arXiv subjects

Nicholas Teh

Publications and source records attributed to Nicholas Teh.

At least 19 recordsLinked to original sources

arXiv2026

Fair Division with Binary Valuations: Characterizations

We consider the fair allocation of indivisible goods with binary valuations. In this setting, the maximum Nash welfare rule, the leximin rule, and all additive welfarist rules with a strictly concave function coincide. We show that for any number of agents, this rule is the only rule that satisfies envy-freeness up to one good, strategyproofness, neutrality, minimal completeness, and invariance under disapproving unassigned goods (IDU). Moreover, we present an alternative characterization for two agents, where we replace IDU with non-redundancy and resource-monotonicity. In both characterizations, all axioms are necessary.

econ.TH↗

arXiv2026

Online Fair Division of Chores with Upgrade Recourse

We study the online fair division of indivisible chores among $n$ agents under upgrade recourse, where earlier assignments may be revised but each chore previously assigned to an agent must be replaced by a distinct chore that she finds no more costly. We show that a simple adaptation of round robin maintains envy-freeness up to one chore (EF1) together with a $(2-1/n)$-approximation of the maximin share (MMS), and that no deterministic algorithm without predictions can do better, even when all agents rank the chores in the same order. This stands in contrast to the allocation of goods, where the optimal MMS factor vanishes as $n$ grows and cannot be achieved together with EF1. Moreover, knowing only the number of chores modulo a period of length $O(n)$ (and nothing about the costs) suffices to improve upon this factor at the final configuration, while the same algorithm retains a factor below $3$ even if this information is wrong. On the other hand, envy-freeness up to any chore (EFX) essentially requires the exact number of chores: uncertainty between two consecutive values already rules out every finite approximation, and even exact knowledge does not allow EFX to be combined with an MMS guarantee better than the trivial factor of $n$.

cs.GT↗

arXiv2026

Learning Lotteries with Minimal Stakeholder Violations from Membership Queries

An AI developer often has several candidate versions of a system and can deploy a mixture of them, sending each request to a randomly chosen version, as in A/B testing. Each stakeholder (e.g., a safety team, a regulator, or a customer with private test data) accepts a mixture exactly when its expected performance (e.g., its rate of harmful replies) meets the stakeholder's own standard, which the developer can learn only by asking whether a proposed mixture is acceptable. We model this as choosing a lottery over $m$ alternatives for $n$ agents, each of whom accepts exactly the lotteries in an unknown halfspace and answers such binary membership queries. While Choo et al. (ICML 2026) sought a lottery that every agent accepts, none may exist when, say, safety and helpfulness requirements conflict, so we minimize the number of rejecting agents. We show that this problem is APX-hard even when all halfspaces are known and all utilities are $0$ or $1$. We then give exact algorithms that run in polynomial time when $m$ is constant, and in quasi-polynomial time when, for a constant $η$, some optimal lottery has no accepting agent within $η$ of its threshold. When only some agents can be queried, optimizing over a uniform sample of agents gives additive and relative guarantees for the whole population, with a sample size that does not depend on $n$, and a linear program gives a polynomial-time method for any $m$. On the other hand, for $m\ge 3$, a deterministic algorithm that queries only $K$ distinct agents cannot guarantee fewer than $n-K$ extra rejections, and this is achievable; in other words, it can guarantee nothing for the agents it does not ask. Small synthetic experiments indicate that our worst-case bounds are conservative on random instances.

cs.LG↗

arXiv2026

Characterizing Maximum Nash Welfare: Envy-Freeness and Proportionality

When allocating indivisible goods, the maximum Nash welfare rule is known to be the unique welfarist rule satisfying envy-freeness up to one good. We strengthen this characterization by showing that envy-freeness up to one good can be replaced by substantially weaker fairness requirements: envy-freeness up to $k$ goods or proportionality up to $k$ goods, for any fixed $k$. Our characterization holds even for profiles with identical goods.

