The Arrow Possibility Theorem (notes by Simon Vicary)

1. Introduction

The Arrow Possibility Theorem is the fundamental theorem in social choice theory. Indeed, by proving the Theorem in 1951 Arrow can be said to have founded the field of social choice. For this reason alone, if for no other, the award of a joint Nobel prize to Arrow, together with Hicks was an insult. Hicks suffered a similar insult too!

The theorem is a difficult one. You will be glad to learn that you will not be expected to know how to prove it. Relatively easy proofs can be found in Sugden and MacKay, if you are curious. Even after nearly 50 years, the simple proofs need a little care if you are to work through them. Our task today is one that also proved difficult, understanding what the theorem actually means, and what its significance is in economics. However, the effort is worthwhile. The theorem has a habit of cropping up in various contexts, and, furthermore, helps us to understand how the economist’s approach to welfare fits in with some of the themes in political philosophy.

Even Arrow himself had some difficulties in the early days. In particular, he referred to the impossibility of constructing what he called a social welfare function, and was criticised for this by a number of economists, notably Little (1952). To gain an initial understanding of the theorem it is probably best to forget such notions and concentrate on the idea of a constitution.

2. The Notion of a Constitution

What do we mean by the term constitution? Real-world constitutions are complicated beasts. They vary according to time and place. Some are highly legalistic, for example that of the United States, some are informal, such as that of the United Kingdom. Custom and implicit understandings probably play a role in all constitutions. However, the legal issue is here less important than what all constitutions do. Whether we are dealing with the United States or the local tiddlywinks league they have this in common:

There is a set of individuals who have some common interest, and have to make decisions as a group which, in principle, affect all members of the group. The individuals have different views as to the best decision to be made in any particular set of circumstances. The Constitution is a device by which the differing preferences of the individuals are put together so as to reach a common decision.

This view of the constitution explains the various terminologies that you might see if you read widely on this topic. Sen, for example, talks of a Collective Choice Rule. MacKay talks of an aggregation device. You may spot some parallels here with Rousseau’s general will, but we defer consideration of this point for the time being.

More formally, we can think of our community having to make a choice between a certain set of alternatives. We will call this set of alternatives the Set of Social States. Hence:

S is the set of social states.

The alternatives themselves are, of course, the elements of set S. We denote these as x, y, z etc. We can then write:

x ∈ S, y ∈ S, z ∉ S

The problem the community faces is this. There is a set of alternative social states S. How does it choose between them? Before proceeding it might be useful to have a firmer idea as to what constitutions actually are. Let us think of some examples.

3. Examples of Constitutions

As some of the issues underlying the theorem relate to developments in political philosophy that started in the 17^th century, we start with two rather whimsical ‘constitutions’.

· Charles I, one member of society, chooses for the group.

· God chooses, via His representative, Charles I.

· All decisions are arrived at by taking a majority vote.

· All decisions are taken by individuals operating in a market system re-inforced by a set of laws to which all individuals have given their consent.

These constitutions are easy to understand, if only in principle. Even so, it is not clear how some of them would work. Where there are many alternatives what exactly do we mean by a majority vote? How could we gain universal agreement to a set of laws? Nevertheless, the general principle should be clear. In particular, note a special case of the general problem. Any voting system collects some information on individual preferences, and puts this information together in some way so as to arrive at a collective decision. A voting system is therefore a special case of a constitution as we have defined it. The Arrow Theorem thus applies to all voting systems. Consider the following, discussed further in Sugden:

· Plurality: The alternative chosen is that alternative which gains the greatest number of individual first preferences. This is how MPs are chosen in the U.K.

