We study the allocation of indivisible goods under conflicting constraints, represented by a graph. In this framework, vertices correspond to goods and edges correspond to conflicts between a pair of goods. Each agent is allocated an independent set in the graph. In a recent work of Kumar et al. (AAMAS, 2024), it was shown that a maximal EF1 allocation exists for interval graphs and two agents with monotone valuations. We significantly extend this result by establishing that a maximal EF1 allocation exists for any graph when the two agents have monotone valuations. To compute such an allocation, we present a polynomial-time algorithm for additive valuations, as well as a pseudo-polynomial time algorithm for monotone valuations. Moreover, we complement our findings by providing a counterexample demonstrating a maximal EF1 allocation may not exist for three agents with monotone valuations; further, we establish NP-hardness of determining the existence of such allocations for every fixed number n >= 3 of agents. All of our results for goods also apply to the allocation of chores.
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The proportional veto principle, which captures the idea that a candidate vetoed by a large group of voters should not be chosen, has been studied for ranked ballots in single-winner voting. We introduce a version of this principle for approval ballots, which we call flexible-voter representation (FVR). We show that while the approval voting rule and other natural scoring rules provide the optimal FVR guarantee only for some flexibility threshold, there exists a scoring rule that is FVR-optimal for all thresholds simultaneously. We also extend our results to multi-winner voting.
Multiwinner Elections have emerged as a prominent area of research with numerous practical applications. Given a set of candidates, C, a set of voters, V, approving a subset of candidates (called approval set of a voter), and an integer k, we consider the problem of selecting a ``good'' committee using Thiele rules. This problem is computationally challenging for most Thiele rules with monotone submodular satisfaction functions, as there is no (1-1/e- epsilon) approximation algorithm in f(k)(|C| + |V|)^(o(k)) time for any fixed epsilon > 0 and any computable function f, and no PTAS even when the length of approval set is two. Skowron designed an approximation scheme running in FPT time parameterized by the combined parameter, size of the approval set, and k. In this paper, we consider a parameter d+k (no d voters approve the same set of d candidates), where d is upper bounded by the size of the approval set (thus, can be much smaller). With respect to this parameter, we design parameterized approximation schemes, a lossy polynomial-time preprocessing method, and show that an extra committee member suffices to achieve the desired score (i.e., 1-additive approximation). Additionally, we resolve an open question by Yang and Wang regarding the fixed-parameter tractability of the problem under the PAV rule with the total score as the parameter, demonstrating that it admits an FPT algorithm.
Strategyproofness in network auctions requires that bidders not only report their valuations truthfully, but also do their best to invite neighbours from the social network. In contrast to canonical auctions, where the value-monotone allocation in Myerson's Lemma is a cornerstone, a general principle of allocation rules for strategyproof network auctions is still missing. We show that, due to the absence of such a principle, even extensions to multi-unit network auctions with single-unit demand present unexpected difficulties, and all pioneering researches fail to be strategyproof. For the first time in this field, we identify two categories of monotone allocation rules on networks: Invitation-Depressed Monotonicity (ID-MON) and Invitation-Promoted Monotonicity (IP-MON). They encompass all existing allocation rules of network auctions as specific instances. For any given ID-MON or IP-MON allocation rule, we characterize the existence and sufficient conditions for the strategyproof payment rules, and show that among all such payment rules, the revenue-maximizing one exists and is computationally feasible. With these results, the obstacle of combinatorial network auction with single-minded bidders is now resolved.
Launching a new blockchain system or application is frequently facilitated by a so called airdrop, where the system designer chooses a pre-existing set of potentially interested parties and allocates newly minted tokens to them with the expectation that they will participate in the system — such engagement, especially if it is of significant level — facilitates the system and raises its value and also the value of its newly minted token, hence benefiting the airdrop recipients. A number of challenging questions befuddle designers in this setting, such as how to choose the set of interested parties and how to allocate tokens to them. To address these considerations we put forward a game theoretic model for such airdrop games. Our model can be used to guide the designer’s choices based on the way the system’s value depends on participation (modeled by a “technology function” in our framework) and the costs that participations incurs. We identify both bad and good equilibria and identify the settings and the choices that can be made where the designer can influence the players towards good equilibria in an expedient manner.