econ.TH↗

arXiv2026

Online Fair Division with Frequency Predictions

We study online fair division of indivisible goods among $n$ agents with frequency predictions: before any good arrives, the algorithm learns the multiset of each agent's values for the goods, but neither the arrival order nor which values of different agents belong to the same good. Neoh et al. (ICML 2026) showed that such predictions allow every share-based guarantee that is achievable offline, and left open whether a positive approximation of envy-freeness up to one good (EF1) is possible. We show that deterministic online algorithms can guarantee $1/2$-EF1 for three agents and $(3-\sqrt5)/2$-EF1 for four agents, whereas for five or more agents, no online algorithm can guarantee any positive approximation of EF1, even with $n+1$ goods and identical predictions. Together with the known result for two agents, a positive approximation of EF1 is therefore possible if and only if $n\leq4$. We also rule out any positive approximation of EFX for three or more agents, even when the valuations take only two distinct positive values. Next, we use the meta-algorithm of Neoh et al. (ICML 2026) to compute an epistemic EFX (EEFX) allocation in polynomial time for any $n$, and hence a $1/(n-1)$-EF$(n-2)$ allocation for $n\geq3$, whereas algorithms that fix a picking sequence in advance can guarantee a positive approximation of EF$k$ if and only if $n\leq k+2$. Finally, for noisy predictions, we give algorithms that return EEFX allocations when the predictions are exact, and guarantee $(1-2\varepsilon+O(\varepsilon^2))$-EEFX under underestimates and, for $n\geq3$, $(1-2(n-1)\varepsilon+O(\varepsilon^2))$-EEFX under overestimates; the linear terms are the best possible, and larger overestimates rule out any positive approximation for such algorithms.

cs.GT↗

arXiv2026

Online Fair Division under an Oblivious Adversary

We study online fair division, where indivisible goods arrive one at a time and must be allocated immediately and irrevocably among $n\ge2$ agents. Against an oblivious adversary that fixes the input in advance, we ask what fairness guarantees randomized online algorithms can achieve. We present the Deficit Allocation (DA) algorithm, which improves the previously known high-probability approximation to PROP1 from inverse-logarithmic to inverse-doubly-logarithmic in $n/δ$, where $δ$ is the failure probability. DA does not require $δ$ as input. With probability at least $1-δ$, a refined variant achieves an arbitrarily close approximation to proportionality when no single item is too valuable relative to an agent's proportional share; without any restriction on item values, it obtains an arbitrarily close approximation to PROP$k$ for $k$ doubly logarithmic in $n/δ$. For $n2^{-2n}\leδ\le1/2$, our bounds on the approximation factor and $k$ are asymptotically tight for the stronger fractional benchmark that divides each item equally among the agents who value it positively. We also prove that algorithms that use only which agents value each item positively, but not by how much, cannot improve upon the logarithmic dependence of uniformly random allocation. However, other fairness notions admit weaker guarantees. For EF1, every randomized online algorithm has an input on which any positive approximation succeeds with probability at most $e^{-Ω(n)}$. For EFX, this probability is at most $1/n!$ with only $n+1$ items, tight within a factor of $(n+1)/2$; for MMS, it can be at most $5/6$. Finally, allocating each item uniformly among the agents who value it positively gives, with probability at least $1-δ$, an arbitrarily close approximation to EF$k$ for $k$ logarithmic in $n/δ$, and this dependence is necessary for this rule.

cs.GT↗

arXiv2026

Online Fair Division with Upgrade Recourse

We study the online fair division of indivisible goods. In the standard model, where assignments are irrevocable, no algorithm can guarantee any positive fraction of the maximin share (MMS). Reassigning earlier goods can restore fairness, but it may take value away from agents who already hold them. We ask how much fairness can be maintained if every reassignment must leave each affected agent at least as well off. We consider two forms of such upgrade recourse. Under item-level upgrades, each good an agent holds must be kept or replaced by a distinct good she values at least as much; under bundle-level upgrades, only her total value must not decrease. For item-level upgrades, we show that the optimal MMS factor is essentially $1/H_n \approx 1/\ln n$, where $H_n$ is the $n$-th harmonic number, and that fixed picking sequences nearly attain it. Since these sequences use only the agents' rankings of individual goods, knowing the actual values cannot improve the optimal factor for up to 33 agents, and improves it by at most a vanishing margin as $n$ grows. Bundle-level upgrades allow substantially stronger guarantees. By permitting several-for-one exchanges, they turn this vanishing factor into a constant: we achieve exact MMS for two agents and more than half of MMS for any number of agents, in both cases together with relaxations of envy-freeness. Finally, we show that predictions about future goods let online algorithms inherit any offline MMS guarantee, losing only an additive term bounded by the prediction error. Under item-level upgrades, however, insisting on exact MMS when predictions are correct leaves no positive guarantee when they are wrong.

