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date: 02 December 2020

# Arrow’s Theorem

• Conal DuddyConal DuddyDepartment of Economics, University College Cork
•  and Ashley PigginsAshley PigginsJ.E. Cairnes School of Business & Economics, National University of Ireland, Galway

### Summary

Kenneth Arrow’s “impossibility” theorem is rightly considered to be a landmark result in economic theory. It is a far-reaching result with implications not just for economics but for political science, philosophy, and many other fields. It has inspired an enormous literature, “social choice theory,” which lies on the interface of economics, politics, and philosophy. Arrow first proved the impossibility theorem in his doctoral dissertation—Social Choice and Individual Values—published in 1951. It is a remarkable result, and had Arrow not proved it, it is unlikely that the theorem would be known today.

A social choice is simply a choice made by, or on behalf of, a group of people. Arrow’s theorem is concerned more specifically with the following problem. Suppose that we have a given set of options to choose from and that each member of a group of individuals has his or her own preference over these options. By what method should we construct a single ranking of the options for the group as a whole? Any such method may be represented mathematically by a “social welfare function.” This is a function that receives as its input the preference ordering of each individual and then generates as its output a social preference ordering.

Arrow defined some properties that would seem to be essential to any reasonable social welfare function. These properties are called “unrestricted domain,” “weak Pareto,” “independence of irrelevant alternatives,” and “non-dictatorship.” Each of these properties, when taken alone, does appear to be very necessary indeed. Yet, Arrow proved that these properties are in fact mutually incompatible. This troubling fact has been central to the study of social choice ever since.

### Introduction

Arrow’s theorem can be given many different interpretations, and it can be proved in various ways. A simple proof is given in this article. The approach uses the classical setting of the theorem, where individual preferences are aggregated into a single social preference. Other social choice frameworks exist, but versions of Arrow’s theorem can be established within each of them.

Prior to Arrow’s work, economists had developed a way of thinking about alternative economic policies (or, more generally, social states), through the device of a Bergson-Samuelson social welfare function. Bergson’s (1938) paper established the function and the central ideas were developed in chapter 8 of Samuelson’s influential Foundations of Economic Analysis (Samuelson, 1947). At an abstract level, a Bergson-Samuelson social welfare function is a function $W$ that maps a set of social states $X$ into the real numbers, for example, we have $W(x)$, $W(y)$, and so on where $x$ and $y$ denote alternative social states. The interpretation is that $W(x)$ is an ordinal index of the “social welfare” arising under state $x$. If $W(x)≥W(y)$ then the social welfare is at least as high in $x$ than in $y$ and similar interpretations can be given to $W(x)=W(y)$ and $W(x)>W(y)$. Whether $x$ corresponds to higher social welfare than $y$ can only be determined by making explicit value judgments that will restrict the form that the $W$ function takes. For example, assume that each of the $n≥2$ individuals in society possesses his or her own ordinal utility function $Ui$ defined over $X$. This is a numerical representation of an individual’s preferences over the alternative states. For example, if an individual prefers $x$ to $y$ (maybe because he would be better off in $x$ than in $y$) then it must be the case that $Ui(x)$ is greater than $Ui(y)$. The $Ui$ function is ordinal in that only the order of preference can be inferred from $Ui(x)$, $Ui(y)$, and so on.

We could adopt an “individualistic” perspective and write

$Display mathematics$(1)

According to equation (1), the ordinal index of social welfare in $x$ is determined by a function $f$ that takes as its input $n$ numbers which correspond to the $n$ ordinal utilities in $x$. Under this formulation, social welfare is determined solely by individual utility. Under certain assumptions about the nature of the social states and the nature of individual preferences, as well as under the assumption that $f$ is increasing in $U1, U2, …, Un$, Samuelson derived conditions under which the social welfare function (1) is maximized subject to certain feasibility constraints.

Arrow noticed that it is possible to generalize the Bergson-Samuelson framework as defined in (1). Note that as the social state is varied, as long as individual preferences over the social states remain the same and the ordinal utility functions representing these preferences are unchanged, then the ordinal ranking of social states from society’s point of view (induced by $W$) must also remain unchanged. If the individuals’ ordinal preferences over the states do in fact change (thus changing the ordinal utility functions representing these preferences), then a new social welfare index must be calculated for each state that will potentially induce a new ordering of the states from society’s point of view.

