Complexity of Deliberative Coalition Formation

TL;DR

This paper analyzes the computational complexity of deliberative coalition formation models in metric spaces, revealing hardness results, potential exponential length of deliberation sequences, and bounds on compromise sizes.

Contribution

It proves the difficulty of finding highly supported proposals, computes the potential exponential length of deliberation sequences, and strengthens bounds on compromise sizes in hypercube models.

Findings

Finding high-support proposals is computationally hard.

Sequences of 2-compromise transitions can be exponentially long.

Lower bounds on compromise size in hypercube models are significantly increased.

Abstract

Elkind et al. (AAAI, 2021) introduced a model for deliberative coalition formation, where a community wishes to identify a strongly supported proposal from a space of alternatives, in order to change the status quo. In their model, agents and proposals are points in a metric space, agents' preferences are determined by distances, and agents deliberate by dynamically forming coalitions around proposals that they prefer over the status quo. The deliberation process operates via k-compromise transitions, where agents from k (current) coalitions come together to form a larger coalition in order to support a (perhaps new) proposal, possibly leaving behind some of the dissenting agents from their old coalitions. A deliberation succeeds if it terminates by identifying a proposal with the largest possible support. For deliberation in d dimensions, Elkind et al. consider two variants of their model: in the Euclidean model, proposals and agent locations are points in R^d and the distance is measured according to ||...||_2; and in the hypercube model, proposals and agent locations are vertices of the d-dimensional hypercube and the metric is the Hamming distance. They show that in the continuous model 2-compromises are guaranteed to succeed, but in the discrete model for deliberation to succeed it may be necessary to use k-compromises with k >= d. We complement their analysis by (1) proving that in both models it is hard to find a proposal with a high degree of support, and even a 2-compromise transition may be hard to compute; (2) showing that a sequence of 2-compromise transitions may be exponentially long; (3) strengthening the lower bound on the size of the compromise for the d-hypercube model from d to 2^{\Omega(d)}.