· Single Transferable Vote: Voters are asked to rank alternatives (candidates) on their ballot paper. If there is a candidate with more than 50% of first preferences, that alternative is chosen. If there is no such alternative, then that alternative with the lowest number of first preferences is eliminated. The votes for that alternative are then re-allocated according to those voters' second preferences. The votes are counted again, and an alternative is chosen if it gains 50% of the vote. If not, the alternative with the least number of "votes" is eliminated, and its votes re-assigned as before. The process continues until an alternative does gain 50% of votes. This method is used by the Labour Party to select its leader, and is also used to select the President of France (although in this case only two rounds of voting are used). In these cases, though, the vote is taken sequentially, and is called the method of exhaustive ballot. In principle, though, it is the same procedure.

· The Borda Count: Here voters are again asked to rank candidates in order of preference. However, in this case votes award points to the candidates. Thus if there are four candidates, each candidate will gain 4 points for a first preference, 3 for a second preference, 2 for a third preference, and 1 for a final preference. (Or perhaps 3,2,1,0). Points are summed, and the winning candidate is the one with the greatest number of points. Avariant of this method is used to determine the winner of the Eurovision Song Contest! (The scoring does not quite correspond to the pattern outlined.)

· Committee Procedure: In this case each alternative is placed in a pair-wise majority vote with each other alternative. That alternative which beats all other alternatives is chosen.

Wikipedia http://en.wikipedia.org/wiki/Voting_system gives you a comprehensive account of voting systems. For this module, you do not need to know all about the various schemes on offer, and to keep things manageable we will focus on the four listed.

There is one interesting property to note about each of these voting schemes, as with any other. Besides making a choice in any given situation, the voting system also (with one possible exception) provides an overall ranking of the alternatives. This is easy to see with the Borda Count, but it is also true of the other schemes, as you will see if you think about it.

4. Arrow's Problem

Armed with our knowledge of constitutions, or voting systems, we can now return to Arrow. It is important at the outset to have a firm grasp of the problem he faced. This is what it was.

Suppose, thought Arrow, that we have a set of individuals. Suppose these individuals have to make a collective choice. That is, they have to choose between alternatives that affect each of them. Suppose too that each individual has a ranking over the alternatives, or the elements of S. How should the group translate these individual rankings into a collective decision?

As we have seen, there are various "constitutions" that make this translation for us. In this sense, the problem can obviously be "solved". Arrow's point, however, was that we are obviously not indifferent as to the way in which individual preferences are to be combined together so as to achieve the collective decision. Consider the simple constitutions I listed earlier. It is obviously an important matter which is chosen. Indeed, ultimately, civil wars and revolutions are concerned with exactly this question.

Arrow therefore asked this question. Suppose we have our individuals each with his or her individual ranking. Suppose we make some value judgements reflecting the sort of restrictions we might want to see placed on the operation of the constitution. What sort of constitution or, if you like, voting system would emerge?

Now we obviously cannot be too demanding. The restrictions we use must be mild and acceptable to any "reasonable" individual. Arrow thought of using four such restrictions. His expectation was that he would be able to say very little about the sort of constitution that would emerge. In fact, what happened was the very opposite. The "desirable" properties he proposed turned out to be impossible to achieve together in any constitution, hence the "Possibility" or "Impossibility" Theorem.

The desirable properties were presented as a set of "Arrow Axioms". The exact presentation varies from writer to writer, but no matter what version you read, the meaning is much the same. We now turn to an exposition of these axioms.

5. The Arrow Problem

First let us clarify the setting of the problem.

Each individual is supposed to have a ranking over the elements of set S. We denote this ranking by ≿_i, so that x≿_i y means that individual i considers alternative x to be at least as good as alternative y. The relation ≿_i incorporates indifference. You will be familiar with this idea from basic consumer theory. The strict preference relation is derived in the usual way, so that we have:

{x ≿_iy, ~ y≿_ix} ⇔ {x≻_iy}

with indifference being defined by:

{x ≿_iy, y≿_ix} ⇔ {x~_iy}

Now we know that ≿_i gives us an ordering over the objects of choice if the following hold:

Completeness: ≿_i y or y≿_i x

Reflexivity: ∀ x ∈ S, x ≿_i x

Transitivity: ∀ x, y, z ∈ S, {x ≿_i y, y ≿_i z} ⇔ x ≿_i z

This means that the individual ranks all the alternatives from the best to the worst. Now Arrow approached the problem in the following way. Starting from the individual orderings, he asked if it would be possible to construct a group ordering. We denote the group ordering simply by ≿ (no subscript). It is in this sense that Sen can talk of a collective choice function ≿ = C(≿₁, ≿₂,..., ≿_n). The task of the constitution therefore was to translate individual rankings into a group ranking, and Arrow tried to investigate the circumstances in which this was possible.

