Search Results (123)

Between 1959 and 1962, Alvin Scodel, J. Sayer Minas, and colleagues conducted some of the earliest laboratory studies of strategic interaction using non-zero-sum games. Working at the margins of economics in the Journal of Conflict Resolution, they documented a striking pattern: subjects frequently chose options that reduced an opponent’s payoff by more than their own, even when mutual cooperation was both individually and collectively optimal. These results—especially the behavior observed in their so-called Game H4, a Harmony Game in which cooperation strictly dominated defection—anticipate a central insight of evolutionary game theory: what matters for adaptation is relative payoff, not absolute gain. This essay reinterprets the Scodel–Minas experiments through a Darwinian lens, arguing that they provide an early empirical challenge to Nash-equilibrium reasoning and to models that evaluate strategies solely in terms of absolute utility. By reconstructing the H4 payoff structure and embedding it within a simple evolutionary framework, I show how small levels of “competitive” behavior can destabilize cooperative equilibria that appear self-evident under standard assumptions. I then revisit three later “puzzles” in the evolution of cooperation—altruistic punishment, the fragility of “win–win” treaties, and rejections in ultimatum bargaining—to ask how differently they might have been framed had the Scodel–Minas findings been part of the canonical experimental literature. Rather than treating these phenomena as surprising anomalies, a historically informed, relative-payoff perspective suggests that they could have been recognized much earlier as natural expressions of an already documented pattern. Full article

There has been significant recent progress in algorithms for approximation of Nash equilibrium in large two-player zero-sum imperfect-information games and exact computation of Nash equilibrium in multiplayer normal-form games. While counterfactual regret minimization and fictitious play are scalable to large games and have convergence guarantees in two-player zero-sum games, they do not guarantee convergence to Nash equilibrium in multiplayer games. We present an approach for exact computation of Nash equilibrium in multiplayer imperfect-information games that solves a quadratically-constrained program based on a nonlinear complementarity problem formulation from the sequence-form game representation. This approach capitalizes on recent advances for solving nonconvex quadratic programs. Our algorithm is able to quickly solve three-player Kuhn poker after removal of dominated actions. Of the available algorithms in the Gambit software suite, only the logit quantal response approach is successfully able to solve the game; however, the approach takes longer than our algorithm and also involves a degree of approximation. Our formulation also leads to a new approach for computing Nash equilibrium in multiplayer normal-form games which we demonstrate to outperform a previous quadratically-constrained program formulation. Full article

The unbridled pursuit of statistical rationality has precipitated a crisis of value rationality in e-commerce ecosystems, leading to algorithmic involution—a dilemma characterized by destructive hyper-competition. To reconcile this theoretical paradox and explore effective governance pathways, this paper constructs a tripartite evolutionary game model involving e-commerce platforms, government regulators, and consumers. Simulation results indicate that high-intensity government deterrence constitutes the necessary stability foundation of hard constraints, while consumer activism acts as the decisive accelerator of the soft environment contingent on high synergistic gains and low information screening costs. Furthermore, a platform’s pivot toward “algorithm for good” is not driven by altruism, but by the rational calibration between short-term extractive gains and long-term benevolent returns. Sensitivity analysis confirms that reducing the ratio of these two factors is the effective lever to speed up system convergence. Finally, effective governance requires restructuring this payoff matrix by establishing dynamic penalty mechanisms and transparent low-cost feedback channels to render ethical algorithmic behavior a dominant strategy in terms of economic rationality. This research aims to guide the e-commerce ecosystem from a zero-sum game of involution toward a sustainable equilibrium of multi-party value co-creation. Full article

As the digitalization and intelligent transformation of power systems accelerates, the Power Internet of Things (PIoT) plays a pivotal role in ensuring efficient energy transmission and real-time regulation. However, this openness and interconnectivity also expose the system to diverse cyber threats, where attackers can disrupt stable power communication and dispatch operations through means such as data tampering, denial-of-service attacks, and control intrusion. To characterize the dynamic adversarial process between attackers and defenders in the PIoT, this paper constructs a zero-sum differential game model for cyber attack–defense strategy selection. To achieve equilibrium in the formulated differential game, optimal control theory is employed to solve the optimization problems of the game participants, thereby deriving the optimal strategies for both attackers and defenders. Finally, simulation results illustrate the evolution of network resource competition between attackers and defenders in the PIoT. The results also demonstrate that our proposed model can effectively and accurately describe the evolution of the system security state and the impact of strategic interactions between attackers and defenders. Full article

