Does Arrow's Theorem apply when choosing a single best candidate?

Question

According to Wiki, Arrow's Impossibility Theorem proves that we cannot create a social welfare function that obeys unanimity, non-dictatorship, and IIA.

However, in real elections, we want to choose just one candidate, rather than ranking them all. Let $f$ be a function that maps a ballot list B to a single ''best'' candidate. Does Arrow's Theorem imply that no $f$ can satisfy unanimity, non-dicatorship, and IIA?

I believe these are natural ways to extend the definitions of unanimity, non-dictatorship, and IIA precisely to a function that chooses a single ''best'' candidate:

• $f$ satisfies unanimity when each ballot prefers $a$ to $b$, then $f(B)\neq b$.
• $f$ satisfies non-dictatorship if $f$ does not simply return the top choice of some fixed voter.
• $f$ satisfies IIA if for any two ballot lists $B_1,B_2$ that have $a,b$ in the same relative positions, then $f(B_1)=a$ or $b$ implies that $f(B_2)=f(B_1)$.

I'm hoping that we can easily show that, e.g., the existence of such an $f$ would imply existence of a corresponding social welfare function $w$, e.g. by defining $w(C, B)$ such that $w(f(C, B)) > w(f(C-\{f(C, B)\}, B)) > \ldots$.

Answer 1

The answer to your question is provided by the Gibbard-Satterthwaite theorem, well known in the mechanism design literature. The theorem states

Suppose there are at least three alternatives and that for each individual any strict ranking of these alternatives is permissible. Then the only unanimous, strategyproof social choice function is a dictatorship. (Benoit 2000)

Actually, early proofs followed the strategy you are outlining and used the Arrow's theorem to prove the G-S theorem. Today, we have direct proofs (I rather like the Benoit (2000) but there are others as well) and we also have a better understanding that the two theorems are actually deeply connected. Reny (2001) provides a single proof of both theorems!

Edit:

To clarify, in the statement of the theorem, by social choice function is meant any function that chooses from the ballot list $B$ a single alternative $a$, i.e. the "best" candidate as you call it. By unanimity is meant something less restrictive then what you propose (but implied by your condition): if each ballot prefers $a$ to all other alternatives, then $a$ is chosen. Dictatorship is as you defined it. Strategyproof simply means that no voter wants to change her ballot $B_i$ given the ballots of all other voters $B_{-i}$. Or $f(B_i,B_{-i}) \succeq_if(\tilde{B}_i,B_{-i})$ for any permissible $\tilde{B}_i$.

Sources:

Benoıt, Jean-Pierre. "The Gibbard–Satterthwaite theorem: a simple proof." Economics Letters 69.3 (2000): 319-322.

Reny, Philip J. "Arrow’s theorem and the Gibbard-Satterthwaite theorem: a unified approach." Economics Letters 70.1 (2001): 99-105.

Answer 2

I am no sure if it is an answer, but it is a bit too long for a comment.

I am slightly buffled by your question because I thought Arrow's impossiblity theorem was based on 'best candidate' type of election.

if 4 people ABCD are choosig between XYZ

A's preference XYZ B's Preference YXZ C's preference ZYX D's preference ZYX

everyone votes f(ABCD) = Z

However, if candidate X

A and B's preference is now YZ YZ C and D's preference is now ZY ZY

so there is no majority with 1/2 prefer X and 1/2 prefer Y

This system is a 'choose your best candidate system'. Unless you force people not to vote, should their favourite candidate die or drop out, this would be the outome of the system, so the best candidate election will fail this IIA criterion.