# Counting in Arrow's theorem

I seem to be really confused with the counting system in Arrow's theorem. Can I have a simple explanation how they determine the outcome? I can't determine the outcome using rules from my notes. It says the roles are 1) If all vote the same that would be the outcome. 2) the ranking of A over B doesn't depend on other candidates and it depends how they are ranked compared to each other. But then what if 1/2 of the votes put A over B and the other half B over A?

What is the algorithm that gives you the out come?

By way of example: Suppose that we have three candidates A B C and two voters. So we get two votes ABC and BAC what's the outcome?

• In your example you have a tie if the orderings are weighted uniformly. Hence either no outcome or a random choice between A and B hopefully, depending on the system. The quintessential information has been given by Henning. Reading and knowing a full proof of Arrow's theorem is cool in general.
– MWL
Nov 14, 2013 at 16:13
• I actually tried to read [a proof](www.cs.elte.hu/~kope/ultrafilter.pdf) but it actually confuses me more since my impression was that the theorem is stating how to determine the result of the election simply by looking how many times A precedes B. The outcome is AB if A precedes B if it more than half of the votes puts A before B. But it seems it doesn't tell really how to count. It says no matter what rule u put u always have a dictator. Still I don't understand how he determines his outcomes on page 37.
– Bob
Nov 14, 2013 at 16:30

Arrow's impossibility theorem is not about any particular way to determine the outcome. The theorem says that no matter which fixed rule you select for transforming voters' preference orderings into a result, at least one of the following strange results can happen:

• One candidate can win over another even though every voter prefers the other candidate to the one who won.
• The ranking between A and B can change even if all voters keep their preferences between A and B constant (in other words, tactical voting is possible).
• There's a single voter whose ballot is the only one that matters for the result.

(with a few additional technical assumptions omitted).

• Can you please tell me if I get it right now? Thank you very much! So Arrow's theorem states that if we have at least 3 candidates and finite number of voters, each of whom ranks the candidates and if the result is also a ranking then no matter how we determine the outcome if we require the two following rules: 1) if all rank the same that is the outcome 2) whether we put A above B only depends on the ranking of A and B and is independent how other people are ranked in compare to A and B. Then we can't resist having a dictator.
– Bob
Nov 14, 2013 at 16:45
• @Sandra: Yes, that sounds about right. Nov 14, 2013 at 16:48
• @Sandra : There are two additional "hidden" condition in Arrow's Theorem, which are often forgotten, but in fact just as important : 3) Social ordering : the outcome must be a complete and transitive ordering. 4) Unrestricted domain : the social ordering function must be defined over all conceivable profiles of preferences. If you want to practice the theorem, it's a great exercise to try to find examples of social ordering function for combination of all the properties but one (e.g [non-dictatorial, Social ordering, pareto efficiency, unrestricted domain, but not independence]). Nov 15, 2013 at 14:55