Electric vehicles (EVs) are pivotal for enhancing the flexibility of power systems, with vehicle-grid integration (VGI) constituting the fundamental mechanism for their participation in grid regulation. VGI relies on multi-source information from EVs, charging infrastructure, traffic network, power grid, and meteorology. However, ineffective data integration mechanisms have resulted in data silos, which impede the realization of synergistic value from multi-source data fusion. To address these issues, this paper develops a quadripartite evolutionary game model that incorporates data providers, data users, government, and data service platforms, overcoming the limitation of traditional tripartite models in fully capturing the complete data value chain. The model systematically examines the cost–benefit dynamics and strategy evolution among stakeholders throughout the data-sharing process. Leveraging evolutionary game theory and Lyapunov stability criteria, sensitivity analyses were conducted on key parameters, including data costs and government subsidies, on the MATLAB platform. Results indicate that multi-source data integration accelerates system convergence and facilitates a multi-party equilibrium. Government subsidies as well as reward and punishment mechanisms emerge as critical drivers of sharing, with an identified subsidy threshold of ε_S = 0.02 for triggering multi-source integration. These key factors can also accelerate system convergence by up to 79% through enhanced subsidies (e.g., reducing stabilization time from 0.29 to 0.06). Importantly, VGI data sharing represents a non-zero-sum game. Well-designed institutional frameworks can achieve mutually beneficial outcomes for all parties, providing quantitatively supported strategies for constructing incentive-compatible mechanisms. Full article

Emergency supplies allocation is a critical task in post-disaster response, as ineffective or delayed decisions can directly lead to increased human suffering and loss of life. In practice, emergency managers must make rapid allocation decisions over multiple periods under incomplete information and highly unpredictable demand, making robust and adaptive decision support essential. However, existing allocation approaches face several challenges: (1) Those traditional approaches rely heavily on predefined uncertainty sets or probabilistic models, and are inherently static, making them unsuitable for multi-period, dynamically allocation problems; and (2) while reinforcement learning (RL) technique is inherently suitable for dynamic decision-making, most existing RL-base approaches assume fixed demand, making them unable to cope with the non-stationary demand patterns seen in real disasters. To address these challenges, we first establish a multi-period and multi-objective emergency supplies allocation problem with demand uncertainty and then formulate it as a two-player zero-sum Markov game (TZMG). Demand uncertainty is modeled through an adversary rather than predefined uncertainty sets. We then propose RESA, a novel RL framework that uses adversarial training to learn robust allocation policies. In addition, RESA introduces a combinatorial action representation and reward clipping methods to handle high-dimensional allocations and nonlinear objectives. Building on RESA, we develop RESA_PPO by employing proximal policy optimization as its policy optimizer. Experiment results with realistic post-disaster data show that RESA_PPO achieves near-optimal performance, with an average gap of only 3.7% in terms of the objective value of the formulated problem, from the theoretical optimum derived by exact solvers. Moreover, RESA_PPO outperforms all baseline methods, including heuristic and standard RL methods, by at least 5.25% on average. Full article

Von Neumann’s minimax theorem defines optimal strategic unpredictability in zero-sum games. Empirical evidence from professional sports has been interpreted as positive behavioral evidence for minimax. In this article, we analyze the strategic optimality of offensive plays in the basketball endgame when a team has a final possession and trails by no more than a single basket. This final moment of the game most closely approximates the simultaneous-move conditions of a game where minimax theory applies. Using comprehensive NBA data from 2010 to 2025, we test for equality of success rates across shooter types (star vs. non-stars) and shot selection (two-point vs. three-point). Our analysis reveals systematic violations of minimax play that have intensified with basketball’s shift to three-pointers and higher expected points. In the final decisive moment of the game, we find that teams systematically overuse three-point shots even though the two-point attempt yields higher field goal percentages. In addition, teams over-rely on star players for the final shot; non-star two-point shots have been the top-performing endgame option in 2022–2025. Full article

