From here on, I am going to break the exposition into short bits, because this is hard, and I do not want to lose anyone. My apologies to those of you who are having no trouble following it.
First of all, notice that Arrow assumes only ordinal preference. This means that there is no way in the proof to take account of intensity of preference, only order of preference. Let me give an example to make this clearer. In 1992, George H. W. Bush, Bill Clinton, and H. Ross Perot ran for the Presidency. There were some devoted followers of Perot who were crazy about him, and almost indifferent between Bush and Clinton, whom they viewed as both beltway politicians. Let us suppose that one of these supporters ranked Perot first, way ahead of the other two, and gave the edge slightly to Bush over Clinton, perhaps because Bush was a Republican. A second Perot supporter might have been rather unhappy with the choices offered that year, but preferred Perot slightly over Bush, while hating Clinton passionately. From Arrow's perspective which is that of ordinal preference, these two voters had identical preferences, namely Perot > Bush > Clinton, and an Arrovian SWF would treat the two individual preference orders as interchangeable.
Now, there are many ways in which citizens in America can give expression to the intensity of their preferences, as political scientists are fond of pointing out. One is simply by bothering to vote. Voter enthusiasm, in a nation half of whose eligible voters routinely fail to go to the polls, is a major determinant of the outcome of elections. A second way is by contributing to campaigns, volunteering for campaign work, and so forth. Yet another way is through a vast array of voluntary organizations dedicated to pursuing some issue agenda or advantaging some economic or regional group. None of this can find expression in the sort of Social Welfare Function Arrow has defined. This is a very important limitation on the method of collective decision that we call voting. Now, there are voting schemes that allow voters to give expression to the intensity of their preferences [such as giving each voter a number of votes, which he or she can spread around among many candidates or concentrate entirely on one candidate], but these too are ruled out by Arrow, who only allows the SWF to take account of individual ordinal preferences.
The second thing to note is that the requirement of completeness placed upon the SWF rules out partial orderings, such as those established by Pareto-Preference. It is often the case that every individual in the society prefers some alternative x to some other alternative y, and if there are a number of such cases, a robust partial ordering might be established that, while not complete, nevertheless allows the society to rank a sizeable number of the available alternatives. This option too is ruled out by Arrow's two axioms. These observations have the virtue of helping us to understand just how restricting a collective decision-making apparatus like majority rule is.
We are now ready to state the four conditions that Arrow defines as somehow capturing the spirit of majoritarian democracy. Arrow's theorem will simply be the proposition that there is no Social Welfare Function, defined as he has in the materials above, that is compatible with all four conditions. In the original form of the proof, the conditions were, as you might expect, called Conditions 1, 2, 3, 4, and 5. In the revised version, which I shall be setting forth here, they are called Conditions 1' [a revised version of Condition 1], Condition 3 [which also is sometimes called the Independence of Irrelevant Alternatives], Condition P [for Pareto], and Condition 5. Here they are. I will tell you now that Condition 3 is the kinky one.
Condition 1': All logically possible rankings of the alternative social states are permitted. This is a really interesting condition. What it says, formally speaking, is that each individual may order the alternatives, x, y, z, ... in any consistent way. What it rules out, not so obviously, is any religious or cultural or other constraint on preference. For example, if among the alternatives are various dietary rules, or rules governing abortions, or rules governing dress, nothing is ruled in or ruled out. The individuals are free to rank alternatives in any consistent manner.
Condition 3: Let R1, R2, ......, Rn and R1', R2', .... Rn' be two sets of individual orderings of the entire set of alternatives x, y, z, .... and let C(S) and C'(S) be the corresponding social choice functions. If, for all individuals i and all alternatives x and y in a given environment S, xRiy if and only if xRi'y, then C(S) and C'(S) are the same.
OK, this is confusing, so let us go through it slowly step by step and figure out what it means. To get to the punch line first, this condition says that the society's eventual identification of best elements in an environment is going to be determined solely by the rankings by the individuals of the alternatives in that environment, and not by the rankings by the individuals of alternatives not in the environment. [Remember, the Environment, S, is a subset of all the possible alternatives.] Now, take the condition one phrase at a time. First of all, suppose we have two different sets of individual rankings of all the alternatives. The first set of rankings is the Ri [there are as many rankings in the set as there are individuals -- namely, the first individual's ranking, R1, the second individual's ranking, R2, and so forth.] The second set of rankings is the Ri', which may be different from the first set.
