Impossibility theorem

Impossibility theorem. The Impossibility theorem, also known as Arrow's impossibility theorem, is a fundamental result in social choice theory, which was first proposed by Kenneth Arrow in his doctoral dissertation at Columbia University and later published in his book Social Choice and Individual Values. This theorem has far-reaching implications for economics, politics, and decision theory, and has been influential in the work of scholars such as Amartya Sen, Joseph Stiglitz, and Daniel Kahneman. The Impossibility theorem has been recognized with numerous awards, including the Nobel Memorial Prize in Economic Sciences, which was awarded to Kenneth Arrow in 1972, along with John Hicks, for their pioneering contributions to general equilibrium theory.

Introduction to Impossibility Theorem

The Impossibility theorem states that it is impossible to design a voting system that satisfies a set of reasonable criteria, including non-dictatorship, unanimity, and independence of irrelevant alternatives, as demonstrated by Arrow's theorem. This result has significant implications for democratic theory, as it suggests that there is no perfect way to aggregate individual preferences into a social choice, a concept that has been explored by scholars such as Jean-Jacques Rousseau, John Stuart Mill, and Robert Dahl. The Impossibility theorem has been applied in a variety of contexts, including elections, referendums, and committee decisions, and has been used to analyze the work of institutions such as the United Nations, the European Union, and the International Monetary Fund. Researchers such as Milton Friedman, Gary Becker, and James Buchanan have also used the Impossibility theorem to inform their work on public choice theory and constitutional economics.

History and Development

The Impossibility theorem was first proposed by Kenneth Arrow in 1951, while he was a graduate student at Columbia University, under the supervision of Harold Hotelling and Albert Hart. Arrow's work built on earlier research by scholars such as Vilfredo Pareto, Francis Ysidro Edgeworth, and Erik Lindahl, who had explored the concept of social welfare functions. The Impossibility theorem was later generalized and extended by researchers such as Amartya Sen, Prasanta Pattanaik, and Yongsheng Xu, who have applied it to a wide range of problems in economics, politics, and philosophy, including the work of Karl Marx, Friedrich Hayek, and John Rawls. The theorem has also been influential in the development of game theory, as seen in the work of John von Neumann, Oskar Morgenstern, and Nash equilibrium.

Statement and Proof

The Impossibility theorem states that there is no social welfare function that satisfies the following conditions: non-dictatorship, unanimity, independence of irrelevant alternatives, and transitivity, as demonstrated by Arrow's theorem. The proof of the theorem involves a series of logical steps, which show that any social welfare function that satisfies these conditions must be either dictatorial or intransitive, a concept that has been explored by scholars such as Abraham Wald, Jacob Marschak, and Leonid Hurwicz. The theorem has been formalized using a variety of mathematical techniques, including model theory, category theory, and measure theory, and has been applied to a wide range of problems in economics, politics, and philosophy, including the work of Imre Lakatos, Paul Feyerabend, and Thomas Kuhn.

Implications and Applications

The Impossibility theorem has significant implications for democratic theory, as it suggests that there is no perfect way to aggregate individual preferences into a social choice, a concept that has been explored by scholars such as Robert Nozick, Michael Sandel, and Martha Nussbaum. The theorem has been applied in a variety of contexts, including elections, referendums, and committee decisions, and has been used to analyze the work of institutions such as the United States Congress, the European Parliament, and the International Court of Justice. Researchers such as Herbert Simon, Oliver Williamson, and Elinor Ostrom have also used the Impossibility theorem to inform their work on organizational theory and institutional economics.

Related Theorems and Results

The Impossibility theorem is related to a number of other results in social choice theory, including the Gibbard-Satterthwaite theorem, the McKelvey theorem, and the Plott theorem, which have been developed by scholars such as Allan Gibbard, Mark Satterthwaite, and Charles Plott. These theorems provide further insights into the limitations of voting systems and the challenges of aggregating individual preferences, a concept that has been explored by scholars such as Kenneth Shepsle, Barbara Sinclair, and David Austen-Smith. The Impossibility theorem has also been influential in the development of game theory, as seen in the work of Reinhard Selten, John Harsanyi, and Roger Myerson.

Criticisms and Controversies

The Impossibility theorem has been subject to a number of criticisms and controversies, including challenges to its assumptions and methodology, as seen in the work of scholars such as Amartya Sen, Prasanta Pattanaik, and Yongsheng Xu. Some researchers have argued that the theorem is too narrow or too broad, and that it does not capture the full complexity of social choice problems, a concept that has been explored by scholars such as Mancur Olson, George Stigler, and Gary Becker. Others have suggested that the theorem can be circumvented or overcome through the use of alternative voting systems or decision-making procedures, a concept that has been explored by scholars such as Donald Saari, Steven Brams, and Peter Fishburn. Despite these criticisms, the Impossibility theorem remains a fundamental result in social choice theory, and continues to influence research in economics, politics, and philosophy, including the work of Nobel laureates such as James Mirrlees, William Vickrey, and George Akerlof. Category:Social choice theory