Recoverability of choice functions and binary relations: some duality results

Banerjee and Pattanaik (1996) proved a theorem that the maximal set with respect to a quasi-ordering can be fully recovered by defining the greatest sets with respect to each and every ordering extension thereof and taking their union. Donaldson and Weymark (1998) proved a theorem that a quasi-ordering can be fully recovered by taking the intersection of all the ordering extensions thereof. These recoverability theorems are obviously related, but their exact relationship has never been clarified in the literature. This paper examines the issue of choice-functional recoverability and relational recoverability in a general framework, and establishes several remarkable duality relationships.Thanks are due to Professors Walter Bossert, Prasanta K. Pattanaik, Clemens Puppe, Amartya K. Sen, and John Weymark, with whom we had several opportunities to discuss this and related issues. The first author also would like to express his gratitude to the Andrew Young School of Policy Studies, Georgia State University, and the Scientific Research Grant for Policy Areas (B) Number 603 from the Ministry of Education, Science and Culture of Japan for their financial support which made this collaborative research possible. We are also grateful to the referees whose comments have led to several improvements of the paper.