We study fair division problems with strategic agents capable of gaining advantages by manipulating their reported preferences. Although several impossibility results have revealed the incompatibility of truthfulness with standard fairness criteria, subsequent works have circumvented this limitation through the incentive ratio framework. Previous studies demonstrate that fundamental mechanisms like Maximum Nash Welfare (MNW) and Probabilistic Serial (PS) for divisible goods, and Round-Robin (RR) for indivisible goods achieve an incentive ratio of 2, implying that no individual agent can gain more than double his truthful utility through manipulation. However, collusive manipulation by agent groups remains unexplored. In this work, we define strong group incentive ratio (SGIR) and group incentive ratio (GIR) to measure the gain of collusive manipulation, where SGIR and GIR are respectively the maximum and minimum of the incentive ratios of corrupted agents. Then, we tightly characterize the SGIRs and GIRs of MNW, PS, and RR. In particular, the GIR of MNW is 2 regardless of the coalition size. Moreover, for coalition size c ≥ 1 , the SGIRs of MNW and PS, and the GIRs of PS and RR are c + 1 . Finally, the SGIR of RR is unbounded for coalition size c ≥ 2 . Our results reveal fundamental differences of these three mechanisms in their vulnerability to collusion.

We study the problem of fairly allocating indivisible goods among n strategic agents. It is well-known that truthfulness is incompatible with any meaningful fairness notions. We bypass the strong negative result by considering the concept of incentive ratio, a relaxation of truthfulness quantifying agents’ incentive to misreport. Previous studies show that Round-Robin, which satisfies envy-freeness up to one good (EF1), achieves an incentive ratio of 2 for additive valuations. In this paper, we explore the incentive ratio achievable by fair mechanisms for various classes of valuations besides additive ones. We first show that, for arbitrary ϵ > 0 , every (1/2 + ϵ ) -EF1 mechanism for additive valuations admits an incentive ratio of at least 1.5. Then, using the above lower bound for additive valuations in a black-box manner, we show that for arbitrary ϵ > 0 , every ϵ -EF1 mechanism for cancelable valuations admits an infinite incentive ratio. Moreover, for subadditive cancelable valuations, we show that Round-Robin, which satisfies EF1, achieves an incentive ratio of 2, and every (φ - 1) -EF1 mechanism admits an incentive ratio of at least φ with φ = (1 + √(5)) / 2 ≈ 1.618 . Finally, for submodular valuations, we show that Round-Robin, which satisfies 1/2 -EF1, admits an incentive ratio of n.

We study the problem of allocating divisible resources among $n$ agents, hopefully in a fair and efficient manner. With the presence of strategic agents, additional incentive guarantees are also necessary, and the problem of designing fair and efficient mechanisms becomes much less tractable. While there are flourishing positive results against strategic agents for homogeneous divisible items, very few of them are known to hold in cake cutting. We show that the Maximum Nash Welfare (MNW) mechanism, which provides desirable fairness and efficiency guarantees and achieves an \emph{incentive ratio} of $2$ for homogeneous divisible items, also has an incentive ratio of $2$ in cake cutting. Remarkably, this result holds even without the free disposal assumption, which is hard to get rid of in the design of truthful cake cutting mechanisms. Moreover, we show that, for cake cutting, the Partial Allocation (PA) mechanism proposed by Cole et al. \cite{DBLP:conf/sigecom/ColeGG13}, which is truthful and $1/e$-MNW for homogeneous divisible items, has an incentive ratio between $[e^{1 / e}, e]$ and when randomization is allowed, can be turned to be truthful in expectation. Given two alternatives for a trade-off between incentive ratio and Nash welfare provided by the MNW and PA mechanisms, we establish an interpolation between them for both cake cutting and homogeneous divisible items. Finally, we study the existence of fair mechanisms with a low incentive ratio in the connected pieces setting. We show that any envy-free cake cutting mechanism with the connected pieces constraint has an incentive ratio of $\Theta(n)$.

