Motivated by practical allocation problems, we study integer splittable congestion games with capacitated resources and player-specific costs. In this setting, each player has an integer weight that has to be split in integer units across multiple resources, each with a capacity limiting the total assigned weight, i.e., its congestion. The latency of a resource is equal to a player-specific constant when its congestion is not greater than the capacity and becomes prohibitive, i.e., equal to ∞, once the capacity is exceeded. We analyze the computational complexity of finding an allocation that optimizes utilitarian social welfare under two cost models (total cost and per-unit cost). Furthermore, we investigate the computation of, speed of convergence to, and efficiency of Nash equilibria.
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Potential games are a fundamental class of games in which pure Nash equilibria are guaranteed to exist, yet computing such equilibria is computationally intractable for several subclasses. This has led to extensive research on computing approximate pure Nash equilibria. In this paper, we study payoff-maximization potential games. For these games, strong approximation guarantees are known only for restricted subclasses, most notably Pd--Flip games. We show that standard approaches based on unilateral improvement moves can fail to provide any finite approximation guarantee even for simple extensions of Pd--Flip games. To overcome this limitation, we propose an algorithmic framework based on coordinated moves by small groups of players, whose approximation guarantee and number of moves are controlled by two natural game parameters, the stretch and the spread, which are bounded for broad classes of games of interest. In the special case of Pd--Flip games, our framework can be configured to recover the existing algorithm, matching its approximation guarantee and the number of moves performed.
Social Law Synthesis (SLS) in strategic environments is a novel multi-unit mechanism design problem, spanning modeling to computational challenges. We derive a method to specify the problem succinctly, reduce payment determination to allocation determination, and design an integer linear programming (ILP)-based algorithm that further reduces allocation to a polynomial-time ILP formulation. This offloads intractability to powerful ILP solvers, yielding a truthful, individually rational, and profit-optimal mechanism.
Modern AI-driven systems rely on large populations of autonomous agents that make decentralized routing decisions under uncertainty, where rare but severe tail latency can critically degrade quality of experience, safety, and reliability. To model agents’ aversion to such tail latency, we study nonatomic selfish routing games in which agents minimize the Conditional Value-at-Risk (CVaR) of path latency. CVaR explicitly captures both the likelihood and severity of tail latency, but its inherent nonadditivity across network edges poses a fundamental challenge. We address it by identifying a worst-case dependence structure—tail risk concentration—under which tail latency across network edges is synchronized. We show that CVaR penalizes this dependence structure and becomes additive under worst-case tail dependence, which enables tight inefficiency analysis. To quantify the resulting inefficiency induced by risk-aversion, we adopt the price of risk aversion (PRA), defined as the worst-case ratio between the total system cost at a risk-averse equilibrium and that at a risk-neutral equilibrium. We show that, for arbitrary latency functions and general network topologies, the PRA under CVaR admits a tight upper bound that grows linearly with both the network size and the maximum edge-level upper-tail cost. We further prove that this bound is tight by constructing a family of Braess-type networks that achieve a matching lower bound. These results provide the first tight worst-case inefficiency bound for CVaR-based selfish routing and offer insights into how tail-risk-averse decision making by autonomous agents amplifies congestion externalities in large-scale multi-agent systems.
We introduce Hedonic Project Games, a model in which agents choose projects with divisible rewards while holding subjective preferences over coalition composition. This framework captures a fundamental trade-off absent from existing models: agents care simultaneously about who they collaborate with and what they work on. We study three stability notions: classical Nash Stability and two refinements, Joining Stability and Leaving Stability, which account for the welfare of both the deviating agent and the affected coalition members. We evaluate the efficiency of stable outcomes using the Price of Anarchy and Price of Stability, comparing the social welfare of stable outcomes to that of an optimal allocation. While stable outcomes may not exist in general, we identify broad and natural preference classes in which stability and efficiency improve significantly. In particular, under monotonic-decreasing preferences in coalition size, Nash and joining stability coincide and are guaranteed to exist, whereas leaving stability may fail. Under per-capita non-decreasing preferences, socially optimal outcomes are always Nash stable and coincide with leaving stability, although equilibrium inefficiency remains unbounded. Experiments on synthetic and real-world data support the theoretical efficiency results.
