Thursday, 1 January 2015

Global Transitivity & the Rawlsian Social Welfare Function.


The Rawlsian social welfare function uses as its measure of social welfare the utility of the worst-off member of society. The following argument can be used to motivate the Rawlsian social welfare function. Imagine a group of individuals who have not yet entered the economy (they are ‘behind the veil of ignorance’) and do not yet know what position they will occupy. That is, they may become rich members of the economy or poor members. If asked what form of social welfare function they would wish the economy to have- an extremely risk-averse individual would propose the Rawlsian.
(From: Rawlsian social welfare function in A Dictionary of Economics)
Unfortunately there would be no way to work out, from information about their individual preferences- i.e. what they would do with their Liberty based on their haecceity, all the things which make them unique as individuals- what precisely the Rawlsian outcome would be. In other words, either the Liberty Principle applies to nothing or its exercise can never be based on what makes an individual unique. Thus, in this thought experiment, no one's Liberty leads to any change in the imposed Rawlsian solution.
 A stupid person gullible enough to be brain-washed by Rawls might briefly think otherwise but would soon get bored and try to bite her own head off.

Arrow (1963), introducing the possibility that individual preference orderings can impact Social orderings, proved that a Rawlsian Social Welfare Function couldn't use information about individual preferences while also satisfying some apparently reasonable or desirable properties and thus questioned the usefulness of 'ideal theories'- including Rawlsian Jutice-as-Fairness avant la lettre.
One workaround, that of D.G Saari, sees Arrow's approach as throwing away information otherwise available from the notion of global transitivity.
More speculatively, as the engaging John Lawrence writes- 'The fineness or coarseness of the grid on which individuals specify their preference profiles determines the amount of information conveyed. Since this grid is traditionally determined by the number of alternatives, there is no such thing as an irrelevant alternative. The problem of social choice, in general, can be viewed as the transmission of information from multiple sources (the individuals) to one receiver (society). Since there are finite information transmission constraints, there will be some probability of error, P(e), regarding the placement or ranking of alternatives both in individual and social profiles. As individual information is increased, P(e) in the social profile can be made to approach zero as closely as desired.' (Lawrence is invoking Shannon's information theoretic notion of 'channel capacity'.)

'However, no one has considered the individual preference orderings themselves to be probabilistic by virtue of the fact that each individual is allowed to specify only a finite amount of information upon which the SWF must then determine a social preference ordering. The uncertainty with regard to the individual’s “true” preferences (those that would be manifested if infinite information were available) leads to possible errors in the social ordering. We show that the P(e) can be made arbitrarily small by increasing the information flow from the individuals. The fact that P(e) will always be non-zero with finite information constraints coincides with Arrow’s [1963] Impossibility Theorem. Although a SWF that provides a social ordering that has even a single error, in the sense that the ordering doesn’t meet Arrow’s conditions and axioms for one pair of alternatives and one domain element (but provides correct solutions in every other case), would traditionally be considered not to exist, we take the approach of allowing errors in the SWF and then trying to minimize them.'

Saari's approach, as discussed here, soon comes a cropper. However, more generally, by thinking about the cost/benefit aspect of holding preferences or participating in their Aggregation, Ideal-type theory resurfaces as cheap and ambiguous (i.e. Knightian) enough to be an undominatable mixed strategy choice provided Global Transitivity is a Scientific Research Program.
Under these circumstances, the Rawlsian SWF doesn't cash out as some sort of more or less paternalistic compulsory insurance scheme. Instead it becomes information theoretic and organizational in Alan Kirman's sense and thus puts a beating heart back into Samuelson General Equilibrium.

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