It will be noted that it may be asking too much of a constitution to provide group rankings, although this is in fact what happens with the voting rules mentioned earlier. As it happens, if we simply insist that the constitution makes rational and consistent choices from any subset the Arrow Theorem is much the same. We will therefore follow the original Arrow approach here, as it is a little simpler. We thus note:

Requirement: The constitution has to provide a group ranking given that each individual in the group has an ordering over any set of social states.

We now state the Arrow Axioms.

6. The Arrow Axioms

Axiom 1 Unrestricted Domain

The constitution must provide a ranking of social alternatives for all logically possible combinations of individual orderings.

The word domain here reminds us of Sen's terminology of a "social choice function", ≿ = C(≿₁, ≿₂,..., ≿_n). The domain of the function is the orderings that the individuals together can record. There must be no restrictions on the preferences that any individual can state, except for the "rationality" axioms of completeness, reflexivity and transitivity.

Axiom 2 The Pareto Principle

x ≻_i y ∀ i ⇒ x ≻ y

Thus, if every single individual considers that alternative x is better than alternative y, then the constitution should choose x over y. This axiom captures the idea that we would want the constitution to reflect the unanimous preference of the individuals in society. Note, however, the very weak version of unanimity used here. Should there be just one individual who is indifferent as between x and y, then there is no restriction as to how the constitution should rank x and y.

The next axiom is the most difficult one, and caused the greatest difficulty when people first studied the theorem. Indeed, even Arrow himself had problems with this axiom. It is difficult to understand, and so before presenting it, a few words of explanation will be helpful.

Suppose {≿₁, ≿₂, ...., ≿_n} and {≿*₁, ≿*₂,...., ≿*_n} are two possible sets of rankings for the individuals in the community. They differ from one another, but in a rather special way. There exists a subset of S, say T such that for all i:

x ≿_i y ⇔ x ≿*_i y for x, y ∈ T

Thus as far as the subset T is concerned, there is no change in any individual's ranking of social states. If this is true the next axiom can be stated as follows:

Axiom 3 The Independence of Irrelevant Alternatives

Suppose we have two sets of rankings as described, with the set T. In this case, ∀ x, y ∈ T, x ≿ y ⇔ x ≿* y

Note in the statement of the axiom that there is no i subscript: as far as the set T is concerned, the constitution ranks the alternatives in exactly the same way as before. This seems natural, in that within set T no individual has changed rankings. However, the two rankings referred to are different. But they differ with respect only to alternatives outside set T. Such alternatives are irrelevant as far as the ranking of alternatives within set T are concerned. Hence the name.

An example may be useful. Suppose S = {x, y, z} and our two rankings are:

Individual 1: x ≻₁ y ≻₂ z

Individual 2: z≻₂ x≻₂ y

Consider now the set T = {x, z}. As far as T is concerned rankings of y are irrelevant. Individuals can change the ranking of alternative y in any way they like, but this must have no effect on the group ranking of x and z.

The axiom as given is in the form Arrow originally gave it. In simple terms it says that the ranking of a set of alternatives T should not depend on how an alternative outside this set (z) is ranked. Suppose this ‘outside’ alternative z is not available. The alternatives within T receive a ranking. Now suppose that z becomes available. Would this affect the ranking of alternatives within T? The answer should be “no”. Suppose everyone thought z was a dreadful choice. This is pretty much equivalent to saying it is unavailable. The outcome will be the same as if this were so. Under Arrow’s version of the axiom the ranking of T will be the same regardless of how z is ranked by individuals, so the ranking of T would be the same as if z were unavailable. Hence the axiom could also be put in another way: if x≿y in set T, and if x, y ∈U. Suppose too that U⊂T. In this case it remains that x≿y. In words, if x wins in a big set, it must also win in a smaller set.