Small and Medium-sized Enterprises (SMEs) face disproportionately high risks from Advanced Persistent Threats (APTs), which often evade traditional cybersecurity measures. Existing frameworks catalogue adversary tactics and defensive solutions but provide limited quantitative guidance for allocating limited resources under uncertainty, a challenge amplified by the growing use of AI in both offensive operations and digital forensics. This paper proposes a game-theoretic model for improving digital forensic readiness (DFR) in SMEs. The approach integrates the MITRE ATT&CK and D3FEND frameworks to map APT behaviors to defensive countermeasures and defines 32 custom DFR metrics, weighted using the Analytic Hierarchy Process (AHP), to derive utility functions for both attackers and defenders. The main analysis considers a non-zero-sum attacker–defender bimatrix game and yields a single Nash equilibrium in which the attacker concentrates on Impact-oriented tactics and the defender on Detect-focused controls. In a synthetic calibration across ten organizational profiles, the framework achieves a median readiness improvement of 18.0% (95% confidence interval: 16.3% to 19.7%) relative to pre-framework baselines, with targeted improvements in logging and forensic preservation typically reducing key attacker utility components by around 15–30%. A zero-sum variant of the game is also analyzed as a robustness check and exhibits consistent tactical themes, but all policy conclusions are drawn from the empirical non-zero-sum model. Despite relying on expert-driven AHP weights and synthetic profiles, the framework offers SMEs actionable, equilibrium-informed guidance for strengthening forensic preparedness against advanced cyber threats. Full article

Peter Lombard’s First Book of the Sentences presents formidable questions concerning the principle of the Son’s generation. Addressing a gap in contemporary scholarship, this article examines Lombard’s foundational exposition of the Father’s power and will to generate. Placing Lombard in dialogue with Thomas Aquinas, this study traces the development of this doctrine across Aquinas’s career, from his commentary on the Sentences to De potentia and the Summa theologiae. Our analysis adopts Aquinas’s own framework to investigate a series of questions: whether generation is an act of nature or will; whether the power to generate is part of omnipotence; whether it is essential or relational; and whether the Son possesses this power. This study finds that Aquinas’s conclusions often converge with Lombard’s intuitions. Both affirm that generation is by nature while simultaneously accompanied by a concomitant will, and that the generative power is rooted in the divine essence. Aquinas’s analysis, however, represents a significant metaphysical development. A key evolution is traced in Aquinas’s understanding of the power to generate, which shifts from being a quasi-natural power distinct from omnipotence to a form of paternal omnipotence. His characterization of this power also matures from being a middle ground between the essential and the relational to being principally essential, signifying the relation of paternity only obliquely. This trajectory toward a firmer grounding in the divine essence is supported by an increasingly refined set of arguments for the Son’s unicity, with principles like the determination of nature and divine simplicity becoming more prominent in his later works. By charting these developments, this article demonstrates how Aquinas builds upon Lombard’s foundational intuitions to construct a more systematic and robust Trinitarian theology. Ultimately, our analysis illuminates the intellectual journey from sound doctrinal intuition to profound metaphysical articulation, where the tenets of faith are secured by a cogent intellectual framework. Our analysis also offers a counter-narrative to contemporary assumptions, challenging modern conceptions of power as a zero-sum game and of freedom as mere arbitrary choice. Full article

Reinforcement learning agents are highly susceptible to adversarial attacks that can severely compromise their performance. Although adversarial training is a common countermeasure, most existing research focuses on defending against single-type attacks targeting either observations or actions. This narrow focus overlooks the complexity of real-world mixed attacks, where an agent’s perceptions and resulting actions are perturbed simultaneously. To systematically study these threats, we introduce the Action and State-Adversarial Markov Decision Process (ASA-MDP), which models the interaction as a zero-sum game between the agent and an adversary attacking both states and actions. Using this framework, we show that agents trained conventionally or against single-type attacks remain highly vulnerable to mixed perturbations. Moreover, we identify a key challenge in this setting: a naive mixed-type adversary often fails to effectively balance its perturbations across modalities during training, limiting the agent’s robustness. To address this, we propose the Action and State-Adversarial Proximal Policy Optimization (ASA-PPO) algorithm, which enables the adversary to learn a balanced strategy, distributing its attack budget across both state and action spaces. This, in turn, enhances the robustness of the trained agent against a wide range of adversarial scenarios. Comprehensive experiments across diverse environments demonstrate that policies trained with ASA-PPO substantially outperform baselines—including standard PPO and single-type adversarial methods—under action-only, observation-only, and, most notably, mixed-attack conditions. Full article