We study the problem of fairly and efficiently allocating a set of items among strategic agents with additive valuations, where items are either all indivisible or all divisible. When items are goods, numerous positive and negative results are known regarding the fairness and efficiency guarantees achievable by truthful mechanisms, whereas our understanding of truthful mechanisms for chores remains considerably more limited. In this paper, we discover various connections between truthful good and chore allocations, greatly enhancing our understanding of the latter via tools from the former. For indivisible chores with two agents, by leveraging the observation that a simple bundle-swapping operation transforms several properties for goods including truthfulness to the corresponding properties for chores, we characterize truthful mechanisms and derive tight guarantees of various fairness notions achieved by truthful mechanisms. Moreover, for homogeneous divisible chores, by generalizing the above transformation to an arbitrary number of agents, we characterize truthful mechanisms with two agents, show that every truthful mechanism with two agents admits an efficiency ratio of 0, and derive a large family of strictly truthful, envy-free (EF), and proportional mechanisms for an arbitrary number of agents. Finally, for indivisible chores with an arbitrary number of agents having bi-valued cost functions, we give an ex-ante truthful, ex-ante Pareto optimal, ex-ante EF, and ex-post envy-free up to one item mechanism, improving the best guarantees for bi-valued instances by prior works.

In today's online advertising markets, it is common for advertisers to set long-term budgets. Correspondingly, advertising platforms adopt budget control methods to ensure that advertisers' payments lie within their budgets. Most budget control methods rely on the value distributions of advertisers. However, due to the complex advertising landscape and potential privacy concerns, the platform hardly learns advertisers' true priors. Thus, it is crucial to understand how budget control auction mechanisms perform under unassured priors. This work answers this problem from multiple aspects. We consider the unassured prior game among the seller and all buyers induced by different mechanisms in the stochastic model. We restrict the parameterized mechanisms to satisfy the budget-extracting condition, which maximizes the seller's revenue by extracting buyers' budgets as effectively as possible. Our main result shows that the Bayesian revenue-optimal mechanism and the budget-extracting bid-discount first-price mechanism yield the same set of Nash equilibrium outcomes in the unassured prior game. This implies that simple mechanisms can be as robust as the optimal mechanism under unassured priors in the budget-constrained setting. In the symmetric case, we further show that all these five (budget-extracting) mechanisms share the same set of possible outcomes. We further dig into the structural properties of these mechanisms. We characterize sufficient and necessary conditions on the budget-extracting parameter tuple for bid-discount/pacing first-price auctions. Meanwhile, when buyers do not take strategic behaviors, we exploit the dominance relationships of these mechanisms by revealing their intrinsic structures.

We study the framework of a dynamic decision-making scenario with resource constraints. In this framework, an agent, whose target is to maximize the total reward under the initial inventory, selects an action in each round upon observing a random request, leading to a reward and resource consumptions that are further associated with an unknown random external factor. While previous research has already established an $\widetilde{O}(\sqrt{T})$ worst-case regret for this problem, this work offers two results that go beyond the worst-case perspective: one for the worst-case gap between benchmarks and another for logarithmic regret rates. We first show that an $\Omega(\sqrt{T})$ distance between the commonly used fluid benchmark and the online optimum is unavoidable when the former has a degenerate optimal solution. On the algorithmic side, we merge the re-solving heuristic with distribution estimation skills and propose an algorithm that achieves an $\widetilde{O}(1)$ regret as long as the fluid LP has a unique and non-degenerate solution. Furthermore, we prove that our algorithm maintains a near-optimal $\widetilde{O}(\sqrt{T})$ regret even in the worst cases and extend these results to the setting where the request and external factor are continuous. Regarding information structure, our regret results are obtained under two feedback models, respectively, where the algorithm accesses the external factor at the end of each round and at the end of a round only when a non-null action is executed.

