An extension to "A subsemigroup of the rook monoid"

A recent paper studied an inverse submonoid $M_n$ of the rook monoid, by representing the nonzero elements of $M_n$ via certain triplets belonging to $\mathbb{Z}^3$. In this short note, we allow the triplets to belong to $\mathbb{R}^3$. We thus study a new inverse monoid $\overline{M}_n$, which is a supermonoid of $M_n$. We point out similarities and find essential differences. We show that $\overline{M}_n$ is a noncommutative, periodic, combinatorial, fundamental, completely semisimple, and strongly $E^*$-unitary inverse monoid

Approximately Stationary Bandits with Knapsacks

Bandits with Knapsacks (BwK), the generalization of the Bandits problem under global budget constraints, has received a lot of attention in recent years. Previous work has focused on one of the two extremes: Stochastic BwK where the rewards and consumptions of the resources of each round are sampled from an i.i.d. distribution, and Adversarial BwK where these parameters are picked by an adversary. Achievable guarantees in the two cases exhibit a massive gap: No-regret learning is achievable in the stochastic case, but in the adversarial case only competitive ratio style guarantees are achievable, where the competitive ratio depends either on the budget or on both the time and the number of resources. What makes this gap so vast is that in Adversarial BwK the guarantees get worse in the typical case when the budget is more binding. While ``best-of-both-worlds'' type algorithms are known (single algorithms that provide the best achievable guarantee in each extreme case), their bounds degrade to the adversarial case as soon as the environment is not fully stochastic. Our work aims to bridge this gap, offering guarantees for a workload that is not exactly stochastic but is also not worst-case. We define a condition, Approximately Stationary BwK, that parameterizes how close to stochastic or adversarial an instance is. Based on these parameters, we explore what is the best competitive ratio attainable in BwK. We explore two algorithms that are oblivious to the values of the parameters but guarantee competitive ratios that smoothly transition between the best possible guarantees in the two extreme cases, depending on the values of the parameters. Our guarantees offer great improvement over the adversarial guarantee, especially when the available budget is small. We also prove bounds on the achievable guarantee, showing that our results are approximately tight when the budget is small

Liquid Welfare guarantees for No-Regret Learning in Sequential Budgeted Auctions

We study the liquid welfare in repeated first-price auctions with budget limited buyers. We focus on first-price auctions, which are commonly used in many settings, and consider liquid welfare, a natural and well-studied generalization of social welfare for the case of budget-limited buyers. We use a behavioral model for the buyers, assuming a learning style guarantee: the resulting utility of each buyer is within a $\gamma$ factor (where $\gamma \ge 1$) of the utility achievable by shading her value with the same factor at each iteration. We show a $\gamma + 1/2 + O(1/\gamma)$ price of anarchy for liquid welfare assuming buyers have additive valuations. This positive result is in stark contrast to repeated second-price auctions, where even with $\gamma=1$, the resulting liquid welfare can be arbitrarily smaller than the maximum liquid welfare. We prove a lower bound of $\gamma$ on the liquid welfare loss under the above assumption in first-price auctions, making our bound asymptotically tight. For the case when $\gamma = 1$ our theorem implies a price of anarchy upper bound that is about $2.4$; we show a lower bound of $2$ for that case. We also give a learning algorithm that the players can use to achieve the guarantee needed for our liquid welfare result. Our algorithm achieves utility within a $\gamma = O(1)$ factor of the optimal utility even when a buyer's values and the bids of the other buyers are chosen adversarially, assuming the buyer's budget grows linearly with time. The competitiveness guarantee of the learning algorithm deteriorates somewhat as the budget grows slower than linearly with time. Finally, we extend our liquid welfare results for the case where buyers have submodular valuations over the set of items they win across iterations with a slightly worse price of anarchy bound of $\gamma + 1 + O(1/\gamma)$ compared to the guarantee for the additive case

Online Resource Sharing via Dynamic Max-Min Fairness: Efficiency, Robustness and Non-Stationarity

