Arrow's impossibility theorem

When Kenneth Arrow proved his theorem in 1950, it inaugurated the modern field of social choice theory, a branch of welfare economics studying mechanisms to aggregate preferences and beliefs across a society. Such a mechanism of study can be a market, voting system, constitution, or even a moral or ethical framework. • Unrestricted domain – the social choice function is a total function over the domain of all possible orderings of outcomes, not just a partial function. • In other words, the system must always make some choice, and cannot simply "give up" when the voters have unusual opinions. • Without this assumption, majority rule satisfies Arrow's axioms by "giving up" whenever there is a Condorcet cycle. • In other words, it is possible for any candidate to defeat any other candidate, given some combination of votes. • This is often replaced with the stronger Pareto efficiency axiom: if every voter prefers over , then should defeat . However, the weaker non-imposition condition is sufficient. Independence A commonly-considered axiom of rational choice is independence of irrelevant alternatives (IIA), which says that when deciding between and , one's opinion about a third option should not affect their decision. to philosopher Sidney Morgenbesser: : Morgenbesser, ordering dessert, is told by a waitress that he can choose between blueberry or apple pie. He orders apple. Soon the waitress comes back and explains cherry pie is also an option. Morgenbesser replies "In that case, I'll have blueberry." Arrow's theorem shows that if a society wishes to make decisions while always avoiding such self-contradictions, it cannot use ranked information alone. == Theorem ==

Intuitive argument Condorcet's example is already enough to see the impossibility of a fair ranked voting system, given stronger conditions for fairness than Arrow's theorem assumes. Suppose we have three candidates (A, B, and C) and three voters whose preferences are as follows: If C is chosen as the winner, it can be argued any fair voting system would say B should win instead, since two voters (1 and 2) prefer B to C and only one voter (3) prefers C to B. However, by the same argument A is preferred to B, and C is preferred to A, by a margin of two to one on each occasion. Thus, even though each individual voter has consistent preferences, the preferences of society are contradictory: A is preferred over B which is preferred over C which is preferred over A. Because of this example, some authors credit Condorcet with having given an intuitive argument that presents the core of Arrow's theorem. ; Pareto efficiency : If alternative \mathbf{a} is preferred to \mathbf{b} for all orderings R_1, \ldots, R_N, then \mathbf{a} is preferred to \mathbf{b} by F(R_1, R_2, \ldots, R_N). and Ariel Rubinstein. The simplified proof uses an additional concept: • A coalition is weakly decisive over (x, y) if and only if when every voter i in the coalition ranks x \succ_i y, and every voter j outside the coalition ranks y \succ_j x, then x \succ y. Thenceforth assume that the social choice system satisfies unrestricted domain, Pareto efficiency, and IIA. Also assume that there are at least 3 distinct outcomes. {{Math proof|proof=Let z be an outcome distinct from x, y. Claim: G is decisive over (x, z). Let everyone in G vote x over z. By IIA, changing the votes on y does not matter for x, z. So change the votes such that x \succ_i y \succ_i z in G and y \succ_i x and y \succ_i z outside of G. By Pareto, y \succ z. By coalition weak-decisiveness over (x, y), x \succ y. Thus x \succ z. \square Similarly, G is decisive over (z, y). By iterating the above two claims (note that decisiveness implies weak-decisiveness), we find that G is decisive over all ordered pairs in \{x, y, z\}. Then iterating that, we find that G is decisive over all ordered pairs in X.}} {{Math proof|proof=Let G be a coalition with size \geq 2. Partition the coalition into nonempty subsets G_1, G_2. Fix distinct x, y, z. Design the following voting pattern (notice that it is the cyclic voting pattern which causes the Condorcet paradox): \begin{align} \text{voters in } G_1&: x \succ_i y \succ_i z \\ \text{voters in } G_2&: z \succ_i x \succ_i y \\ \text{voters outside } G&: y \succ_i z \succ_i x \end{align} (Items other than x, y, z are not relevant.) Since G is decisive, we have x \succ y. So at least one is true: x \succ z or z \succ y. If x \succ z, then G_1 is weakly decisive over (x, z). If z \succ y, then G_2 is weakly decisive over (z, y). Now apply the field expansion lemma.