These considerations allow (1) to be recast in terms of what is known as an “Arrow” social welfare function (equation [2]). Let individual $i$’s underlying preference ordering be denoted by $Ri$. Mathematically speaking, $Ri$ is a reflexive, transitive, and complete binary relation over $X$. Writing $xRiy$ means that individual $i$ either prefers $x$ to $y$ or is indifferent between them. A profile of individual preference orderings, $R1, R2, …, Rn$, is a list of $n$ orderings, one per person. Associated to each of these profiles is a social preference ordering $R$, which again is reflexive, transitive, and complete. Therefore, an Arrow social welfare function can be written

$Display mathematics$(2)

Note that every Bergson-Samuelson social welfare function $f$ in (1) corresponds to an Arrow social welfare function $g$ in (2). The converse need not be the case since our assumptions are not sufficient to guarantee that individual preference orderings can be numerically represented (as is required by [1]).

Notice the general nature of the Arrow framework. The set of alternatives $X$ is commonly taken in economics to be a set of social states, with each state denoting a complete description of all relevant aspects of the state of affairs prevailing in society. Alternative interpretations of $X$ are possible. In the standard political science interpretation, $X$ could denote a set of candidates for election and thus (2) would describe a voting rule under which voter preferences over alternative candidates are aggregated to determine a collective (social) preference over the candidates. In what follows, the elements of $X$ are referred to as alternatives.

Arrow’s motivation when developing the impossibility theorem was to think about how social choices can be made when individual preferences conflict. People may differ in their preferences over alternatives, and so we might wonder if these conflicting preferences can be aggregated in a “reasonable” way into a coherent social preference by using some social choice mechanism (like a voting rule). Arrow’s theorem says that if the social choice mechanism is required to satisfy certain conditions, then this is an impossible task. The methods of social choice used in actual societies must, therefore, violate at least one of Arrow’s requirements. Political decision-making perspectives on Arrow’s theorem usually focus on how it shows the impossibility of a political choice that satisfies the general welfare or that there is no collective will (Riker, 1982).

However, Arrow’s theorem does not say that it is impossible to make systematic judgments about social welfare (induced through different economic or social policies). It says that any mechanism for doing this must violate at least one of the assumptions made in the theorem. The theorem does not, therefore, show that there cannot be a Bergson-Samuelson social welfare function—only that this function cannot be derived under the conditions stated by Arrow.

### Examples of $g$ Functions

The best way of introducing Arrow’s theorem is by giving several examples of actual $g$ functions. The simplest is the following variation on the plurality rule. Each individual gives a score of 1 to those alternatives he or she prefers the most (according to their ordering); all other alternatives are given a score of zero. A score here can be thought of as a “vote,” where each individual will vote for at least one alternative, and he or she may vote for more than one if there are multiple alternatives jointly tied at the top of the ordering. Scores are added up across individuals, and society regards one alternative as socially preferred to another if and only if it has a larger total score (and socially indifferent if they have the same total score).

This is an extremely simple function. To see how it works, consider the following example. Suppose that there are two individuals and three alternatives labeled $x$, $y$, and $z$. If both individuals rank $x$ as the unique best alternative and $z$ as the unique worst alternative, then the social ordering of alternatives under this rule will place $x$ at the top (with a score of 2), with $y$ and $z$ tied in second place (both receiving a total score of zero). It is straightforward to see how this particular $g$ function will work at different profiles. However, despite its simplicity, there is something clearly anomalous about the outcome described here. In the example, all of the individuals strictly prefer $y$ to $z$, and yet society is indifferent between $y$ and $z$.

Another $g$ function is given by the Borda rule. Suppose that $X$ contains $m$ alternatives and assume for simplicity that all of the individuals rank these $m$ alternatives from best to worst with no ties/indifferences. If an individual ranks alternative $x$ as best, then alternative $x$ is awarded $m$ points under the Borda rule; if $y$ is second best it is awarded $m−1$ points; and so on. Like the plurality rule, each alternative receives a score from each individual based on its place in that individual’s ordering, and we simply sum the scores to determine the social ranking. For example, suppose that individual 1 strictly prefers $x$ to $y$ and $y$ to $z$ (and hence $x$ to $z$ by the transitivity requirement), and individual 2 strictly prefers $y$ to $z$ and $z$ to $x$ (and hence $y$ to $x$). Under the Borda rule, society ranks $y$ as best (with a total score of 5), $x$ second (with a total score of 4), and $z$ worst (with a total score of 3). If society was required to make a choice on the basis of this social ordering, then the Borda rule would recommend choosing $y$.

