The relationship between the normalized gradient addition mechanism and quadratic voting

Quadratic voting and the normalized gradient addition mechanism are both social choice mechanisms that confront individuals with quadratic budget constraints, but they are applicable in different contexts. Adapting one or both to apply to the same context, this paper explores the relationship between these two mechanisms in three contexts: marginal adjustments of continuous policies, simultaneous voting on many public choices, and voting on a single public choice accompanied by private monetary consequences. In the process, we provide some formal analysis of quadratic voting when (instead of money) votes are paid for with abstract tokens that are equally distributed by the mechanism designer.