Way back in 1996 the BBC brought out a series of books, edited by Griff Rhys Jones, entitled The Nation’s Favourite 100 Poems. Other books that followed in the series gave the nation’s 100 favourite 20^th century poems, 100 favourite love poems, comic poems etc. The nation’s favourite poem on this exercise was Kipling’s If… This was written in 1912, but it was not the nation’s favourite 20^th century poem. Hence was a violation of independence! (The ranking of 20^th century poems was not invariant with respect to the ranking of poems written before 1900.)

Suppose you go into a café, and see just 2 drinks on the menu, tea and coffee. The conversation goes like this:

You: A coffee, please

Waiter: We do have cocoa. It’s just not on the menu.

You: Oh well in that case I’ll have tea.

If you find this conversation strange then you may feel there is some argument for the independence axiom.

Axiom 4 Non-Dictatorship

~∃ i such that ∀ x, y ∈ S, x ≻_i y ⇒ x ≻ y

If the statement in the axiom were true, then there would be an individual whose preference between any two alternatives automatically, and regardless of the preference of anyone else, determined group preference. In other words, the group or constitutional ranking always corresponds to that of one single individual. This is hardly a desirable feature of any constitution. A natural term for such an individual is "dictator", and it seems a very mild requirement of any constitution, that it should not be dictatorial.

These are the four axioms. We can now state the Theorem.

7. The Arrow Theorem

Statement of the Theorem

It is not possible to construct a constitution that satisfies all four Arrow Axioms and at the same time delivers a group ordering of social alternatives from the individual orderings.

Proof

Actually, you don't need to know the proof, but it might be useful to outline one very common way in which the Theorem is proved.

Step 1: The Decisive Group.

First define a decisive group. A group of individuals is said to be decisive for alternative x as opposed to alternative y, if, whenever every single member of the group prefers x to y then the constitution must rank x above y, regardless of the preferences of any other members of the community. Note the difference between decisiveness and dictatorship. A dictator is a single person who is decisive over all alternatives.

A decisive group must exist. From the Pareto Principle, it should be clear that the whole community is a decisive group.

Next assume that there is a decisive individual. Using the ordering requirement, the Pareto Principle, Independence, and Unrestricted Domain, it can be proved that if an individual is decisive over a pair of alternatives, then that individual is decisive over all alternatives, and is consequently a dictator.

Finally, take a decisive group with more than one individual. Using again the ordering requirement, the Pareto Principle, Independence of Irrelevant Alternatives, and Unrestricted Domain, it can be shown that for any decisive group with more than one individual there must exist a smaller decisive group.

Hence there must exist a decisive individual, who by the previous conclusion must be a dictator. Q.E.D.

This proof illustrates one way of looking at the Theorem. I could put it as a sort of catechism:

· Do you accept the need for an ordering of social alternatives? (Yes)

· Do you accept the desirability of having an unrestricted domain? (Yes)

· Do you accept the unanimity principle? (Yes)

· Do you agree that independence of irrelevant alternatives is desirable? (Well, perhaps!)

In this case, as a matter of logic, I can now say that you believe in dictatorship! Notice therefore how devastating the Theorem is. Apparently innocuous and mild requirements lead to dictatorship. It appears to be impossible to construct a constitution with such mild and "reasonable" properties as embodied in the Arrow axioms. Naturally, people were at first rather worried by Arrow's findings, and an academic industry was set up, trying to find out ways around the dilemma. One obvious route lay in the axioms themselves. They might look reasonable and mild, but as the Theorem showed, there was more to them than met the eye. So perhaps we need to look at them again. This is what we shall now do.