}} By Pareto, the entire set of voters is decisive. Thus by the group contraction lemma, there is a size-one decisive coalition—a dictator. Proofs using the concept of the pivotal voter originated from Salvador Barberá in 1980. The proof given here is a simplified version based on two proofs published in Economic Theory. Setup Assume there are n voters. We assign all of these voters an arbitrary ID number, ranging from 1 through n, which we can use to keep track of each voter's identity as we consider what happens when they change their votes. Without loss of generality, we can say there are three candidates who we call A, B, and C. (Because of IIA, including more than 3 candidates does not affect the proof.) We will prove that any social choice rule respecting unanimity and independence of irrelevant alternatives (IIA) is a dictatorship. The proof is in three parts: • We identify a pivotal voter for each individual contest (A vs. B, B vs. C, and A vs. C). Their ballot swings the societal outcome. • We prove this voter is a partial dictator. In other words, they get to decide whether A or B is ranked higher in the outcome. • We prove this voter is the same person, hence this voter is a dictator. Part one: There is a pivotal voter for A vs. B Consider the situation where everyone prefers A to B, and everyone also prefers C to B. By unanimity, society must also prefer both A and C to B. Call this situation profile[0, x]. On the other hand, if everyone preferred B to everything else, then society would have to prefer B to everything else by unanimity. Now arrange all the voters in some arbitrary but fixed order, and for each i let profile i be the same as profile 0, but move B to the top of the ballots for voters 1 through i. So profile 1 has B at the top of the ballot for voter 1, but not for any of the others. Profile 2 has B at the top for voters 1 and 2, but no others, and so on. Since B eventually moves to the top of the societal preference as the profile number increases, there must be some profile, number k, for which B first moves above A in the societal rank. We call the voter k whose ballot change causes this to happen the pivotal voter for B over A. Note that the pivotal voter for B over A is not, a priori, the same as the pivotal voter for A over B. In part three of the proof we will show that these do turn out to be the same. Also note that by IIA the same argument applies if profile 0 is any profile in which A is ranked above B by every voter, and the pivotal voter for B over A will still be voter k. We will use this observation below. Part two: The pivotal voter for B over A is a dictator for B over C In this part of the argument we refer to voter k, the pivotal voter for B over A, as the pivotal voter for simplicity. We will show that the pivotal voter dictates society's decision for B over C. That is, we show that no matter how the rest of society votes, if pivotal voter ranks B over C, then that is the societal outcome. Note again that the dictator for B over C is not a priori the same as that for C over B. In part three of the proof we will see that these turn out to be the same too. In the following, we call voters 1 through k − 1, segment one, and voters k + 1 through N, segment two. To begin, suppose that the ballots are as follows: • Every voter in segment one ranks B above C and C above A. • Pivotal voter ranks A above B and B above C. • Every voter in segment two ranks A above B and B above C. Then by the argument in part one (and the last observation in that part), the societal outcome must rank A above B. This is because, except for a repositioning of C, this profile is the same as profile k − 1 from part one. Furthermore, by unanimity the societal outcome must rank B above C. Therefore, we know the outcome in this case completely. Now suppose that pivotal voter moves B above A, but keeps C in the same position and imagine that any number (even all!) of the other voters change their ballots to move B below C, without changing the position of A. Then aside from a repositioning of C this is the same as profile k from part one and hence the societal outcome ranks B above A. Furthermore, by IIA the societal outcome must rank A above C, as in the previous case. In particular, the societal outcome ranks B above C, even though Pivotal Voter may have been the only voter to rank B above C. By IIA, this conclusion holds independently of how A is positioned on the ballots, so pivotal voter is a dictator for B over C. Part three: There exists a dictator In this part of the argument we refer back to the original ordering of voters, and compare the positions of the different pivotal voters (identified by applying parts one and two to the other pairs of candidates). First, the pivotal voter for B over C must appear earlier (or at the same position) in the line than the dictator for B over C: As we consider the argument of part one applied to B and C, successively moving B to the top of voters' ballots, the pivot point where society ranks B above C must come at or before we reach the dictator for B over C. Likewise, reversing the roles of B and C, the pivotal voter for C over B must be at or later in line than the dictator for B over C. In short, if kX/Y denotes the position of the pivotal voter for X over Y (for any two candidates X and Y), then we have shown : kB/C ≤ kB/A ≤ kC/B. Now repeating the entire argument above with B and C switched, we also have : kC/B ≤ kB/C. Therefore, we have : kB/C = kB/A = kC/B and the same argument for other pairs shows that all the pivotal voters (and hence all the dictators) occur at the same position in the list of voters. This voter is the dictator for the whole election. Stronger versions Arrow's impossibility theorem still holds if Pareto efficiency is weakened to the following condition: ; Non-imposition : For any two alternatives a and b, there exists some preference profile such that is preferred to by . == Interpretation and practical solutions ==

