arXiv subjects

Xiaolin Bu

Publications and source records attributed to Xiaolin Bu.

11 recordsLinked to original sources

arXiv2026

Bayesian Fair Division: Truthfulness in Picking Sequence with Correlated Valuations

Sequential allocation mechanisms contain a class of widely studied mechanisms (e.g., round-robin) in the fair division of indivisible goods, where agents take turns picking items in a predefined picking order. It is known that the sequential allocation mechanisms are not truthful: when an agent's most preferred item is not valued by others, the agent may manipulate the mechanism by choosing to defer picking that item and instead competing for another slightly less preferred item that is valued by others. Two underlying reasons are that each agent has perfect knowledge of the others' valuations, and each item's value to each agent can differ significantly. Will the mechanism be more truthful when each agent only has partial information about the others' valuations, which are known to be roughly consistent? This naturally motivates the study of the Bayesian fair division model. In this paper, we answer this question affirmatively for two agents. Under the Bayesian model, we precisely characterize the extent of this ``rough consistency'' that incentivizes agents' truth-telling. In particular, we show that for the case of two agents, when the valuations are positively correlated, truth-telling forms a Bayesian Nash equilibrium under the sequential allocation mechanisms. However, we show that truthfulness fails to extend to the setting with more than two agents. For more than two agents, we reveal a new type of manipulation that is different from the above-mentioned manipulation that defers a highly valued but less competitive item. Our result reveals a fundamental limitation on the truthfulness of sequential mechanisms.

cs.GT↗

arXiv2026

Auctions with Contract Design

We consider a new auction model where the bidders' utilities and the auctioneer's revenue depend on a quality factor of the transaction determined by costly and strategic investments of the bidders. Applications of our model include ad auctions, government concessions and crowdsourcing contests. Crucially, these quality-enhancing efforts made by the bidders are often sunk costs incurred prior to the allocation, creating a fundamental moral hazard problem where the risk of losing the auction discourages investments. In this paper, we study the design of revenue-maximizing contracts integrated into auctions: the auctioneer commits to a transfer rule that rewards the winner for the ex-post realized quality of the transaction to incentivize higher effort. Our new framework is a natural generalization of both the auction theory and the principal-agent model. We consider both the second-price and the first-price auctions. We show that natural symmetric Bayes Nash equilibria exist in both auctions. Assuming these natural equilibria are played by the bidders and the number of bidders is large, we study linear contracts and derive the optimal reward factor of the transfer rule that maximizes the auctioneer's revenue. As the main result, we show that the optimal reward factor converges to the auctioneer's marginal benefit from the quality, as the number of bidders grows. That is, it is optimal for the auctioneer to fully pass through the quality value to the winner. This observation is largely independent of the auction rule used: we derive a revenue equivalence theorem showing that the revenue remains the same as long as symmetric Bayes Nash equilibria exist. Lastly, by quantitatively comparing with the standard auctions where no quality reward is used, we show that the use of contracts effectively improves the revenue by incentivizing high investments from the bidders.

cs.GT↗

arXiv2026

Best-of-Both-Worlds Fairness for Mixed Goods and Chores

We study the fundamental problem of fairly dividing indivisible items among agents with additive utilities. In our model, an item can be a good yielding non-negative utilities to some agents and simultaneously a chore yielding negative utilities to others. We take the best-of-both-worlds perspective and our goal is to construct a randomized allocation that is exactly fair ex ante while also being supported on ex post approximately fair allocations. The fairness notions examined in this paper are envy-freeness (EF) and its well-known relaxation envy-freeness up to one item (EF1). Our main result is that ex-ante EF and ex-post EF1 can be achieved simultaneously. To achieve this, we introduce a novel probabilistic Hall-type matrix decomposition that intricately correlates the fractional assignments of goods and chores. We resolve this decomposition problem by combining continuous minimax duality -- via Sion's minimax theorem -- with carefully designed biased flow networks.

cs.GT↗

arXiv2025

Truthful and Almost Envy-Free Mechanism of Allocating Indivisible Goods: the Power of Randomness

