Unimodal Domains, Anti-exchange Properties, and Coalitional Strategy-proofness of Voting Rules

It is shown that simple and coalitional strategy-proofness of a voting rule on the full unimodal domain of a convex idempotent interval space (X, I) are equivalent properties if (X, I) satisfies interval anti-exchange, a basic property also shared by a large class of convex geometries including -but not reducing totrees and linear geometries. Therefore, strategy-proof location problems in a vast class of networks fall under the scope of that proposition. It is also established that a much weaker minimal anti-exchange property is necessary to ensure equivalence of simple and coalitional strategy-proofness in that setting. An immediate corollary to that result is that such ‘unimodal’equivalence fails to hold both in certan median interval spaces including those induced by bounded distributive lattices that are not chains, and in certain non-median interval spaces including those induced by non-trivial Hamming graphs. Thus, anti-exchange properties of the relevant interval space provide a powerful general common principle that explains the varying relationship between simple and coalitional strategy-proofness of voting rules for full unimodal domains across different interval spaces, both median and non-median. MSC 2010 Classification: 05C05, 52021, 52037 JEL Classification number : D71