# Arrow's impossibility theorem

In social choice theory, Arrow's impossibility theorem, the general possibility theorem or Arrow's paradox is an impossibility theorem stating that when voters have three or more distinct alternatives (options), no ranked voting electoral system can convert the ranked preferences of individuals into a community-wide (complete and transitive) ranking while also meeting a specified set of criteria: unrestricted domain, non-dictatorship, Pareto efficiency and independence of irrelevant alternatives (IIA)

So, I'm looking for an example where the preferences satisfy unrestricted domain, non dictatorship, IIA and is an ordering, but not meet the Pareto criteria.

Let the set of alternatives be $A = \left\{a_1,a_2,...,a_k\right\}$. Let the number of players be $n$. Let the set of preference orderings over $A$ be $\mathcal{P}$. Then the set of preference profiles is the Cartesian product $$\mathcal{P}^n = \times_{i=1}^{n} \mathcal{P}.$$ Let us denote the preference ordering $$a_1 \succ a_2 \succ ... \succ a_k$$ by $p^*$. Define the Social Choice Function $F$ by $$\forall p \in \mathcal{P}^n: F(p) = p^*.$$ This SCF $F$ clearly has universal domain, is not a dictatorship, and is also independent of irrelevant alternatives. (Or any alternatives for that matter.)

• Hey, thanks. But I am unable to understand the functioning of this social choice rule. How is it not a dictatorship and how does it not meet Pareto criteria? – LUCIFER Dec 10 '17 at 1:07
• @LUCIFER Could you please define what you think a dictatorship SCF is? And can you please also define the Pareto property for SCFs? – Giskard Dec 10 '17 at 7:28
• Dictatorship is when there is an individual such that if he prefers x over y, then so does the society irrespective of what the society feels. – LUCIFER Dec 10 '17 at 21:00
• And Pareto property is that if all individuals prefer x over y, then so should the society. It is about how society preserves total agreement. – LUCIFER Dec 10 '17 at 21:01
• @LUCIFER If you understand these then surely you see how my SCF fulfills neither condition. I will improve notation a bit. – Giskard Dec 10 '17 at 21:16

The Pareto criterion has two effects: It guarantees that every ranking can occur as a social ranking and it connects social rankings to individual rankings. If one drops the Pareto criterion but keeps the assumption that every social ranking is possible, one obtains a generalization in which every SCG corresponds to a dictatorship or an anti-dictatorship in which the social ranking is exactly opposite to the ranking of a specific individual, an anti-dictator. This is known as Wilson's theorem, originally from:

Wilson, Robert. "Social choice theory without the Pareto principle." Journal of Economic Theory 5.3 (1972): 478-486.

• "keeps the assumption that every social ranking is possible" That is not an assumption in this question though. – Giskard Dec 11 '17 at 14:03