We consider dynamic auctions for finding Walrasian equilibria in markets with indivisible items and strong gross substitutes valuation functions. Each price adjustment step in these auction algorithms requires finding an inclusion-wise minimal maximally overdemanded set or an inclusion-wise minimal maximally underdemanded set at the current prices. Both can be formulated as a submodular function minimization problem. We observe that minimizing this submodular function corresponds to a polymatroid sum problem, and using this viewpoint, we give a fast and simple push-relabel algorithm for finding the required sets. This improves on the previously best running time of Murota, Shioura and Yang (ISAAC 2013). Our algorithm is an adaptation of the push-relabel framework by Frank and Miklós (JJIAM 2012) to the particular setting. We obtain a further improvement for the special case of unit-supplies. We further show the following monotonicity properties of Walrasian prices: both the minimal and maximal Walrasian prices can only increase if supply of goods decreases, or if the demand of buyers increases. This is derived from a fine-grained analysis of market prices. We call packing prices a price vector such that there is a feasible allocation where each buyer obtains a utility maximizing set. Conversely, by covering prices we mean a price vector such that there exists a collection of utility maximizing sets of the buyers that include all available goods. We show that for strong gross substitutes valuations, the component-wise minimal packing prices coincide with the minimal Walrasian prices and the component-wise maximal covering prices coincide with the maximal Walrasian prices. These properties in turn lead to the price monotonicity results.

We consider the problem of fairly allocating a set of indivisible items under the criteria of the maximin share guarantee. Specifically, we study approximation of maximin share allocations under hereditary set system valuations, in which each valuation function is based on the independent sets of an underlying hereditary set system. Using a lone divider approach, we show the existence of 1/2-approximate MMS allocations, improving on the 11/30 guarantee of Li and Vetta [ 37 ]. Moreover, we prove that (2/3 + ε)-approximate MMS allocations do not always exist in this model for every ε > 0, an improvement from the recent 3/4 + ε result of Li and Deng [ 36 ]. Our existence proof is constructive but does not directly yield a polynomial-time approximation algorithm. However, we show that a 2/5-approximate MMS allocation can be found in polynomial time, given valuation oracles. Finally, we show that our existence and approximation results transfer to a variety of problems within constrained fair allocation, improving on existing results in some of these settings.

Consider a single auctioneer who wants to sell multiple units of distinct indivisible items to bidders with private valuations. The set of feasible allocations is constrained to integer base points, which are the integer points of an integer base polyhedron. Each bidder’s valuation is the integer restriction of a sum of nondecreasing, concave single-parameter functions. This seemingly abstract setting is of theoretical relevance and has various interesting applications. In this context, we develop an ascending auction that implements a social welfare-maximizing allocation, charges Vickrey–Clarke–Groves prices, relies only on a single price, is ex-post incentive-compatible, and satisfies unconditional winner privacy. The auction has a polynomial running time in the number of bidders, items, and units; in the case of linear separable valuations it runs even in strongly polynomial time, thereby improving on the literature. Moreover, by relaxing unconditional winner privacy, the auction can be made fully polynomial in the number of bidders, items, units, and integer breakpoints of bidders’ valuations. If we assume that bidders are unit-demand , then our auction is dominant-strategy incentive-compatible and (weakly) group strategy-proof, much like deferred acceptance auctions.

We study truthful mechanisms for allocation problems in graphs, both for the minimization (i.e., scheduling) and maximization (i.e., auctions) setting. The minimization problem is a special case of the well-studied unrelated machines scheduling problem, in which every given task can be executed only by two pre-specified machines in the case of graphs or a given subset of machines in the case of hypergraphs. This corresponds to a multigraph whose nodes are the machines and its hyperedges are the tasks. This class of problems belongs to multidimensional mechanism design, for which there are no known general mechanisms other than the VCG and its generalization to affine minimizers. We propose a new class of truthful mechanisms that have significantly better performance than affine minimizers in many settings. Specifically, we provide upper and lower bounds for truthful mechanisms for general multigraphs, as well as special classes of graphs such as stars, trees, planar graphs, k -degenerate graphs, and graphs of a given treewidth. We also consider the objective of minimizing or maximizing the L p -norm of the values of the players, a generalization of the makespan minimization that corresponds to p = ∞, and extend the results to any p > 0.

