TLDR: The paper introduces a new method to measure conflict in Random Permutation Sets (RPS), which handle uncertainty with order information. Existing methods fail to capture conflicts arising from differing element orders. Inspired by Rank-Biased Overlap, the proposed “non-overlap-based conflict measure” accounts for order, prioritizes top-ranked elements (top-weightedness), allows focusing on specific depths (arbitrary truncation), and includes an adjustable parameter for decision-maker persistence. This provides a more accurate and flexible tool for managing conflicts in ordered uncertain information fusion.
Uncertainty is a fundamental aspect of information, and various mathematical theories have been developed to model and reason with it. One such theory is the Dempster-Shafer theory (DST), which allows for modeling ignorance by assigning beliefs to sets of possibilities. Recently, a new formalism called Random Permutation Set (RPS) theory has emerged, extending DST to incorporate order information, which is crucial in many real-world scenarios like time sequences, causality, or action sequences.
While RPS theory offers a powerful way to handle uncertain information with inherent order, a significant challenge has been accurately measuring the conflict between different pieces of evidence represented by permutation mass functions (PMFs). Traditional conflict measures in DST primarily focus on whether the intersection of two sets is empty. If sets overlap, they are often considered non-conflicting. However, in RPS, even if two ordered sets contain the same elements, their differing orders can represent a substantial conflict in the underlying information or decision-maker’s propensity.
A new research paper, titled “A Non-overlap-based Conflict Measure for Random Permutation Sets” by Ruolan Cheng, Yong Deng, and Enrique Herrera-Viedma, addresses this critical gap. The authors propose a novel method to quantify conflict in RPS that specifically accounts for the order of elements. This approach is inspired by the Rank-Biased Overlap (RBO) measure, a technique commonly used to compare indefinite ordered lists.
The paper delves into the nature of conflicts in RPS from two perspectives: Random Finite Set (RFS) and DST. It emphasizes the DST viewpoint, where the order of elements within a focal set signifies a qualitative propensity, meaning higher-ranked elements hold more importance. For instance, if a PMF suggests “omega1 then omega2,” it implies a stronger belief transfer tendency towards omega1 first.
The proposed conflict measure, unlike existing methods, does not simply check for empty intersections. Instead, it quantifies the “non-overlap” between permutations at various depths. This leads to several unique and valuable properties:
Top-Weightedness
The method naturally prioritizes elements at the top of the permutation. This means that inconsistencies in the ordering of higher-ranked elements will result in a more significant conflict measure than inconsistencies occurring further down the list. This aligns with the intuition that initial preferences or observations often carry more weight.
Arbitrary Truncation
Decision-makers can choose to focus on a specific “depth” of the permutation. For example, they might only care about the first three most important elements and disregard any inconsistencies beyond that point. This flexibility allows for tailoring the conflict measurement to the specific needs and priorities of an application.
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Adjustable Parameter for Persistence
The method includes an adjustable parameter, ‘p’, which controls how quickly the influence of elements diminishes with depth. This parameter can be interpreted as the “persistence” of a decision-maker. A smaller ‘p’ means the decision-maker is less persistent, focusing heavily on the top-ranked elements, while a larger ‘p’ implies more consideration for elements further down the list. This parameter also allows the measure to handle permutations of “infinite depth,” where the conflict value will converge to a stable state.
Through numerical examples, the authors demonstrate that their new measure effectively captures order conflicts that previous methods would overlook, often yielding a zero conflict value even when significant ordering differences exist. This advancement provides a more accurate and nuanced tool for understanding and managing conflicts in multi-source information fusion tasks that involve ordered uncertainty.
This research marks a significant step forward in the theoretical development of Random Permutation Sets, offering a robust framework for processing and combining uncertain information where the sequence of events or preferences is paramount. For more technical details, you can refer to the full research paper available at arXiv:2510.16001.
