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The centuries-old paradox of voting is that majorities sometimes prefer x to y, y to z, and z to x - a cycle. The discovery of the sources and consequences of such cycles, under majority rule and countless other regimes, constitutes much of the mathematical theory of voting and social choice. This book explores the big questions posed by the paradox of voting: positive questions about how to predict outcomes and explain observed stability, and normative questions about how to hold elections, how to take account of preference intensities, the relevance of social welfare to social choice, and challenges to formal 'rationality', individual and social. The overall lesson is that cycles are facts, ubiquitous, and consequential in non-obvious ways, not puzzles to be solved, much less maladies or misfortunes to be avoided or regretted.

Master of the Game: Henry Kissinger and the Art of Middle East Diplomacy, by Martin Indyk. New York: Knopf, 2021. 633 pages. $35.

The widespread use in legislative studies of the one-dimensional model and its median-stability consequence raises a question: Do stability and one-dimensionality rest on evidence drawn from observed votes? They do not and cannot. I prove that every possible legislative history is compatible with a transitive majority preference (hence stability), and except in very special circumstances with a cyclic majority preference (hence instability) as well: observed votes can never refute and almost never confirm stability. One-dimensionality fares worse: any legislative history is compatible with the one-dimensional model if it includes no two votes with overlapping pairs of alternatives, but otherwise, I show, it is almost certainly incompatible with the model, even in those rare cases that ensure transitivity. Voting evidence aside, the one-dimensional model is unduly restrictive, and arguments in its defense do not survive scrutiny. Adapted from the source document.