Infinitely repeated games support equilibrium concepts beyond those present in one-shot games (e.g., cooperation in the prisoner's dilemma). Nonetheless, repeated games fail to capture our real-world intuition for settings with many anonymous agents interacting in pairs. Repeated games with restarts, introduced by Berker and Conitzer, address this concern by giving players the option to restart the game with someone new whenever their partner deviates from an agreed-upon sequence of actions. In their work, they studied symmetric games with symmetric strategies. We significantly extend these results, introducing and analyzing more general notions of equilibria in asymmetric games with restarts. We characterize which goal strategies players can be incentivized to play in equilibrium, and we consider the computational problem of finding such sequences of actions with minimal cost for the agents. We show that this problem is NP-hard in general. However, when the goal sequence maximizes social welfare, we give a pseudo-polynomial time algorithm.
We study the distortion of one-sided and two-sided matching problems on the line. In the one-sided case, n agents need to be matched to n items, and each agent's cost in a matching is their distance from the item they were matched to. We propose an algorithm that is provided only with ordinal information regarding the agents' preferences (each agent's ranking of the items from most- to least-preferred) and returns a matching aiming to minimize the social cost with respect to the agents' true (cardinal) costs. We prove that our algorithm simultaneously achieves the best-possible approximation of 3 (known as distortion) with respect to a variety of social cost measures which include the utilitarian and egalitarian social cost. In the two-sided case, where the agents need be matched to n other agents and both sides report their ordinal preferences over each other, we show that it is always possible to compute an optimal matching. In fact, we show that this optimal matching can be achieved using even less information, and we provide bounds regarding the sufficient number of queries.
In this work, we consider the design of Non-Obviously Manipulable (NOM) mechanisms, mechanisms that bounded rational agents may fail to recognize as manipulable, for two relevant classes of succinctly representable Hedonic Games: Additively Separable and Fractional Hedonic Games. In these classes, agents have cardinal scores towards other agents, and their preferences over coalitions are determined by aggregating such scores. This aggregation results in a utility function for each agent, which enables the evaluation of outcomes via the utilitarian social welfare. We first prove that, when scores can be arbitrary, every optimal mechanism is NOM; moreover, when scores are limited in a continuous interval, an optimal mechanism that is NOM exists. Given the hardness of computing optimal outcomes in these settings, we turn our attention to efficient and NOM mechanisms. To this aim, we first prove a characterization of NOM mechanisms that simplifies the class of mechanisms of interest. Then, we design a NOM mechanism returning approximations that asymptotically match the best-known approximation achievable in polynomial time. Finally, we focus on discrete scores, where the compatibility of NOM with optimality depends on the specific values. Therefore, we initiate a systematic analysis to identify which discrete values support this compatibility and which do not.
We focus on the design of algorithms for finding equilibria in 2-player zero-sum games. Although it is well known that such problems can be solved by a single linear program, there has been a surge of interest in recent years for simpler algorithms, motivated in part by applications in machine learning. Our work proposes such a method, inspired by the observation that the duality gap (a standard metric for evaluating convergence in min-max optimization problems) is a convex function for bilinear zero-sum games. To this end, we analyze a descent-based approach, variants of which have also been used as a subroutine in a series of algorithms for approximating Nash equilibria in general non-zero-sum games. In particular, we study a steepest descent approach, by finding the direction that minimises the directional derivative of the duality gap function. Our main theoretical result is that the derived algorithms achieve a geometric decrease in the duality gap until we reach an approximate equilibrium. Finally, we complement this with an experimental evaluation, which provides promising findings. Our algorithm is comparable with (and in some cases outperforms) some of the standard approaches for solving 0-sum games, such as OGDA (Optimistic Gradient Descent/Ascent), even with thousands of available strategies per player.
