Impartial Rank Aggregation

TL;DR

This paper explores impartial rank aggregation functions that produce fair rankings independent of individual input positions, achieving optimal properties like full rank, monotonicity, and weak unanimity with polynomial-time mechanisms.

Contribution

It introduces the design of impartial rank aggregation mechanisms that satisfy optimal properties such as full rank, monotonicity, and weak unanimity, with proven optimality and polynomial-time implementation.

Findings

Impartial mechanisms can achieve full rank and monotonicity for n ≥ 4.

Weak unanimity can replace monotonicity for n ≥ 5.

All mechanisms are implementable in polynomial time.

Abstract

We study functions that produce a ranking of $n$ individuals from $n$ such rankings and are impartial in the sense that the position of an individual in the output ranking does not depend on the input ranking submitted by that individual. When $n \geq 4$, two properties concerning the quality of the output in relation to the input can be achieved in addition to impartiality: individual full rank, which requires that each individual can appear in any position of the output ranking; and monotonicity, which requires that an individual cannot move down in the output ranking if it moves up in an input ranking. When $n \geq 5$, monotonicity can be dropped to strengthen individual full rank to weak unanimity, requiring that a ranking submitted by every individual must be chosen as the output ranking. Mechanisms achieving these results can be implemented in polynomial time. Both results are best possible in terms of their dependence on $n$. The second result cannot be strengthened further to a notion of unanimity that requires agreement on pairwise comparisons to be preserved.