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Recent work on incorporating strong-correlation (sc) corrections into the scLH22t local hybrid functional [A. Wodyński and M. Kaupp, J. Chem. Theory Comput. 18, 6111–6123 (2022)] used a hybrid procedure, applying a strong-correlation factor derived from the reverse Becke–Roussel machinery of the KP16/B13 and B13 functionals to the nonlocal correlation term of a local hybrid functional. Here, we show that adiabatic-connection factors for strong-correlation-corrected local hybrids (scLHs) can be constructed in a simplified way based on a comparison of semi-local and exact exchange-energy densities only, without recourse to exchange-hole normalization. The simplified procedure is based on a comparative analysis of Becke's B05 real-space treatment of nondynamical correlation and that in LHs, and it allows us to use, in principle, any semi-local exchange-energy density in the variable used to construct local adiabatic connections. The derivation of competitive scLHs is demonstrated based on either a modified Becke–Roussel or a simpler Perdew–Burke–Ernzerhof (PBE) energy density, leading to the scLH23t-mBR and scLH23t-tPBE functionals, which both exhibit low fractional spin errors while retaining good performance for weakly correlated situations. We also report preliminary attempts toward more detailed modeling of the local adiabatic connection, allowing a reduction of unphysical local maxima in spin-restricted bond-dissociation energy curves (scLH23t-mBR-P form). The simplified derivations of sc-factors reported here provide a basis for future constructions and straightforward implementation of exchange-correlation functionals that escape the zero-sum game between low self-interaction and static-correlation errors. [ABSTRACT FROM AUTHOR]

Professor Peter Sudhölter, a renowned mathematician and economist, passed away on April 2, 2024, after a courageous battle with cancer. Throughout his career, he made significant contributions to the field of cooperative game theory, publishing 61 articles and one book. He obtained his PhD in mathematics from the University of Bielefeld in 1988 and went on to become a full professor at the University of Southern Denmark in 2013. Sudhölter's research focused on various topics, including cooperative games, non-transferable utility games, and the characterization of equilibrium concepts. His work has had a lasting impact on the field and will continue to influence future research. [Extracted from the article]

In this paper, an optimisation problem for the monotone mean-variance (MMV) criterion is considered from the perspective of the insurance company. The MMV criterion is an amended version of the classical mean-variance (MV) criterion which guarantees the monotonicity of the utility function. With this criterion we study the optimal investment and reinsurance problem which is formulated as a zero-sum game between the insurance company and an imaginary player. We apply the dynamic programming principle to obtain the corresponding Hamilton–Jacobi–Bellman–Isaacs (HJBI) equation. As the main conclusion of this paper, by solving the HJBI equation explicitly, the closed forms of the optimal strategy and the value function are obtained. Moreover, the MMV efficient frontier is also provided. At the end of the paper, a numerical example is presented. [ABSTRACT FROM AUTHOR]

A generalized class of the discrete game of timing is solved as a finite zero-sum game defined on a symmetric lattice of the unit square. The game is a progressive discrete silent duel whose kernel is skew-symmetric, and the players, referred to as duelists, have identical shooting accuracy functions featured with an accuracy proportionality factor a and a power constant β describing nonlinearity of the shooting accuracy. As the duel starts, time moments of possible shooting become denser by a geometric progression. Apart from the duel beginning and end time moments, every following time moment is the partial sum of the respective geometric series. Due to the skew-symmetry, both the duelists have the same optimal strategies and the game optimal value is 0. As the duelist has a single bullet, there is no reason for considering solutions in mixed strategies, if any, with non-singleton supports. If a ⩾ 2 β − 1 , the duelist's optimal strategy is the middle of the duel time span; otherwise, the duel solution may be not a pure strategy solution. The case of quadratic accuracy for β = 2 is further considered, which is reasoned by that linearly developing real-time processes are quite rare, and the quadratic accuracy is a non-linearity pattern that makes an interaction model more reliable by slightly diminishing the duelist's confidence. Thus, the case of a < 2 β − 1 is thoroughly studied for pure strategy solutions. A boundary case of a is found, by which the duel has four pure strategy solutions which are of the time moment preceding the duel end moment and the duel end moment itself. • Discrete game of timing is solved as a finite zero-sum game on a symmetric lattice. • Duelist is featured with shooting accuracy proportionality and nonlinearity factors. • The breaking point of the accuracy is found above which pure strategy solutions exist. • Pure strategy solutions are found below the breaking point for the quadratic accuracy. [ABSTRACT FROM AUTHOR]

In this research, we investigate the horse racing game, a two-player game with six strategies. Our objective is to develop a theoretical framework for the game by employing the process of quantization. The Nash equilibrium (NE) of the quantum horse racing game is analyzed through an examination of the payoff matrix, and the results are compared to those of the classical horse racing game. Our findings demonstrate that the quantum horse racing game features both pure and mixed strategy NE, whereas the classical horse racing game possesses only mixed strategy NE. Additionally, our results indicate that the quantum horse racing game can mitigate the risks associated with the zero-sum game while outperforming the classical game. [ABSTRACT FROM AUTHOR]