We study the problem of fairly allocating indivisible goods among strategic agents. Amanatidis et al. show that truthfulness is incompatible with any meaningful fairness notions. Thus we adopt the notion of incentive ratio, which is defined as the ratio between the largest possible utility that an agent can gain by manipulation and his utility in honest behavior under a given mechanism. We select four of the most fundamental mechanisms in the literature on discrete fair division, which are Round-Robin, a cut-and-choose mechanism of Plaut and Roughgarden, Maximum-Nash-Welfare and Envy-Graph Procedure, and obtain extensive results regarding the incentive ratios of them and their variants. For Round-Robin, we establish the incentive ratio of $2$ for additive and subadditive cancelable valuations, the unbounded incentive ratio for cancelable valuations, and the incentive ratios of $n$ and $\lceil m / n \rceil$ for submodular and XOS valuations, respectively. Moreover, the incentive ratio is unbounded for a variant that provides the $1/n$-approximate maximum social welfare guarantee. For the algorithm of Plaut and Roughgarden, the incentive ratio is either unbounded or $3$ with lexicographic tie-breaking and is $2$ with welfare maximizing tie-breaking. This separation exhibits the essential role of tie-breaking rules in the design of mechanisms with low incentive ratios. For Maximum-Nash-Welfare, the incentive ratio is unbounded. Furthermore, the unboundedness can be bypassed by restricting agents to have a strictly positive value for each good. For Envy-Graph Procedure, both of the two possible ways of implementation lead to an unbounded incentive ratio. Finally, we complement our results with a proof that the incentive ratio of every mechanism satisfying envy-freeness up to one good is at least $1.074$, and thus is larger than $1$ by a constant.

We study the problem of dynamically allocating T indivisible items to n agents with the restriction that the allocation is fair all the time. Due to the negative results to achieve fairness when allocations are irrevocable, we allow adjustments to make fairness attainable with the objective to minimize the number of adjustments. For restricted additive or general identical valuations, we show that envy-freeness up to one item (EF1) can be achieved with no adjustments. For additive valuations, we give an EF1 algorithm that requires O(mT) adjustments, improving the previous result of O(nmT) adjustments, where m is the maximum number of different valuations for items among all agents. We further impose the contiguity constraint on items such that items are arranged on a line by the order they arrive and require that each agent obtains a consecutive block of items. We present extensive results to achieve either proportionality with an additive approximate factor (PROPa) or EF1, where PROPa is a weaker fairness notion than EF1. In particular, we show that for identical valuations, achieving PROPa requires Θ(nT) adjustments. Moreover, we show that it is hopeless to make any significant improvement for either PROPa or EF1 when valuations are nonidentical. Our results exhibit a large discrepancy between the identical and nonidentical cases in both contiguous and noncontiguous settings. All our positive results are computationally efficient.

Contextual Bandits with Knapsacks (CBwK) is a fundamental and essential framework for modeling a dynamic decision-making scenario with resource constraints. Under this framework, an agent selects an action in each round upon observing a request, leading to a reward and resource consumption that are further associated with an unknown external factor. The agent's target is to maximize the total reward under the initial inventory. While previous research has already established an $\widetilde{O}(\sqrt{T})$ worst-case regret for this problem, this work offers two results that go beyond the worst-case perspective, one for worst-case locations, and another for logarithmic regret rates. We start by demonstrating that the unique-optimality and degeneracy of the fluid LP problem, which is both succinct and easily verifiable, is a sufficient condition for the existence of an $\Omega(\sqrt{T})$ regret lower bound. To supplement this worst-case location result, we merge the re-solving heuristic with distribution estimation skills and propose an algorithm that achieves an $\widetilde{O}(1)$ regret as long as the fluid LP has a unique and non-degenerate solution. This condition is mild as it is satisfied for most problem instances. Furthermore, we prove our algorithm maintains a near-optimal $\widetilde{O}(\sqrt{T})$ regret even in the worst cases, and extend these results to the setting where request and external factor are continuous. Regarding information, our regret results are obtained under two feedback models, respectively, where the algorithm accesses the external factor at the end of each round and at the end of a round only when a non-null action is executed.

