Since the publication of Arrow's (1951) impossibility theorem, much effort has been spent on the analysis and rationalizability of committee decision making. The traditional approach to this problem considers rules which aggregate individual binary relations to a binary relation for society. In this survey we call such rules, which are assumed to be defined on profiles of individual rankings, by the name social decision functions. In Aleskerov (1999) can be found a property that social decision functions are required to satisfy. This property is called locality. In Arrow's original work it was called independence of irrelevant alternatives. Social Decision functions which satisfy locality are called local social decision functions. Aleskerov (1999) not only contains a state of the art survey of local social decision functions, but several original contributions to the literature as well. However, Aleskerov does not restrict the domain of social decision functions to be profiles of individual rankings. In different characterization of individual rankings as a subset and usually as a strict subset. It is well known in the theory of axiomatic choice theory that a characterization valid on a given domain may fail to hold on a subdomain. Our purpose in this survey is to show that such is not the case with local social decision functions. It is necessary to justify the domain we have chosen for our survey. Social sciences in general and economic theory in particular, has never confronted any major problem while representing individual preferences by a strict ranking. It is only the issue concerning social preferences by a strict ranking which has been at the centre of the debate concerning aggregation of preferences in social choice theory. Thus the domain comprising profiles of individual rankings is consistent with the demands of economic theory and yet highlights the problems that arise very naturally in social aggregation procedures.