This paper aims to present alternative characterizations for different types of set-valued robustness concepts. Equivalent scalar representations for various set order relations are derived when the sets are the union of sets. Utilizing these findings in conjunction with image space analysis, specific isolated sets are defined for different notions of robust solutions. These isolated sets serve as the basis for deriving both necessary and sufficient robust optimality conditions. The validity of the results is demonstrated through several illustrative examples. Additionally, the paper concludes with an application of our present approach to two-player zero-sum matrix games. [ABSTRACT FROM AUTHOR]

The potential emergence of artificial general intelligence (AGI) systems has sparked intense debate among researchers, policymakers, and the public due to their potential to surpass human intelligence in all domains. This note argues that for an AI to be considered "general", it should achieve superhuman performance not only in zero-sum games but also in general-sum games, where winning or losing is not clearly defined. In this note, I propose a game-theoretic framework that captures the strategic interactions between a representative human agent and a potential superhuman machine agent. Four assumptions underpin this framework: Superhuman Machine, Machine Strategy, Rationality, and Strategic Unpredictability. The main result is an impossibility theorem, establishing that these assumptions are inconsistent when taken together, but relaxing any one of them results in a consistent set of assumptions. This note contributes to a better understanding of the theoretical context that can shape the development of superhuman AI. • A game between a human vs an Artificial General Intelligence system is introduced. • An AI system is general if it is successful in both zero-sum and general-sum games. • Assumptions are Superhuman AI, Machine Strategy, Rationality, and Unpredictability. • An impossibility theorem shows that these assumptions lead to inconsistency. • The result is tight; relaxing any of the assumptions resolves the inconsistency. [ABSTRACT FROM AUTHOR]

This study proposes an organizational mechanism that links ideology to the use of terrorism in mass dissident campaigns. Ideology affects the level of competition among factions within mass dissident campaigns by shaping whether actors see their interactions as a positive- or zero-sum game. We identify ideological diversity within a campaign and the degree to which ideologies embrace the principle of pluralism as key factors affecting the intensity of factional competition and, consequently, the occurrence of terrorism. We introduce new data on the ideologies of campaigns from the Nonviolent and Violent Campaigns and Outcomes 2.0 dataset and use causal mediation analysis to test our proposed mechanism. We find that greater ideological diversity within a campaign increases the likelihood of terrorism by increasing factional competition. We also find that the presence of a pluralist ideology is associated with a lower likelihood of terrorism by the lowering of factional competition. By shedding light on the mechanisms that link ideology to terrorism, this study helps advance our understanding of why dissident groups might decide to use terrorism tactics within the context of a campaign of mass resistance. [ABSTRACT FROM AUTHOR]

In this paper, we give an example of a statement concerning two-player zero-sum games which is undecidable, meaning that it can neither be proven or disproven by the standard axioms of mathematics. Earlier work has shown that there exist "paradoxical" two-player zero-sum games with unbounded payoffs, in which a standard calculation of the two players' expected utilities of a mixed strategy profile yield a positive sum. We show that whether or not a modified version of this paradoxical situation, with bounded payoffs and a weaker measurability requirement, exists is an unanswerable question. Our proof relies on a mixture of techniques from set theory and ergodic theory. [ABSTRACT FROM AUTHOR]

This article develops distributed optimal control policies via Q‐learning for multi‐agent systems (MASs) by solving dual games. According to game theory, first, the distributed consensus problem is formulated as a multi‐player non‐zero‐sum game, where each agent is viewed as a player focusing only on its local performance and the whole MAS achieves Nash equilibrium. Second, for each agent, the anti‐disturbance problem is formulated as a two‐player zero‐sum game, in which the control input and external disturbance are a pair of opponents. Specifically, (1) an offline data‐driven off‐policy for distributed tracking algorithm based on momentum policy gradient (MPG) is developed, which can effectively achieve consensus of MASs with guaranteed l2$$ {l}_2 $$‐bounded synchronization error. (2) An actor‐critic‐disturbance neural network is employed to implement the MPG algorithm and obtain optimal policies. Finally, numerical and practical simulation results are conducted to verify the effectiveness of the developed tracking policies via MPG algorithm. [ABSTRACT FROM AUTHOR]

