Every Bit Helps: Achieving the Optimal Distortion with a Few Queries

A fundamental task in multi-agent systems is to match n agents to n alternatives (e.g., resources or tasks). This is often done by eliciting agents' ordinal rankings over the alternatives rather than their exact numerical utilities. While this simplifies elicitation, the incomplete information leads to inefficiency, captured by a worst-case measure called distortion. Recent work shows that making just a few cardinal utility queries per agent can significantly improve the distortion, with Amanatidis et al. (2024) achieving O(√n) distortion with two queries per agent. We generalize their result by achieving O(n^(1/λ)) distortion with λ queries per agent, for any constant λ, which is optimal up to a constant factor given a previous lower bound by Amanatidis et al. (2022). We extend this finding to the general social choice problem of selecting one of m alternatives based on n agents' preferences, achieving O((min{n, m})^(1/λ)) distortion with λ queries per agent, for any constant λ, which is also optimal given prior results. Thus, our work settles open questions regarding the optimal distortion achievable with a fixed number of cardinal value queries in both settings.