Harsanyi’s Social Aggregation Theorem

Abstract

In the earlier chapters, it has been assumed that social states are characterized by complete certainty in the sense that they are fully observable by the individuals under consideration. Consequently, each state may be regarded as a certain prospect. An individual can therefore order alternative social states using his preferences without any ambiguity. Thus, the decisions taken by the individuals are taken in an environment of certainty. But when states are affected by uncertainty, the decision criterion may be of a different type. To understand this, consider two farmers for whom the extent of rainfall has a very high impact on their crop outputs from their respective lands. Rainfall conditions may be subdivided into the following categories: (i) flood, (ii) optimum, (iii) hardly sufficient, (iv) less than hardly sufficient, and (v) drought. Each of these categories represents a circumstance of nature. In such a situation, each farmermaximizes his expected (von Neumann–Morgenstern) utility function. A natural question that arises in this context is the following: howare the individual utilities aggregated to arrive at a social utility? John C.Harsanyi (1955) made an excellent recommendation along this line. Harsanyi assumed at that outset that individual and social preferences fulfill the expected utility axioms and these preferences are portrayed by von Neumann–Morgenstern utility functions. The set of alternatives on which individual and social preferences are defined is constituted by the lotteries bred from a finite set of well-defined basic prospects. By including a Pareto principle within the framework, Harsanyi demonstrated that social utility function can be expressed as an affine combination of individual utility functions. In other words, given that the origin of the social utility function has been appropriately normalized, social utility comes to be a weighted sum of individual utilities. This relationship is referred to as theHarsanyi social aggregation theorem (Weymark 1991, 1994).