8. Interpretation of the Axioms

(a) The Ordering Requirement

One obvious point one can make is that we may not be overly concerned with achieving a social ordering of alternatives. It is true that the voting systems that we mentioned earlier do in fact order alternatives, but our main interest in having a vote is to come to a collective decision. We may simply want a decision that reflects certain rationality requirements as far as the alternative adopted is concerned, without having to worry about rejected alternatives. As I will explain presently this escape route was not really possible for the precise problem Arrow had set himself. Even if we put this to one side, however, this route is in fact closed to us. Imposing standard "rationality" requirements on the actual decision (eg if x wins in set S, then x must also win in set T where T is contained in S), produces effectively the same result as Arrow found.

One line of enquiry was to modify the ordering requirement. This was suggested by Sen in 1969. He suggested quasi-transitivity rather than full transitivity in the requirement we make of the constitution. Effectively, this means than indifference (a tie) may not show transitivity, but strict preference ≻ must still show transitivity. It is not too difficult to show that quasi-transitivity can be consistent with the four Arrow axioms. Sen gave the example of the "Pareto Extension Rule": alternatives x and y are tied (judged indifferent) if they are Pareto non-comparable, and x ≻ y iff x ≿_i y for all i, with x≻_i y for some i. This seems a nice way out, except that shortly afterwards Gibbard pointed out that if we keep the Arrow axioms and require just quasi-transitivity then there must exist an oligarchy. In other words, weakening the rationality requirement ends dictatorship, but it does not really dispense with the fundamental problem that Arrow posed.

(b) Unrestricted Domain

If the rationality (or ordering requirement) is not causing the trouble, then perhaps we need to look a little more critically at the axioms, starting with Unrestricted Domain. Recall the meaning of the axiom: people can state any ordering they like, and the constitution has to produce its own ordering for any preferences thus declared. One could argue that this is too strong. Some preferences are so strange and unlikely that they can, as an empirical matter, be ignored. We will see a variant of this idea later in the module, and as a practical matter, it may have some force. It is, though, rather difficult to see what can be made of this point. Who decides what constitutes a strange and rare set of preferences, and how could we ever get hard empirical evidence as to how rare a particular ordering might be?

In all of this, there is a danger. The purpose of this axiom was to reflect citizen sovereignty. The constitution was to be regarded solely as a vehicle for realising the values and preferences of the separate individuals in society. If we abandon Unrestricted Domain, then there is a sense in which the constitution is imposing preferences on the community. An example of this would be the old Soviet Union, in which one could vote for the Communist Party, or go into another booth to vote against the Communist Party. Other parties were not recognised, and people's preferences over them were ignored. I hope you find this practice objectionable. Other violations of unrestricted domain are more open to debate. You may like to think of some contemporary and less stark examples.

(c) The Pareto Principle

There seems little to say about unanimity. If we are considering a constitution or a voting system, it would be very strange if it did not have the property that when everybody prefers x to y, then x is ranked above y, or that y cannot be chosen whenever x is available. It is doubtful that anyone would accept such a way of making collective decisions.

Even so, one or two comments may be in order. Consider the doctrine of the divine right of kings. Underlying this is the view that individuals, and by extension, communities should do the will of God. Christian doctrine makes it clear that the will of man is different from the will of God, and so we might expect that at times every one will want to follow a course different from the will of God. If one argues that the right thing to do is in fact the will of God then in this case one is proposing a violation of the Pareto Principle.

You may find this argument quaint, but it can be dressed in more contemporary clothes. Suppose that there are certain courses of action that are morally wrong. If one were to accept this, does it not then become appropriate to embody these considerations into the constitution? Such considerations might, for example, lie behind the fact that anti-abortion legislation was introduced into the Irish constitution in 1983. A stronger example is this: what moral status would the unanimous decisions of a group of Nazis have?

Suffice it to say that the do exist circumstances in which one might want to question the applicability of the Pareto Principle. These are, however, likely to be exceptions, and we therefore pass on to the next axiom.