The increased use of social media has triggered a visual shift in digital political communication. One central strategy is visual (self-) personalization and many studies have analyzed why and to what extent political communication is focused on individuals. Yet, most research has concentrated on personalization, leaving aside the question of whether it comes at the expense of policy issues or whether people are brought into focus as facilitators of such issues. We fill this research gap by analyzing how visual (self-) personalization on Instagram pages of parties and politicians during election campaigns has changed over time and whether it comes at the cost of addressing policy issues. We use data from three quantitative content analyses of Instagram posts published by parliamentary parties (n = 7) and top candidates (n = 18) in the 'hot phases' before three national German elections (N = 2313), covering all nationwide elections since Instagram has become a common campaign tool. It is shown that personal depictions are very common in Instagram campaigns of parties and candidates while addressing policy issues in posts varies between elections. Using a multilevel binomial regression model, we show that addressing policy issues is negatively related to personalization but further analysis shows that it is not a zero-sum game at the cost of addressing policy issues. Our results provide insights into how the presentation of politics might change when image- and video-based platforms continue to gain importance in political communication. [ABSTRACT FROM AUTHOR]

Porous media reconstruction is quite significant in the fields of environmental supervision, oil and natural gas engineering, biomedicine and material engineering. The traditional numerical reconstruction methods such as multi-point statistics (MPS) are based on the statistical characteristics of training images (TIs), but the reconstruction quality may be unsatisfactory and the process is time-consuming. Recently, with the rapid development of deep learning, its powerful ability in predicting features has been used to reconstruct porous media. Generative adversarial network (GAN) is one of the generative methods of deep learning, which is derived from the two-person zero-sum game through the confrontation between the generator and discriminator. However, the traditional GAN cannot pay special attention to the effective features in learning, and the degradation problem easily occurs with the increase of layer numbers. Besides, high-resolution (HR) and large FOV (field of view) are usually contradictory for physical imaging equipment. Therefore, in practical experiments, due to the limitations of the resolution of imaging equipment and the sample size, it is difficult to obtain large-scale HR images of porous media physically. At this point, numerical super-resolution (SR) reconstruction seems to be a cost-efficient way. In this paper, residual networks and attention mechanisms are combined with single-image GAN (SinGAN), which can learn the structural characteristics of porous media from a low-resolution (LR) 3D image, and then reconstruct 3D HR or large-scale images of porous media. Comparison to some other numerical methods has proved that our method can reconstruct high-quality HR images with practicability and effectiveness. [ABSTRACT FROM AUTHOR]

This paper investigates the robust H∞$$ {H}_{\infty } $$ optimal formation control for the position subsystem of quadrotor unmanned aerial vehicles (UAVs) subject to external disturbances and collision constraints. To prevent collision with both members of the formation and external obstacles, a collision avoidance potential function is constructed using relative position and velocity information. The basic bounded control input can ensure the stable flight and collision avoidance of a quadrotor UAV system. Based on the approximate dynamic programming (ADP) framework and two‐player zero‐sum differential game theory, the H∞$$ {H}_{\infty } $$ optimal controller is designed to further enhance the control performance of the system. The optimal value function is approximated by a single layer neural network, which avoids solving complex nonlinear Hamilton‐JacobiIsaac (HJI) equation. The stability of the closed loop system is proved. The effectiveness of the robust H∞$$ {H}_{\infty } $$ optimal formation controller based on learning is validated through MATLAB simulation in two distinct scenarios. [ABSTRACT FROM AUTHOR]

This paper considers an infinitely repeated three-player zero-sum game with two-sided incomplete information, in which an informed player plays two zero-sum games simultaneously at each stage against two uninformed players. This is a generalization of the model in Aumann et al. (Repeated games with incomplete information. MIT Press, New York, 1995) of two-player zero-sum one-sided incomplete information games. Under a correlated prior, the informed player faces the problem of how to optimally disclose information among two uninformed players in order to maximize his long-term average payoffs (i.e., undiscounted payoffs). Our objective is to understand the adverse effects of "information spillover" from one game to the other in the equilibrium payoff set of the informed player. We provide conditions under which the informed player can fully overcome such adverse effects and characterize equilibrium payoffs. In a second result, we show how the effects of information spillover on the equilibrium payoff set of the informed player might be severe. Finally, we compare our findings on the equilibrium-payoff set of the informed player with those of Bayesian Persuasion models with multiple receivers. [ABSTRACT FROM AUTHOR]

The most difficult decision problems arise when several parties with several criteria must reach a consensus. This problem can be modelled as a game with vector-valued payoffs. If the players are allowed to use mixed strategies, there can be many Nash equilibria, and therefore many outcomes. The role of negotiation is to choose a specific outcome, or to restrict the set of outcomes to a small subset. One promising approach to negotiation support is scalarization of the vector payoff function. Here we apply Germeier scalarizing function, also known as the Rawlsian function, to mixed-strategy multicriteria games. After developing the mathematical background, we extend to these games the principle of Best Guaranteed Value, the value that a player may count on regardless of the other players' actions. We suggest that a good outcome for negotiation in a multicriteria game is a Nash equilibrium outcome that provides each player with the payoffs that are better than its Best Guaranteed Value. We describe all such outcomes, thereby defining a new negotiation support mechanism. [ABSTRACT FROM AUTHOR]

