We consider a game theoretic variant of an interval scheduling problem. Every job is associated with a length, a weight, and a color. Each player controls all the jobs of a specific color, and needs to decide on a processing interval for each of its jobs. Jobs of the same color can be processed simultaneously by the machine. A job is covered if the machine is configured to its color during its whole processing interval. The goal of the machine is to maximize the sum of weights of all covered jobs, and the goal of each player is to maximize the sum of weights of covered jobs from its color. The study of this game is motivated by several applications like antenna scheduling for wireless networks. We first show that given a strategy profile of the players, the machine scheduling problem can be solved in polynomial time. We then study the game from the players’ point of view. We analyze the existence of Nash equilibria, its computation, and inefficiency. We distinguish between instances of the classical interval scheduling problem, in which every player controls a single job, and instances in which color sets may include multiple jobs.
We study the problem of minimizing metric distortion in multi-winner elections, where a committee of size k is selected from a set of candidates based on voters’ ordinal preferences. We assume that voters and candidates are embedded on a line metric, and social cost is determined by the underlying metric distances. The distortion of a voting rule is the worst-case ratio between the social cost of the elected committee and an optimal committee. Previous work has focused on the q -cost model, in which a voter’s cost is given by the distance to their q th closest committee member. Here, we study the additive cost , where a voter’s cost is the sum of distances to all committee members. We introduce the Polar Comparison Rule and analyze its distortion under utilitarian additive cost. We show that it achieves a distortion upper bound of almost 2.33 for all committee sizes k > 4, improving upon the previously best-known upper bound of 3. Moreover, for k = 2 and k = 3, we establish tight distortion bounds of 2.41 and 2.33, respectively. We also derive lower bounds that depend on the parity of k and analyze the behavior of distortion for small and large committee sizes. Finally, we extend our results to the egalitarian additive cost.
Lotteries are a prevalent form of gambling between a seller and buyers. Designing a lottery requires a model of how buyers make decisions when confronted with uncertain outcomes. Cumulative prospect theory (CPT) is a descriptive model that captures people’s propensity to overestimate extreme events and their different attitudes toward gains and losses. In this study, we design a lottery that maximizes the seller’s profit when the buyers’ decision-making adheres to the CPT framework. The main difficulty is the nonconvexity of the CPT framework, which we overcome by reformulating the problem as a three-level optimization problem and characterizing its optimal solution. Based on the analysis, we propose a linear-time algorithm that computes the optimal lottery. Furthermore, we present an efficient algorithm applicable to a broader setting with a ticket price constraint. This is the first study to employ the CPT framework in designing an optimal lottery with more than two outcomes.
A college applicant submits costly applications to a set of colleges while uncertain which colleges will admit her. If she knows the application costs, her utility of attending each college, and her probability of being admitted to each college, which subset of colleges should she apply to maximize her expected net payoff? The problem is an instance of non-monotone submodular maximization. There are two principal variants in the literature, one where the events of getting into different colleges are independent, and the other where they are correlated. The main results in the literature handle cases where all application costs are the same. We provide exact and approximate algorithms for the problem when college application costs vary.
We introduce a new game-theoretical solution concept for settings where players have probabilistic information about preferences described as distributions about von Neumann-Morgenstern utility functions. This solution concept extends the Markov-Conley-Chains (MCCs) and alpha-Rank algorithm to Bayesian games, providing a tractable method for analyzing strategic interactions without requiring precise knowledge of players' utility functions. The approach avoids the limitations of Nash equilibrium, such as multiple equilibria, by ranking strategies based on their probabilities of success. We apply the method to well-known bimatrix games and the Boston matching mechanism, offering a novel framework for analyzing the outcomes of games with probabilistic preferences.
Congestion games allow to model competitive resource sharing in various distributed systems. Pure Nash equilibria, that are stable outcomes of a game, could be far from being socially optimal. Our goal is to identify combinatorial structures that limit the inefficiency of equilibria. This question has been mainly investigated for congestion games defined over networks. Instead, we focus on symmetric matroid congestion games, where the strategies of every player are the bases of a given matroid. We derive new upper bounds on the Price of Anarchy (PoA) of congestion games defined over k-uniform matroids and paving matroids with delay functions in class 𝒟 . For both affine and polynomial delay functions, our bounds indicate that the inefficiency of pure Nash equilibria is limited by these combinatorial structures.