(d) Independence of Irrelevant Alternatives

This is the most difficult of the Arrow axioms, and has been the source of much confusion. Nevertheless, it has now been firmly established what the axiom is actually doing, even if its ultimate desirability is still a matter of debate. To see what the axiom does we can return to the example given earlier.

Individual 1: x ≻₁ y ≻₁ z x ≻*₁ z ≻*₁ y

Individual 2: z ≻₂ x ≻₂ y z ≻*₂ y ≻*₂ x

Now consider the change in preferences, and in particular how the constitution might rank x and z. Recall that the Independence axiom says that rankings over y, an irrelevant alternative, have no consequences for the rankings over x and z. Notice that neither individual has changed rankings over x and z, in the sense that both before and after the change 1 prefers x to z, and 2 prefers z to x. Hence, if we follow the independence axiom, if x(z) is chosen over z(x) before the change in preferences, then x(z) must still be chosen over z(x) after the change. There is nothing in the axioms as they stand to tell us how to choose between the two, however.

One might, though, argue in the following way. Initially, y lies between x and z for individual 1, whilst x and z are adjacent for individual 2. This suggests that 1's feelings in favour of x are stronger than 2's in favour of z. Thus x should be chosen. Similar reasoning indicates, though, that after the change, z should be chosen, contrary to what the independence axiom requires. The reasoning we used is, therefore not consistent with the independence axiom.

What this suggests is that intensity of preference should have no role in social choice, and this indeed is one feature of independence that was established fairly soon after Arrow presented his theorem. This is clearly more restrictive than one would like. In practice we often do allow for intensity. Consider going drinking with your friends. You feel strongly that you want to go to the Union Bar, your friend has a very mild preference for going to the Haworth. In practice you are likely to choose the Union. On the other hand, in constructing a voting system, how precisely can we measure intensity of preference, and compare one person's intense feelings with another? In fact, it is a possible objection to any scheme that violates independence that it may be assigning quite inappropriate intensity measures to individual preference.

Arrow missed this point in his original account, and one might ask then why he introduced the idea. There are two points, one of which I will deal with now, the other I will deal with presently.

Arrow himself was worried about the possibility of manipulability of the constitution. That is, an individual could by voting dishonestly alter the outcome in his/her favour. The way independence might do this can be seen like this. Independence means that outside alternatives are irrelevant to the choice among alternatives within any given set. But this includes sets of only two alternatives, as we discussed earlier ({x, z}). Social choice can then be reduced to simple binary choice, where each alternative is paired off with each other alternative, and the resulting decision over the pair is the same decision as over the whole set S. Now if the choice is really between just two alternatives, then no individual has any incentive other than to vote honestly. Thus as social choice is determined through binary choice no individual will be dishonest in his/her voting behaviour.

The issue, as we shall see, is a little more complicated than this, and we shall return to this question later on in the module. For the present, it will suffice simply to note the matter.

The obvious question to ask at this stage is: why bother? Is there are reason why we would want the independence axiom? The answer partly comes from the way Arrow originally thought about the problem. His problem was to construct a constitution such that better states were ranked above states that were not so good. But to decide on whether state x is actually better than state y we look only at the characteristics of the two states. (Think of the tea coffee example earlier). If this is our purpose, it is difficult to see why the characteristics or availability of a third state z has any bearing at all on how we should compare x and y.

The lesson is that a voting system violates independence (as many do), we may need to be a little cautious in inferring from that that the ranking of alternatives really represents a social welfare ranking. This insight has some implications for how we think of issues of current concern. How much reliance, for example, should we place on the league tables that pervade the public sector these days? As a matter of logic they cannot be relied upon to provide a true ranking of the quality of schools hospitals etc. Consider also the view that having won an election a government has a mandate to carry out its manifesto commitments. This is a convention of the UK political system, but the ranking of that comes out of a vote will not necessarily confirm to a true welfare ranking of the parties (whatever this means, if anything).