We study the problem of fairly and truthfully allocating $m$ indivisible items to $n$ agents with additive preferences. Specifically, we consider truthful mechanisms outputting allocations that satisfy EF$^{+u}_{-v}$, where, in an EF$^{+u}_{-v}$ allocation, for any pair of agents $i$ and $j$, agent $i$ will not envy agent $j$ if $u$ items were added to $i$'s bundle and $v$ items were removed from $j$'s bundle. Previous work easily indicates that, when restricted to deterministic mechanisms, truthfulness will lead to a poor guarantee of fairness: even with two agents, for any $u$ and $v$, EF$^{+u}_{-v}$ cannot be guaranteed by truthful mechanisms when the number of items is large enough. In this work, we focus on randomized mechanisms, where we consider ex-ante truthfulness and ex-post fairness. For two agents, we present a truthful mechanism that achieves EF$^{+0}_{-1}$ (i.e., the well-studied fairness notion EF$1$). For three agents, we present a truthful mechanism that achieves EF$^{+1}_{-1}$. For $n$ agents in general, we show that there exists a truthful mechanism that achieves EF$^{+0}_{-O(\sqrt{n})}$. On the negative side, when considering the stronger notion EF$_{-v}^{+u}$X, we show that it cannot be achieved by any randomized truthful mechanism for any $u, v$, and any fixed number of agents. We further consider fair and truthful mechanisms that also satisfy the standard efficiency guarantee: Pareto-optimality. We provide a mechanism that simultaneously achieves truthfulness, EF$1$, and Pareto-optimality for bi-valued utilities (where agents' valuation on each item is either $p$ or $q$ for some $p>q\geq0$). For tri-valued utilities (where agents' valuations on each item belong to $\{p,q,r\}$ for some $p>q>r\geq0$) and any $u,v$, we show that truthfulness is incompatible with EF$^{+u}_{-v}$ and Pareto-optimality even for two agents.

cs.GT↗

arXiv2025

Logarithmic Comparison-Based Query Complexity for Fair Division of Indivisible Goods

We study the problem of fairly allocating $m$ indivisible goods to $n$ agents, where agents may have different preferences over the goods. In the traditional setting, agents' valuations are provided as inputs to the algorithm. In this paper, we adopt the query model, which has been widely considered for other similar problems (such as matching [Nis21], graph isomorphism [OS18], and equilibrium in game [Bab16]), and apply it to the fair division problem. In particular, we consider a new \emph{comparison-based query model}, where the algorithm presents two bundles of goods to an agent and the agent responds by telling the algorithm which bundle she prefers. We investigate the query complexity for computing allocations with several fairness notions, including \emph{proportionality up to one good} (PROP1), \emph{envy-freeness up to one good} (EF1), and \emph{maximin share} (MMS). Our main result is an algorithm that computes an allocation that satisfies both PROP1 and $\frac12$-MMS within $O(\log m)$ queries with a constant number of $n$ agents. For identical and additive valuation, we present an algorithm for computing an EF1 allocation within $O(\log m)$ queries with a constant number of $n$ agents. To complement the positive results, we show that the lower bound of the query complexity for any of the three fairness notions is $Ω(\log m)$ even with two agents.

cs.GT↗

arXiv2025

Fair Division with Allocator's Preference

We study the fair allocation of indivisible resources among agents. Most prior work focuses on fairness and/or efficiency among agents. However, the allocator, as the resource owner, may also be involved in many scenarios (e.g., government resource allocation, heritage division, company personnel assignment, etc). The allocator inclines to obtain a fair or efficient allocation based on her preference over the items and to whom each item is allocated. We propose a model and study two problems: 1) Find an allocation fair to both agents and allocator; 2) Maximize allocator's efficiency under agents' fairness. We consider the two fundamental fairness criteria: envy-freeness and proportionality. For the first problem, we study the existence of an allocation that is envy-free up to $c$ goods (EF-$c$) or proportional up to $c$ goods (PROP-$c$) from both the agents' and the allocator's perspectives, called doubly EF-$c$ or doubly PROP-$c$. When the allocator's utility only depends on the items (not recipients), we prove that a doubly EF-$1$ allocation always exists. For the general setting where the allocator has a preference over the items and to whom each item is allocated, a doubly EF-$1$ allocation always exists for two agents, a doubly PROP-$2$ allocation always exists for personalized bi-valued valuations, and a doubly PROP-$O(\log n)$ allocation always exists. For the second problem, we give (in)approximability results with asymptotically tight bounds in most settings. When agents' valuations are binary, maximizing the allocator's social welfare while ensuring agents' fairness criteria of PROP-$c$ (with a general number of agents) and EF-$c$ (with a constant number of agents) are both polynomial-time solvable for any integer $c$. Strong inapproximability holds for most of the other settings (general valuations, EF-$c$, etc).

cs.GT↗

arXiv2025

Approximability Landscape of Welfare Maximization within Fair Allocations

Fair allocation of indivisible goods studies allocating $m$ goods among $n$ agents in a fair manner. While fairness is a fundamental requirement in many real-world applications, it often conflicts with (economic) efficiency. This raises a natural and important question: How can we identify the most welfare-efficient allocation among all fair allocations? This paper answers from the perspective of computational complexity. Specifically, we study the problem of maximizing utilitarian social welfare under two widely studied fairness criteria: envy-freeness up to any item (EFX) and envy-freeness up to one item (EF1). We examine both normalized and unnormalized valuations, where normalized valuations require that each agent's total utility for all items is identical. The key contributions of this paper can be summarized as follows: (i) we sketch the complete complexity landscape of welfare maximization subject to fair allocation constraints; and (ii) we provide interesting bounds on the price of fairness for both EFX and EF1. Specifically: (1) For $n=2$ agents, we develop polynomial-time approximation schemes (PTAS) and provide NP-hardness results for EFX and EF1 constraints; (2) For $n>2$ agents, under EFX constraints, we design algorithms that achieve approximation ratios of $O(n)$ and $O(\sqrt{n})$ for unnormalized and normalized valuations, respectively. These results are complemented by asymptotically tight inapproximability results. We also obtain similar results for EF1 constraints; (3) When the number of agents is a fixed constant, we show that the optimal solution can be computed in polynomial time by slightly relaxing the fairness constraints, whereas exact fairness leads to strong inapproximability; (4) Furthermore, our results imply the price of EFX is $Θ(\sqrt{n})$ for normalized valuations, which is unknown in the literature.