Hedonic games are a central model of coalition formation, yet most of their general subclasses are marked by negative results: stable outcomes often fail to exist, and deciding their existence is typically computationally hard. The most notable exception is the class of hedonic games with the common ranking property (HGCRP), a nontrivial subclass that guarantees both the existence and tractability of stable partitions. The common ranking property implies the top-coalition property, which in turn implies the weak top-coalition property, forming a natural hierarchy of increasingly general domains. In this paper, we extend the frontier of strong existence and tractability results beyond the HGCRP to hedonic games with the top-coalition property (HGTCP) and hedonic games with the weak top-coalition property (HGWTCP). We show that every HGWTCP instance admits a strong individually stable (SIS) partition that can be computed in polynomial time, and prove that our polynomial-time algorithm generates a partition that is both SIS and Pareto optimal (PO) if preferences are strict. These results show that strong stability and tractability results persist well beyond the common ranking framework. We further show that a contractually Nash stable (CNS) partition may fail to exist even in HGTCP, revealing a sharp contrast with HGCRP, where the existence of partitions that are both CNS and PO is guaranteed. Taken together, these results provide a comprehensive characterization of stability and efficiency in top-coalition-based hedonic games, mapping how stability guarantees evolve from HGCRP to HGWTCP.
Real-world multi-agent systems, from traffic coordination to resource allocation, are often modeled as general-sum games where individual incentives conflict with collective welfare. In these settings, the central challenge is not merely finding an equilibrium, but selecting socially desirable outcomes among many suboptimal Nash equilibria. Standard deep multi-agent reinforcement learning (MARL) methods struggle with this problem, as value-decomposition approaches are constrained by monotonicity assumptions and policy-gradient methods often converge to stable but socially inefficient equilibria. To address this limitation, we propose Phi-Actor-Critic (Phi-AC), a framework that leverages swap regret minimization to steer learning toward high-welfare correlated equilibria (CE). To make counterfactual regret estimation tractable in deep MARL, Phi-AC employs a centralized attention critic that predicts vector-valued regrets in a single forward pass, avoiding computationally expensive counterfactual simulations. We further introduce a Lagrangian-based equilibrium selection mechanism that optimizes social welfare while enforcing stability through regret constraints. Experiments on matrix games, Multi-Agent Particle Environments (MPE), and the Melting Pot Harvest scenario demonstrate that Phi-AC learns efficient and stable coordination strategies across diverse mixed-motive settings while maintaining high collective return and competitive fairness.
We study the question of existence and fast computation of fair and efficient allocations of indivisible resources among agents with additive valuations. As such allocations may not exist for arbitrary instances, we ask if they exist for typical or random instances, meaning when the utility values of agents for the resources are drawn from certain distributions. In this paper, we extend the previously studied formal models of this problem to non-identical items. We assume that every item is associated with a distribution U_j, and every agent's utility value for the item is drawn independently from U_j. We show that envy-free fair and maximum social welfare efficient allocations exist with high probability in the asymptotic setting, meaning when the number of agents n and items m are large. Further, we show that when m = Ω(n log n), then by only sampling O(log m) or O((log m)^2) utility values per item instead of all the n, we can compute these allocations in Õ(m) time. Finally, we simulate our algorithms on randomly generated instances and show that even for small instances, we suffer small multiplicative losses in the fairness and efficiency guarantees and converge to fully optimal guarantees quickly.
The Kidney Exchange Problem is a prominent challenge in healthcare and economics, arising in the context of organ transplantation. It has been extensively studied in artificial intelligence and optimization. In a kidney exchange, a set of donor-recipient pairs and altruistic donors are considered, with the goal of identifying a sequence of exchanges—comprising cycles or chains starting from altruistic donors—such that each donor provides a kidney to the compatible recipient in the next donor-recipient pair. These exchanges create a network of transplants aimed at maximizing the total number, t, of successful transplants. Due to constraints in medical resources, limits are often imposed on the lengths of these cycles and chains. Recently, this problem was deterministically solved in O* (14.34ᵗ) time (IJCAI 2024). In this paper, we introduce the representative set technique for the Kidney Exchange Problem, showing that the problem can be deterministically solved in O* (6.855ᵗ) time.