(e) Non-dictatorship

Like the Pareto Principle, there is comparably little to say about Non-dictatorship. It is a very mild and, one suspects, acceptable restriction on social choice. The only point worth mentioning is that we should be dealing with genuine social choice. That is, the alternatives under consideration really do affect everyone. If the issue concerned the colour of my pullover, or even more, my pyjamas, then dictatorship (by me) is probably (I hope) acceptable.

9. The Significance of the Theorem

It is clear that Arrow's Theorem is extremely important in collective decision-making. Its exact status, however, is even now a matter of some debate. Behind this lurks the view in some quarters that perhaps it is not so important after all. Such a view comes out in many of Buchanan's extensive writings. To clarify matters, it may be as well to stress that there are two ways of interpreting what Arrow achieved. Both seem to have been in Arrow's mind, but we need to draw a sharp distinction. We take each in turn

(a) Voting Systems

This is not controversial. Arrow's theorem applies to any voting system we might care to think about. In this sense there will always be some "flaw" in any voting system proposed. That there were problems had been recognised long before Arrow. Charles Dodgson (do you know who he was?) wrote about some difficulties in the 19th century, but the earliest widely cited finding is that of the Marquis de Condorcet in 1785. Arrow's major contribution was to shown that all of the problems previously discussed are special cases of a more fundamental problem. This alone makes the theorem important. Condorcet's paradox of voting is a standard way of illustrating the Arrow Problem, we will now follow convention.

Suppose we have three individuals A, B, and C, who have to choose between three alternatives x, y, z. Suppose their preferences are as follows:

	First Preference	Second Preference	Third Preference
Individual A	x	y	z
Individual B	y	z	x
Individual C	z	x	y

Suppose we decide to use Committee Procedure to arrive at our social ranking. As explained earlier, this involves taking each pair of possible alternatives, and taking a vote. Thus x is ranked above y if and only if a majority of voters prefers (votes for) x in its pair-wise contest with y. The alternative that beats all other alternatives is the one chosen, and is called the Condorcet Choice. More generally, it is any alternative not dominated by any other alternative.

Let us now see what happens in the above example. Individuals A and C prefer x to y, so x "beats" y by two votes to one, and we could write x ≻ y. Similarly, as A and B prefer y to z, we have y ≻ z. By the transitivity requirement we must now have x ≻ z. However, as inspection of the table reveals, as B and C prefer z to x what we actually have is z ≻ x. Hence the constitution does not have the transitivity property. This problem was pointed out by Condorcet in 1785. We can now say that committee procedure is, as all voting schemes, subject to the Arrow Theorem. Our example shows that it cannot guarantee transitivity. It is a useful exercise to think a little further about committee procedure, and to establish which Arrow axioms it does conform to. The exercise could be repeated for any voting scheme you might like to consider. It is possible that some schemes establish transitivity. In this case they must violate at least one of the Arrow axioms. Look at the other schemes I gave you at the beginning of the lecture, and see how they fit in to the Arrow Theorem.

One small point about the paradox of voting may be worth noting. You can see from the example that the Condorcet choice does not exist. Each alternative is dominated by another alternative. Even if we do not want a ranking, and simply want to make a rational choice from the possibilities in front of us, the Arrow problem still remains. This illustrates what I said earlier. The requirement on the constitution to provide us with a ranking of alternatives is not essential to the problem that Arrow uncovered.

