Many societal decision problems lie in high-dimensional continuous spaces not amenable to the voting techniques common for their discrete or single-dimensional counterparts. These problems are typically discretized before running an election or decided upon through negotiation by representatives. We propose a algorithm called Iterative Local Voting for collective decision-making in this setting. In this algorithm, voters are sequentially sampled and asked to modify a candidate solution within some local neighborhood of its current value, as defined by a ball in some chosen norm, with the size of the ball shrinking at a specified rate. We first prove the convergence of this algorithm under appropriate choices of neighborhoods to Pareto optimal solutions with desirable fairness properties in certain natural settings: when the voters' utilities can be expressed in terms of some form of distance from their ideal solution, and when these utilities are additively decomposable across dimensions. In many of these cases, we obtain convergence to the societal welfare maximizing solution.We then describe an experiment in which we test our algorithm for the decision of the U.S. Federal Budget on Mechanical Turk with over 2,000 workers, employing neighborhoods defined by various L-Norm balls. We make several observations that inform future implementations of such a procedure.

A major challenge in crowdsourcing evaluation tasks like labeling objects, grading assignments in online courses, etc., is that of eliciting truthful responses from agents in the absence of verifiability. In this paper, we propose new reward mechanisms for such settings that, unlike many previously studied mechanisms, impose minimal assumptions on the structure and knowledge of the underlying generating model, can account for heterogeneity in the agents' abilities, require no extraneous elicitation from them, and furthermore allow their beliefs to be (almost) arbitrary. These mechanisms have the simple and intuitive structure of an output agreement mechanism: an agent gets a reward if her evaluation matches that of her peer, but unlike the classic output agreement mechanism, this reward is not the same across evaluations, but is inversely proportional to an appropriately defined popularity index of each evaluation. The popularity indices are computed by leveraging the existence of a large number of similar tasks, which is a typical characteristic of these settings. Experiments performed on MTurk workers demonstrate higher efficacy (with a $p$-value of $0.02$) of these mechanisms in inducing truthful behavior compared to the state of the art.