Potential Games

In decentralised priority coordination, agents announce priority levels and a shared resource serves them in decreasing order, as at an unsignalised intersection; the levels form the decision layer of a hierarchical controller. Such interactions are routinely replaced by a potential game, i.e.\ by a common objective, for analysis and design. This paper determines what that surrogate misses, using the Hodge decomposition of the incentives into a potential component, which a common objective can represent, and a harmonic component, which it cannot. For the linear payoff, both components are obtained in closed form on every conflict graph and for every deterministic tie-breaking protocol: in common units, the harmonic energy is the number of conflicts and the potential energy adds the number of adjacent pairs of conflicts. Consequently, for every rationality parameter, the best common-objective model of the agents' choice log-odds, weighted uniformly over unilateral moves, has a relative squared error of at least $1/(d_{\max}+1)$, where $d_{\max}$ is the largest number of conflicts of one agent; for an eight-vehicle intersection it is exactly one fifth, for any number of priority levels. Invisible to strict-improvement dynamics, the missed component is, under low-rationality log-linear learning with uniform revision and to leading order, the stationary probability current, and its energy sets the entropy-production rate. Payoff design cannot remove it: on the complete conflict graph of $N$ agents, under a total-order protocol and with at least three priority levels, every nonconstant rank-based payoff leaves a relative error of at least $1/N$, with equality exactly for affine payoffs.

We study the problem of computing the tightest $α$-potential approximation of a general-sum game over continuous action spaces, within a prescribed class of potential functions and when each player has access only to its own utility function. The difficulty is twofold: the approximation error involves a worst-case search over an infinite set of unilateral deviations, and the required utility information is distributed across players. For a linear-in-parameters potential class, we use an exact finite-tuple reformulation that separates the problem into a global outer search over deviation tuples and distributed convex inner problems. We develop a primal--dual inner oracle tailored to this structure and establish a uniform one-sided accuracy guarantee. This oracle can be combined with global outer search to obtain an end-to-end guarantee on the outer optimization error. We also develop a projected zeroth-order outer method as a computationally lighter alternative for higher-dimensional problems. Numerical experiments illustrate the accuracy--computation tradeoff between the two outer-search methods and show that the proposed optimization framework can improve upon analytical $α$-potential constructions.

We study noncooperative multi-vehicle games among heterogeneous autonomous vehicles, where each vehicle adopts a decentralized closed-loop policy based on its own state, and optimizes an objective that depends on other vehicles through potentially asymmetric interaction weights. We develop an $α$-potential game framework that reduces the computation of an approximate Nash equilibrium (NE) to the minimization of a single auxiliary $α$-potential function. We explicitly construct this $α$-potential, establish the existence of its minimizers, and characterize the equilibrium approximation error $α$ in terms of interaction asymmetry. We further introduce vehicle-specific scaling to reduce the effective interaction asymmetry, thereby tightening the equilibrium approximation and, in important cases, recovering an exact NE despite asymmetric interactions. We also derive social-efficiency guarantees for the potential-selected policies, revealing how the interaction structure shapes worst-case efficiency. Numerical experiments demonstrate the flexibility of the framework in capturing heterogeneous vehicle interactions, collision and obstacle avoidance, lane changing and overtaking under different traffic configurations, and priority-based intersection crossing.

Clustered federated learning benefits from organizing heterogeneous participants into coalitions that train coalition-specific models, but such clustering is sustainable only if participants prefer their assigned coalition and the required transfers are affordable. We develop a transferable-surplus model separating learning benefit, system cost, participant cost, and monetary transfers; an allocation rule converts coalition surplus into hedonic preferences, and weak budget feasibility guarantees nonnegative retained coordinator surplus. For symmetric pairwise allocations the induced game is an exact potential game: a Nash-stable partition exists, every strict better-response process converges, and with destination consent accepted better responses reach an individually stable partition. We characterize feasibility of bounded pair incentives and verify the exponentially many budget constraints in polynomial oracle time when retained slack is submodular. Decomposing welfare into participant potential and retained slack yields additive and multiplicative price-of-stability guarantees, the latter asymptotically tight; exact balance gives welfare-optimal stability only on the pairwise-representable class, and budget feasibility alone permits unbounded welfare loss. Global potential maximization equals weighted maximum-agreement correlation clustering, and approximation followed by stabilization satisfies an end-to-end welfare bound governed by retained slack and negative-edge mass, attained by an explicit construction. In a preregistered five-seed CIFAR-10 study the mechanism reaches the certified estimated-table welfare optimum on every primary instance, equal-surplus sharing has no Nash-stable outcome on three, and pairwise validation gain gives far more reliable pair signs than gradient alignment.