(b) Social Welfare

The second interpretation of the Theorem was the one Arrow had particularly in mind. He wished to see how one might construct a social welfare function. In this module we follow Paretian value judgements we have the social welfare function W = W(U1, U2,.....,Un) with the derivative of W with respect to each U_i being positive (The Benthamite-Utilitarian untility function is a special case of this). The question one should always raise when confronted with such an animal is to ask "Where does it come from?" One answer to this could be that each individual has his or her own social welfare function. This may or may not be Paretian. Whether this is so or not, it seems a little strange to think of people walking around with some equation of this sort in their head. On the other hand bear in mind that people do have strong views about politics and social affairs. The social welfare function is in this sense merely a formal way of modelling the foundations of the undoubtedly complex welfare judgements that people make all the time. Supposing peoples' social preferences therefore can in some sense be captured by a social welfare function, what follows? First of all an individual's social welfare function will enable that individual to rank the social states within any set S. Thus the relation ≿_i which gave us the starting point for the Arrow analysis can stem from each individual's view of social welfare. It may, of course, simply reflect that individual's selfish preferences (which could themselves be treated as a form of social welfare function). The Arrow problem could then be cast in the following way:

Given each individual has his or her own special view as to what constitutes social welfare, and given that individual views might differ, can we somehow mix together the individual views of social welfare to construct a group welfare function? At this point the procedure has some similarity to Rousseau’s general will. The (big) difference lies in the fact that there is no presumption that there is a true collective out there waiting to be found. Arrow’s view of the problem, unlike Rousseau, is firmly individualist. This group function derived could then be used by the government to make decisions on any matter that affects the whole community. The answer to the question is, if we accept the desirability of the Arrow axioms, simple: “No, it is not possible in general for a well-behaved group welfare function be constructed.” Various comments follow:

· It follows from the Arrow Theorem that there is no "objective" way of assessing any economic or social policy. Normally, in economics, we assume that any agent is maximising some objective function, utility for individuals, profit for firms and so on. However, for groups this cannot be true. Social "welfare" cannot have the same status for the group as utility does for the individual. It is not therefore open to any commentator to argue against a policy simply on "efficiency" grounds. Such a view conveniently ignores the fact the person may be adopting a very special and personal view of social welfare.

· As an example of this point, consider the practice in much of economics of giving an objective function to any group, be it a trade union, the "government", or even a family/ household. An example might the indifference curve macro-economists draw between unemployment and inflation on the Phillips Curve diagram. There is always a point you can make about this. Where does the function come from? How has the writer solved the Arrow problem? Does the government's utility represent only the electoral function (eg probability of the government being re-elected) of the party in office?

· In political discussion one often hears the phrase "public interest". Whilst such a phrase may not be totally vacuous, the Arrow problem directs our attention to a serious problem. Public interest can only mean some amalgam of individual interests, but as we have seen this is precisely what is impossible under the conditions that Arrow proposed. The suspicion might well be that "public interest" is acting as a cover for some private interest.

· In connection with the Independence Axiom, it might be worth pointing out that if social state x is judged as "better" than y, it would seem rather odd if this judgement should be altered by people's views on other possible states. Given the special problem Arrow faced the independence axiom makes a good deal of sense.

· What does one do in response to the Arrow Theorem? One can look further at the axioms and think of new axioms. This has generated a lot of academic articles, but it is unclear whether this literature takes us very far beyond Arrow. We will briefly touch on some of the more interesting ideas in some of the later topics in the module.

· Another strand of thinking has been to look again at unanimity. In a sense this has an old pedigree, going back to the social contract theorists of the 17th and 18th centuries. The idea here is that people give unanimous consent the method by which a community makes decisions. One may not agree with actual decisions made, but as one has given one's consent (in principle) to the constitution, one accepts the collective decision when this happens. How unanimity might be achieved is another question, but for this the work of Buchanan and Tullock (1962) and Rawls (1971) gives some hint. Again we will touch on some of these ideas in the weeks ahead.

· Finally, a comment on Rousseau’s general will. As it is not possible to construct a ranking of alternatives that reflects in some way the collective interest (“general will”) of the group, if this is to be based on the preferences of individuals, there is a suggestion from Arrow’s result that the whole notion of a general will involves an imposition on the community.

10. Conclusion

I hope I have convinced you that the Arrow Theorem is of central concern in collective decision-making. We will not be pursuing all the themes it opens up, but it has a way of popping up in different contexts. For this reason, if for no other, it is worth learning well!