%27s_impossibility_theorem" rel="nofollow noreferrer">Arrow’s Impossibility Theorem</a>. To do that, I have been relying on <a href="https://en.wikipedia.org/wiki/Arrow%27s_impossibility_theorem" rel="nofollow noreferrer">Wikipedia’s proof by decisive coalition</a>, as well as on the proof provided by Amartya Sen in Sen &amp; Maskin’s book “<a href="https://cup.columbia.edu/book/the-arrow-impossibility-theorem/9780231526869" rel="nofollow noreferrer">The Arrow Impossibility Theorem</a>”. Arrow’s Theorem states the following:</p>
<p><em>If there are at least thee alternatives and the voter set is finite, there exists no <strong>social welfare function</strong> satisfying <strong>Pareto optimality</strong>, <strong>independence of irrelevant alternatives</strong>, <strong>unrestricted domain</strong> and <strong>non-dictatorship</strong>.</em></p>
<p>For the relevant definitions, see <a href="https://en.wikipedia.org/wiki/Arrow%27s_impossibility_theorem" rel="nofollow noreferrer">Wikipedia</a>.</p>
<p>I can see that majority voting, in the presence of just two alternatives, satisfies all of Arrow’s conditions. What I cannot see is which steps in the proofs I quoted above rely on the existence of at least three alternatives.</p>
<p><strong>Could anyone help me see which steps in the proofs mentioned above require the existence of at least three alternatives?</strong></p>

CC BY-SA 4.0
2024-06-12T14:08:29.113Z
In the wiki page, the proof of the group contraction lemma starts with: ``Fix distinct $x$ , $y$ , $z$''. Clearly if there are only two alternatives, this is impossible to do.
tdm
CC BY-SA 4.0
2024-06-12T14:13:46.513Z
@tdm, And even before that, the proof of the field expansion lemma starts with "Let $z$ be an outcome distinct from $x$, $y$."
VARulle
CC BY-SA 4.0
2024-06-12T15:02:47.923Z
@tdm Post your comment as an answer or close the Q?
Giskard
<microeconomics><mathematical-economics><theory><social-choice>
Where does Arrow’s Impossibility Theorem standard proof rely on the existence of at least three alternatives?
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# Where does Arrow’s Impossibility Theorem standard proof rely on the existence of at least three alternatives?

(Asked by: username_0 on 2024-06-12)

I am trying to understand the proof of the world-famous [Arrow’s Impossibility Theorem](https://en.wikipedia.org/wiki/Arrow%27s_impossibility_theorem). To do that, I have been relying on [Wikipedia’s proof by decisive coalition](https://en.wikipedia.org/wiki/Arrow%27s_impossibility_theorem), as well as on the proof provided by Amartya Sen in Sen & Maskin’s book “[The Arrow Impossibility Theorem](https://cup.columbia.edu/book/the-arrow-impossibility-theorem/9780231526869)”. Arrow’s Theorem states the following:

*If there are at least thee alternatives and the voter set is finite, there exists no **social welfare function** satisfying **Pareto optimality**, **independence of irrelevant alternatives**, **unrestricted domain** and **non-dictatorship**.*

For the relevant definitions, see [Wikipedia](https://en.wikipedia.org/wiki/Arrow%27s_impossibility_theorem).

I can see that majority voting, in the presence of just two alternatives, satisfies all of Arrow’s conditions. What I cannot see is which steps in the proofs I quoted above rely on the existence of at least three alternatives.

**Could anyone help me see which steps in the proofs mentioned above require the existence of at least three alternatives?**

**Tags:** microeconomics, mathematical-economics, theory, social-choice
> **Comments:**
> **username_1**: In the wiki page, the proof of the group contraction lemma starts with: ``Fix distinct $x$ , $y$ , $z$''. Clearly if there are only two alternatives, this is impossible to do.  
> **username_2**: @tdm, And even before that, the proof of the field expansion lemma starts with "Let $z$ be an outcome distinct from $x$, $y$."  
> **username_3**: @tdm Post your comment as an answer or close the Q?  

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## 0 answers

