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The theory of choice, individual and social, was mainly developed by economists, with crucial contributions from psychologists, political scientists, sociologists, mathematicians, and philosophers.
Individual choice concerns the selection by an individual of alternatives from a set. In standard microeconomic theory, the individual is supposed to have a preference over a set (or a utility function, that is, a numerical representation of the preference). A standard behavioral assumption asserts that the individual selects the best alternatives according to his or her preference. This implies that the preference and the set of alternatives have appropriate mathematical properties. This is the case, for instance, if the set of alternatives is finite (the number of alternatives is a positive integer) and the preference is a weak ordering (a ranking of the alternatives from the most preferred to the least preferred with possible ties). When the set of alternatives is the standard budget set of microeconomics, the selection is still possible when appropriate topological assumptions are made on the weak ordering and the space of goods. The selected alternatives are the demand set. If there is a single alternative, it is the individual demand. It will depend, given a preference, on the budget set that is defined by the individual’s wealth and the prices. For a given wealth, as a consequence, demand depends on prices. The behavioral maximization assumption is illuminatingly discussed by Amartya Sen (2002).
Although in microeconomics the standard direction is from preference (or utility) to choice (or demand), revealed preference theory reverses this direction. It is alleged that choice is observable, but preference is not. In revealed preference theory, choice is supposed to reveal preference. More precisely, if choice satisfies suitable consistency properties, one can retrieve preference. As an example of such a consistency condition, imagine that you are making a choice in a department store that includes a food department. Your choice in the entire store that happens to be food must be identical to the selection of food you would make if you visited only the food department. Given this kind of consistency condition, it is possible to retrieve a preference that is a weak ordering.
Uncertainty in individual choice differs whether it is objective uncertainty, à la John von Neumann (1903–1957) and Oskar Morgenstern (1902–1977), or subjective uncertainty, à la Leonard Savage (1917–1971). This entry will discuss only objective uncertainty. In this case, the recourse to utility functions is imperative. The set of alternatives is the set of probabilities over prospects—say, lotteries if the prospects are prizes. The individual has a preference given, for instance, by a weak ordering over the set of lotteries. With a utility function representing a weak ordering (which is possible given appropriate conditions), the only property of the real numbers one can use is the ordering property (“greater than or equal to”). The utility functions are said to be ordinal. They are unique up to a strictly increasing transformation. Over lotteries (with further assumptions), one obtains a utility function (called the von NeumannMorgenstern utility function) that satisfies the expected utility hypothesis: The utility of a lottery is equal to the sum of the utilities of the prizes weighted by the probabilities. For instance, in a lottery with two prizes, a bicycle and a car, if the probability to win the bicycle is .99 and the probability to win the car is .01, the utility of the lottery is equal to .99 times the utility of the bicycle plus .01 times the utility of the car. When the expected utility hypothesis is satisfied, the utility function is unique up to an affine positive transformation, and differences of utility become meaningful because these differences can be compared according to the “greater than or equal to” relation. Such utility functions are called cardinal. They are used as the basic element of decision theory under risk, where some further assumptions are made on the utility function (concavity, derivability and properties of derivatives).
Social choice is about the selection of alternatives made by a group of individuals. There are obviously two aspects of social choice corresponding to its double origin: voting and social ethics. Although there were precursors in antiquity and medieval times, the birth of social choice theory is generally attributed to the Marquis de Condorcet (1743–1794) and Jean-Charles Borda (1733–1799), two French scholars, at the end of the eighteenth century. The tremendous modern development of this theory stems from the works of Kenneth Arrow and Duncan Black (1908–1991). Individuals are supposed to have preferences over a set of alternatives. Since these preferences are generally conflicting, one must construct rules to obtain a synthetic (or social) preference or a social choice. Arrow’s (im)possibility theorem asserts that there does not exist any rule satisfying specified properties. On the other hand, Black’s analysis demonstrates that majority rule generates a social preference provided that some homogeneity of individual preferences (single-peakedness) is assumed.
Arrow’s book ( 1963) established the formalism in which social choice theory has developed since then. Two major results are due to Sen and to Allan Gibbard and Mark Satterthwaite. Sen showed the impossibility of having a rule that admits a minimal level of liberty in the society (the group of individuals) and a principle of unanimity (according to which the social preference or the social choice must respect the unanimous preferences of the individuals). Gibbard and Satterthwaite proved independently that there was no rule that was immune to the strategic behavior of individuals—that is, there are situations in which it is advantageous for an individual to reveal a preference that is not his or her sincere preference.
When individual preferences are over uncertain prospects and are represented by von NeumannMorgenstern utility functions, John Harsanyi (1920– 2000) showed that the utilitarianism doctrine could be revived. In a rather caricatural way, the utilitarianism doctrine asserts that social utility is the sum of individual utilities and that social utility has to be maximized. In some of his works, Harsanyi provided a scientific foundation for a kind of weighted utilitarianism. Harsanyi’s utilitarianism is often opposed to the liberal egalitarianism of John Rawls (1921–2002).
A major trend of recent research on voting theory is about scoring systems (e.g., the plurality rule used in the United States and Great Britain or Borda’s rule). Donald Saari’s contributions to scoring rules are a major advance in social choice and voting theory, with important possible applications.
- Arrow, Kenneth J.  1963. Social Choice and Individual Values. 2nd ed. New York: Wiley.
- Arrow Kenneth J. 1984. Collected Papers, 1: Social Choice and Justice. Oxford: Blackwell.
- Arrow Kenneth J. 1984. Collected Papers, 3: Individual Choice under Certainty and Uncertainty. Oxford: Blackwell.
- Black, Duncan. 1958. The Theory of Committees and Elections. Cambridge, U.K.: Cambridge University Press.
- Gaertner, Wulf. 2006. A Primer in Social Choice Theory. Oxford: Oxford University Press.
- Harsanyi, John C. 1976. Essays on Ethics, Social Behavior, and Scientific Explanation. Dordrecht, Netherlands: Reidel.
- Kahneman Daniel, and Amos Tversky, eds. 2000. Choices, Values, and Frames. Cambridge, U.K.: Cambridge University Press.
- Kreps, David M. 1988. Notes on the Theory of Choice. Boulder, CO: Westview.
- Saari, Donald. 1995. Basic Geometry of Voting. Berlin: Springer.
- Sen, Amartya K. 1970. Collective Choice and Social Welfare. San Francisco, CA: Holden-Day.
- Sen, Amartya K. 2002. Rationality and Freedom. Cambridge, MA: Harvard University Press.
- Taylor, Alan D. 2005. Social Choice and the Mathematics of Manipulation. Cambridge, U.K.: Cambridge University Press.
- von Neumann, John, and Oskar Morgenstern. 1953. The Theory of Games and Economic Behavior. 3rd ed. Princeton, NJ: Princeton University Press.
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