cs.GT↗

arXiv2024

Best-of-Both-Worlds Fair Allocation of Indivisible and Mixed Goods

We study the problem of fairly allocating either a set of indivisible goods or a set of mixed divisible and indivisible goods (i.e., mixed goods) to agents with additive utilities, taking the best-of-both-worlds perspective of guaranteeing fairness properties both ex ante and ex post. The ex-post fairness notions considered in this paper are relaxations of envy-freeness, specifically, EFX for indivisible-goods allocation, and EFM for mixed-goods allocation. For two agents, we show that there is a polynomial-time randomized algorithm that achieves ex-ante envy-freeness and ex-post EFX / EFM simultaneously. For $n$ agents with bi-valued utilities, we show there exist randomized allocations that are (i) ex-ante proportional and ex-post EFM, and (ii) ex-ante envy-free, ex-post EFX, and ex-post fractionally Pareto optimal.

cs.GT↗

arXiv2023

EFX Allocations Exist for Binary Valuations

We study the fair division problem and the existence of allocations satisfying the fairness criterion envy-freeness up to any item (EFX). The existence of EFX allocations is a major open problem in the fair division literature. We consider binary valuations where the marginal gain of the value by receiving an extra item is either $0$ or $1$. Babaioff et al. [2021] proved that EFX allocations always exist for binary and submodular valuations. In this paper, by using completely different techniques, we extend this existence result to general binary valuations that are not necessarily submodular, and we present a polynomial time algorithm for computing an EFX allocation.

cs.CE↗

arXiv2023

On Existence of Truthful Fair Cake Cutting Mechanisms

We study the fair division problem on divisible heterogeneous resources (the cake cutting problem) with strategic agents, where each agent can manipulate his/her private valuation in order to receive a better allocation. A (direct-revelation) mechanism takes agents' reported valuations as input and outputs an allocation that satisfies a given fairness requirement. A natural and fundamental open problem, first raised by [Chen et al., 2010] and subsequently raised by [Procaccia, 2013] [Aziz and Ye, 2014] [Branzei and Miltersen, 2015] [Menon and Larson, 2017] [Bei et al., 2017] [Bei et al., 2020], etc., is whether there exists a deterministic, truthful and envy-free (or even proportional) cake cutting mechanism. In this paper, we resolve this open problem by proving that there does not exist a deterministic, truthful and proportional cake cutting mechanism, even in the special case where all of the following hold: 1) there are only two agents; 2) agents' valuations are piecewise-constant; 3) agents are hungry. The impossibility result extends to the case where the mechanism is allowed to leave some part of the cake unallocated. We also present a truthful and envy-free mechanism when each agent's valuation is piecewise-constant and monotone. However, if we require Pareto-optimality, we show that truthful is incompatible with approximate proportionality for any positive approximation ratio under this setting. To circumvent this impossibility result, motivated by the kind of truthfulness possessed by the I-cut-you-choose protocol, we propose a weaker notion of truthfulness: the proportional risk-averse truthfulness. We show that several well-known algorithms do not have this truthful property. We propose a mechanism that is proportionally risk-averse truthful and envy-free, and a mechanism that is proportionally risk-averse truthful that always outputs allocations with connected pieces.

cs.GT↗

arXiv2022

Fair Division with Prioritized Agents

We consider the fair division problem of indivisible items. It is well-known that an envy-free allocation may not exist, and a relaxed version of envy-freeness, envy-freeness up to one item (EF1), has been widely considered. In an EF1 allocation, an agent may envy others' allocated shares, but only up to one item. In many applications, we may wish to specify a subset of prioritized agents where strict envy-freeness needs to be guaranteed from these agents to the remaining agents, while ensuring the whole allocation is still EF1. Prioritized agents may be those agents who are envious in a previous EF1 allocation, those agents who belong to underrepresented groups, etc. Motivated by this, we propose a new fairness notion named envy-freeness with prioritized agents "EFPrior", and study the existence and the algorithmic aspects for the problem of computing an EFPrior allocation. With additive valuations, the simple round-robin algorithm is able to compute an EFPrior allocation. In this paper, we mainly focus on general valuations. In particular, we present a polynomial-time algorithm that outputs an EFPrior allocation with most of the items allocated. When all the items need to be allocated, we also present polynomial-time algorithms for some well-motivated special cases.

cs.GT↗