Again, this is an extremely simple function. However, just like the plurality rule, it can be problematic. Imagine now that we repeat this exercise with individual 2 changing only her $z$ versus $x$ pairwise ranking. The Borda rule now declares a tie between $x$ and $y$ (both with total scores of 5), which seems anomalous as no individual’s $x$ versus $y$ pairwise ranking has changed. The anomaly is even more striking when we consider social choice. In the original situation, $y$ is the recommended social choice. If we now remove $z$ from the set of alternatives, then we end up with a tie between $x$ and $y$. These choices appear irrational. If $y$ is uniquely chosen when the choice set is ${x,y,z}$ then $y$ should still be uniquely chosen when the choice set contracts to ${x,y}$. Similarly, if $x$ is rejected in favor of $y$ when the choice set is ${x,y,z}$, then $x$ should still be rejected in favor of $y$ when the choice set is ${x,y}$.

We may wish to consider (or invent) other $g$ functions that might overcome the defects identified with the plurality and Borda rules. However, what Arrow’s theorem says (informally speaking) is that when there are at least three alternatives and at least two individuals, then there is in fact no $g$ function that can always satisfy a certain set of natural requirements. Attempting to discover (or invent) a $g$ function that simultaneously meets these requirements is a hopeless task. This is a striking discovery.

Before going on to consider the modern statement of Arrow’s theorem (which differs slightly from Arrow’s original), it is worth emphasizing the importance of having at least three alternatives for Arrow’s theorem to hold. As can be seen from the examples, if there are just two alternatives, then clearly none of the anomalies described could arise. In fact, when there are exactly two alternatives (social choice is uninteresting when there is one alternative), then both the plurality rule and the Borda rule produce the same social ranking as the well-known majority rule. Under the majority rule, $x$ is at least as good as $y$ from society’s point of view if and only if the number of individuals who consider $x$ to be at least as good as $y$ is greater than or equal to the number of individuals who consider $y$ to be at least as good as $x$. A strong case can be made for using the majority rule when there are exactly two alternatives, as was demonstrated in an important paper by May (1952). May demonstrates that in the two-alternative setting, the majority rule is the only $g$ function that satisfies some requirements which are stronger than the corresponding Arrow conditions.

Unfortunately, the problem that arises for the majority rule is that is ceases to be a $g$ function when there are three or more alternatives. Suppose that there are three individuals and individual 1 strictly prefers $x$ to $y$ and $y$ to $z$ (hence $x$ to $z$), individual 2 strictly prefers $y$ to $z$ and $z$ to $x$ (hence $y$ to $x$), and individual 3 strictly prefers $z$ to $x$ and $x$ to $y$ (hence $z$ to $y$). The majority social preference relation will cycle; a majority strictly prefers $x$ to $y$, a majority strictly prefers $y$ to $z$, and a majority strictly prefers $z$ to $x$. Cycling is ruled out by the requirement that the $g$ function associates a single social preference ordering to every profile of individual preferences. This is known as the “majority voting paradox” or “Condorcet’s paradox.”

### Arrow’s Conditions

Arrow presents a set of conditions that a reasonable $g$ function ought to satisfy. He then proves that no $g$ function can satisfy all of them. This article describes these assumptions in turn before presenting a proof of the theorem. Writing $xPiy$ means $xRiy$ and not $yRix$. Again, the absence of the subscript denotes the fact that we are referring to the social preference relation. Note that we present here the modern version of Arrow’s conditions as stated, for example, by Sen (2017, pp. 282–288). These differ slightly from Arrow’s original conditions. However, they correspond closely to the conditions stated in the added chapter of the second edition of Arrow’s book published in 1963.

Unrestricted Domain (UD): The domain of the $g$ function must include every logically possible preference profile.

This condition says that the $g$ function must allow as an input any logically possible profile of individual preference orderings. No ordering is inadmissible for any individual and thus individuals are allowed to hold any preference ordering that they like.

Weak Pareto (WP): For every profile of individual orderings, $(R1, R2, …, Rn)$, in the domain of $g$ and for all $x$ and $y$ in $X$, if $xPiy$ for every individual $i$, then $xPy$.

This condition says that if every individual strictly prefers alternative $x$ to alternative $y$, then society must strictly prefer $x$ to $Y$.

Independence of Irrelevant Alternatives (IIA): Consider any two profiles of preference orderings, $(R1, R2, …, Rn)$ and $(R′1, R′2, …, R′n)$, in the domain of $g$ and any two alternatives $x$ and $y$. If, for every individual $i$, [$xRiy$ if and only if $xR′iy$] and [$yRix$ if and only $yR′ix$], then [$xRy$ if and only if $xR′y$] and [$yRx$ if and only if $yR′x$] where the social orderings $R$ and $R′$ correspond to the preference profiles $(R1, R2, …, Rn)$ and $(R′1, R′2, …, R′n)$, respectively.

