How mathematical impossibility changed welfare economics: a history of Arrow’s impossibility theorem. (English) Zbl 1422.91028
This is a highly interesting account of the history of Kenneth Arrow’s impossibility theorem and related topics. It emphasizes the importance of Arrow’s mathematical education, in particular his familiarity with the theory of ordered sets. Indeed, the concept of an ordered set provided a natural description of the preferences of an individual or society, and enabled Arrow to unify the study of welfare economics with the theory of voting. Thus, Arrow’s impossibility theorem can be interpreted in two alternative ways: Either as the nonexistence of an ideal voting procedure, or the nonexistence of a suitable social welfare function mapping the preferences of all individuals to a corresponding social ordering of the given social states.
The article also discusses the close relation between the work of Arrow and Duncan Black. For example, both rediscovered an old paradox due to Condorcet, and both of them have almost simultaneously proved a theorem on determining the winner in cases where the preferences of individual voters correspond to single peaked curves.
The final parts of the paper provide a comprehensive overview of the reception of Arrow’s theorem by his contemporaries, which was both praising and critical; Black’s negative opinion is described in detail.
The article is very well written and will be an important source of information for readers interested in voting theory, mathematical economics, and history of mathematics.
The article also discusses the close relation between the work of Arrow and Duncan Black. For example, both rediscovered an old paradox due to Condorcet, and both of them have almost simultaneously proved a theorem on determining the winner in cases where the preferences of individual voters correspond to single peaked curves.
The final parts of the paper provide a comprehensive overview of the reception of Arrow’s theorem by his contemporaries, which was both praising and critical; Black’s negative opinion is described in detail.
The article is very well written and will be an important source of information for readers interested in voting theory, mathematical economics, and history of mathematics.
Reviewer: Antonín Slavík (Praha)
MSC:
| 91-03 | History of game theory, economics, and finance |
| 06-03 | History of ordered structures |
| 01A60 | History of mathematics in the 20th century |
| 91B14 | Social choice |
| 91B12 | Voting theory |
| 91B15 | Welfare economics |
Keywords:
Arrow’s impossibility theorem; social choice; welfare economics; order relations; voting theory; Kenneth Arrow; Duncan Black; Condorcet paradoxReferences:
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