How can individual agents coordinate their actions in a distributed manner to achieve a shared objective? This question arises across various systems-economic, technical, and sociological-all of which face common challenges such as scalability, heterogeneity, and conflicting individual and collective goals. In economic markets, these challenges are mitigated by the use of a common currency, which enables participants to coordinate their actions toward efficient outcomes. This raises the question of whether similar mechanisms, such as a common currency, can be applied to other systems, including technical and sociological contexts. In this paper, we explore this idea within the context of social media, where communities form around shared interests. We propose that social support (in the form of likes, shares, and comments) functions as a currency that coordinates the actions of users in content markets. We investigate two core questions: (1) Can social support serve as a currency that shapes the production and sharing of content, and (2) What role do influencers play in coordinating content creation and dissemination? Through formal modeling and analysis, we demonstrate that social support can act as an efficient coordination mechanism, similar to money in economic markets. Influencers play a dual role in aggregating content and acting as proxies for information, helping content producers navigate large markets. Our findings suggest that while social support as a currency leads to efficient outcomes in ideal markets, imperfections in information introduce a "price of influence," resulting in suboptimal outcomes. However, as content markets grow, this price diminishes, and social welfare approaches optimal levels. These insights offer a framework for understanding coordination in distributed environments, with potential applications to both sociological and technical systems, including multi-agent AI systems.
In job-scheduling games, each job is a selfish player that selects a machine to minimize its own completion time. Coordination mechanisms are employed to reduce the inefficiency of equilibria that result from such decentralized decision-making. This paper contributes to the extensive body of research on coordination mechanisms by investigating their application to unrelated parallel machines, where each machine may use its own scheduling policy to determine the processing order of assigned jobs. Since pure Nash equilibria (NE) are not guaranteed to exist in this setting, we identify and characterize several classes of instances-motivated by real-world applications-in which a NE is guaranteed to exist. For each such class, we design an algorithm to compute a NE, prove the convergence of best-response dynamics, and analyze the inefficiency of equilibria with respect to the makespan. In addition, we study two fundamental problems: (1) computing a NE schedule with low makespan, and (2) selecting, given a matrix of processing times, machine-specific scheduling policies that guarantee the existence of a NE with low makespan. For both problems, we establish computational hardness results.
We address auctions in two-sided markets with budget constraints on buyers, a fundamental setting also crucial for applications such as display advertising. Our goal is to design efficient mechanisms that satisfy dominant strategy incentive compatibility, individual rationality, and budget balance. To overcome the limitations of impossibility theorems, we assume prior knowledge of sellers' valuations and focus on liquid welfare, an efficiency objective that takes budgets into account. Our contributions are twofold: First, we improve the efficiency guarantees of the polyhedral clinching auction by Hirai and Sato (2022). Second, using the reduction method of Dütting et al. (2021), we extend the mechanism to an efficient single-sample mechanism for budget-constrained auctions, providing the budget extension of their results. Notably, our results hold even under polymatroid constraints and apply to both divisible and indivisible goods.
In the Course Allocation problem, there are a set of students and a set of courses at a given university. University courses may have different numbers of credits, typically related to different numbers of learning hours, and there may be other constraints such as courses running concurrently. Our goal is to allocate the students to the courses such that the resulting matching is stable, which means that no student and course(s) have an incentive to break away from the matching and become assigned to one another. We study several definitions of stability and for each we give a mixture of polynomial-time algorithms and hardness results for problems involving verifying the stability of a matching, finding a stable matching or determining that none exists, and finding a maximum size stable matching. We also study variants of the problem with master lists of students, and lower quotas on the number of students allocated to a course, establishing additional complexity results in these settings.
We study symmetric bimatrix games that also have the common-payoff property, i.e., the two players receive the same payoff at any outcome of the game. Due to the symmetry property, these games are guaranteed to have symmetric Nash equilibria, where the two players play the same (mixed) strategy. While the problem of computing such symmetric equilibria in general symmetric bimatrix games is known to be intractable, namely PPAD-complete, this result does not extend to our setting. Indeed, due to the common-payoff property, the problem lies in the lower class CLS, ruling out PPAD-hardness. In this paper, we show that the problem remains intractable, namely it is CLS-complete. On the way to proving this result, as our main technical contribution, we show that computing a Karush-Kuhn-Tucker (KKT) point of a quadratic program remains CLS-hard, even when the feasible domain is a simplex.
Allocating indivisible goods is a ubiquitous task in fair division. We study additive welfarist rules, an important class of rules which choose an allocation that maximizes the sum of some function of the agents' utilities. Prior work has shown that the maximum Nash welfare (MNW) rule is the unique additive welfarist rule that guarantees envy-freeness up to one good (EF1). We strengthen this result by showing that MNW remains the only additive welfarist rule that ensures EF1 for identical-good instances, two-value instances, as well as normalized instances with three or more agents. On the other hand, if the agents' utilities are integers, we demonstrate that several other rules offer the EF1 guarantee, and provide characterizations of these rules for various classes of instances.
Super-stability and strong stability are properties of a matching in the stable matching problem with ties. In this paper, we introduce a common generalization of super-stability and strong stability, which we call non-uniform stability. First, we prove that we can determine the existence of a non-uniformly stable matching in polynomial time. Next, we give a polyhedral characterization of the set of non-uniformly stable matchings. Finally, we prove that the set of non-uniformly stable matchings forms a distributive lattice.