cs.GT↗

arXiv2026

Minimizing Cumulative Envy in Allocating a Sequence of Items

We study temporal fair division with indivisible goods that arrive sequentially and must be allocated irrevocably. In contrast to the usual online model, we assume that valuations and future arrivals are known in advance, and ask how unfairness evolves during the process. We introduce \emph{cumulative maximum envy}: the sum, over all rounds, of the maximum pairwise envy at that round. Equivalently, this is the area under the worst-envy curve, and it captures both the magnitude and the duration of envy. For a fixed arrival order, we show that the corresponding decision problem is strongly NP-complete and that minimizing this objective admits no constant-factor approximation unless P = NP, even under identical valuations and even under binary valuations. We complement these hardness results with a dynamic program that gives pseudopolynomial-time solvability for a constant number of agents, polynomial-time algorithms in further restricted settings, and an FPTAS for fixed $n$ under identical integer valuations. We then study a sequencing variant where the algorithm may choose the arrival order. This variant remains NP-complete even for two agents with identical valuations; however, a simple greedy algorithm achieves a $3/2$-approximation for $n=2$ agents, an $n/(n-1)$-approximation for any number of agents, and an additive guarantee depending on the maximum value of any good.

cs.GT↗

arXiv2026

Proportional Representation in Temporal Voting with Ranked Preferences

We study proportional representation in temporal voting, where one candidate is selected in each round. While prior work has focused on approval ballots, we consider ranked preferences, which may change over time. A natural approach treats each voter's top candidates as approved, but the right cutoff may differ across voters and rounds. We therefore require proportionality to hold for every admissible choice of cutoffs, whether fixed and common, common but varying across rounds, or set individually for each voter in each round. Combining these interpretations with temporal versions of justified representation (JR), proportional JR (PJR), extended JR (EJR), and proportionality for solid coalitions (PSC) gives us a hierarchy of axioms. We ask which of these axioms can be guaranteed, and with how much knowledge of the future. Unlike with approval ballots, no version of EJR can be guaranteed, and for the other axioms, flexibility in the cutoffs comes at a price. With a fixed common cutoff, JR, PJR, and PSC can be guaranteed, but only by rules that see all preferences in advance. Once the cutoff may vary across rounds, even such rules cannot guarantee JR or PSC for groups that agree in only some rounds. For groups that agree in every round, however, knowing only the number of rounds suffices for PJR in polynomial time, and PSC needs no knowledge of the future at all. Under individual cutoffs, no version of JR or PJR can be guaranteed, yet a rule as simple as serial dictatorship achieves PJR up to an additive loss that no rule can improve on, however much it knows. Natural preference restrictions restore exact guarantees. Finally, we show that checking our axioms is often coNP-complete; but perhaps surprisingly, a stronger axiom can be easier to check.

cs.GT↗

arXiv2026

Full Justified Representation in Temporal Voting: Efficient Computation and Verification

In temporal voting, a fixed set of voters makes a collective decision in each of several rounds, and proportionality requires that groups with shared interests be represented across these decisions. Full justified representation (FJR) is among the strongest proportionality axioms known to be satisfiable in this setting, since it allows different members of a group to be satisfied by different selected candidates. However, whether an FJR outcome can be computed in polynomial time has remained open. We resolve this question by introducing temporal FJR+, a strengthening of both FJR and extended justified representation+ (EJR+) that can be computed and verified in polynomial time. The axiom measures a group's agreement in each round by the largest number of its members who approve a common candidate, and requires some member to attain the integer part of the group's proportional share of its best attainable average satisfaction, with a shortfall of at most one unit when this share is an integer. We show that verification reduces to counting approvals among the voters below each satisfaction threshold. For computation, we observe that local search for Proportional Approval Voting (PAV) can stop at an outcome violating FJR. Somewhat counterintuitively, the remedy is to subtract a small multiple of the minimum voter satisfaction from the PAV score: every FJR+ violation then admits a single-round change whose improvement guarantees polynomial running time. We further show that temporal FJR+ is strictly stronger than FJR and EJR+ combined, and coincides with lower quota on party-list preferences. Finally, temporal FJR+ and multiwinner FJR+ are incomparable in general, but coincide when approvals are static and each candidate-round pair is treated as a separate candidate.

cs.GT↗

arXiv2026

Maximin Share Allocations Beyond Weakly Lexicographic Valuations

We study the fair division of indivisible goods among agents with nonnegative additive valuations. An agent's maximin share (MMS) is the value she can guarantee by partitioning the goods into as many bundles as there are agents and receiving a least-valued bundle. We prove the existence of exact MMS allocations for a valuation domain that generalizes several few-valued domains as well as weakly lexicographic valuations. For each agent, the lower-valued goods, after a positive rescaling, may take values in $\{0,1,d,d+1\}$, $\{0,1,2,2e\}$, or $\{0,1,2,3,4\}$, where $d,e\ge 2$ are integers. Above these goods, the valuation may contain arbitrarily many value classes, subject to the condition that a single good in each class is worth at least the total value of all strictly lower-valued goods. The value classes, scaling factors, and parameters may vary across agents. We give an algorithm that computes an MMS allocation in $O(nm\log^2(2+m))$ time, where $n$ is the number of agents and $m$ is the number of goods.

