Beyond the worst case: Distortion in impartial culture electorates

TL;DR

The paper investigates average distortion in impartial culture electorates when agents have random valuations drawn from a common distribution, asking how few cardinal queries can substantially reduce social-welfare loss caused by ordinal voting. It proves a universal $avdist=\Omega(m)$ lower bound in the average-case but shows that a single query per agent suffices to achieve constant average distortion in binary valuations, via the Mean mechanism, and extends this with randomized threshold-based mechanisms (RtMean) to general distributions, achieving $avdist=O(\log m + \log(\sigma^2/\mu^2))$. It further explores worst-case distortion, introducing RtSearch to obtain $O(\log m)$ worst-case distortion with $O(\log m)$ queries, and provides lower bounds showing that constant distortion in worst-case requires $\Omega(\log m)$ queries and that 1-query deterministic rules have $\Omega(\sqrt{m})$ distortion. Overall, the work draws a bridge between worst-case and average-case analyses, highlighting the potential of limited cardinal information to dramatically improve social-welfare outcomes in voting. The findings have implications for designing practical mechanisms that judiciously acquire cardinal data to substantially improve welfare without extensive querying.

Abstract

{\em Distortion} is a well-established notion for quantifying the loss of social welfare that may occur in voting. As voting rules take as input only ordinal information, they are essentially forced to neglect the exact values the agents have for the alternatives. Thus, in worst-case electorates, voting rules may return low social welfare alternatives and have high distortion. Accompanying voting rules with a small number of cardinal queries per agent may reduce distortion considerably. To explore distortion beyond worst-case conditions, we use a simple stochastic model according to which the values the agents have for the alternatives are drawn independently from a common probability distribution. This gives rise to so-called {\em impartial culture electorates}. We refine the definition of distortion so that it is suitable for this stochastic setting and show that, rather surprisingly, all voting rules have high distortion {\em on average}. On the positive side, for the fundamental case where the agents have random {\em binary} values for the alternatives, we present a mechanism that achieves approximately optimal average distortion by making a {\em single} cardinal query per agent. This enables us to obtain slightly suboptimal average distortion bounds for general distributions using a simple randomized mechanism that makes one query per agent. We complement these results by presenting new tradeoffs between the distortion and the number of queries per agent in the traditional worst-case setting.

Figures (2)

• Figure 1: An example for our lower bound construction in Theorem \ref{['thm:omega_log_m_lower_bound_non_stochastic']} that illustrates the way in which the agents' valuations are defined. The alternative $\widehat{a}$ picked by the mechanism $\mathcal{M}$ is marked as a black box in every agent's ranking. We assume that $\mathcal{M}$ queried the positions corresponding to the dashed boxes. In this example, the mechanism only queries the first position in a bucket which is without loss of generality. The gray areas correspond to buckets in which all alternatives have high values. White areas correspond to buckets in which alternatives have low values, either because the bucket contains the winning alternative $\widehat{a}$ or because mechanism $\mathcal{M}$ queried the value of an alternative in the bucket.
• Figure 2: An example of our lower bound construction in Theorem \ref{['thm:0/1_valued_1_query_lower_bound']} where $m = 18$. Hence, we set $t=4$ and $n=t^2=16$ resulting in the profile $P$ shown above. In every agent's ranking (horizontal bars), the example mentions the top-ranked alternative. Additionally, we marked alternatives $11$ and $12$ in every ranking in order to showcase the symmetry of $P$. Alternatives $17$ and $18$ are shown for two agents of every cohort. Notice that these alternatives appear in the exact same positions of every agent's ranking.

Theorems & Definitions (39)

• Theorem 1
• proof
• Claim 2
• proof
• Lemma 3: Chernoff bound, upper tail
• Definition 4: implied social welfare
• Definition 5: mechanism Mean
• Theorem 6
• proof
• Lemma 7