This article, titled "Case Material: 'Patricia'", discusses the therapy process with a 38-year-old woman named Patricia who sought help to understand her anger towards others. Patricia had a difficult childhood, feeling unwanted by her mother and experiencing sexual molestation by two teenage boys. Throughout therapy, Patricia exhibited anger, sarcasm, and a tendency to sabotage positive interactions. The therapist reflects on their own experiences and challenges in working with Patricia, including a breast cancer diagnosis. Towards the end of therapy, Patricia began to show progress and acceptance of her traumatic past. The article highlights the complex dynamics and milestones in the therapeutic relationship. [Extracted from the article]

Bilingualism is linked to both enhanced and hampered performance in various cognitive measures, yet the extent to which these bilingual advantages and disadvantages co-occur is unclear. To address this gap, we perform a systematic review and two quantitative analyses. First, we analyze results from 39 studies, obtained through the PRISMA method. Less than 50% of the studies that show up as results for the term "bilingual disadvantage" report exclusively a disadvantage, that shows bilinguals performing worse than monolinguals in a task. A Bayesian analysis reveals robust evidence for bilingual effects, but no evidence for differences in the proportion of advantages and disadvantages, suggesting that when results from different cognitive domains such as executive functions and verbal fluency are analyzed together, bilingual effects amount to a zero-sum game. This finding was replicated by repeating the analysis, using the datasets of two recent meta-analyses. We propose that the equilibrium we observe between positive and negative outcomes may not be accidental. Contrary to widespread belief, advantageous and disadvantageous effects are not stand-alone outcomes in free variation. We reframe them as the connatural components of a dynamic trade-off, whereby enhanced performance in one cognitive measure is offset by an incurred cost in another domain. [ABSTRACT FROM AUTHOR]

To attenuate the effect of disturbances on control performance, a multi‐step adaptive critic control (MsACC) framework is developed to solve zero‐sum games for discrete‐time nonlinear systems. The MsACC algorithm utilizes multi‐step policy evaluation to obtain the solution of the Hamilton–Jacobi–Isaac equation, which is faster than that of the one‐step policy evaluation. The convergence rate of the MsACC algorithm is adjustable by varying the step size of the policy evaluation. In addition, the stability and convergence of the MsACC algorithm are proved under certain conditions. In order to realize the MsACC algorithm, three neural networks are established to approximate the control input, the disturbance input, and the cost function, respectively. Finally, the effectiveness of the MsACC algorithm is verified by two simulation examples, including a linear system and a nonlinear plant. [ABSTRACT FROM AUTHOR]

This paper studies zero-sum splitting games with finite sets of states. Players dynamically choose a pair of martingales { p t , q t } t , in order to control a terminal payoff u (p ∞ , q ∞) . A first part introduces the notion of "Mertens–Zamir transform" of a real-valued matrix and use it to approximate the solution of the Mertens–Zamir system for continuous functions on the square [ 0 , 1 ] 2 . A second part considers the general case of finite splitting games with arbitrary correspondences containing the Dirac mass on the current state: building on Laraki and Renault (Math Oper Res 45:1237–1257, 2020), we show that the value exists by constructing non Markovian ε -optimal strategies and we characterize it as the unique concave-convex function satisfying two new conditions. [ABSTRACT FROM AUTHOR]

This paper rescues Deirdre McCloskey's ideas about Rhetoric in Economics and some issues left out in subsequent discussions. There is a parallel between the ideas of McCloskey and Bruno Latour, and the complementarities of their perspectives are very relevant for the development of Economics. Some reflections are here offered to improve the practice of Economics, which are related to the social constructions of facts in the field, to the internalization of political aspects, and to the importance of alternating between fact and fiction in a dynamic that does not result in a zero-sum game, underlining the characteristic of Economics as being inherently rhetorical. [ABSTRACT FROM AUTHOR]

We consider a finite-horizon two-person zero-sum differential game in which the system dynamics is described by a linear differential equation with a Caputo fractional derivative and the goals of the players' control are to minimize and maximize a quadratic terminal-integral cost function, respectively. We present conditions for the existence of a game value and obtain formulas for players' optimal feedback control strategies with memory of motion history. The results are based on the construction of a solution to an appropriate Hamilton–Jacobi equation with fractional coinvariant derivatives under a natural right-end boundary condition. [ABSTRACT FROM AUTHOR]

An attack–defense model is considered in which the defense party at each point of defense has several lines and uses a target allocation of its forces. The average damage caused by attack forces breaking through all defense points is used as a criterion for attack effectiveness. The problem of optimizing the defense force distribution over defense lines is solved, and optimal strategies of the parties in a zero-sum game are found. [ABSTRACT FROM AUTHOR]

