# Arrow impossibility theorem and social choice.

I have read the Arrow impossibility theorem in Foundations of Mathematical Economics(Michael Carter). It is just too difficult to understand.

So, does Arrow'theorem mean that there is always a dictator in our society,regardless of any system of election ?

Can you recommend me some books that delve deeper into this matter ?

I thank you very much for your answer.

• What do you want to understand? The proof? – Shuhao Cao Jun 19 '13 at 15:27
• Arrow's theorem says "you can have all three nice properties at once". That means that if you have a voting system which satisfies two out of the three propreties, then it cannot satisfy the third. If you have a good supply of voting system examples, I advise you to work out which ones satsify which of these three criteria. You should be able to find examples which satsify any two and not the third. The students I was working with were working out of the social choice chapters in For All Practical Purposes (but I can't recommend this text very strongly...) – rschwieb Jun 19 '13 at 16:21
• Off topic, but Foundations of Mathematical Economics is a great book. Kudos on working through it. Definitely worth the effort. – CommonerG Jul 2 '13 at 13:07
• No problem man, but the important weakness of the book is the typo and the lack of "intelligent" examples to illustrate the concepts. Something like this Arrow theory are very weak-written and hard to understand! (to an obtuse person like me) – SON TO Jul 2 '13 at 15:36

## 3 Answers

Arrow's Theorem doesn't say that you will always have a dictator. It says if you wanted to preserve two other "nice" properties in a voting system (I'll say later what those are) of your design, you can't have them without also having a dictator (can't have your cake and eat it too sort of thing).

Suppose there are $n$ voters and a set of alternatives, $S$, that they are voting on. Alternatives can be whether to paint the city hall green, blue or purple, whether to choose Joe, Bill, Susie, or Alice for class president etc. The only restriction here for the theorem to hold is that you have to be voting on three or more alternatives. Now suppose each of the voters $i \in \{1, \ldots, n\}$ has a personal ranking over the alternatives, P_i (e.g., $P_i = (Susie, Bill, Joe, Alice)$). Assume that these personal rankings are strict, so there are no ties in preferences (if there are ties, break them randomly and theorem works). Now $F$ is a social welfare function if it maps $\Pi_iP \to P$, where $P$ is the set of all possible rankings of the alternatives (so if there are $k$ alternatives, all permutations of $(1, \dots,, k)$). So think of this social welfare function as taking in everyone's ballots and deciding based on its input what society's ranking of the outcomes will be.

Now Arrow's theorem says we can't have all three of the following properties of our social welfare function be true:

1. If $aP_ib$ $\forall i$, $aF(\Pi_iP_i)b$. ($aPb$ means under ranking $P$, $a$ holds a higher position. So this condition says if everyone likes $a$ better than $b$, the ranking that the social welfare function spits out will also respect that relative ranking of the two outcomes; seems natural, no?)

2. If $\Pi_iP_i$ is one set of strict rankings over the alternatives by the $n$ voters, and $\Pi_iR_i$ is another, and moreover, alternatives $a$ and $b$ have the same relative ranking under $R_i$ as they do under $S_i$, then $F(\Pi_iP_i)$ and $F(\Pi_iR_i)$ have the same relative rankings of $a$ and $b$.

3. There is no dictator:, i.e, $F$ is not a projection map. So the social choice function does not merely spit out the preference list submitted by the $ith$ voter for some $i$.

• It's funny: the first line of the answer I had been working on had '"nice" properties' quotes and all, also :) – rschwieb Jun 19 '13 at 16:19
• It's quite difficult, is it not ? – SON TO Jun 19 '13 at 17:10
• @SON TO I saw the theorem's proof in a class taught by Phil Reny, and perhaps his way of showing it is a bit nonstandard. Here is a link to his proof, and maybe it'll help: home.uchicago.edu/~preny/papers/arrow-gibbard-satterthwaite.pdf – user68432 Jun 20 '13 at 3:27
• @rschwieb :) that's kinda funnny – user68432 Jun 20 '13 at 3:31
• Nevertheless, I thank you very much... – SON TO Jun 20 '13 at 14:06

Arrow's Impossibility Theorem states that no voting system can satisfy a set of conditions at a time. One of these conditions is that of being non-dictatorial. The way the theorem is usually proven is by showing that if a system satisfies all other conditions, then it is dictatorial. This is equivalent to the statement that those conditions, plus non-dictatorial, can never be satisfied simultaneously. This does not imply that every voting system is dictatorial, only that if it is not dictatorial it must violate at least one of the other conditions.

Maybe this reference could help you understand the proof, if that's what you're looking for. There are many books on Social Choice Theory out there (that's what you should look for) but I don't know which one to recommend.

H. Moulin, Axioms of cooperative decision making. Cambridge University Press, 1988.

V.I.Danilov, A.I.Sotskov, Social Choice Mechanisms, Springer, 2002.