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Following "Barberà et al. (1991, Econometrica 59:595-609)", we study rules (or social choice functions) through which agents select a subset from a set of objects. We investigate domains on which there exist nontrivial strategy-proof rules. We establish that the set of separable preferences is a maximal domain for the existence of rules satisfying strategy-proofness and no-vetoerWe would like to thank Professors Barbera, Masso, Neme, and Sonnenschein for helpful discussions. We are also grateful to an associate editor, two anonymous reviewers, and participants at the 10th International Meeting of the Society for Social Choice and Welfare in Moscow, the 1st MOVE-ISER joint workshop at Universitat Autonoma de Barcelona, and a seminar at Kyoto University for helpful comments. Hatsumi and Serizawa acknowledge support from the Japan Society for the Promotion of Science through the Research Fellowship for Young Scientists 22-4996 and the Grant-in-Aid for Scientific Research 22330061, respectively. Berga acknowledges the support from the Spanish Ministry of Science and Innovation through the grants SEJ2007-60671 and ECO2010-16353, and from Generalitat de Catalunya through the grant SGR2009-0189. She also acknowledges the Research Recognition Programme of the Barcelona GS

A maximal domain for strategy-proof and no-vetoer rules in the multi-object choice model