The paper is a continuation of author's research on the existence of an ε-equilibrium in the sense of piecewise program strategies in two-person zero-sum games associated with a nonlinear nonautonomous controlled differential equation in a Banach space and a general cost functional. As in the preceding paper on this subject, the main result consists in sufficient conditions for an ε-equilibrium. The difference is that this time we investigate a game without discriminating the players or fixing a Volterra chain. The results are illustrated as applied to a game associated with a nonlinear pseudoparabolic partial differential equation governing the evolution of an electric field in a semiconductor. [ABSTRACT FROM AUTHOR]

In this article, we present an iterative algorithm based on differential dynamic programming (DDP) for finite‐horizon two‐person zero‐sum differential games. The technique of DDP is used to expand the Hamilton–Jacobi–Isaacs (HJI) partial differential equation into higher‐order differential equations. Using value function and saddle point approximations, the DDP expansion is transformed into algebraic matrix equation in integral form. Based on the algebraic matrix equation, a DDP iterative algorithm is developed to learn the solution to the differential games. Strict proof is proposed to guarantee the iterative convergences of the value function and saddle point. The new algorithm is fundamentally different from existing results, in the sense that it overcome the technical obstacle to address the time‐varying behavior of HJI partial differential equation. Simulation examples are given to demonstrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

We consider a general type of non-Markovian impulse control problems under adverse non-linear expectation or, more specifically, the zero-sum game problem where the adversary player decides the probability measure. We show that the upper and lower value functions satisfy a dynamic programming principle (DPP). We first prove the dynamic programming principle (DPP) for a truncated version of the upper value function in a straightforward manner. Relying on a uniform convergence argument then enables us to show the DPP for the general setting. Following this, we use an approximation based on a combination of truncation and discretization to show that the upper and lower value functions coincide, thus establishing that the game has a value and that the DPP holds for the lower value function as well. Finally, we show that the DPP admits a unique solution and give conditions under which a saddle point for the game exists. As an example, we consider a stochastic differential game (SDG) of impulse versus classical control of path-dependent stochastic differential equations (SDEs). [ABSTRACT FROM AUTHOR]

A singular system, assumed to possess both regularity and freedom from impulses, is categorized as a causal system. Noncausal systems (NSs) are a class of singular systems anticipated to exhibit regularity. This study focuses on investigating zero-sum games (ZSGs) in the context of NSs. We introduce recurrence equations grounded in Bellman's optimality principle. The saddle-point solution for multistage two-player ZSGs can be obtained by solving these recurrence equations. This methodology has demonstrated its effectiveness in addressing two-player ZSGs involving NSs. Analytical expressions that characterize saddle-point solutions for two types of two-player ZSGs featuring NSs, encompassing both linear and quadratic control scenarios, are derived in this paper. To enhance clarity, we provide an illustrative example that effectively highlights the utility of our results. Finally, we apply our methodology to analyze a ZSG in the realm of environmental management, showcasing the versatility of our findings. [ABSTRACT FROM AUTHOR]

The Colonel Blotto game is one of the most classical zero-sum games, with diverse applications in auctions, political elections, etc. We consider the discrete two-battlefield Colonel Blotto Game, a basic case that has not yet been completely characterized. We study three scenarios where at least one player's resources are indivisible (discrete), and compare them with a benchmark scenario where the resources of both players are arbitrarily divisible (continuous). We present the equilibrium values for all three scenarios, and provide a complete equilibrium characterization for the scenario where both players' resources are indivisible. Our main finding is that, somewhat surprisingly, the distinction between continuous and discrete strategy spaces generally has no effect on players' equilibrium values. In some special cases, however, the larger continuous strategy space when resources are divisible does bring the corresponding player a higher equilibrium value than when resources are indivisible, and this effect is more significant for the stronger player who possesses more resources than for the weaker player. [ABSTRACT FROM AUTHOR]

The well-known H ∞ control design gives robustness to a controller by rejecting perturbations from the external environment, which is difficult to do for completely unknown affine nonlinear systems. Accordingly, the immediate objective of this paper is to develop an on-line real-time synergetic learning algorithm, so that a data-driven H ∞ controller can be received. By converting the H ∞ control problem into a two-player zero-sum game, a model-free Hamilton–Jacobi–Isaacs equation (MF-HJIE) is first derived using off-policy reinforcement learning, followed by a proof of equivalence between the MF-HJIE and the conventional HJIE. Next, by applying the temporal difference to the MF-HJIE, a synergetic evolutionary rule with experience replay is designed to learn the optimal value function, the optimal control, and the worst perturbation, that can be performed on-line and in real-time along the system state trajectory. It is proven that the synergistic learning system constructed by the system plant and the evolutionary rule is uniformly ultimately bounded. Finally, simulation results on an F16 aircraft system and a nonlinear system back up the tractability of the proposed method. • A model-free Hamiltonian–Jacobi–Isaacs equation is derived for H ∞ control problems. • A real-time on-line evolutionary rule for tuning neural network weights is developed. • The system and neural network approximation errors achieve synergetic learning. [ABSTRACT FROM AUTHOR]