Arrow's theorem establishes that no ranked voting rule can always satisfy independence of irrelevant alternatives, but it says nothing about the frequency of spoilers. This led Arrow to remark that "Most systems are not going to work badly all of the time. All I proved is that all can work badly at times." Attempts at dealing with the effects of Arrow's theorem take one of two approaches: either accepting his rule and searching for the least spoiler-prone methods, or dropping one or more of his assumptions, such as by focusing on rated voting rules. Thus, Condorcet proved a weaker form of Arrow's impossibility theorem long before Arrow, under the stronger assumption that a voting system in the two-candidate case will agree with a simple majority vote. However, a well-defined Condorcet winner does exist if the distribution of voters is rotationally symmetric or otherwise has a uniquely-defined median. In most realistic situations, where voters' opinions follow a roughly-normal distribution or can be accurately summarized by one or two dimensions, Condorcet cycles are rare (though not unheard of). Holliday and Pacuit devised a voting system that provably minimizes the number of candidates who are capable of spoiling an election, albeit at the cost of occasionally failing vote positivity (though at a much lower rate than seen in instant-runoff voting). This opens up the possibility of finding another social choice procedure that satisfies independence of irrelevant alternatives. Arrow's theorem can thus be considered a special case of Harsanyi's utilitarian theorem and other utility representation theorems like the VNM theorem, which show rational behavior requires consistent cardinal utilities. While Arrow's theorem does not apply to graded systems, Gibbard's theorem still does: no voting game can be straightforward (i.e. have a single, clear, always-best strategy). Meaningfulness of cardinal information Arrow's framework assumed individual and social preferences are orderings or rankings, i.e. statements about which outcomes are better or worse than others. Taking inspiration from the behavioralist approach, some philosophers and economists rejected the idea of comparing internal human experiences of well-being. Arrow originally agreed with these position, rejecting the meaningfulness of cardinal utilities, Similarly, Amartya Sen first claimed interpersonal comparability is necessary for IIA, but later came to argue in favor of cardinal methods for assessing social choice, arguing they would only require "rather limited levels of partial comparability" to hold in practice. Other scholars have noted that interpersonal comparisons of utility are not unique to cardinal voting, but are instead a necessity of any non-dictatorial choice procedure, with cardinal voting rules making these comparisons explicit. David Pearce identified Arrow's original nihilist interpretation with a kind of circular reasoning, with Hildreth pointing out that "any procedure that extends the partial ordering of must involve interpersonal comparisons of utility." Similar observations have led to implicit utilitarian voting approaches, which attempt to make the assumptions of ranked procedures more explicit by modeling them as approximations of the utilitarian rule (or score voting). In psychometrics, there is a general consensus that self-reported ratings (e.g. Likert scales) are meaningful and provide more information than pure rankings, as well as showing higher validity and reliability. Cardinal rating scales (e.g. Likert scales) provide more information than rankings alone. A review by Kaiser and Oswald found that ratings were more predictive of important decisions (such as international migration and divorce) than even standard socioeconomic predictors like income and demographics, writing that "this feelings-to-actions relationship takes a generic form, is consistently replicable, and is fairly close to linear in structure. Therefore, it seems that human beings can successfully operationalize an integer scale for feelings". suggesting human behavior can cause IIA failures even if the voting method itself does not. However, past research has typically found such effects to be fairly small, and such psychological spoilers can appear regardless of electoral system. Balinski and Laraki discuss techniques of ballot design derived from psychometrics that minimize these psychological effects, such as asking voters to give each candidate a verbal grade (e.g. "bad", "neutral", "good", "excellent") and issuing instructions to voters that refer to their ballots as judgments of individual candidates. In spatial (n-dimensional ideology) models of voting, this can be relaxed to require only 1-e^{-1} (roughly 64%) of the vote to prevent cycles, so long as the distribution of voters is well-behaved (quasiconcave). Infinite populations Fishburn shows all of Arrow's conditions can be satisfied for uncountably infinite sets of voters given the axiom of choice; however, Kirman and Sondermann demonstrated this requires disenfranchising almost all members of a society (eligible voters form a set of measure 0), leading them to refer to such societies as "invisible dictatorships". == Common misconceptions ==

Arrow's theorem is not related to strategic voting, which does not appear in his framework, == See also ==