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AXIOMATIC SOCIAL CHOICE THEORY
Think of a community that has to choose from a number of alternatives. In a democratic setting, the alternative that is chosen by the community (the social choice) should depend on the preferences of the individuals. It is unlikely that the individuals in the community are unanimous in their ranking of these alternatives. Can procedures be devised for social decision-making that respect individual preferences in the presence of diversity of opinion? It is well known that some procedures lead to difficulties. As long ago as 1785, the French thinker Marquis de Condorcet had noted the inconsistencies that majority voting can lead to. Suppose our community consists of three individuals 1, 2 and 3, who have to decide between three alternatives x, y and z. Let the preferences of the individuals be as below:
Then, in a vote between x and y, x defeats y; likewise y defeats z, but paradoxically z defeats x. So this procedure can lead to intransitivities, and manipulating the order of voting can make any of the alternatives emerge as the "socially chosen" one. This fact has long been exploited by people who set the agenda for meetings. For instance, if you want to make y win, set the agenda so that first a vote is taken between x and z. In this round, z wins and x is eliminated. Then take a vote between z and y. Obviously y will be the ultimate winner. Problems of this kind are not peculiar to majority voting. Mathematical reasoning shows that some inconsistency or the other must arise in all procedures for making social choices. This fact is far from apparent. Most people would find it difficult to believe that it is even possible to use mathematics to think about such social and political questions. And yet, not only is a mathematical formulation possible, but it actually provides completely unexpected insights. The level of mathematics used is elementary. What is exemplified is the way the use of mathematics as a language forces clear and logical thinking about a problem.
Sen extended Arrows work in many directions. One of Sens major results concerns the difficulties of reconciling libertarian ideals and considerations of efficiency. To understand this, we look first at Arrows theorem. Arrow's Impossibility Theorem Suppose there is a set X of social alternatives under consideration by a community of N individuals, each of whom can order these alternatives according to his preferences. The individuals are assumed to have what are called rational preferences, that is each individual can rank any two alternatives, and if alternative x is better than y, and y is better than z, then x must be regarded as better than z. A social choice function is a procedure for choosing one alternative from the set X, given the preferences of all the individuals. We assume that the social choice function has unrestricted domain, that is the procedure should work for any possible configuration of rational individual preferences. Ideally this social choice function should satisfy some minimal conditions corresponding to our notions of a "good" procedure. The first of these is what is called the Pareto condition: if for a pair of alternatives x and y, everybody strictly prefers x to y, then y should not be chosen. The second condition is monotonicity. Suppose an alternative x is the social choice under a particular profile of preferences. Consider a different profile where for all individuals any alternative y, which was inferior to x under the original profile, continues to remain inferior to x. Then x should be the social choice under the new preference profile also, because its desirability relative to any y has not decreased for any individual. An individual is said to be a dictator for a given social choice function if, whenever he prefers alternative x to y, then y is not chosen, no matter what the preferences of the other individuals are. That is, the alternative chosen "socially" is always the one that this individual wants. A social choice function that admits a dictator is called dictatorial. Evidently we would like our social choice function to be non-dictatorial. Arrow proved a result that is equivalent to the following:
The Liberal Paradox Liberalism can mean many things, but at the very least it should entail the following: some choices are purely personal, and an individual should be left free to decide on them. What colour you paint your bedroom walls should be left entirely to you, no matter what the opinions of others are. The decisive voice concerning this choice should be yours: if you want pink walls, they should be pink, irrespective of how offensive others find this colour. If you prefer sleeping on your back, others should not be able to compel you to sleep on your side. Or, if at a restaurant you want to eat idlis, other customers should not be able to force you to eat uppuma. A mild requirement of liberalism, then, would be that each individual should have a decisive say over at least one personal choice. An even weaker notion is that there should be at least two individuals who have a decisive say over at least one pair of choices each. Sen calls the latter minimal liberalism: he points out that cutting down the number of individuals with such freedom to one "would permit even a complete dictatorship, which is not very liberal". On the other hand, economists have long regarded the Pareto principle an essential requirement of "efficiency": if for a pair of alternatives x and y, everybody strictly prefers x to y, then social arrangements should not be such that y is chosen. Further, a social decision function (a procedure for combining individual preferences into social preferences) should have unrestricted domain: it should work for all possible profiles of individual preferences. Also, social preferences should not be intransitive: if the society prefers alternative x to y, and y to z, then x should be preferred to z. Sen showed that there cannot be a social decision function that meets all these requirements simultaneously: he called this result "the impossibility of a Paretian liberal". A simple illustration might run as follows. The social choice is between three alternatives: Mr A sees the controversial film Fire, Mr B does so, or nobody sees it. Call these a, b, and c respectively. Mr B, the guardian of public morals, prefers most that nobody watches the film, next that he himself suffers the screening (being less impressionable), and last that Mr A be allowed to be corrupted by the film. So B prefers c to b, and b to a. As Sen remarks "prudes & prefer to be censors rather than being censored". Mr A is altogether different. He prefers b to a, and a to c. His logic is that above all he would prefer someone to see the film rather than no one; and if forced to choose between himself and Mr B, he would rather Mr B widened his outlook. "Minimal liberalism" asks that A and B be decisive over at least one pair of alternatives each. In the choice between nobody seeing the film (c) and only B being forced to see it (b), liberalism requires that Bs preferences should guide the social choice. Likewise, as between a and c, As preferences should be decisive: A should be allowed to see the film rather than preventing everyone from doing so. Hence, socially a should be preferred to c, and c should be preferred to b. Then by transitivity a should be socially preferred to b. But, both A and B prefer b to a. So the resulting social choice violates the Pareto principle! Related Project : Information Revelation
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