We examine the Target–Attacker–Defender (TAD) pursuit-evasion game. This is a two team, zero-sum differential game consisting of the aforementioned three agents obeying simple motion. The game terminates at the first time t f when the Attacker achieves point capture of either the Target or Defender. The Attacker chases the Target in such a manner as to minimise the distance between itself and the Target, minus the distance between itself and the Defender at termination time; and the Target and Defender work as a team to achieve the exact opposite goal. This paper introduces and verifies the most elegant characterisation of the state-feedback Nash equilibrium. Whereas previous methods in the literature segregated the state space into at least three separate regions, we introduce a new unifying paradigm given by the Holographic Principle, and conjecture that this principle is more broadly applicable to a large class of simple motion pursuit-evasion games. [ABSTRACT FROM AUTHOR]

Network attack and defence games are gradually becoming a new approach through which to study the protection of infrastructure networks such as power grids and transportation networks. Uncertainty factors, such as the subjective decision preferences of attackers and defenders, are not considered in existing attack and defence game studies for infrastructure networks. In this paper, we introduce, respectively, the attacker's and defender's expectation value, rejection value, and hesitation degree of the target, as well as construct an intuitionistic fuzzy goal-based attack and defence game model for infrastructure networks that are based on the maximum connectivity slice size, which is a network performance index. The intuitionistic fuzzy two-player, zero-sum game model is converted into a linear programming problem for solving, and the results are analysed to verify the applicability and feasibility of the model proposed in this paper. Furthermore, different situations, such as single-round games and multi-round repeated games, are also considered. The experimental results show that, when attacking the network, the attacker rarely attacks the nodes with higher importance in the network, but instead pays more attention to the nodes that are not prominent in the network neutrality and median; meanwhile, the defender is more inclined to protect the more important nodes in the network to ensure the normal performance of the network. [ABSTRACT FROM AUTHOR]

How would human players negotiate a spatial array with a randomly behaving partner? In an interactive, online, zero-sum grid game, we tested whether and how players create areas and place sequences by coloring-in places in a 20 × 20 matrix (261 human adults, 261 machine partners, N = 522). Several indicators showed that human players followed an intentional strategy rather than coloring in places ad libitum. First, human players built a larger largest cluster from places than the machine. Second, they were sensitive to task instructions and could modify their strategy accordingly. When instructed to build an area, they built a bigger one. When set under time pressure, they colored-in faster and only had time to build a smaller cluster from places. Third, the most compelling human strategy to create an area was lining up pathways (place sequencing). Despite these differences, the resulting spatial field structure was similar in human and machine players. [ABSTRACT FROM AUTHOR]

In this paper, a novel parallel learning framework is developed to solve zero-sum games for discrete-time nonlinear systems. Briefly, the purpose of this study is to determine a tentative function according to the prior knowledge of the value iteration (VI) algorithm. The learning process of the parallel controllers can be guided by the tentative function. That is to say, the neighborhood of the optimal cost function can be compressed within a small range via two typical exploration policies. Based on the parallel learning framework, a novel dichotomy VI algorithm is established to accelerate the learning speed. It is shown that the parallel controllers will converge to the optimal policy from contrary initial policies. Finally, two typical systems are used to demonstrate the learning performance of the constructed dichotomy VI algorithm. [ABSTRACT FROM AUTHOR]

To eliminate the requirement of the precise model for model-based control methods, an improved modelfree H∞ control method is designed for batch processes with unknown dynamics and disturbances. Firstly, both the zero-sum game value function and the Q-function are presented as two-dimensional (2D) forms, and their relation is analyzed to obtain the model-free Bellman equation. Secondly, an on policy 2D game Q-learning method is proposed for learning the optimal gains of the designed H∞ controller. On this basis, the behavior control policy and the behavior disturbance policy are individually applied by developing an off-policy 2D game Q-learning method. Subsequently, the strict proof about the convergence and the unbiasedness of the off-policy approach are given. Finally, the simulation results of injection velocity manifest the validity and effectivity of the proposed algorithm. [ABSTRACT FROM AUTHOR]

This article, published in the Journal of Museum Education, explores the concept of engaging critically with and learning from the past, particularly in the context of Holocaust education. It discusses the debates surrounding the singularity of the Holocaust and its relevance in today's society, as well as the tension between the specific horrors of the Holocaust and the universal lessons that can be learned from it. The article also addresses the challenges faced by educators in discussing the Holocaust and other forms of hatred and prejudice in a productive and inclusive manner. Overall, the article emphasizes the importance of open dialogue, questioning assumptions, and engaging with the public in difficult discussions about history and structures of hatred. [Extracted from the article]