This condition requires that, if the individual orderings change but everyone’s ranking of a pair of alternatives remains unchanged, then the social ranking of those two alternatives must remain unchanged though the social ranking over other pairs of alternatives may change. To emphasize how this condition works, in the earlier example describing the “anomaly” that arises under the Borda rule, it would violate IIA for the social preference between $x$ and $y$ to change whenever no individual’s $x$ versus $y$ pairwise ranking changes. IIA has proved to be the most controversial and variously interpreted of the Arrow conditions.

Non-dictatorship (ND): There does not exist any individual $k$ such that for all alternatives $x$ and $y$ and for every profile of individual orderings $(R1, R2, …, Rn)$ in the domain of $g$, if $xPky$ then $xPy$.

This condition says that there should not be any individual such that, whenever he or she strictly prefers any alternative $x$ to any other alternative $y$, society must strictly prefer $x$ to $y$, irrespective of other people’s preferences.

Arrow’s theorem can now be stated: If there are at least two individuals and at least three alternatives in $X$, then there does not exist any g function (as defined in equation [2]) that simultaneously satisfies UD, WP, IIA, and ND.

Although each of Arrow’s conditions looks reasonable in isolation, combining them leads to a counterintuitive conclusion. To emphasize this point, notice the following property about any $g$ function satisfying Arrow’s assumptions. It is called “strict-ranking neutrality.” The property shows, first, that if everyone holds a strict preference over any pair of alternatives then society cannot be indifferent between these alternatives. Second, if everyone’s strict ranking of $x$ and $y$ is the same as their ranking of $z$ and $w$, then society’s strict ranking of $x$ and $y$ must be the same as its ranking of $z$ and $w$. The first part of this property is rather counterintuitive. For example, if there are two individuals and one strictly prefers $x$ to $y$ while the other strictly prefers $y$ to $x$, then social indifference can never be the outcome. Intuitively, however, a tie might be a natural outcome in certain circumstances.

Similarly, the second part of the property ignores how strongly these pairwise preferences are held. Individual 1 may have a strong preference for $x$ over $y$ (measured, perhaps, by the number of alternatives that are between $x$ and $y$ in his or her ordering) while individual 2’s preference for $y$ over $x$ might be mild. It seems reasonable then for society to prefer $x$ to $y$. However, this means that society must prefer $z$ to $w$ when individual 1 mildly prefers $z$ to $w$, while individual 2 strongly prefers $w$ to $z$. Again, this seems counterintuitive.

The proof of the first part of this property is straightforward, and it may help readers to understand how Arrow’s conditions combine together to produce puzzling conclusions. Take any three alternatives, $x$, $y$, and $z$. By UD, the following three preference profiles exist. First take a profile, $(R1, R2, …, Rn)$, in which some individuals strictly prefer $x$ to $y$ and everyone else strictly prefers $y$ to $x$. Second, take another profile $(R′1, R′2, …, R′n)$ in which everyone who strictly prefers $x$ to $y$ at $(R1, R2, …, Rn)$, strictly prefers $z$ to $y$ at $(R′1, R′2, …, R′n)$, and everyone who strictly prefers $y$ to $x$ at $(R1, R2, …, Rn)$ strictly prefers $y$ to $z$ at $(R′1, R′2, …, R′n)$. The third profile $(R1, R2, …, Rn)$ retains these features of $(R1, R2, …, Rn)$ and $(R′1, R′2, …, R′n)$ but adds the assumption that everyone strictly prefers $x$ to $z$. Therefore, at $(R1, R2, …, Rn)$, individuals either strictly prefer $x$ to $z$ and $z$ to $y$, or they strictly prefer $y$ to $x$ and $x$ to $z$.

$R$ denotes the social preference relation at $(R1, R2, …, Rn)$, $R′$ denotes the social preference relation at $(R′1, R′2, …, R′n)$, and so on. From the definition of $x$, we must have either $xRy$ or $yRx$. Without loss of generality, assume that $yRx$. From WP, we have $xPz$. IIA yields $yRx$ and so by transitivity $yPz$. IIA implies that $yP′z$. We now show that given our assumption (that $yRx$), it cannot in fact be the case that $xRy$. By way of contradiction, assume that $xRy$ also holds. Then we can construct a new profile (R**…) from $(R1, R2, …, Rn)$ by simply reversing everyone’s $x$ versus $z$ preference (i.e., they now all strictly prefer $z$ to $x$ rather than $x$ to $z$). No other pairwise preference is affected by this change. Given $xRy$, IIA implies that $xRy$ and WP implies that $zPx$, and so transitivity implies that $zPy$. This would then imply, via IIA, that $zP′y$ and we already know that $yP′z$. This is a contradiction. Therefore, it cannot be the case that if we assume $yRx$ then $xRy$ can also hold. Therefore, $yPx$. Of course, we could have assumed from the beginning that $xRy$ (instead of $yRx$) and a similar argument would imply that $xPy$ (and $zP′y$). This completes the proof of the first implication of strict-ranking neutrality.

