Almost optimal manipulation of a pair of alternatives
Jacek Szybowski, Konrad Kułakowski, Sebastian Ernst
TL;DR
The paper tackles the problem of assessing how easily a decision-maker can manipulate a ranking derived from pairwise comparisons. It develops a method to compute the closest additive PCM that makes two selected alternatives tie in priority, using an orthogonal projection onto a tie space defined by the equality of the two alternatives' row averages, with the result applicable to multiplicative PCMs via logarithmic mapping. A key theoretical contribution is that the projection preserves all other weights while the tied alternatives receive equalized weights equal to their original average, and a practical measure, EMI, quantifies manipulation ease as the distance between the original and manipulated PCMs. The approach offers a concrete metric to evaluate the vulnerability of PCM-based decision processes and provides a foundation for detecting and studying manipulation, with potential extensions to incomplete PCMs and gradient-based methods.
Abstract
The role of an expert in the decision-making process is crucial, as the final recommendation depends on his disposition, clarity of mind, experience, and knowledge of the problem. However, the recommendation also depends on their honesty. But what if the expert is dishonest? Then, the answer on how difficult it is to manipulate in a given case becomes essential. In the presented work, we consider manipulation of a ranking obtained by comparing alternatives in pairs. More specifically, we propose an algorithm for finding an almost optimal way to swap the positions of two selected alternatives. Thanks to this, it is possible to determine how difficult such manipulation is in a given case. Theoretical considerations are illustrated by a practical example.