Partial Dependence Plots (PDPs) visualize how changes in a single feature affect the average model prediction. They are widely used in practice to interpret decision tree ensembles and other machine learning models. Joint-PDPs extend this idea to pairs of features, revealing their combined effect. Partial Dependence Interaction Values (PDIVs) measure feature interactions. The Any-Order-PDIVs task computes these interactions for every feature subset across all rows of the dataset. We introduce WOODELF++, a unified and efficient approach for computing all these useful explainability tools on decision tree ensembles, building on WOODELF, an algorithm for efficient SHAP computation. By deriving suitable metrics over pseudo-Boolean functions, WOODELF++ can compute PDPs (exact and approximate), Joint-PDPs, and Any-Order-PDIVs in a unified framework. Our method delivers substantial complexity improvements over the state of the art, including an exponential gain for Any-Order-PDIVs. Additionally, we introduce and efficiently compute Full PDPs, which leverage the model’s split thresholds to faithfully capture its behavior across all possible feature values. WOODELF++ is implemented in pure Python and supports GPU acceleration. On a dataset with 400,000 rows, WOODELF++ computes PDP and Joint-PDP up to 6x faster than the state of the art and up to five orders of magnitude faster than scikit-learn. For Any-Order-PDIVs, the gap is even larger: WOODELF++ computes all interaction values in 5 minutes, while the state of the art is estimated to require over 1,000,000 years.
Understanding when fair allocation mechanisms lead to stable coalition structures is fundamental for designing robust multi-agent systems. We investigate the compatibility between stable allocations in transferable utility (TU) games and the stability of coalition structures in induced hedonic games, where agents' preferences over coalitions are derived from payoffs assigned by a fixed allocation rule applied to all subgames of the TU game. We analyze FX-FE strong Nash stability (SNS) in induced hedonic games, which captures a strong form of robustness under free-exit and free-entry conditions. We show that any efficient allocation rule ensuring core membership for the grand coalition induces a hedonic game that guarantees the existence of an FX-FE strong Nash stable partition. Examining the Shapley value, we further show that when it lies outside the core, the resulting hedonic game may or may not possess FX-FE strongly Nash stable partitions, highlighting a sensitive interaction between Shapley-based fairness and coalition-level stability. Our framework bridges SHAP methodology from explainable AI with hedonic coalition theory, providing theoretical foundations for understanding when fair allocation mechanisms shape coalition-level incentives in a way that ensures strategic stability in team formation and multi-agent systems.
We study multi-agent contracts, in which a principal delegates a task to multiple agents and incentivizes them to exert effort. Prior research has mostly focused on maximizing the principal’s utility, often resulting in highly disparate payments among agents. Such disparities among agents may be undesirable in practice, for example, in standardized public contracting or worker cooperatives where fairness concerns are essential. Motivated by these considerations, our objective is to quantify the tradeoff between maximizing the principal's utility and equalizing payments among agents, which we call the price of non-discrimination. Our first result is an almost tight bound on the price of non-discrimination, which scales logarithmically with the number of agents. This bound can be improved to a constant by allowing some relaxation of the non-discrimination requirement. We then provide a comprehensive characterization of the tradeoff between the level of non-discrimination and the loss in the optimal utility.
We initiate the study of online contracts, which integrate the game-theoretic considerations of economic contract theory, with the algorithmic and informational challenges of online algorithm design. Our starting point is the classic online setting with preemption, in which a hiring principal faces a sequence of adversarial agent arrivals. Upon arrival, the principal must decide whether to tentatively accept the agent to their team, and whether to dismiss previous tentative choices. Dismissal is irrevocable, giving the setting its online decision-making flavor. In our setting, the agents are rational players: once the team is finalized, a game is played where the principal offers contracts (performance-based payment schemes), and each agent decides whether or not to work. Working agents reward the principal, and the goal is to choose a team that maximizes the principal's utility. Our main positive result is a 1/2-competitive algorithm when agent rewards are additive, which matches the best-possible competitive ratio. Our algorithm is randomized and this is necessary, as we show that no deterministic algorithm can attain a bounded competitive ratio. Moreover, if agent rewards are allowed to exhibit combinatorial structure known as XOS, even randomized algorithms might fail. En route to our competitive algorithm, we develop the technique of balance points, which can be useful for further exploration of online contracts in the adversarial model.
Subgame solving is a technique for scaling algorithms to large games by locally refining a precomputed blueprint strategy during gameplay. While straightforward in perfect-information games where search starts from the current state, subgame solving in imperfect-information games must account for hidden states and uncertainty about the opponent's past strategy. Gadget games were developed to ensure that the improved subgame strategy is robust against any possible opponent's strategy in a zero-sum game. Gadget games typically contain infinitely many Nash equilibria. We demonstrate that while these equilibria are equivalent in the gadget game, they yield vastly different performance in the full game, even when facing a rational opponent. We propose gadget game sequential equilibria as the preferred solution concept. We introduce modifications to the sequence-form linear program and counterfactual regret minimization that converge to these refined solutions with only mild additional computational cost. Additionally, we provide several new insights into the surprising superiority of the resolving gadget game over the max-margin gadget game. Our experiments compare different Nash equilibria of gadget games in several standard benchmark games, showing that our refined equilibria consistently outperform unrefined Nash equilibria, and can reduce the exploitability of the overall strategy by more than 50%.