The proof of the second implication proceeds as follows. Take a profile, $(R1, R2, …, Rn)$, in which some individuals strictly prefer $x$ to $y$, and everyone else strictly prefers $y$ to $x$. Second, take another profile $(R′1, R′2, …, R′n)$ in which everyone who strictly prefers $x$ to $y$ at $(R1, R2, …, Rn)$, strictly prefers $z$ to $w$ at $(R′1, R′2, …, R′n)$, and everyone who strictly prefers $y$ to $x$ at $(R1, R2, …, Rn)$ strictly prefers $w$ to $z$ at $(R′1, R′2, …, R′n)$. We derived $yPx$ in the proof of the first implication. Also, we know from the first implication that $zP′w$ or $wP′z$. By way of contradiction, we shall assume that $zP′w$. A third profile $(R1, R2, …, Rn)$ retains these features of $(R1, R2, …, Rn)$ and $(R′1, R′2, …, R′n)$ but adds the assumption that the individuals who strictly prefer $x$ to $y$ at $(R1, R2, …, Rn)$ in fact strictly prefer $x$ to $z$, $z$ to $w$ and $w$ to $y$. Similarly, we assume that all of the other individuals strictly prefer $w$ to $y$, $y$ to $x$ and $x$ to $z$. By IIA, we have $yPx$, WP implies $xPz$, and IIA implies $zPw$. By transitivity, $yPw$ which contradicts WP. This contradiction proves the second implication of strict-ranking neutrality.

### A Simple Proof of Arrow’s Theorem

The proof given here is based on the popular “pivotal voter” technique of Barberá (1980) and Geanakoplos (2005). The proof proceeds by assuming that a $g$ function exists that satisfies UD, WP and IIA. It is then shown that this function must violate ND. Let $a$ and $b$ denote two alternatives and let us begin by supposing that all individuals strictly prefer $b$ to $a$. Note that throughout this proof, any preference profile can be postulated, as is permitted by the UD assumption. The $g$ function must output a social preference ordering at every one of these profiles. At this profile, to satisfy the weak Pareto criterion, the collective preference must be $bPa$. Now suppose that we arrange the individuals in a line by ascending order of seniority. We take each person in turn, beginning with the youngest, and we convince that person to change his or her preference from $bPia$ to $aPib$. One by one, we reverse the preferences of the individuals in this way until the collective ranking changes from $bPa$ to $aRb$. This change in the collective preference must happen at some step in the sequence because, if it does not happen, then the weak Pareto criterion will be violated at the end of the sequence. Let us say that this change in the collective preference happens when the $ith$ individual switches his or her preference. For simplicity, let us refer to this individual simply as $i$.

This situation, in which individual $i$ is pivotal, is described by the following two small tables. In the first table we see that all individuals who are younger than $i$ prefer $a$ to $b$, while individual $i$ and all individuals older than $i$ prefer $b$ to $a$. The social preference in this case is $bPa$, as is indicated to the right of the table. Then, in the second table, we see that individual $i$ has switched from strictly preferring $b$ to $a$ to strictly preferring $a$ to $b$. This results in a change to the social preference.

$Display mathematics$(3)
$Display mathematics$(4)

Using (3) and (4), together with Arrow’s conditions, we will show that $i$ must be a dictator.

The first step is to show that $i$ must be pivotal in this way for all pairs of alternatives, not just for $a$ and $b$. Let $x$ and $y$ be two other alternatives and let us return to the situation described in (3). Now we place $x$ just below $a$ and place $y$ just above $b$ in each person’s preference, so that we have the following.

$Display mathematics$

This means that the individuals younger than $i$ have preference ordering $aPxPyPb$, while $i$ and individuals who are senior to $i$ have preference ordering $yPbPaPx$. The weak Pareto condition implies that $aPx$ and that $yPb$. And, since no individual has changed his or her preference over $a$ and $b$, IIA implies that the collective preference over $a$ and $b$ remains $bPa$. Therefore, we have $yPbPaPx$. Note that this implies $yPx$. Now let us remove alternatives $a$ and $b$. Since no individual has changed his or her preference over $x$ and $y$, IIA implies that the collective preference over $x$ and $y$ remains $yPx$. So, we have the following.

$Display mathematics$(5)

In this way, we have been able to take (3) and replace $a$ and $b$ with $x$ and $y$ to obtain (5). Now let us take (4) and use a similar technique. This time we place $x$ just above $a$ and place $y$ just below $b$ in each individual’s preference, so that we have the following.