We study the allocation of shared resources over multiple rounds among competing agents, via a dynamic max-min fair (DMMF) mechanism: the good in each round is allocated to the requesting agent with the least number of allocations received to date. Previous work has shown that when an agent has i.i.d. values across rounds, then in the worst case, she can never get more than a constant strictly less than $1$ fraction of her ideal utility -- her highest achievable utility given her nominal share of resources. Moreover, an agent can achieve at least half her utility under carefully designed `pseudo-market' mechanisms, even though other agents may act in an arbitrary (possibly adversarial and collusive) manner. We show that this robustness guarantee also holds under the much simpler DMMF mechanism. More significantly, under mild assumptions on the value distribution, we show that DMMF in fact allows each agent to realize a $1 - o(1)$ fraction of her ideal utility, despite arbitrary behavior by other agents. We achieve this by characterizing the utility achieved under a richer space of strategies, wherein an agent can tune how aggressive to be in requesting the item. Our new strategies also allow us to handle settings where an agent's values are correlated across rounds, thereby allowing an adversary to predict and block her future values. We prove that again by tuning one's aggressiveness, an agent can guarantee $\Omega(\gamma)$ fraction of her ideal utility, where $\gamma\in [0, 1]$ is a parameter that quantifies dependence across rounds (with $\gamma = 1$ indicating full independence and lower values indicating more correlation). Finally, we extend our efficiency results to the case of reusable resources, where an agent might need to hold the item over multiple rounds to receive utility

Karma: Resource Allocation for Dynamic Demands

The classical max-min fairness algorithm for resource allocation provides many desirable properties, e.g., Pareto efficiency, strategy-proofness and fairness. This paper builds upon the observation that max-min fairness guarantees these properties under a strong assumption -- user demands being static over time -- and that, for the realistic case of dynamic user demands, max-min fairness loses one or more of these properties. We present Karma, a generalization of max-min fairness for dynamic user demands. The key insight in Karma is to introduce "memory" into max-min fairness -- when allocating resources, Karma takes users' past allocations into account: in each quantum, users donate their unused resources and are assigned credits when other users borrow these resources; Karma carefully orchestrates exchange of credits across users (based on their instantaneous demands, donated resources and borrowed resources), and performs prioritized resource allocation based on users' credits. We prove theoretically that Karma guarantees Pareto efficiency, online strategy-proofness, and optimal fairness for dynamic user demands (without future knowledge of user demands). Empirical evaluations over production workloads show that these properties translate well into practice: Karma is able to reduce disparity in performance across users to a bare minimum while maintaining Pareto-optimal system-wide performance.Comment: Accepted for publication in USENIX OSDI 202

Robust Pseudo-Markets for Reusable Public Resources

We study non-monetary mechanisms for the fair and efficient allocation of reusable public resources, i.e., resources used for varying durations. We consider settings where a limited resource is repeatedly shared among a set of agents, each of whom may request to use the resource over multiple consecutive rounds, receiving utility only if they get to use the resource for the full duration of their request. Such settings are of particular significance in scientific research where large-scale instruments such as electron microscopes, particle colliders, or telescopes are shared between multiple research groups; this model also subsumes and extends existing models of repeated non-monetary allocation where resources are required for a single round only. We study a simple pseudo-market mechanism where upfront we endow each agent with a budget of artificial credits, proportional to the fair share of the resource we want the agent to receive. The endowments thus define for each agent her ideal utility as that which she derives from her favorite allocation with no competition, but subject to getting at most her fair share of the resource across rounds. Next, on each round, and for each available resource item, our mechanism runs a first-price auction with a selective reserve, wherein each agent submits a desired duration and a per-round-bid, which must be at least the reserve price if requesting for multiple rounds; the bidder with the highest per-round-bid wins, and gets to use the item for the desired duration. We consider this problem in a Bayesian setting and show that under a carefully chosen reserve price, irrespective of how others bid, each agent has a simple strategy that guarantees she receives a $1/2$ fraction of her ideal utility in expectation. We also show this result is tight, i.e., no mechanism can guarantee that all agents get more than half of their ideal utility