## Abstract

The mathematical properties of the methods used in different countries and organisations to vote for their legislatures or executives are examined, with examples to illustrate the merits and problems that arise. We meet Arrow’s celebrated Impossibility Theorems, which show that there can be no ‘best’ method to choose a winner from more than two candidates, and discuss the inevitability of tactical voting. How the USA has chosen its House of Representatives illustrates problems of logic, involving simple arithmetical ideas. We describe Simpson’s Paradox, and note how failure to appreciate its existence can lead to flawed conclusions. Medical tests for drug use, or the presence of cancers, are not infallible; we look at the balance between false positives, and failure to detect actual cases. The Gini Index measures inequality in income or wealth; we note this, and several other measures, make comparisons between them, and point to links with *metric spaces*.

## References and Further Reading

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*Is it all about the Tails? The Palma measure of Income Inequality*. (Working paper 343, September 2013, Centre for Global Development)Google Scholar - Gibbard A (1973) Manipulation of voting schemes: a general result.
*Econometrika*41(4) pages 587–601Google Scholar - Hodge J K and Klima R E (2005) The Mathematics of Voting and Elections: A Hands-On Approach. American Mathematical SocietyGoogle Scholar
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*Significance*1(2) pages 73–75Google Scholar - Satterthwaite M A (1975) Strategy-proofness and Arrow’s conditions: Existence and Correspondence Theorems for voting procedures and social welfare functions.
*Journal of Economic Theory*10(2) pages 187–217Google Scholar - Taylor A D and Pacelli A M (2008) Mathematics and Politics: Strategy, Voting, Power and Proof. SpringerGoogle Scholar