$Display mathematics$

This time, the weak Pareto criterion implies that $xPa$ and that $bPy$. And IIA implies that $aRb$. Therefore, we have $xPaRbPy$ which implies that $xPy$. Now let us remove $a$ and $b$. By IIA, we have the following.

$Display mathematics$(6)

Comparing the outcomes at (5) and (6), we see that the $ith$ individual is pivotal for $x$ and $y$, just as he or she is for $a$ and $b$. The crucial point is the following. Whenever the individuals who are younger than $i$ disagree with the individuals who are older than $i$, over any pair of alternatives, then the social preference will be the same as the preference of individual $i$.

Note that this argument involved four alternatives: $a$, $b$, $x$, and $y$. If the entire set of alternatives contains only three items, then a slightly altered version of the argument can be used to come to the same conclusion. We omit that version of the argument here.

Now we come to the second stage of the proof. Let $c$ be some third alternative, distinct from $a$ and $b$, and consider the following profile.

$Display mathematics$

Here, the younger individuals prefer $c$ to both $a$ and $b$. But we do not specify their preferences over $a$ and $b$. Symmetrically, the older individuals prefer both $a$ and $b$ to $c$, but their preferences over $a$ and $b$ are left unspecified. Note that individual $i$ sides with the younger individuals when comparing $b$ and $c$ but then sides with the older individuals when comparing $a$ and $c$. Hence, the social preference must have $aPc$ and $cPb$. Therefore, we have $aPcPb$. This implies that $aPb$.

Now let us remove $c$. By IIA, we have the following.

$Display mathematics$(8)

Here, the preferences of all individuals other than $i$ over $a$ and $b$ are unspecified. What we see here is that, whenever individual $i$ prefers $a$ to $b$, the social preference must be $aPb$, irrespective of the preferences of the other individuals. This argument can be applied to any pair of alternatives to show that individual $i$ must have this dictatorial power over all pairs.

### Circumventing the Impossibility Theorem

Arrow’s theorem is concerned with Arrow social welfare functions ($g$ functions as in [2]). Like any function, an Arrow social welfare function has a domain and a co-domain. The domain is the set of all possible inputs for the function. For an Arrow social welfare function, each possible profile of preferences is an element of the domain. The co-domain is the set of all acceptable outputs from the function. For an Arrow social welfare function, each possible social preference ordering is an element of the co-domain.

The domain and the co-domain that are specified in Arrow’s theorem are not germane to every problem of social choice. The domain that is specified is the one that contains every logically possible combination of individual preference orderings. But consider, for example, the problem of distributing 50 chocolates to two individuals. It is logically possible that a person’s first choice is to receive all 50 chocolates, second choice is to receive 32 chocolates, third choice is to receive 45 chocolates, and that this person’s preference continues in this haphazard way. These preferences are hardly sensible. An impossibility theorem that requires us to posit unnatural preference orderings like this would not be very compelling. On the other hand, if the alternatives that we are comparing vary on many dimensions or in ways that are very subjective, then we may find that every possible ordering of those alternatives is plausible. In this case, the unrestricted domain of Arrow’s theorem is appropriate.

The suitability of the co-domain that is specified in Arrow’s theorem is also context-sensitive. Arrow’s co-domain contains every weak ordering of the alternatives. This means that an acceptable outcome is a ranking from best to worst (with ties allowed) with every alternative included. But, in some contexts, it may be acceptable that some alternatives are not ranked. And some other kinds of ambiguity, as discussed later, may also be acceptable. In these cases, we must consider a co-domain different from the one in Arrow’s theorem.

Since the publication of Arrow’s theorem, there has been a great deal of literature on the logical consequences of changing the domain or the co-domain that were specified in that theorem. Each of these approaches to circumventing the theorem are discussed in turn.

#### Domain Restrictions

Note that the domain for an Arrow social welfare function may contain a great many profiles. For example, just four voters and four alternatives produce over 30 million possible profiles. An Arrow social welfare function must associate an outcome to every one of these profiles and will hopefully do so in a way that satisfies Arrow’s conditions. We have seen that this is an impossible task. Yet, it may become possible if we reduce the size of the domain by excluding some profiles. For this reason, an important area of research in response to Arrow’s theorem has focused on what are called “domain restrictions.” Researchers seek to understand when and how possibility may emerge when the unrestricted domain of Arrow’s theorem is replaced with a smaller domain.