We address the problem of selecting k representative nodes from a network, aiming to simultaneously achieve two objectives: identifying the most influential nodes and ensuring that the selection proportionally reflects the diversity within the network. We propose a general approach to accomplish this by combining ideas from network science and computational social choice. Notably, our algorithms depend only on the connections between nodes and do not utilize any additional information that would explicitly identify groups of nodes. We analyze them theoretically, and demonstrate their effectiveness through a series of experiments.
Coalition formation studies how to partition a set of agents into disjoint coalitions based on their preferences. In this paper, we consider the class of additively separable hedonic games and study the setting in which agents' valuations of each other evolve over time. Since rearranging coalitions can be costly, our goal is to find a stable partition close to the current one, where distance is measured by the number of agents that must change coalition. We study this problem for four stability notions based on single-agent deviations: Nash, individual, contractual Nash, and contractual individual stability. For all four, deciding whether a close stable partition exists is NP-complete, even under severe restrictions on the valuations and even when only a single valuation changes. On the positive side, we give polynomial-time algorithms for the two contractual notions under restricted symmetric valuations, and show that over long sequences of updates, these algorithms maintain stability with constant average reconfiguration cost.
In approval-based budget division, the task is to allocate a divisible resource to the candidates based on the voters' approval preferences over the candidates. For this setting, Brandl et al. (2021) have shown that no distribution rule can be strategyproof, efficient, and fair at the same time. In this paper, we aim to circumvent this impossibility theorem by focusing on approximate strategyproofness. To this end, we analyze the incentive ratio of distribution rules, which quantifies the maximum multiplicative utility gain of a voter by manipulating. While it turns out that several classical rules have a large incentive ratio, we prove that the Nash product rule (NASH) has an incentive ratio of 2, thereby demonstrating that we can bypass the impossibility of Brandl et al. by relaxing strategyproofness. Moreover, we show that an incentive ratio of 2 is optimal within three natural classes of rules and that the positive result for the Nash product rule even holds when voters may report arbitrary concave utility functions. Finally, we complement our results with an experimental analysis.
A common criticism of liquid democracy within the relevant academic literature is that delegation cycles can occur, seemingly resulting in unused voting power. Yet, practitioners argue that delegation cycles are not only unproblematic but are even formed intentionally by participants. To bring theory closer to reality, we introduce a model that captures this strategic behavior under uncertainty. We study the existence, structure and quality of Nash equilibria, revealing that delegation cycles naturally emerge. To complement these findings, we perform computational experiments using best-response dynamics.
Many multiagent systems depend on collective decisions made by self-interested agents, which raises deep questions about coalition formation and stability. We study social choice with endogenous, outcome-contingent transfers, where agents voluntarily form contracts that redistribute utility depending on the collective decision, allowing for fully strategic coalition formation. We show that under consensus rules, individually rational strong Nash equilibria (IR–SNE) always exist, implementing welfare-maximizing outcomes with feasible transfers, and provide a simple, efficient algorithm to construct them. For more general anonymous, monotonic, and resolute rules, we identify necessary conditions for viable deviations, significantly limiting the possibility of destabilizing coalitions. By bridging cooperative and noncooperative perspectives, our approach shows that transferable utility can achieve core-like stability, restoring efficiency and budget balance even where classical impossibility results generally apply. Overall, this framework offers a practical and robust way to coordinate large-scale strategic multiagent systems.
We study a robust scheduling problem on identical machines in which machines may fail after the initial assignment. The goal is to compute an initial schedule together with a recovery strategy that minimizes the post-failure makespan, under the constraint that jobs on available machines cannot be moved. We consider two failure models: a strong adversary, which selects the failed machines, and a weak adversary, which selects only the number of failures. For up to k failures, we give algorithms with robustness ratios 1.618 for the strong adversary and 1.5 for the weak adversary. For the single-failure case, k = 1, we obtain best possible ratios 1.387 and 1.281, respectively, matching our lower bounds.