<p>Let's say there are individuals 1 and 2, and alternatives A, B, C, and D. Society uses the Rawlsian SWF and thus ranks alternatives according to their maximal rank within individuals' rankings. Denote society's preferences by <span class="math-container">$\succ^*$</span>.</p>
<p>If the individual rankings are:</p>
<p>1: A <span class="math-container">$\succ$</span> B <span class="math-container">$\succ$</span> C <span class="math-container">$\succ$</span> D</p>
<p>2: B <span class="math-container">$\succ$</span> D <span class="math-container">$\succ$</span> A <span class="math-container">$\succ$</span> C,</p>
<p>then B <span class="math-container">$\succ^*$</span> A.</p>
<p>If you change the ranking of 1 such that</p>
<p>1: C <span class="math-container">$\succ$</span> D <span class="math-container">$\succ$</span> A <span class="math-container">$\succ$</span> B</p>
<p>2: B <span class="math-container">$\succ$</span> D <span class="math-container">$\succ$</span> A <span class="math-container">$\succ$</span> C,</p>
<p>then A <span class="math-container">$\succ^*$</span> B. Thus, society's ranking of A vs. B has changed, while each individual's ranking of A vs. B stayed the same. Hence this SWF fails IIA.</p>

CC BY-SA 4.0
2021-03-16T23:33:00.600Z
2021-03-16T23:33:00.600Z
VARulle
<p>Independence of irrelevant alternatives prevents you from using the information needed to implement a Rawlsian SWF; the information who is society's worst-off cannot be used.</p>
<p>Indeed, the relevant information is not even specified in a profile of preferences.</p>

CC BY-SA 4.0
2021-03-16T12:49:24.007Z
2021-03-16T12:49:24.007Z
Michael Greinecker
<p>given Arrow impossibility theorem the only social welfare function that satisfies unrestricted domain, pareto and the independence of irrelevant alternatives is Dictatorship. However I was wondering which one of these criteria the Rawlsian SWF fails. I was suspecting Independence of irrelevant alternatives, however I can't come up with a proof.</p>
<p>Thanks in advance for your help.</p>

<welfare-economics><social-welfare><social-choice>
Rawlsian SWF and Arrow Impossibilty Theorem
Hustler885
# Rawlsian SWF and Arrow Impossibilty Theorem

(Asked by: username_0 on 2021-03-16)

given Arrow impossibility theorem the only social welfare function that satisfies unrestricted domain, pareto and the independence of irrelevant alternatives is Dictatorship. However I was wondering which one of these criteria the Rawlsian SWF fails. I was suspecting Independence of irrelevant alternatives, however I can't come up with a proof.

Thanks in advance for your help.

**Tags:** welfare-economics, social-welfare, social-choice

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## 2 answers

### Answer by username_1 on 2021-03-16. Score: 2 | Accepted answer

Let's say there are individuals 1 and 2, and alternatives A, B, C, and D. Society uses the Rawlsian SWF and thus ranks alternatives according to their maximal rank within individuals' rankings. Denote society's preferences by $\succ^\*$.

If the individual rankings are:

1: A $\succ$ B $\succ$ C $\succ$ D

2: B $\succ$ D $\succ$ A $\succ$ C,

then B $\succ^\*$ A.

If you change the ranking of 1 such that

1: C $\succ$ D $\succ$ A $\succ$ B

2: B $\succ$ D $\succ$ A $\succ$ C,

then A $\succ^\*$ B. Thus, society's ranking of A vs. B has changed, while each individual's ranking of A vs. B stayed the same. Hence this SWF fails IIA.


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### Answer by username_2 on 2021-03-16. Score: 5

Independence of irrelevant alternatives prevents you from using the information needed to implement a Rawlsian SWF; the information who is society's worst-off cannot be used.

Indeed, the relevant information is not even specified in a profile of preferences.


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