Perhaps the most important domain restriction requires that individual preferences be single-peaked. To illustrate what a single peaked preference is, let us take an example. Suppose that there are five candidates standing in an election and that these candidates can be located along the familiar left–right political spectrum. Figure 1 gives an example of a single-peaked preference ordering over these five candidates and an example of a preference ordering that is not single-peaked.

The circles connected by a solid line represent a preference ordering that has a single peak. This individual strictly prefers Moderate right to Centrist, Centrist to Hard right, Hard right to Moderate left, and Moderate left to Hard left. Intuitively, as we move away in any particular direction from the individual’s most preferred option (which corresponds to the peak), then we move down the individual’s ordering. The squares connected by a dashed line represent a preference ordering that is double-peaked and not single-peaked. This preference ordering has two peaks, one at Moderate right and the other at Moderate left.

A single-peaked domain is one that admits only preference orderings that are single peaked relative to some common ordering of the alternatives. This “common ordering” is provided by the left–right political spectrum in the previous example. In other contexts, the alternatives may be arranged by order of how expensive they are or by how geographically distant they are.

Even when candidates are arranged along a left–right spectrum, it does not follow that all voters must have single-peaked preferences. However, if the basis of the common ordering is sufficiently important to all individuals in a group, be it left–right political alignment or geographic distance, then it may be reasonable to expect that every individual will have a single-peaked preference relative to that common ordering.

A very important result due to Black (1948) says that majority rule (which in general may cycle) always produces a transitive social preference whenever individual preferences are all single-peaked. Since majority rule satisfies all of Arrow’s conditions but fails to be a proper $g$ function (because of cycling), we can escape from impossibility by restricting the domain of the Arrow social welfare function by admitting only preferences that are single-peaked.

#### Co-domain Expansions

Domain restriction is just one approach we may explore to find escape routes from impossibility. Instead of restricting the domain, another approach is to expand the co-domain. The co-domain is the set of all acceptable outputs from the Arrow social welfare function. Hence, an expanded co-domain means that we are accepting of a greater variety of outputs from the $g$ function. By being more permissive in this way, we may find that an Arrow social welfare function emerges that can satisfy all of Arrow’s conditions.

We can expand the co-domain drastically by accepting cyclic preferences as valid outcomes. An example of a cyclic preference is $aPbPcPa$. If we are accepting of outcomes such as this, then majority rule satisfies Arrow’s conditions, even if we retain the original domain of Arrow’s theorem. However, in many contexts, a social preference that forms a cycle may simply be nonsensical or of no practical use. So it is worthwhile to consider ways in which we may have a co-domain that is somewhat more permissive than Arrow’s, without going so far as to accept cyclic preferences as valid outcomes.

Arrow required the outcome to be a weak ordering of the alternatives. A weak ordering is a binary relation that is complete and transitive. In the literature on co-domain expansions, as opposed to domain restrictions, both properties of weak orderings have come under scrutiny.

Gibbard (1969/2014) and Sen (1969) allow the social preference to be “quasi-transitive” instead of requiring full transitivity. Quasi-transitivity requires logical consistency among the strict preferences that society has but permits inconsistencies that involve indifference. For example, suppose that society prefers $a$ to $b$, is indifferent between $b$ and $c$, and is indifferent between $a$ and $c$. How can $c$ be equally as good as both $a$ and $b$ when $a$ is better than $b$? Yet, this is permitted by quasi-transitivity. In this way, quasi-transitivity would seem to allow for a certain kind of ambiguity in the social preference.

Gibbard (1969/2014) takes the framework of Arrow’s theorem but expands the co-domain by requiring only quasi-transitivity instead of full transitivity. He shows that we do in fact escape from impossibility through this co-domain expansion. However, an “oligarchy” is the only kind of Arrow social welfare function that can satisfy Arrow’s conditions in this case. This means that there must be a group of people who have the power to veto any social preference. That is, if any member of the oligarchy prefers $a$ to $b$, then the social preference may not rank $b$ above $a$. As such, if the oligarchy is small, the situation is hardly much better than dictatorship, and, if the oligarchy is large, society will be extremely indecisive (i.e., the Arrow social welfare function will produce widespread, perhaps universal, social indifference).

Weymark (1984) takes a different approach to expanding the co-domain of Arrow’s framework. Instead of replacing transitivity with quasi-transitivity, he simply removes the requirement that the social preference be complete. But Weymark finds that, here too, it is only by having an oligarchy that we can satisfy Arrow’s conditions. We must either suffer the tyranny of a small group of people with veto power or accept the paralyzing indecisiveness of a large group of people with veto power. There is a technical difference between Gibbard’s (1969/2014) oligarchy and Weymark’s, but they are substantially very similar.

