We study envy-free cake cutting with strategic agents, where each agent may manipulate his private information in order to receive a better allocation. We focus on piecewise constant utility functions and consider two scenarios: the general setting without any restriction on the allocations and the restricted setting where each agent has to receive a connected piece. We show that no deterministic truthful envy-free mechanism exists in the connected piece scenario, and the same impossibility result for the general setting with some additional mild assumptions on the allocations. Finally, we study a large market model where the economy is replicated and demonstrate that truth-telling converges to a Nash equilibrium.
A classic theorem of Euclidean geometry asserts that any noncollinear set of n points in the plane determines at least n distinct lines. Chen and Chvátal conjectured that this holds for an arbitrary finite metric space, with a certain natural definition of lines in a metric space.We prove that in any metric space with n points, either there is a line containing all the points or there are at least Ω( √ n) lines. This is the first polynomial lower bound on the number of lines in general finite metric spaces. In the more general setting of pseudometric betweenness, we prove a corresponding bound of Ω(n 2/5 ) lines. When the metric space is induced by a connected graph, we prove that either there is a line containing all the points or there are Ω(n 4/7 ) lines, improving the previous Ω(n 2/7 ) bound. We also prove that the number of lines in an n-point metric space is at least n/5w, where w is the number of different distances in the space, and we give an Ω(n 4/3 ) lower bound on the number of lines in metric spaces induced by graphs with constant diameter, as well as spaces where all the positive distances are from {1, 2, 3}.
We study the problem of fairly dividing a heterogeneous resource, commonly known as cake cutting and chore division, in the presence of strategic agents. While a number of results in this setting have been established in previous works, they rely crucially on the free disposal assumption, meaning that the mechanism is allowed to throw away part of the resource at no cost. In the present work, we remove this assumption and focus on mechanisms that always allocate the entire resource. We exhibit a truthful and envy-free mechanism for cake cutting and chore division for two agents with piecewise uniform valuations, and we complement our result by showing that such a mechanism does not exist when certain additional constraints are imposed on the mechanisms. Moreover, we provide bounds on the efficiency of mechanisms satisfying various properties, and give truthful mechanisms for multiple agents with restricted classes of valuations.
We study the online allocation problem under a roommate market model introduced in [Chan et al., 2016]. Consider a fixed supply of n rooms and a list of 2n applicants arriving online in random order. The problem is to assign a room to each person upon her arrival, such that after the algorithm terminates, each room is shared by exactly two people. We focus on two objectives: (1) maximizing the social welfare, which is defined as the sum of valuations that applicants have for their rooms, plus the happiness value between each pair of roommates; (2) satisfying the stability property that no small group of people would be willing to switch roommates or rooms. We first show a polynomial-time online algorithm that achieves a constant competitive ratio for social welfare maximization. We then extend it to the case where each room is assigned to c > 2 people, and achieve a competitive ratio of Ω(1/c 2 ). Finally, we show both positive and negative results in satisfying various stability conditions in this online setting.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.