In the Stable Roommates problem, we seek a stable matching of the agents into pairs, in which no two agents have an incentive to deviate from their assignment. It is well known that a stable matching is unlikely to exist, but a stable partition always does and provides a succinct certificate for the unsolvability of an instance. Furthermore, apart from being a useful structural tool to study the problem, every stable partition corresponds to a stable half-matching, which has applications, for example, in sports scheduling and time-sharing. We establish new structural results for stable partitions and show how to enumerate all stable partitions and the cycles included in such structures efficiently. We also adapt optimality criteria from stable matchings to stable partitions and give complexity and approximability results for the problems of computing such "fair" and "optimal" stable partitions. Through this research, we contribute to a deeper understanding of stable partitions from a combinatorial point of view, as well as the computational complexity of computing "fair" or "optimal" stable half-matchings in practice, closing the gap between integral and fractional stable matchings and paving the way for further applications of stable partitions to unsolvable instances and computationally hard stable matching problems.
We study the classic single-item auction setting of Myerson, but under the assumption that the buyers’ values for the item are distributed over finite supports. Using strong LP duality and polyhedral theory, we rederive various key results regarding the revenue-maximizing auction, including the characterization through virtual welfare maximization and the optimality of deterministic mechanisms, as well as a novel, generic equivalence between dominant-strategy and Bayesian incentive compatibility. Inspired by this, we abstract our approach to handle more general auction settings, where the feasibility space can be given by arbitrary convex constraints, and the objective is a linear combination of revenue and social welfare. We characterize the optimal auctions of such systems as generalized virtual welfare maximizers, by making use of their KKT conditions, and we present an analogue of Myerson’s payment formula for general discrete single-parameter auction settings. Additionally, we prove that total unimodularity of the feasibility space is a sufficient condition to guarantee the optimality of auctions with integral allocation rules. Finally, in the full version of our paper, we demonstrate this KKT approach by applying it to a setting where bidders are interested in buying feasible flows on trees with capacity constraints, and provide a combinatorial description of the (randomized, in general) optimal auction.
The seminal result of Nash in game theory states that any normal-form game has a Nash equilibrium if each player can randomize their strategy. The assumption that players can randomize arbitrarily is non-trivial, as true randomness might be scarce or costly and humans are known to have difficulty generating truly random sequences. In a repeated game, the assumption that players are unconstrained in their capability to randomize their strategies is particularly strong if the amount of random bits required to play the repeated game scales linearly with the number of repetitions. We identify conditions on a normal-form game under which, if players have a limited capability to randomize, certain Nash equilibria of its finitely repeated version cannot be played. We provide a complete characterization of normal-form games for which there exists Nash equilibria of its finitely repeated version using O(1) randomness, closing an open question posed by Budinich and Fortnow [3] (EC ’11) and Hubáček, Naor and Ullman [8] (SAGT ’15, TCSys ’16). Moreover, we prove a 0–1 law for randomness in repeated games, showing that any repeated game either has O(1)-randomness Nash equilibria, or all of its Nash equilibria require (n) randomness. Our techniques are general and naturally characterize the payoff space of sublinear-entropy equilibria, and could be of independent interest to the study of players with other bounded capabilities in repeated games.
We study a market mechanism that sets edge prices to incentivize strategic agents to efficiently share limited network capacity. In this market, agents form coalitions, with each coalition sharing a unit capacity of a selected route and making payments to cover edge prices. Our focus is on the existence and computation of market equilibrium, where challenges arise from the interdependence between coalition formation among strategic agents with heterogeneous preferences and route selection that induces a network flow under integral capacity constraints. To address this interplay between coalition formation and network capacity utilization, we introduce a novel approach based on combinatorial auction theory and network flow theory. We establish sufficient conditions on the network topology and agents' preferences that guarantee both the existence and polynomial-time computation of a market equilibrium. Additionally, we identify a particular market equilibrium that maximizes utilities for all agents and the outcome is equivalent to the classical Vickrey-Clarke-Groves mechanism. Furthermore, we extend our results to multi-period settings and general networks, showing that when the sufficient conditions are not met, an equilibrium may still exist but requires more complex, path-based pricing mechanisms that set differentiated prices based on agents' preference parameters.
In this paper, we investigate the k -Facility Location Problem on the line within the Bayesian Mechanism Design framework and analyze the percentile mechanisms, a class of truthful mechanisms that locates the facilities based on the order of the agents’ reports. We first connect the k -FLP to the Wasserstein projection problems and use this connection to retrieve the limit of the ratio between the expected cost of a percentile mechanism and the expected optimal cost. Moreover, we characterize its limit and convergence speed. We infer an upper bound on the Bayesian approximation ratio when n > k , contrasting the classic worst-case analysis where percentile mechanisms have an unbounded approximation ratio whenever k > 2. This allows us to introduce criteria to determine which percentile mechanism is better suited to address a given agent distribution. We then establish the existence of an optimal percentile mechanism and characterize it via a system of k equations. Finally, we estimate the optimality loss that occurs if we retrieve the optimal percentile mechanism using an approximation of the agents’ distribution. All results hold for the Social, Maximum, and l p costs.
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 mechanisms that are truthful and 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 $p=\infty$, and extend the results to any $p>0$.