In this paper, we study a fair resource scheduling problem involving the assignment of a set of interval jobs among a group of heterogeneous machines. Each job is associated with a release time, a deadline, and a processing time. A machine can process a job if the entire processing period falls within the release time and deadline of the job. Each machine can process at most one job at any given time, and different jobs yield different utilities for the machines. The goal is to find a fair and efficient schedule of the jobs. We discuss the compatibility between envy-freeness up to any item (EFX) and various efficiency concepts. Additionally, we present polynomial-time algorithms for various settings.
We study the complexity of candidate control in participatory budgeting elections. The goal of constructive candidate control is to ensure that a given candidate wins by either adding or deleting candidates from the election (in the destructive setting, the goal is to prevent a given candidate from winning). We show that such control problems are NP-hard to solve for many participatory budgeting voting rules, including Phragmén and Equal-Shares, but there are natural cases with polynomial-time algorithms. We also argue that control by deleting candidates is a useful tool for assessing the performance (or, strength) of initially losing projects, and we support this view with experiments on real-life PB instances.
We study a temporal voting model where voters have dynamic preferences over a set of public chores---projects that benefit society, but impose individual costs on those affected by their implementation. We investigate the computational complexity of optimizing utilitarian and egalitarian welfare. Our results show that while optimizing the former is computationally straightforward, minimizing the latter is computationally intractable, even in very restricted cases. Nevertheless, we identify several settings where this problem can be solved efficiently, either exactly or by an approximation algorithm. We also examine the effects of enforcing temporal fairness and its impact on social welfare, and analyze the competitive ratio of online algorithms. We then explore the strategic behavior of agents, providing insights into potential malfeasance in such decision-making environments. Finally, we discuss a range of fairness measures and their suitability for our setting.
Estimating player utilities from observed equilibria is crucial for many applications. Existing approaches to tackle this problem are either limited to specific games or do not scale well with the number of players. Our work addresses these issues by proposing a novel utility estimation method for general multi-player non-cooperative games. Our main idea consists in reformulating the inverse game problem as an inverse variational inequality problem and in selecting among all utility parameters consistent with the data, the so-called incenter. We show that the choice of the incenter can produce parameters that are most robust to the observed equilibrium behaviors. However, its computation is challenging, as the number of constraints in the corresponding optimization problem increases with the number of players and the behavior space size. To tackle this challenge, we propose a loss function-based algorithm, making our method scalable to games with many players or a continuous action space. Furthermore, we show that our method can be extended to incorporate prior knowledge of player utilities, and that it can handle inconsistent data, i.e., data where players do not play exact equilibria. Numerical experiments on three game applications demonstrate that our methods outperform the state of the art. The code, datasets, and supplementary material are available at https://github.com/cuilvye/Incenter-Project.
Coalition formation involves self-organized coalitions generated through strategic interactions of autonomous selfish agents. In online learning of coalition structures, agents' preferences toward each other are initially unknown before agents interact. Coalitions are formed iteratively based on preferences that agents learn online from repeated feedback resulting from their interactions. In this paper, we introduce online learning in coalition formation through the lens of distributed decision-making, where self-interested agents operate without global coordination or information sharing, and learn only from their own experience. Under our selfish perspective, each agent seeks to maximize her own utility. Thus, we analyze the system in terms of Nash stability, where no agent can improve her utility by unilaterally deviating. We devise a sample-efficient decentralized algorithm for selfish agents that minimize their Nash regret, yielding approximately Nash stable solutions. In our algorithm, each agent uses only one utility feedback per round to update her strategy, but our algorithm still has Nash regret and sample complexity bounds that are optimal up to logarithmic factors.
We study the problem of fairly allocating a set of indivisible items among a set of agents. We consider the notion of (approximate) maximin share (MMS) and we provide an improved lower bound of 1/2 (which is tight) for the case of subadditive valuations when the number of agents is at most four. We also provide a tight lower bound for the case of multiple agents, when they are equipped with one of two possible types of valuations. Moreover, we propose a new model that extends previously studied models in the area of fair division, which will hopefully give rise to further research. We demonstrate the usefulness of this model by employing it as a technical tool to derive our main result, and we provide a thorough analysis for this model for the case of three agents. Finally, we provide an improved impossibility result for the case of three submodular agents.