We study the query complexity of finding the set of all Nash equilibria \(\mathcal {X}_\ast \times \mathcal {Y}_\ast \) in two-player zero-sum matrix games. Fearnley and Savani [18] showed that for any randomized algorithm, there exists an n × n input matrix where it needs to query Ω ( n 2 ) entries in expectation to compute a single Nash equilibrium. On the other hand, Bienstock et al. [5] showed that there is a special class of matrices for which one can query O ( n ) entries and compute its set of all Nash equilibria. However, these results do not fully characterize the query complexity of finding the set of all Nash equilibria in two-player zero-sum matrix games. In this work, we characterize the query complexity of finding the set of all Nash equilibria \(\mathcal {X}_\ast \times \mathcal {Y}_\ast \) in terms of the number of rows n of the input matrix \(A \in \mathbb {R}^{n \times n} \) , row support size \(k_1 := |\bigcup \limits _{x \in \mathcal {X}_\ast } \text{supp}(x)| \) , and column support size \(k_2 := |\bigcup \limits _{y \in \mathcal {Y}_\ast } \text{supp}(y)| \) . We design a simple yet non-trivial randomized algorithm that returns the set of all Nash equilibria \(\mathcal {X}_\ast \times \mathcal {Y}_\ast \) by querying at most O ( nk 5 · polylog( n )) entries of the input matrix \(A \in \mathbb {R}^{n \times n} \) in expectation, where k ≔ max{ k 1 , k 2 }. This upper bound is tight up to a factor of poly( k ), as we show that for any randomized algorithm, there exists an n × n input matrix with min { k 1 , k 2 } = 1, for which it needs to query Ω ( nk ) entries in expectation in order to find the set of all Nash equilibria \(\mathcal {X}_\ast \times \mathcal {Y}_\ast \) .

Budget-feasible procurement has been a major paradigm in mechanism design since its introduction by Singer [ 28 ]. An auctioneer (buyer) with a strict budget constraint is interested in buying goods or services from a group of strategic agents (sellers). In many scenarios, it makes sense to allow the auctioneer to only partially buy what an agent offers, e.g., an agent might have multiple copies of an item to sell, might offer multiple levels of a service, or may be available to perform a task for any fraction of a specified time interval. Nevertheless, the focus of the related literature has been on settings in which each agent’s services are either fully acquired or not at all. A reason for this is that in settings with partial allocations, such as the ones mentioned, there are strong inapproximability results (see, e.g., Anari et al. [ 5 ], Chan and Chen [ 10 ]). Under the mild assumption of being able to afford each agent entirely, we are able to circumvent such results. We design a polynomial-time, deterministic, truthful, budget-feasible, \((2+\sqrt {3})\) -approximation mechanism for the setting in which each agent offers multiple levels of service and the auctioneer has a valuation function that is separable concave, i.e., it is the sum of concave functions. We then use this result to design a deterministic, truthful, and budget-feasible O (1)-approximation mechanism for the setting in which any fraction of a service can be acquired, again for separable concave objectives. For the special case in which the objective is the sum of linear valuation functions, we improve the best known approximation ratio for the problem from \((3+\sqrt {5})/2\) (by Klumper and Schäfer [ 19 ]) to 2. This establishes a separation between this setting and its indivisible counterpart.

