Arrow’s impossibility theorem and voting schemes Arrow’s impossibility theorem states that there is no procedure for aggregating individual preference orderings into a collective preference ordering that satisfies certain apparently desirable axioms. This has often been taken to mean that there is no such thing as a good voting system. Actual voting systems, however, do not typically attempt to generate a (complete) social ordering. The goal, rather, is to simply select one candidate who will govern — and the task of the ranking the others, while perhaps interesting, is not of immediate practical importance. This raises a simple question: does Arrow’s result extend to preference aggregation procedures that choose a single candidate based on a profile of individual preferences?

To make this more precise, let me (informally) restate Arrow’s axioms as adapted to the problem at hand:

Unrestricted domain. The procedure chooses a best candidate for every possible list of individual orderings.

Weak Pareto. If all voters prefer candidate A to candidate B, then candidate B is not elected.

Non-dictatorship. There is no voter who alone determines which candidate is elected (regardless of the other voters’ preference orderings).

Independence of Irrelevant Alternatives. If candidate A is elected when opposed to a set of rival candidates, then A will still be elected when opposed by some subset of this set of rival candidates.

(Notice that the transitivity axiom cannot, it would seem, be adapted to this context.)

Question: does there exist a voting scheme that satisfies all of the axioms listed above?
The answer is no in the following sense. 

Suppose there are $N = \{1,\dots,n\}$ individuals, and $M$ alternatives, where $|M|\geq 3$.

Let $\mathcal{P}$ be the space of strict preferences over $M$, each individual has preferences $P_i \in \mathcal{P}$, and let $P = \{P_1,\dots,P_n\}$ be the profile of preferences.

We look for a social choice function $f:\mathcal{P}^N \rightarrow M$.


$f$ satsifies Unanimity (Weak Pareto) if all individuals rank $a$ first, then $f(P) = a.$
$f$ is Strategy-Proof if it is a dominant strategy to tell the truth about your preference.
$f$ is Dictatorial if there exists an individual $i$ such that whenever $f(P) = a$, then $a$ is ranked first according to $i$.



  Theorem (Gibbard–Satterthwaite) A social choice function $f$ is Unanimous and Strategy-Proof if and only if it is Dictatorial.