# 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 $$f:\mathcal{P}^N \rightarrow M$$.

1. $$f$$ satsifies Unanimity (Weak Pareto) if all individuals rank $$a$$ first, then $$f(P) = a.$$

2. $$f$$ is Strategy-Proof if it is a dominant strategy to tell the truth about your preference.

3. $$f$$ 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.

• Seems like $f$ should map from the full space of profiles to the space of alternatives? Feb 6 '20 at 7:14
• The answer should definitely end up at Gibbard-Satterthwaite, but right now the link between IIA and strategy-proofness is not really explored, it is not clear why one could "exchange" one for the other. Feb 6 '20 at 7:16
• Yes it would be great if you could explain why (if?) IIA and strategy proofness are equivalent. Notice that strategy proofness is not directly assumed in the problem posed.
– user17900
Feb 6 '20 at 11:55
• @Giskard you're absolutely correct, excuse my lazy notation. I've edited my answer. As for the connection between strategy-proofness and IIA, I'm not sure either. In fact, I wonder if the definition in the question is correct in this context. Feb 6 '20 at 19:44
• @WalrasianAuctioneer I'm not sure what you mean by 'correct'! We can always ask the question: does there exist a voting scheme that satisfies such and such axioms (and this list of axioms need not include strategyproofness -- indeed Arrow's original axioms do not include this, at least explicitly)
– user17900
Feb 6 '20 at 19:55

The axioms in Arrow's impossibility theorem (regarding social choice functionals) are about the preference profiles, so should an analogous theorem regarding social choice functions. The IIA axiom that you list is about the set of alternatives, not the profile of preferences.

Chapter 21.E in MWG proves a result for social choice function analogous to Arrow's impossibility result for social choice functional. There, the counterpart of IIA is a monotonicity condition, which requires that the social choice function be invariant with respect to changes in the preference profile that only affects the lower contour set of the currently chosen alternative.

Just as IIA requires that a social welfare functional's ranking of alternatives $$x$$ and $$y$$ remain unchanged when their relative rankings across two profiles of preferences are unchanged, so monotonicity requires that a social welfare function selects the same alternative $$x$$ as long as its relative position against the other alternatives remain the same across two preference profiles.

Formally, monotonicity can be defined as follows:

Suppose $$f(\succsim_1,\dots,\succsim_I)=x$$. If for all individual $$i$$ and all $$y\ne x$$, a different preference profile $$(\succsim_1',\dots,\succsim_I')$$ is such that $$x\succsim_iy \Rightarrow x\succsim_i'y$$, then $$f(\succsim_1',\dots,\succsim_I')=x$$.

Then, Proposition 21.E.1 of MWG says:

Suppose that the number of alternatives is at least three and that the domain of admissible preference profiles is either $$\mathcal A=\mathbb R^I$$ or $$\mathcal A=\mathcal P^I$$. Then every weakly Paretian and monotonic social choice function $$f:\mathcal A\to X$$ is dictatorial.