The article discusses a case study of a patient named Patricia and explores the emotions and challenges experienced by the analyst in their therapeutic relationship. The analyst reflects on the dynamics of jealousy and envy, the moments of relational satisfaction in analysis, and the difficulties of dealing with hateful transference. The article also touches on the analyst's feelings of disappointment and curiosity, as well as the importance of patience and self-care in the therapeutic process. The author concludes by expressing astonishment and admiration for the transformative potential of analysis. [Extracted from the article]

Static reduction of information structures is a method that is commonly adopted in stochastic control, team theory, and game theory. One approach entails change of measure arguments, which has been crucial for stochastic analysis and has been an effective method for establishing existence and approximation results for optimal policies. Another approach entails utilization of invertibility properties of measurements, with further generalizations of equivalent information structure reductions being possible. In this paper, we demonstrate the limitations of such approaches for a wide class of stochastic dynamic games and teams, and present a systematic classification of static reductions for which both positive and negative results on equivalence properties of equilibrium solutions can be obtained: (i) those that are policy-independent, (ii) those that are policy-dependent, and (iii) a third type that we will refer to as static measurements with control-sharing reduction (where the measurements are static although control actions are shared according to the partially nested information structure). For the first type, we show that there is a bijection between Nash equilibrium policies under the original information structure and their policy-independent static reductions, and establish sufficient conditions under which stationary solutions are also isomorphic between these information structures. For the second type, however, we show that there is generally no isomorphism between Nash equilibrium (or stationary) solutions under the original information structure and their policy-dependent static reductions. Sufficient conditions (on the cost functions and policies) are obtained to establish such an isomorphism relationship between Nash equilibria of dynamic non-zero-sum games and their policy-dependent static reductions. For zero-sum games and teams, these sufficient conditions can be further relaxed. In view of the equivalence between policies for dynamic games and their static reductions, and closed-loop and open-loop policies, we also present three classes of multistage games and teams with partially nested information structures, where we establish connections between closed-loop, open-loop, and control-sharing Nash and saddle point equilibria. By taking into account a playerwise concept of equilibrium, we introduce two further classes of "playerwise" static reductions: (i) independent data reduction under which the policy-independent reduction holds through players and time, and (ii) playerwise (partially) nested independent reduction under which measurements are independent through players but (partially) nested through time for each player. [ABSTRACT FROM AUTHOR]

Since the seminal PPAD-completeness result for computing a Nash equilibrium even in two-player games, an important line of research has focused on relaxations achievable in polynomial time. In this paper, we consider the notion of an\varepsilon -well-supported Nash equilibrium, where \varepsilon \in [0, 1] corresponds to the approximation guarantee. Put simply, in an \varepsilon -well-supported equilibrium, every player chooses with positive probability actions that are within \varepsilon of the maximum achievable payoff against the other player's strategy. Ever since the initial approximation guarantee of 2/3 for wellsupported equilibria, which was established more than a decade ago, the progress on this problem has been extremely slow and incremental. Notably, the small improvements to 0.6608, and finally to 0.6528, were achieved by algorithms of growing complexity. Our main result is a simple and intuitive algorithm that improves the approximation guarantee to 1/2. Our algorithm is based on linear programming and in particular on exploiting suitably defined zero-sum games that arise from the payoff matrices of the two players. As a byproduct, we show how to achieve the same approximation guarantee in a query-efficient way. [ABSTRACT FROM AUTHOR]

Summary: In this article, a novel event‐triggered zero‐sum differential game through value iteration (VI)‐based adaptive fuzzy control is proposed for modular robot manipulators (MRMs) with uncertain disturbance. The subsystem dynamic of MRMs is established by joint torque feedback (JTF) technology. An identifier based on fuzzy logic model (FLM) is established to identify unknown system dynamics. Then, the trajectory tracking control issue of MRMs with uncertain disturbance is transformed into a two‐player zero‐sum differential game. On the basis of event‐triggered mechanism and VI algorithm, the event‐triggered Hamilton–Jacobi–Issacs (HJI) equation is approximately solved through establishment of actor‐critic structure. Different from the traditional adaptive dynamic programming (ADP) algorithm, this scheme is based on adaptive fuzzy optimal control through an actor‐critic structure, which refers to adaptation of both optimal cost function and optimal control policy. Moreover, tracking errors of closed‐loop MRM system are demonstrated to be uniformly ultimately bounded (UUB) under Lyapunov stability theorem. At last, experimental examples and comparisons show the reliability and effectiveness of this method. [ABSTRACT FROM AUTHOR]

In this paper, we present two-timescale neurodynamic optimization approaches to distributed minimax optimization. We propose four multilayer recurrent neural networks for solving four different types of generally nonlinear convex–concave minimax problems subject to linear equality and nonlinear inequality constraints. We derive sufficient conditions to guarantee the stability and optimality of the neural networks. We demonstrate the viability and efficiency of the proposed neural networks in two specific paradigms for Nash-equilibrium seeking in a zero-sum game and distributed constrained nonlinear optimization. [ABSTRACT FROM AUTHOR]

