Various voting rules are based on ranking the candidates by scores induced by aggregating voter preferences. A winner (respectively, unique winner) is a candidate who receives a score not smaller than (respectively, strictly greater than) the remaining candidates. Examples of such rules include the positional scoring rules and the Bucklin, Copeland, and Maximin rules. When voter preferences are known in an incomplete manner as partial orders, a candidate can be a possible/necessary winner based on the possibilities of completing the partial votes. Past research has studied in depth the computational problems of determining the possible and necessary winners and unique winners.
These problems are all special cases of reasoning about the range of possible positions of a candidate under different tiebreakers. We investigate the complexity of determining this range, and particularly the extremal positions. Among our results, we establish that finding each of the minimal and maximal positions is NP-hard for each of the above rules, including all positional scoring rules, pure or not. Hence, none of the tractable variants of necessary/possible winner determination remain tractable for extremal position determination. Tractability can be retained when reasoning about the top-$k$ positions for a fixed $k$. Yet, exceptional is Maximin where it is tractable to decide whether the maximal rank is $k$ for $k=1$ (necessary winning) but it becomes intractable for all $k>1$.
Incomplete preferences are likely to arise in real-world preference aggregation scenarios. This paper deals with determining whether an incomplete preference profile is single-peaked. This is valuable information since many intractable voting problems become tractable given single-peaked preferences. We prove that the problem of recognizing single-peakedness is \NP-complete for incomplete profiles consisting of partial orders. Despite this intractability result, we find several polynomial-time algorithms for reasonably restricted settings. In particular, we give polynomial-time recognition alg…
UNSW Sydney（新南威尔士大学悉尼分校）；Institute for Logic, Language and Computation (ILLC), University of Amsterdam（阿姆斯特丹大学逻辑、语言与计算研究所）；Université Paris Dauphine-PSL（巴黎多芬-PSL大学）
We study settings in which agents with incomplete preferences need to make a collective decision. We focus on a process of majority dynamics where issues are addressed one at a time and undecided agents follow the opinion of the majority. We assess the effects of this process on various consensus notions -- such as the Condorcet winner -- and show that in the worst case, myopic adherence to the majority damages existing consensus; yet, simulation experiments indicate that the damage is often mild. We also examine scenarios where the chair of the decision process can control the existence (or t…
Why AI Safety Requires Uncertainty, Incomplete Preferences, and Non-Archimedean Utilities
School of Computer Science and Statistics, Trinity College Dublin（特里尼蒂大学计算机科学与统计学学院）；Trinity Quantum Alliance, Trinity College Dublin（特里尼蒂量子联盟）；Management in Networked and Digital Societies (MINDS) Department, Kozminski University（科津斯基大学网络化与数字化社会管理系）
Preferences play a key role in determining what goals/constraints to satisfy when not all constraints can be satisfied simultaneously. In this paper, we study how to synthesize preference satisfying plans in stochastic systems, modeled as an MDP, given a (possibly incomplete) combinative preference model over temporally extended goals. We start by introducing new semantics to interpret preferences over infinite plays of the stochastic system. Then, we introduce a new notion of improvement to enable comparison between two prefixes of an infinite play. Based on this, we define two solution conce…
To mitigate the imbalance in the number of assignees in the Hospitals/Residents problem, Goko et al. [Goko et al., Maximally Satisfying Lower Quotas in the Hospitals/Residents Problem with Ties, Proc. STACS 2022, pp. 31:1--31:20] studied the Hospitals/Residents problem with lower quotas whose goal is to find a stable matching that satisfies lower quotas as much as possible. In their paper, preference lists are assumed to be complete, that is, the preference list of each resident (resp., hospital) is assumed to contain all the hospitals (resp., residents). In this paper, we study a more general…
School choice is the two-sided matching market where students (on one side) are to be matched with schools (on the other side) based on their mutual preferences. The classical algorithm to solve this problem is the celebrated deferred acceptance procedure, proposed by Gale and Shapley. After both sides have revealed their mutual preferences, the algorithm computes an optimal stable matching. Most often in practice, notably when the process is implemented by a national clearinghouse and thousands of schools enter the market, there is a quota on the number of applications that a student can subm…