This paper investigates a swarming strength allocation Bayesian game approach under incomplete information to address the high-value targets protection problem of swarming unmanned systems. The swarming strength allocation Bayesian game model is established by analyzing the non-zero sum incomplete information game mechanism during the protection process, considering high-tech and low-tech interception players. The model incorporates a game benefit quantification method based on an improved Lanchester equation. The method regards massive swarm individuals as a collective unit for overall cost calculation, thus avoiding the curse of dimensionality from increasing numbers of individuals. Based on it, a Bayesian Nash equilibrium solving approach is presented to determine the optimal swarming strength allocation for the protection player. Finally, compared with random allocation, greedy heuristic, rule-based assignment, and Colonel Blotto game, the simulations demonstrate the proposed method’s robustness in large-scale strength allocation. Full article

A finite-horizon zero-sum linear-quadratic differential game with non-homogeneous dynamics is considered. The key feature of this game is as follows. The cost of the control of the minimizing player (the minimizer) in the game’s cost functional is much smaller than the cost of the control of the maximizing player (the maximizer) and the cost of the state variable. This smallness is due to a positive small multiplier (a small parameter) for the quadratic form of the minimizer’s control in the integrand of the cost functional. Two cases of the game’s cost functional are studied: (i) the current state cost in the integrand of the cost functional is a positive definite quadratic form; (ii) the current state cost in the integrand of the cost functional is a positive semi-definite (but non-zero) quadratic form. The latter case has not yet been considered in the literature devoted to the analysis of cheap control differential games. For each of the aforementioned cases, an asymptotic approximation (by the small parameter) of the solution to the considered game is derived. It is established that the property of the aforementioned state cost (positive definiteness/positive semi-definiteness) has an essential effect on the asymptotic analysis and solution of the differential equations (Riccati-type, linear, and trivial), appearing in the solvability conditions of the considered game. The cases (i) and (ii) require considerably different approaches to the derivation of the asymptotic solutions to these differential equations. Moreover, the case (ii) requires developing a significantly novel approach. The asymptotic solutions of the aforementioned differential equations considerably differ from each other in cases (i) and (ii). This difference yields essentially different asymptotic solutions (saddle point and value) of the considered game in these cases, meaning it is of crucial importance to distinguish cases (i) and (ii) in the study of various theoretical and real-life cheap control zero-sum linear-quadratic differential games. The asymptotic solutions of the considered game in cases (i) and (ii) are compared with each other. An academic illustrative example is presented. Full article

In this work, an adaptive dynamic programming (ADP)-based event-triggered infinite-horizon ($H_{\infty }$) control algorithm is proposed for high-precision speed regulation of permanent magnet synchronous motors (PMSMs). The $H_{\infty }$ control problem of PMSM can be formulated as a two-player zero-sum differential game, and only a single critic neural network is needed to approximate the solution of the Hamilton–Jacobi–Isaacs (HJI) equations online, which significantly simplifies the control structure. Dynamically balancing control accuracy and update frequency through adaptive event-triggering mechanism significantly reduces the computational burden. Through theoretical analysis, the system state and critic weight estimation error are rigorously proved to be uniform ultimate boundedness, and the Zeno behavior is theoretically precluded. The simulation results verify the high accuracy tracking capability and the strong robustness of the algorithm under both load disturbance and shock load, and the event-triggering mechanism significantly reduces the computational resource consumption. Full article

The tethered space net robot offers an effective solution for active space debris removal due to its large capture envelope. However, most existing studies overlook the evasive behavior of non-cooperative targets. To address this, we model an orbital pursuit–evasion game involving a tethered net and propose a game theory-based leader–follower tracking control strategy. In this framework, a virtual leader—defined as the geometric center of four followers—engages in a zero-sum game with the evader. An adaptive dynamic programming method is employed to handle input saturation and compute the Nash Equilibrium strategy. In the follower formation tracking phase, a synchronous distributed model predictive control approach is proposed to update all followers’ control simultaneously, ensuring accurate tracking while meeting safety constraints. The feasibility and stability of the proposed method are theoretically analyzed. Additionally, a body-fixed reference frame is introduced to reduce the capture angle. Simulation results show that the proposed strategy successfully captures the target and outperforms existing methods in both formation keeping and control efficiency. Full article