Loosely speaking, an oligarchy is formed by spreading the power of a dictator over multiple individuals. When there is a dictator, the social preference is $aPb$ if that is the dictator’s preference. In the case of an oligarchy, on the other hand, the social preference is $aPb$ if and only if that preference is held by every oligarch. So the emergence of oligarchy as an alternative to dictatorship is not a very positive result. Indeed, the results of Gibbard (1969/2014) and Weymark (1984) may be presented as impossibility theorems since an oligarchy would rarely be an acceptable method of social choice.

### Reaction From Political Science and Further Reading

In addition to the formal responses to Arrow’s theorem described previously, it is also worth considering the reaction from political scientists. Political scientists emphasize that there are two important aspects to democracy: aggregation and deliberation. Aggregation is usually achieved through voting in elections, and Arrow’s theorem clearly applies to the problem of aggregation. Elections enable society to make social choices when individual preferences conflict. Importantly, where individual preferences come from is not central to the theory of aggregation. They are simply the inputs that, when combined with a method of aggregation, determine the output (the election winner, or set of winners, or ranking of candidates, depending on the context).

Arrow’s theorem has led some to conclude that the aggregative aspect of democracy (voting) is not as valuable as it might first appear. According to this view, elections matter in that they restrain the behavior of politicians by subjecting them to periodic electoral tests. They are not though, in general, a way of discovering the “will of the people.” This view is most often associated with the work of Riker (1982) in a radical interpretation of Arrow’s theorem. Riker argued that Arrow’s theorem shows that “populist democracy” in which elected officials implement the will of the people (or the majority will) in their decision making is impossible. Mackie (2003) is, in part, a response to, and critique of, Riker.

The importance of the deliberative aspect is associated with theorists like Habermas (1996), among others. Habermas argues that public discussion and debate makes people reflect on their preferences. Deliberation is another name for this process of reflection. One possible consequence of deliberation is that people’s preferences may change. Some even go so far as to suggest that everyone in society will hold the same post-deliberation preferences, thus making the problem of social choice trivial. This view is expressed by Elster (1986). He says that under deliberation “there would not be any need for an aggregation mechanism, since a rational discussion would tend to produce unanimous preferences” (p. 112).

It is fair to say that “deliberationists” are more optimistic than Riker (1982) about democracy. For deliberationists, a democracy has certain procedural virtues that go beyond voting. For example, Gutmann and Thompson (2004) define a deliberative democracy as a

form of government in which free and equal citizens (and their representatives), justify decisions in a process in which they give one another reasons that are mutually acceptable and generally accessible, with the aim of reaching conclusions that are binding in the present on all citizens but open to challenge in the future. (p. 7)

Viewed this way, democracy involves a dynamic process of open and transparent debate, the aim of which is to lead to understandable social choices being made. An attempt at a reconciliation between these two traditions has been made by Dryzek and List (2003).

We conclude by giving some guidance on further reading. Arrow’s book is essential reading. The second edition of 1963 is the most commonly cited edition as it includes an additional chapter by Arrow titled “Notes on the Theory of Social Choice” in which Arrow discusses the origin and history of social choice theory and presents a new proof of the impossibility theorem, which is the modern version presented here. The additional chapter also includes some thoughts and reflections on the possibility of social choice. A brief account of the impossibility theorem can be found in Arrow’s own entry on “Arrow’s Theorem” in the New Palgrave Dictionary of Economics (Arrow, 2008). Amartya Sen’s book Collective Choice and Social Welfare, first published in 1970, is a major contribution and a classic of social choice theory. The book arguably brought social choice theory into the mainstream of economics. The book has been updated with 11 new chapters (Sen, 2017). The book alternates between non-mathematical and mathematical chapters, and the non-mathematical chapters give valuable intuitive discussions of the concepts developed in the accompanying mathematical chapters. Sen himself has been a major figure in social choice theory, and his collection of papers on the topic is worth reading in its entirety (Sen, 1997).

For political scientists, a very accessible introduction to Arrow’s theorem and the subsequent literature is Penn (2015). Additionally, the two volumes of Austen-Smith and Banks (2000, 2005) discuss advanced topics in social choice theory (including Arrow’s theorem) that are relevant to political scientists. A technical survey of results in social choice theory is Sen (1986), and Arrow’s theorem in particular is treated by Campbell and Kelly (2002). There is a large literature on social choice with interpersonal comparisons, which has not been touched on here. A good introduction to that literature is Blackorby, Donaldson, and Weymark (1984). On interpretations of Arrow’s theorem in political science, Riker (1982), Patty and Penn (2014) and Miller (2019) are recommended. Finally, the writings of Saari (2001a, 2001b, 2008) contain important discussions of Arrow’s theorem.

#### References

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