Distributed constraint optimization problems (DCOPs) provide a popular framework for distributed decision making under limited communication, but many real-world instances are too large to solve monolithically. We address this challenge from two complementary directions. We revisit the connection between DCOPs and potential games, and adapt modern online learning algorithms for equilibrium finding to DCOPs. We show that these algorithms are competitive with representative incomplete DCOP algorithms. We then turn to decomposition frameworks for large-scale DCOPs, motivated by large-scale decentralized satellite scheduling. We propose a new framework that separates a DCOP into two interacting subproblems: a high-level meta-DCOP for task allocation, and independent local optimization problems for scheduling. To couple the two levels, we develop a novel iterative pricing method that updates the meta-level utilities using feedback from the local optimizers. Combining our online learning methods with our iterative pricing framework, we obtain near-optimal performance on real-world decentralized satellite scheduling problem instances, fulfilling over 99% of observation requests compared with 87% for state-of-the-art baselines.

The mean mechanism is known to be non-incentive-compatible, namely, rational players are incentivized to misreport their values. Despite this game-theoretic issue, the mean mechanism is prevalent in practice due to its other desirable properties. We give a full characterization of pure Nash equilibria--how the players will misreport--for the affine mechanism, of which the mean is a special case. Furthermore, we characterize both complete-information and Bayesian games under the affine mechanism. Our results highlight the inevitability of extreme exaggeration in such games.

This paper studies coalition formation as a decentralized dynamical process driven by unilateral exit-and-join decisions. Agents evaluate local moves using the Aumann-Dreze value, so payoffs are computed within the agent's current coalition rather than through a globally negotiated coalition structure. The resulting model links cooperative payoff allocation with noncooperative best-response behavior: a terminal partition is precisely a coalition structure with no admissible, individually profitable exit-and-join deviation. We establish equilibrium characterizations, identify conditions under which the dynamics admit scalar Lyapunov or exact-potential representations, and analyze how switching and acceptance costs shape local stability. Numerical experiments test finite-time stabilization, cost sensitivity, and a special convex-game benchmark.

We cast the design of parameterized quantum circuits as a four-player potential game whose state is a circuit directed acyclic graph (DAG) and whose players encode trainability, non-stabilizerness, task performance, and hardware cost. Per-player restricted action sets factorize the move space into append, remove, retype, and rewire operations; a block-coordinate $\varepsilon$-Nash residual $δ_\text{Nash}$ certifies that no single player can improve unilaterally. A single weight sweep on MaxCut $K_4$ traces a Pareto frontier from a Clifford endpoint $(M_2/n,\langle H\rangle)=(0,4.00)$ to a non-Clifford endpoint $(0.48,3.30)$. On three four-qubit hardware topologies (heavy-hex, $2\times 2$ grid, Rydberg all-to-all), Nash search achieves the highest mean potential; on the $2\times 2$ grid Nash reaches the theoretical ceiling $Φ_\text{max}=4.10$ on two of five seeds while the simulated-annealing baseline does so on one; paired Wilcoxon tests over five seeds cannot reject the null on any single topology ($p\ge 0.22$). On LiH/STO-3G, seeding Nash from a 58-gate Givens-doubles ansatz produces a 48-operation, depth-25 circuit retaining $97.7\%$ of the correlation energy while simultaneously reducing gate count, increasing non-stabilizerness, and controlling trainability. The framework is complementary to energy-only searches such as ADAPT-VQE and k-UpCCGSD, which reach chemical accuracy with fewer operations but do not optimize the other three axes.

Connected autonomous vehicles (CAVs), which represent a significant advancement in autonomous driving technology, have the potential to greatly increase traffic safety and efficiency through cooperative decision-making. However, existing methods often overlook the individual needs and heterogeneity of cooperative participants, making it difficult to transfer them to environments where they coexist with human-driven vehicles (HDVs).To address this challenge, this paper proposes an adaptive potential game (APG) cooperative driving framework. First, the system utility function is established on the basis of a general form of individual utility and its monotonic relationship, allowing for the simultaneous optimization of both individual and system objectives. Second, the Shapley value is introduced to compute each vehicle's marginal utility within the system, allowing its varying impact to be quantified. Finally, the HDV preference estimation is dynamically refined by continuously comparing the observed HDV behavior with the APG's estimated actions, leading to improvements in overall system safety and efficiency. Ablation studies demonstrate that adaptively updating Shapley values and HDV preference estimation significantly improve cooperation success rates in mixed traffic. Comparative experiments further highlight the APG's advantages in terms of safety and efficiency over other cooperative methods. Moreover, the applicability of the approach to real-world scenarios was validated through field tests.

