Preference diversity orderings

This paper surveys approaches to preference diversity measurement. Applying preference diversity axiomatics, a generalization of the Alcalde-Unzu and Vorsatz (2016) criterion, is developed. It is shown that all previously used indices violate this criterion. Two new indices (geometric mean based and leximaxbased)are developed that satisfy a new criterion. Leximax-based orders act as a polarization index and are compared with Can et al.’s (2015) polarization index. The paper concludes by formulating a new open question: the preference profile reconstruction conjecture

Quantifying consensus of rankings based on q-support patterns

Rankings, representing preferences over a set of candidates, are widely used in many information systems, e.g., group decision making and information retrieval. It is of great importance to evaluate the consensus of the obtained rankings from multiple agents. An overall measure of the consensus degree provides an insight into the ranking data. Moreover, it could provide a quantitative indicator for consensus comparison between groups and further improvement of a ranking system. Existing studies are insufficient in assessing the overall consensus of a ranking set. They did not provide an evaluation of the consensus degree of preference patterns in most rankings. In this paper, a novel consensus quantifying approach, without the need for any correlation or distance functions as in existing studies of consensus, is proposed based on a concept of q-support patterns of rankings. The q-support patterns represent the commonality embedded in a set of rankings. A method for detecting outliers in a set of rankings is naturally derived from the proposed consensus quantifying approach. Experimental studies are conducted to demonstrate the effectiveness of the proposed approach

Reinforcement Learning for Impartial Games and Complex Combinatorial Optimisation Problems

AlphaZero-style reinforcement learning algorithms have succeeded in various games, from traditional board games to advanced video games. However, their integration into combinatorial games, particularly impartial games, presents inherent challenges. A pivotal aspect of these challenges stems from the parity function's role in determining the winning strategies and the fact that they do not exploit the underlying structures of games besides the game rules. Within the domain of combinatorial optimisation, the task of finding large Condorcet domains is remarkably complex, marked by numerous local optima and deep-rooted mathematical structures. Notably, the most phenomenal method hinged mainly on Fishburn's alternating scheme, an approach intricately tied to the parity function. We demonstrate the intrinsic complexity of Condorcet domain generation by showcasing that diverse AI learning paradigms, from deep reinforcement learning and genetic algorithms to local search algorithms, tend to get stuck in some of the numerous local optima. Thus, a genuinely novel approach is needed. The main contribution of the thesis is to present a novel heuristic approach inspired by AlphaZero-style reinforcement learning but using significant expert domain knowledge and heavily relying on various databases accessed during the search. The Condorcet Domain Library, a research software written in C++ for high execution speed and providing Python interfaces for fast prototyping, was initially developed to implement the algorithm. It has evolved into a flexible library with various functionalities, underpinning multiple impactful research projects. In collaboration with mathematicians, further computational ideas, combined with our new algorithms and the library, led to new results for Condorcet domains, including the construction of a set-alternating scheme leading to a set of large CDs that were hitherto unknown and the discovery of new CDs with distinct properties, some of which are intriguingly linked to multi-agent systems. Furthermore, one of the most prominent results is the discovery of new record-breaking Condorcet domains on the number of alternatives ranging from 9 to 24