Discuss some of the difficulties involved in aggregating individual preferences to obtain a preference ordering for the society. (Arrows impossibility theorem)

It is good for the title to only ask for some difficulties, because, there are many indeed. So many, that the question has never been resolved. Nobody has even attempted to put a possible value on the maximum utility that the society can achieve (for example, as a multiple of the current).

Even though the question is difficult, it not even sufficient. Getting the preference ordering is just the first step that should be followed by theories of dynamics: how is the best way to get into the maximum preference position. Still, the term social preference is used widely in politics, and numerous institutions have been set up to improve the welfare of the society. They must then have an idea which position is preferred to current one. Thus the main importance of the social choice at present is not to obtain a preference ordering for every possible case, but to compare possible situations to current one, and if the possible ones are more preferred, then find ways to move towards them.

Preference ordering has some strong assumptions attached, without discussing them, I cannot do a proper critique. Firstly, preferences are measured by utility – there is a certain amount of utility assigned to every bundle of goods. When 2 bundles have the same utility, then they are said to lie on the same indifference curve. Indifference curves are well-behaved and convex to the origin.

Social choice is then concerned with the indifference curves of the society – how to aggregate individual utilities to for a utility of the society. This is quite a difficult task, as one cannot measure the absolute magnitudes of utility. However, in aggregating individual utilities one must should some weights on them, for the system to be fair, thus we need some magnitudes. However, it has been also argued that there should be no weights – utility is maximized when no individual prefers to be in the position of someone else. I will discuss these issues with reference to liberty later. Thus one can immediately see how problems start to rise with the measure. There are also interpretational issues at stake. Are we talking about social preference in a sense that the society is better of at state x than at state y? Or are we trying to order individual preferences to maximise the total well-being in the society? These are not quite the same, because latter can be done by judgement of a group, a voting procedure, whereas the former takes all individual utilities into account and maximizes the total utility. Whereas the former is likely going to result in a more fair measure, the later is more realistic, and easier to calculate and achieve.

Besides the mechanical problems of achieving a preference ordering, there is the problem of existence of an unbiased ordering. This is called the Arrows theorem, and it states that none of the social preference functions can satisfy four reasonable criteria at the same time. While the proof of Arrows theorem is beyond this essay, I would like to dedicate the remainder to exploring each of these four conditions and how they can be relaxed, so that, at least in theory, an unbiased preference function could exist.

First condition is the independence of irrelevant articles (I), which states that when two bundles of goods have nothing in common, then a change of preferences in one bundle must not affect the goods in the other bundle. Second condition states that the unanimous strict preference over a pair must be reflected in the final choice (P). This can be expressed in several ways, most elegant example, I think, is the fact that when each individual either prefers x or is indifferent between goods x and y, then y must never be chosen by a society, when x is present.

Unrestricted domain is also a condition (U). That implies that the function must exist for all possible preferences of the individuals, even the most extreme ones. And lastly there should be no dictators (D), such that someone can, by selecting x, make the society always select x.

Condition (I) seems logical by definition, however people do not always follow simple honest voting procedures. Gibbard Satterth-Waite theorem says that it is possible to manipulate every game involving more than 3 players. People will start trading votes and using strategies to vote for things that they do not particularly want, in the hope that other people want them. Thus a completely irrelevant item might become a bargaining object, and someone else’s preferences over that object will influence the decisions of the subject. In this case it is extremely hard to predict a unique social preference curve. Vote cycling will also occur as the preferences will not be transitive anymore – it is always possible to co-operate. For example, when players a,b,c have following preferences over xyz

Then when everyone ranks their preferences honestly Y will be chosen and Z will be second. However, when A and C do vote trading and A will vote XZY and C will vote ZXY then Z will be chosen first and X second, which is better for A and C. Then Y will find it beneficial to co-operate etc.

Social classes are a way out of this dilemma – it is grouping goods together and saying XYZ is equally preferred in a society. However, this would conflict with the unanimous pair assumption.

Sen has done major work in exploring the possibilities to relax the unrestricted domain assumption in order for a social utility function to exist. Although the work is highly theoretical it is logical in a sense that finding a social preference ordering for just a limited set of indifferences is a big step towards finding a complete preference ordering. Just a qualification whether a marginally different set of goods is better for the society or not is the only thing normally needed in policy construction. Thus the full domain assumption is not that vital.

No dictatorship assumption should be fair. However, it is often not sufficient, as oligarchic groups should also be avoided. They are groups of individuals that are able to “force” their will through. However, one could take the oligarchic issue to an extreme, and say an oligarchic group consisting of n-1 members in an n member society should also not exist! This is surely not necessary. Further issues in dictatorship arise when the concept of liberty and merit are introduced. It might be sometimes fair for one individual to affect the society, when he has done something good for the society. A concept of fairness could be introduced to the system to avoid dictatorship – that means no member of the society would want to posses the bundles of any other member. However, this means taking away from people – thus decreasing their liberty. Also how should one deal with disabled people – they would like to be instead of anyone, but they cannot in practice. One must not lose sight with practice when dealing with these problems. Fairness and liberty can not exist together completely.

Many further issues beyond Arrow’s theorem, and also Arrow’s theorem extensions have been invented since the original article by Arrow in 1951 to deal with the social choice. One of the main one is compensation. A society is in Pareto optimal position when no one could be made better off without making other people worse off. Thus if someone is made better of then they should compensate for the people who are made worse off. But the compensation is hard to achieve in practice. What if no compensation occurs – are we still in a socially preferable position or not?

And finally, not much attention has been devoted to the utility concept itself. There are many more things that society has to value except for utility. One example is the economic sustainability of the sociological system. A sociologically more preferred system should be more sustainable in the long term (like for the next generation). However, assuming that people do not derive utility from the well-being of their children, this position will not be of increased sum of personal utilities. Philanthropy is a further example of this sort. Thus even when we have established the sociological preference curve that orders individual preference curves, we must start questioning whether the individual preferences are the only composite of sociological preference curve. Popper’s principles say that in “good” science, we should be aggregate from the individuals to the whole. However, neo-Kahnian school has argued that this does not apply to social sciences. One must essentially understand the people involved - people are not aggregatable. This is a completely different topic on its own. It is very likely, however, that the social preference ordering depends on very many other things than just individual preferences.