Iterative Local Voting for Collective Decision-making in Continuous Spaces

Abstract

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 L 1 , L 2 and L ∞ balls. We make several observations that inform future implementations of such a procedure.

Truthful Aggregation of Budget Proposals

“…This body of work includes positive results for weak versions of incentive compatibility, but impossibilities for obtaining full incentive compatibility. Garg et al [19] perform a Mechanical Turk study exploring preference structure in a high-dimensional continuous setting similar to ours.…”

Representative Committees of Peers

“…One stream of the literature on multiple referenda deals with the dependency among issues in voters' preferences (see, e.g., Lang & Xia, 2016); or with restrictions on the valid outcomes, as in judgement aggregation (Pauly & Van Hees, 2006). We follow a different stream of the literature, making the simplifying assumption that preferences of each voter are fully captured by their position in some metric space (in our case, the binary cube with the Hamming distance) (Border & Jordan, 1983;Procaccia & Tennenholtz, 2009;Meir, Procaccia, & Rosenschein, 2012;Goel, Krishnaswamy, & Munagala, 2017;Anshelevich, Bhardwaj, Elkind, Postl, & Skowron, 2018;Garg, Kamble, Goel, Marn, & Munagala, 2019).…”

Iterative Deliberation via Metric Aggregation

“…There are other works that consider iterative deliberation processes: e.g., Fain et al [8] consider a process in which, in each iteration, two agents negotiate and move slightly closer to each other's point in the space; Elkind et al [7] consider a process of deliberation in a metric space, concentrating on coalitions that may form around compromise points in the metric space; and Garg et al [14] consider a model in which all agents are moving in the confined radius of a ball around their compromise point. There are also works that consider deliberation and aim at capturing the internal mechanics of deliberation [2,18,3]; we, however, similarly to Elkind et al [7], abstract away the internal mechanism of deliberation and concentrate on the possibility of reaching consensus by deliberation.…”