We consider prophet inequalities under general downward-closed constraints. In a prophet inequality problem, a decision-maker sees a series of online elements with values and needs to decide immediately and irrevocably whether or not to select each element upon its arrival, subject to an underlying feasibility constraint. Traditionally, the decision-maker’s expected performance has been compared to the expected performance of the prophet , i.e., the expected offline optimum. We refer to this measure as the Ratio of Expectations (or, in short, RoE ). However, a major limitation of the RoE measure is that it only gives a guarantee against what the optimum would be on average, while, in theory, algorithms still might perform poorly compared to the realized ex-post optimal value. Hence, we study alternative performance measures. In particular, we suggest the Expectation of Ratio (or, in short, EoR ), which is the expectation of the ratio between the value of the algorithm and the value of the prophet. This measure yields desirable guarantees, e.g., a constant EoR implies achieving a constant fraction of the ex-post offline optimum with constant probability. Moreover, in the single-choice setting, we show that the EoR is equivalent (in the worst case) to the probability of selecting the maximum, a well-studied measure in the literature. However, the probability of selecting the maximum does not generalize meaningfully to combinatorial constraints (beyond single-choice), since its direct equivalent is the probability of selecting an optimal overall set. We, thus, introduce the EoR as a cardinal variant of the probability of selecting the maximum, which extends naturally to combinatorial settings. Our main goal is to understand the relation between RoE and EoR in combinatorial settings. Specifically, we establish two reductions: For every feasibility constraint, the RoE and the EoR are at most a constant factor apart on worst-case instances. Additionally, we show that the EoR is a stronger benchmark than the RoE in that, for every instance (feasibility constraint and product distribution), the RoE is at least a constant fraction of the EoR but not vice versa. Both these reductions imply a wealth of EoR results in multiple settings where RoE results are known.

This article provides threshold policies with tight guarantees for online selection with convex cost (OSCC). In OSCC, a seller wants to sell some asset to a sequence of buyers with the goal of maximizing her profit. The seller can produce additional units of the asset, but at non-decreasing marginal costs. Each time, a buyer arrives and offers a price. The seller must make an immediate and irrevocable decision in terms of whether to accept the offer and produce/sell one unit of the asset to this buyer. The goal is to develop an online algorithm that selects a subset of buyers to maximize the seller’s profit, namely, the total selling revenue minus the total production cost. Our main result is the development of a class of simple threshold policies that are logistically simple and easy to implement but have provable optimality guarantees among all deterministic algorithms. We also derive a lower bound on competitive ratios of randomized algorithms and prove that the competitive ratio of our threshold policy asymptotically converges to this lower bound when the total production output is sufficiently large. Our results generalize and unify various online search, pricing, and auction problems, and provide a new perspective on the impact of non-decreasing marginal costs on real-world online resource allocation problems.

We study the formation of stable outcomes via simple dynamics in cardinal hedonic games, where the valuations of agents change over time depending on the history of the coalition formation process. Specifically, we analyze situations where members of a coalition decrease their valuation for a leaving agent ( resentment ) or increase their valuation for a joining agent ( appreciation ). We show a series of convergence results for dynamics for resentful or appreciative agents that do not hold for classic dynamics. In particular, resentment turns out to be a strong stability-driving force. We complement our theoretical analysis with simulations that shed some light on the average running time of the dynamics and on the structure of the produced outcomes. From an algorithmic perspective, we obtain general hardness results for determining the fastest convergence time, results that also carry over to classic dynamics under static valuation functions. Finally, we explore a related model for preference updates where resentment is expressed by the deviator. We find a more nuanced picture but still a broad possibility of convergence.

We study the problem of fairly allocating a set of indivisible goods among agents with matroid rank valuations: every good provides a marginal value of 0 or 1 when added to a bundle and valuations are submodular. We present a simple algorithmic framework, called General Yankee Swap , that can efficiently compute allocations that maximize any justice criterion (or fairness objective) satisfying some mild assumptions. Along with maximizing a justice criterion, General Yankee Swap is guaranteed to maximize utilitarian social welfare, ensure strategyproofness and use at most a quadratic number of valuation queries. We show how General Yankee Swap can be used to compute allocations for five different well-studied justice criteria: (a) Prioritized Lorenz dominance, (b) Maximin fairness, (c) Weighted leximin, (d) Max weighted Nash welfare, and (e) Max weighted p -mean welfare. In particular, this framework provides the first polynomial time algorithms to compute weighted leximin, max weighted Nash welfare and max weighted p -mean welfare allocations for agents with matroid rank valuations. We also extend this framework to the setting of binary chores — items with marginal values -1 or 0 — and similarly show that it can be used to maximize any justice criteria satisfying some mild assumptions.