cs.GT↗

arXiv2026

Closing Gaps in Online Fair Division

We study the online fair division of indivisible items, where items arrive one at a time and must be allocated immediately and irrevocably. We address three central open questions in the literature. First, we show that for every $n\ge 2$ agents, every fixed $k\ge 1$, and every $α\in(0,1]$, no online algorithm can guarantee $α$-PROP$k$ against an adaptive adversary. This remains true even when the total number of goods is known in advance, all values lie in $[0,1]$, and every good is positively valued by at most two agents. The impossibility extends to a broad range of standard envy-based, proportionality-based, and share-based fairness notions. We also establish an analogous impossibility for chores. Second, in the setting with predictions, a lightweight form of future information, the maximum item value, was previously known to guarantee only $1/n$-PROP1, leaving open whether the dependence on $n$ is necessary. We give a deterministic $19/30$-PROP1 algorithm against adaptive adversaries that does not require knowing the total number of goods. Given an additional upper bound $κ\in[2,n]$ on the number of agents who value any good positively, the guarantee improves to $\max \{ 19/30, n/(n+κ) \}$. The guarantee remains a positive constant under any fixed one-sided prediction error below one. When predictions are exact and the total number of goods $m\ge n\log n$ is known in advance, a deterministic algorithm achieves the same $19/30$-PROP1 factor together with $O(\sqrt{m\log n/n})$ maximum additive envy after normalizing each agent's values by their maximum item value. Third, against a non-adaptive adversary, we determine the tight high-probability PROP1 guarantee of the classical Like rule, which assigns each good uniformly among the agents who value it positively. Its guarantee improves when fewer agents value the same good, unlike uniform random allocation.

cs.GT↗

arXiv2026

Strengthening Proportionality in Participatory Budgeting with Additive Utilities

Proportional representation is a central goal in participatory budgeting, where voters select public projects subject to a shared budget. Full justified representation (FJR) accounts for groups whose members may value different projects, but computing an FJR outcome is strongly NP-hard for additive utilities. We show that FJR up to one project can be achieved in polynomial time for arbitrary nonnegative additive utilities and project costs. Our approach extends Residual-Budget Greedy and satisfies a stronger fractional axiom that can also be verified in polynomial time. For approval utilities, this axiom coincides with FJR+, providing exact FJR in that setting. We further show that every feasible completion of the algorithm's selected set preserves the representation guarantee, providing flexibility in allocating the remaining budget. In particular, we give a completion that ensures priceability at the original budget. For unit-cost projects, we strengthen the representation guarantee to the Droop quota and combine it with priceability whenever there are enough positively valued projects to fill the budget.

cs.GT↗

arXiv2026

Weighted Fair Division of Indivisible Mixed Manna

We study weighted fair division of indivisible mixed manna under additive valuations. First, we resolve the general existence open question for weighted envy-freeness up to one item (WEF1), and show that every instance with arbitrary positive entitlements admits a complete WEF1 allocation computable in polynomial time. We then show that existence does not imply any welfare guarantee, i.e., the utilitarian price of WEF1 is infinite, even for two unweighted agents with normalized valuations, common item signs, and singleton values in a fixed four-value set; a welfare-maximizing WEF1 allocation in the construction is fractionally Pareto optimal. Second, suppose each agent $i$ has a number $a_i>0$ such that their valuation for any item is $-a_i$, $0$, or $a_i$. Then, for arbitrary entitlements, a weighted maximin share (WMMS) allocation always exists, is computable in polynomial time, and can be chosen to be fractionally Pareto optimal. An exact formula for each WMMS value leads to a polynomial-time flow algorithm. In this class, every WEF1 allocation satisfies a best possible additive WMMS guarantee whose loss depends on the agent's entitlement relative to the largest entitlement. Thus maximum entitlement agents receive exact WMMS and, under equal entitlements, every WEF1 allocation is also MMS-fair. Allowing a second positive magnitude can violate exact WMMS, while unrestricted entitlement ratios rule out any fixed multiplicative WMMS guarantee compatible with WEF1 for chores.