We investigate multi-organizational scheduling problems, building upon the framework introduced by Pascual et al. in 2009. In this setting, multiple organizations each own a set of identical machines and sequential jobs with distinct processing times. The challenge lies in optimally assigning jobs across organizations’ machines to minimize the overall makespan while ensuring no organization’s performance deteriorates. To formalize this fairness constraint, we introduce individual rationality, a game-theoretic concept that guarantees each organization benefits from participation. Our analysis reveals that finding an individually rational schedule with minimum makespan is ΘP2-hard, placing it in a complexity class strictly harder than both NP and coNP. We further extend the model by considering an alternative objective: minimizing the sum of job completion times, both within individual organizations and across the entire system. The corresponding decision variant proves to be NP-complete. Through comprehensive parameterized complexity analysis of both problems, we provide new insights into these computationally challenging multi-organizational scheduling scenarios.
We study coalition formation in the framework of hedonic games. These games model the problem of partitioning a set of agents having a preference order over the coalitions they can be part of. A partition is called popular if it does not lose a majority vote among the agents against any other partition. Unfortunately, hedonic games need not admit popular partitions. We go further and settle the complexity of the existence problem concerning popularity in additively separable and fractional hedonic games by showing that it is Sigma_2^p-complete in both cases. We are thus the first work that proves a completeness result of popularity for the second level of the polynomial hierarchy.
When allocating indivisible items to agents, it is known that the only strategyproof mechanisms that satisfy a set of rather mild conditions are constrained serial dictatorships: given a fixed order over agents, at each step the designated agent chooses a given number of items (depending on her position in the sequence). Agents who come earlier in the sequence have a larger choice of items; however, this advantage can be compensated by a higher number of items received by those who come later. How to balance priority in the sequence and number of items received is a nontrivial question. We use a previous model, parameterized by a mapping from ranks to scores, a social welfare functional, and a distribution over preference profiles. For several meaningful choices of parameters, we show that the optimal sequence can be computed exactly in polynomial time or approximated using sampling. Our results hold for several probabilistic models on preference profiles, with an emphasis on the Plackett-Luce model. We conclude with experimental results showing how the optimal sequence is impacted by various parameters.
We consider the problem of resolving the envy of a given initial allocation by adding elements from a pool of goods. We give a characterization of the instances where envy can be resolved by adding an arbitrary number of copies of the items in the pool. From this characterization, we derive a polynomial-time algorithm returning a respective solution if it exists. If the number of copies or the total number of added items are bounded, the problem becomes computationally intractable even in various restricted cases. We perform a parameterized complexity analysis, focusing on the number of agents and the pool size as parameters. Notably, although not every instance admits an envy-free solution, our approach allows us to efficiently determine, in polynomial time, whether a solution exists—an aspect that is both theoretically interesting and far from trivial.
Correlated equilibria are a standard solution concept in game theory and generalize Nash equilibria. In a 2-player non-cooperative game in which player i has action set A_i, a correlated equilibrium is a self-enforcing probability distribution σ over A_1 * A_2. Specifically, when a strategy profile (s_1, s_2) in A_1 * A_2 is sampled according to σ, each player i can observe their own component s_i, but not the other player's component. Knowing s_i and σ, player i cannot increase their expected payoff by defecting and playing a strategy s'_i different from s_i. Correlated equilibria are ubiquitous and crucial in mechanism design, including in the design of blockchain-based protocols which aim to incentivize honest behavior. A correlated equilibrium depends on a centralized and impartial oracle, often called the ''external signal'' in game theory literature, to sample a strategy profile and disclose each player's component to them, while keeping the other player's component secret. However, there is currently no trustless method to achieve this on the blockchain without centralization or relying on trusted third-parties. In this work, we address this challenge and provide two novel protocols, one based on oblivious transfer and the other based on zkSNARKs to replace the public signal with a smart contract. We prove that our approaches are secure and provide the desired privacy properties of a correlated equilibrium, while also being efficient in terms of gas usage and thus affordable in practice.