Paradoxes of Fair Division

article published in academic journalParadoxes, if they do not define a field, render its problems intriguing and often perplexing, especially insofar as the paradoxes remain unresolved. Voting theory, for example, has been greatly stimulated by the Condorcet paradox, which is the discovery by the Marquis de Condorcet that there may be no alternative that is preferred by a majority to every other alternative, producing so-called cyclical majorities. Its modern extension and generalization is Arrow's theorem, which says, roughly speaking, that a certain set of reasonable conditions for aggregating individuals' preferences into some social choice are inconsistent. In the last fifty years, hundreds of books and thousands of articles have been written about these and related social-choice paradoxes and theorems, as well as their ramifications for voting and democracy. Hannu Nurmi provides a good survey and classification of voting paradoxes and also offers advice on "how to deal with them." There is also an enormous literature on fairness, justice, and equality, and numerous suggestions on how to rectify the absence of these properties or attenuate their erosion. But paradoxes do not frame the study of fairness in the same way they have inspired social-choice theory