This paper deals with a robust H∞$$ {H}_{\infty } $$ tracking problem with a discounted factor. A new auxiliary system is established in terms of norm‐bounded time‐varying uncertainties. It is shown that the robust discounted H∞$$ {H}_{\infty } $$ tracking problem for the auxiliary system solves the original problem. Then, the new robust discounted H∞$$ {H}_{\infty } $$ tracking problem is represented as a well‐known zero‐sum game problem. Moreover, the robust tracking Bellman equation and the robust tracking Algebraic Riccati equation (RTARE) are inferred. A lower bound of a discounted factor for stability is obtained to assure the stability of the closed‐loop system. Based on the auxiliary system, the system is reshaped in a new structure that is applicable to Reinforcement Learning methods. Finally, an online Q‐learning algorithm without the knowledge of system matrices is proposed to solve the algebraic Riccati equation associated with the robust discounted H∞$$ {H}_{\infty } $$ tracking problem for the auxiliary system. Simulation results are given to verify the effectiveness and merits of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, a novel adaptive critic control method is designed to solve an optimal H ∞ tracking control problem for continuous nonlinear systems with nonzero equilibrium based on adaptive dynamic programming (ADP). To guarantee the finiteness of a cost function, traditional methods generally assume that the controlled system has a zero equilibrium point, which is not true in practical systems. In order to overcome such obstacle and realize H ∞ optimal tracking control, this paper proposes a novel cost function design with respect to disturbance, tracking error and the derivative of tracking error. Based on the designed cost function, the H ∞ control problem is formulated as two-player zero-sum differential games, and then a policy iteration (PI) algorithm is proposed to solve the corresponding Hamilton–Jacobi–Isaacs (HJI) equation. In order to obtain the online solution to the HJI equation, a single-critic neural network structure based on PI algorithm is established to learn the optimal control policy and the worst-case disturbance law. It is worth mentioning that the proposed adaptive critic control method can simplify the controller design process when the equilibrium of the systems is not zero. Finally, simulations are conducted to evaluate the tracking performance of the proposed control methods. [ABSTRACT FROM AUTHOR]

10.3390/math10081212 8 Ibragimov G., Kuchkarova S., Hasim R.M., Pansera B.A. Differential Game for an Infinite System of Two-Block Differential Equations. References 1 Ibragimov G., Kazimirova R., Pansera B.A. Evasion Differential Game of Multiple Pursuers and One Evader for an Infinite System of Binary Differential Equations. The study of differential games has practical applications in the analysis and resolution of conflict issues by the use of differential equations. [Extracted from the article]

Witnessing the pro-Western shifts in the post-Soviet countries, Russia has failed to develop policies that would attract the former Soviet satellites without keeping them in its sphere of influence via hard-power. On the other hand, Western values, political influence and institutions have penetrated Eurasian countries causing Russia's relative decline. This structural dynamic not only entraps Russia in a zero-sum game with the West but also triggers Russian threat perceptions on the unit level. Russian strategic culture emphasizes anxieties about loss of sovereignty or power shifts in the Russian regime that could mean the end of the current establishment, border security, and great power status resurrection. A combination of these threat perceptions and structural shifts manifests in the Thucydides Trap, an ancient dilemma behind the preventive war, in the post-Soviet space. [ABSTRACT FROM AUTHOR]

This paper takes issue with the widespread claim that positional competitions are zero-sum games. It shows how the notions of ‘positional good’ and ‘positional competition’ have changed in meaning and how this has resulted in conceptual confusion in discussions amongst economists and philosophers. I argue that the Zero-Sum Claim is hardly ever true when it comes to the novel understanding of positionality that currently dominates the philosophical literature. I propose dropping the Zero-Sum Claim and construing positional competitions as win-lose. This is conceptually clearer and deepens our understanding and ethical evaluation of these important competitions in contemporary societies. [ABSTRACT FROM AUTHOR]

We consider two-player zero-sum differential games of fixed duration, where the running payoff and the dynamics are both linear in the controls of the players. Such games have a value, which is determined by the unique viscosity solution of a Hamilton–Jacobi-type partial differential equation. Approximation schemes for computing the viscosity solution of Hamilton–Jacobi-type partial differential equations have been proposed that are valid in a more general setting, and such schemes can of course be applied to the problem at hand. However, such approximation schemes have a heavy computational burden. We introduce a discretized and probabilistic version of the differential game, which is straightforward to solve by backward induction, and prove that the solution of the discrete game converges to the viscosity solution of the partial differential equation, as the discretization becomes finer. The method removes part of the computational burden of existing approximation schemes. [ABSTRACT FROM AUTHOR]