cs.GT↗

arXiv2026

Online House Allocation with Subsidy

House allocation is a fundamental problem in which each agent is assigned exactly one house. While the classical model assumes that all houses are available before the allocation is computed, many practical settings require decisions to be made as houses become available over time. We introduce the online house allocation problem, where houses arrive sequentially and the algorithm must maintain an allocation without knowledge of future arrivals. Unlike online fair division, the one-house-per-agent constraint makes recourse an inherent part of the problem, as accepting a newly arrived house may require reassigning previously allocated houses. We study online house allocation under subsidy-based fairness, where monetary subsidies eliminate envy among agents. We show that envy-freeability can always be maintained online using bounded recourse and that reassignment chains of length linear in the number of agents are unavoidable in the worst case. In contrast, minimizing the total subsidy is fundamentally harder: no deterministic online algorithm against an adaptive adversary, and no randomized online algorithm against a non-adaptive adversary, admits a bounded competitive ratio, even for two agents and four houses. We complement these impossibilities by showing that exact online subsidy minimization is possible whenever there is at most one extra house beyond the number of agents, and that this guarantee is best possible with respect to the number of extra houses. Finally, we develop learning-augmented algorithms that recover the offline optimum under accurate predictions while providing explicit robustness guarantees when predictions are inaccurate.

cs.GT↗

arXiv2026

Temporal Fair Division of Indivisible Mixed Manna: Tractable Settings

We study temporal fair division of indivisible mixed manna. Items arrive over time and must be allocated irrevocably; an item may be a good for some agents, a chore for others, and neutral for the rest. We require the cumulative allocation after every round to be envy-free up to one item (TEF1). Although deciding whether a TEF1 allocation exists is NP-hard even for goods, we identify several tractable settings. First, with at most $k$ item types, an online cyclic rule guarantees EF$\lceil k/2\rceil$ after every item arrival. Thus, every instance with at most two types admits an online TEF1 allocation; moreover, when the numbers of agents and types are fixed, TEF1 existence can be decided in polynomial time. Second, under agreement after agent-specific scaling, provided that the scaling factors are known before arrivals begin, an online rule produces an allocation that is EF1 and Pareto optimal after every item arrival. Third, for a two-part arrival sequence with common rankings, we give a rule that is EF1 after every item arrival. Fourth, when the number of agents is fixed and values are bounded integers, we give an exact pseudo-polynomial algorithm for deciding TEF1 existence. Finally, for goods, every TEF1 allocation gives each agent at least $1/n$ of her maximin share after every round; this factor is tight even for identical valuations and two rounds. Deciding whether an exact temporal maximin-share allocation exists is NP-hard for both goods and chores, even with identical valuations, two agents, and two rounds.

cs.GT↗

arXiv2026

Strengthening Full Justified Representation: Efficient Verification and Computation

Full justified representation (FJR) is among the strongest known satisfiable proportionality axioms for approval-based committee elections. Recent work has shown that an FJR committee can be found in polynomial time, but verifying whether a given committee satisfies FJR remains coNP-complete. We introduce FJR+, a strict strengthening of FJR and EJR+ that can be verified and satisfied in polynomial time. We then analyze the Residual-Budget Greedy (RBG) algorithm and prove that it selects a partial committee such that every size-$k$ completion satisfies FJR+. This freedom allows us to use sequential Phragmén to obtain a priceable completion. The resulting rule always satisfies FJR+ and the sub-core, and it is priceable whenever at least $k$ candidates receive an approval. We also obtain a Droop-quota version of FJR+. Finally, we extend FJR+ to approval-based participatory budgeting with arbitrary project costs. A project-specific version of RBG computes this property in polynomial time and can be continued to a priceable outcome satisfying a cost-based version of the sub-core.

cs.GT↗

arXiv2026

Online Fair Division with Budget Constraints

We study an online variant of discrete fair division under generalized assignment budget constraints. Goods arrive one at a time and must be assigned irrevocably to a feasible agent or to charity, which holds all unallocated goods, while fairness is evaluated only against budget-feasible subsets of every recipient's bundle. We first show that, without additional structure, no deterministic online algorithm can guarantee any fixed approximation to feasible envy-freeness, even in highly symmetric instances. We then identify bounded density spread as a structural condition that restores meaningful guarantees, obtaining approximation algorithms for arbitrary item sizes and showing that, under common valuations and sufficiently small goods, these guarantees can be strengthened to an optimal deterministic frontier. We further study resource augmentation, where the online algorithm is allowed slightly larger budgets than the fairness benchmark, and characterize the resulting improvement in the achievable guarantees. Finally, we develop a learning-augmented framework based on predicting joint value-size types, proving consistency under perfect predictions, robustness to prediction error, and showing that separate predictions of value and size marginals are insufficient to recover strong fairness guarantees.

cs.GT↗