Now, separate out some subset of alternatives, which we will call the Environment S, and focusing only on the alternatives in S, take a look at the way in which the individuals rank those alternatives, ignoring how they rank any of the alternatives left out of S. If the two sets of individual orderings, Ri and Ri', are exactly the same for the alternatives in S, then when the Social Welfare Function cranks out a social ranking, R, based on the individual orderings Ri and a social ranking, R', based on the individual orderings Ri', Condition 3 stipulates that the set of best elements [The Social Choice set] will be the same for R and for R'.
Whew, that still isn't very clear, is it? So let us ask the obvious question: What would this Condition rule out? Here is the answer, in the form of an elaborate example. Just follow along.
Suppose that in the 1992 presidential election, there are just three voters, whom we shall call 1, 2, and 3. Also, suppose there are a total of four eligible candidates: George H. W. Bush, Bill Clinton, H. Ross Perot, and me. Now suppose there are two alternative sets of the rankings of these four candidates by individuals 1, 2, and 3.
Ri: Individual 1: Wolff > Clinton > Bush > Perot
Individual 2: Bush > Perot > Wolff > Clinton
Individual 3: Wolff > Clinton > Bush > Perot
Ri': Individual 1: Clinton > Bush > Perot > Wolff
Individual 2: Bush > Perot > Clinton > Wolff
Individual 3: Clinton > Bush > Perot > Wolff
The crucial thing to notice about these two alternative sets of rankings is that they are identical with regard to the environment S = (Bush, Clinton, Perot). The only difference between the two sets is that in the second set, Wolff has been moved to the bottom of everyone's list. [The voters find out I am an anarchist.]
Now let us consider the following Social Welfare Function: For each individual ranking, assign 10 points to the first choice, 7 points to the second choice, 3 points to the third choice, and 2 points to the fourth choice. Then, for any Environment, S, selected from the totality of available alternatives, determine the social ranking by adding up all of the points awarded to each alternative by the individual rankings. Got it?
Go ahead and carry out that exercise. If you do, you will find that for the first set of rankings, the Ri, and for the Environment S = (Bush, Clinton, Perot). the SWF gives 16 points to Clinton, 16 points to Bush, and 11 points to Perot. So, C(S), the society's decision as to which candidates are at the top, is (Clinton, Bush), because they each have the same number of points, namely 16. But if you now carry out the same process with regard to the second set of individual rankings, the Ri', and the same Environment S, you will discover that the SWF assigns 23 points to Clinton, 24 points to Bush, and 13 points to Perot, which means that C'(S) is (Bush). So the social choice in the Environment S has changed, despite the fact that the relative rankings of the elements in S have not changed, because of a change in the rankings of an element not in S, namely Wolff. And this is just what Condition 3 rules out. It says that the Social Welfare Function cannot be one that could produce a result like this.
All of us are familiar with this sort of problem from sports meets or the Olympics. When we are trying to decide which team or country has done best, we have to find some way to add up Gold medals and Silver medal and Bronze medals, and maybe fourth and fifth places as well. And, as we all know, you get different results, depending on how many points you award for each type of medal. Arrow's Condition 3 rules out SWFs like that.
Condition P: If xPiy for all i, then xPy. This just says that if everyone strongly prefers x to y, so does the society. This is a very weak constraint on the SWF.
Condition 5: The Social Welfare Function is not dictatorial.
Remember the definition of "dictatorial" above. This rules out "l'état c'est moi" as a Social Welfare Function.
So, we have the definitions, etc., and we have the four Conditions that Arrow imposes on a Social Welfare Function. Remember that a Social Welfare Function is defined as a mapping that produces a social ranking that satisfies Axioms I and II. Now Arrow is ready to state his theorem. It is quite simple:
There is no Social Welfare Function that satisfies the four Conditions.