In Euclidean Uniform Facility Location, the input is a set of clients in $\mathrm{R}^{d}$ and the goal is to place facilities to serve them, so as to minimize the total cost of opening facilities plus connecting the clients. We study the classical setting of dynamic geometric streams, where the clients are presented as a sequence of insertions and deletions of points in the grid $\{1,ldots\,\Delta \}^{d}$, and we focus on the high-dimensional regime, where the algorithm’s space complexity must be polynomial (and certainly not exponential) in $d \cdot \log \Delta$.We present a new algorithmic framework, based on importance sampling from the stream, for $O(1)$-approximation of the optimal cost using only poly $(d\cdot\log\Delta)$ space. This framework is easy to implement in two passes, one for sampling points and the other for estimating their contribution. Over random-order streams, we can extend this to a one-pass algorithm by using the two halves of the stream separately. Our main result, for arbitrary-order streams, computes $O(d^{1.5})$-approximation in one pass by using the new framework but combining the two passes differently. This improves upon previous algorithms that either need space exponential in d or only guarantee $O(d\cdot\log^{2}\Delta)$-approximation, and therefore our algorithms for high-dimensional streams are the first to avoid the $O(\log\Delta)$ factor in approximation that is inherent to the widely-used quadtree decomposition. Our improvement is achieved by employing a geometric hashing scheme that maps points in $\mathbb{R}^{d}$ into buckets of bounded diameter, with the key property that every point set of small-enough diameter is hashed into at most poly $(d)$ distinct buckets.Finally, we complement our results with a proof that every streaming 1.085-approximation algorithm requires space exponential in poly $(d \cdot log \Delta)$, even for insertion-only streams.

In today’s online advertising markets, it is common for an advertiser to set a long-period budget. Correspondingly, advertising platforms adopt budget control methods to ensure that an advertiser’s payment is within her budget. Most budget control methods rely on the value distributions of advertisers. However, the platform hardly learns their true priors due to the complex environment advertisers stand in and privacy issues. Therefore, it is essential to understand how budget control auction mechanisms perform under unassured priors. This paper gives a two-fold answer. First, we introduce the bid-discount method into first-price auction. We show that the resulting auction mechanism exhibits desirable revenue maximization and computation properties. Second, we compare this mechanism with other four in the prior manipulation model, where an advertiser can arbitrarily report a value distribution to the platform. These four mechanisms include the optimal mechanism satisfying budget-constrained IC, first-price/second-price mechanisms with the widely-studied pacing method, and bid-discount second-price mechanism. We consider two different settings varying on whether the seller knows the reported value distributions before choosing the auction mechanism. When the reported priors are pre-known to the seller, we show that the bid-discount first-price auction we introduce dominates the other four mechanisms concerning the platform’s revenue. On the other hand, when the seller has no information on the reported priors before committing to a mechanism, we show a surprising strategic-equivalence result between this mechanism and the optimal auction. When buyers are further symmetric, we establish a wide equivalence among variants of first-price and second-price auctions. Based on these findings, we provide a thor-ough understanding of prior dependency in repeated auctions with budgets. The bid-discount first-price

In the problem of (binary) contextual bandits with knapsacks (CBwK), the agent receives an i.i.d. context in each of the $T$ rounds and chooses an action, resulting in a random reward and a random consumption of resources that are related to an i.i.d. external factor. The agent's goal is to maximize the accumulated reward under the initial resource constraints. In this work, we combine the re-solving heuristic, which proved successful in revenue management, with distribution estimation techniques to solve this problem. We consider two different information feedback models, with full and partial information, which vary in the difficulty of getting a sample of the external factor. Under both information feedback settings, we achieve two-way results: (1) For general problems, we show that our algorithm gets an $\widetilde O(T^{\alpha_u} + T^{\alpha_v} + T^{1/2})$ regret against the fluid benchmark. Here, $\alpha_u$ and $\alpha_v$ reflect the complexity of the context and external factor distributions, respectively. This result is comparable to existing results. (2) When the fluid problem is linear programming with a unique and non-degenerate optimal solution, our algorithm leads to an $\widetilde O(1)$ regret. To the best of our knowledge, this is the first $\widetilde O(1)$ regret result in the CBwK problem regardless of information feedback models. We further use numerical experiments to verify our results.