Algebraic Evaluation Theorems

TL;DR

This paper introduces algebraic evaluation (AE), a new method for assessing juror accuracy that outperforms majority voting by requiring fewer assumptions, providing precise evaluations, and detecting independence failures, with implications for AI safety.

Contribution

It develops an algebraic evaluation theorem for juror performance, offering a more accurate, assumption-light alternative to majority voting with practical and safety applications.

Findings

AE handles jurors less than 50% accurate.

AE provides high-precision evaluation with uncertainty bounds.

AE detects failures in error independence assumptions.

Abstract

Majority voting (MV) is the prototypical ``wisdom of the crowd'' algorithm. Theorems considering when MV is optimal for group decisions date back to Condorcet's 1785 jury \emph{decision} theorem. The same error independence assumption underlying the theorem can be used to prove a jury \emph{evaluation} theorem that does purely algebraic evaluation (AE) of juror performance based on a batch of their decisions. Three or more binary jurors are enough to obtain the only two possible statistics of their correctness on a test they took. AE is superior to MV in three ways. First, its empirical assumptions are looser and can handle jurors less than 50\% accurate in making decisions. Second, it has point-like precision in evaluating them given its assumption of error independence. This precision enables a multi-accuracy approach that has higher labeling accuracy than MV and comes with empirical uncertainty bounds. And, third, it is self-alarming about the failure of its error independence assumption. Experiments using demographic data from the American Community Survey confirm the practical utility of AE over MV. Two implications of the theorem for AI safety are discussed - a principled way to terminate infinite monitoring chains (who grades the graders?) and the super-alignment problem (how do we evaluate agents doing tasks we do not understand?).