In data-sensitive domains such as healthcare, cross-silo federated learning (CFL) allows organizations to collaboratively train AI models without sharing raw data. However, practical CFL deployments are inherently coopetitive, in which organizations cooperate during model training while competing in downstream markets. In such settings, training contributions, including data volume, quality, and diversity, can improve the global model yet inadvertently strengthen rivals. This dilemma is amplified by non-IID data, which leads to asymmetric learning gains and undermines sustained participation. While existing competition-aware CFL and incentive-design approaches reward organizations based on marginal training contributions, they fail to account for the costs of strengthening competitors. In this paper, we introduce CoCoGen+, a coopetition-compatible data generation and incentivization framework that jointly models non-IID data and inter-organizational competition while endogenizing GenAI-based synthetic data generation as a strategic decision. Specifically, CoCoGen+ formulates each training round as a weighted potential game, where organizations strategically decide how much synthetic data to generate by balancing learning performance gains against computational costs and competition-caused utility losses. We then provide a tractable equilibrium characterization and derive implementable generation strategies to maximize social welfare. To promote long-term collaboration, we integrate a payoff redistribution-based incentive mechanism to compensate organizations for their contributions and competition-caused utility degradation. Experiments on varying learning tasks validate the feasibility of CoCoGen+. The results show how non-IID data, competition intensity, and incentives shape organizational strategies and social welfare, while CoCoGen+ outperforms baselines in efficiency.

Decentralized air traffic management systems offer a scalable alternative to centralized control, but often assume high levels of cooperation. In practice, such assumptions frequently break down since airspace sectors operate independently and prioritize local objectives. We address the problem of sector overload in decentralized air traffic management by proposing a regulated decentralized protocol that models self-interested behaviors based on best response dynamics. Each sector adjusts the departure times of flights under its control to reduce its own congestion, without requiring centralized joint optimization. A tunable cooperativeness factor models the degree to which each sector accounts for overload in other sectors, while a minimal admissibility rule prevents local updates from creating new overloads. We prove that the proposed protocol satisfies a potential game structure, ensuring that best response dynamics converge to a pure Nash equilibrium under this restriction. In addition, we identify a sufficient condition under which an overload-free solution corresponds to a global minimizer of the potential function. Numerical experiments using 24 hours of European flight data demonstrate that the proposed algorithm substantially reduces overload even with only minimal cooperation between sectors, while maintaining scalability and achieving solution quality comparable to the centralized benchmark.

Public resource allocation involves distributing resources, including urban infrastructure, energy, and transportation, which are typically limited in capacity, to meet social demands. In real-world scenarios, resources are typically limited in capacity, which makes coordination among multiple resources essential. However, existing methods often optimize resource movements in an isolated manner and do not explicitly account for capacity-aware collaboration under spatio-temporal dynamics. To address this limitation, we introduce the Collaborative Public Resource Allocation (CPRA) problem, and propose a Game-Theoretic Spatio-Temporal Reinforcement Learning (GSTRL) framework to solve it. Our contributions are twofold: 1) We formulate CPRA as a potential game and construct the potential function based on the objective function of CPRA, laying a theoretical foundation for approximating the Nash equilibrium of this NP-hard problem; and 2) Our GSTRL framework effectively captures the spatio-temporal dynamics of the overall system. We evaluate GSTRL on two real-world datasets, where experiments show its superior performance. Our source codes are available at https://github.com/thunderlrr/GSTRL.

In federated learning (FL), a central server typically allocates training efforts to clients. However, from a market-oriented perspective, clients may independently choose their training efforts based on rational self-interest. To study this setting, we propose a potential game framework in which each client's payoff is determined by its individual effort and the rewards provided by the server. The rewards are influenced by the collective efforts of all clients and can be modulated by a reward factor. We first establish the existence of Nash equilibria (NEs) and then investigate their uniqueness in a stationary setting. We show that the NEs depend nonlinearly on the reward factor and exhibit a nonsmooth transition at a critical value, where the stationary potential loses strict curvature, leading to nonunique NEs and a jump between low-effort and high-effort branches. Furthermore, we prove the convergence of the best-response algorithm for computing NEs in our FL game. Finally, we apply the clients' rational efforts derived from the NEs to FL training with various datasets and models, thereby validating the effectiveness of the identified critical reward factor. The source code is available at https://github.com/DCN-FAU-AvH/FL-Potential-Game