# Arrow's Dictator

Arrow's Impossibility Theorem is often presented as a negative result in graduate classes. I wonder if the 'dictator' is taken too literally. If the assumptions of the theorem hold, there exists an individual $$i$$ such that the social preference $$\succcurlyeq$$ is equivalent to $$i$$'s individual preferences $$\succcurlyeq_i$$, irregardless of the preferences of all other individuals. However, if $$i$$ had different preferences, they might not be the dictator, it might be another individual. It seems to me that the theorem is simply stating that someone got lucky and will get exactly what they want and not that they are literally a dictator, since with other preferences they wouldn't be.

Everyone's preferences were taken into account when the selection of dictator was made, so why is the dictator result presented as a negative finding? (see MWG, Kreps (2013), Jehle and Reny etc.)

• What thoughts you need here? Arrow does not even describes different political regimes in his model so trivially he does not mean dictator in a sense autocratic ruler. The person who gets what he wants is a dictator because they get to dictate what the final outcome is overriding preferences of other people. Also please note this is not a discussion forum, can you please formulate some sort of question that can be (at least potentially) given definitive answer?
– 1muflon1
Commented Nov 23, 2022 at 19:50
• Reworded it for you - hope that clears it up. Ex-post (after collecting everyone's preferences through a voting mechanism) it may appear they are overriding the preferences of others but ex-ante their preferences were taken into account when choosing the social preference (it indirectly influences who gets what they want). So I'm asking why it's seen as a negative result
– Fića
Commented Nov 23, 2022 at 22:15
• I think you have a wrong conception of what the social welfare functions in Arrow's theorem are. The statement "However, if i had different preferences, they might not be the dictator, it might be another individual." is not correct. Commented Nov 23, 2022 at 22:37
• Oh really.. Okay, let me have a think about why it's not correct. Thank you
– Fića
Commented Nov 23, 2022 at 22:52

The object of Arrow's theorem are social welfare functions, which maps profiles of strict preference relations to to preference relations, $$f:\mathcal{P}^n\to\mathcal{R}$$. There are some variations in the literature but they are not essential for the issue. We say that $$f$$ is dictatorial if there is some $$i$$ such that $$f(\succeq_1,\succeq_2,\ldots,\succeq_n)=\succeq_i$$ for every profile. So if the preferences of $$i$$ differ in two profiles, the selected social ordering will also differ. A dictator is not just someone who happens to have the same preferences as the selected ordering at some profile. This has to hold at ever profile.