Xianwen Shi (University of Toronto)

“Monotonic Norms and Orthogonal Issues in Multi-Dimensional Voting”

Abstract: We study issue-by-issue voting and robust mechanism design in multi-dimensional frameworks where privately informed agents have preferences induced by general norms. We uncover the deep connections between dominant strategy incentive compatibility (DIC) on the one hand, and several geometric/functional analytic concepts on the other. Our main results are: 1) Marginal medians are DIC if and only if they are calculated with respect to a basis such that the norm is orthant-monotonic in the associated coordinate system. 2) Equivalently, marginal medians are DIC if and only if they are computed with respect to coordinates determined by a basis such that, for any vector in the basis, any linear combination of the other vectors is Birkho§ -James orthogonal to it. 3) We show how semi-inner products and normality provide an analytic method that can be used to Önd all DIC marginal medians. 4) As an application, we derive all DIC marginal medians for lp spaces of any Önite dimension, and show that they do not
depend on p (unless p = 2).