Non-zero Sum Research Articles

Adaptive event‐triggered safety control for multiplayer mixed zero‐sum game with partial inputs delay

AbstractThis paper proposes an adaptive safety control method applicable to a multiplayer mixed zero‐sum (MZS) game of nonlinear systems with partial inputs delay. Firstly, a framework is introduced involving N players, where player 1 and player N form a zero‐sum (ZS) game, and player 1 and players 2 to N‐1 form nonzero‐sum (NZS) games, with some players experiencing time delays. Subsequently, the system's value function is augmented with a control barrier function (CBF) to ensure that the system's state remains within a safe region. Secondly, to approximate Nash equilibrium solutions, the study employs adaptive dynamic programming (ADP) and utilizes a critic‐only neural network (NN) to approximate optimal solutions. Diverging from traditional time‐trigger methods, computational and communication load reduction is achieved by introducing a state‐related event trigger condition. The stability of the system is then meticulously analyzed using the Lyapunov theorem. Finally, to validate the effectiveness of the proposed method, the study provides a simulation example demonstrating its performance. In summary, this research introduces an efficient adaptive safety control method for addressing multiplayer MZS games with partial inputs delay, incorporating CBFs, ADP, and state‐related event triggering.

Sums of squares in function fields over henselian discretely valued fields

Let n∈N and let K be a field with a henselian discrete valuation of rank n with hereditarily euclidean residue field. Let F/K be a function field in one variable. It is known that every sum of squares is a sum of 3 squares. We determine the order of the group of nonzero sums of 3 squares modulo sums of 2 squares in F in terms of equivalence classes of certain discrete valuations of F of rank at most n. In the case of function fields of hyperelliptic curves of genus g, K.J. Becher and J. Van Geel showed that the order of this quotient group is bounded by 2n(g+1). We show that this bound is optimal. Moreover, in the case where n=1, we show that if F/K is a hyperelliptic function field such that the order of this quotient group is 2g+1, then F is nonreal.

Integral reinforcement learning-based event-triggered optimal tracking control for modular robot manipulators via non-zero-sum game

Abstract Under an event-triggered mechanism, a non-zero-sum (NZS) game optimal tracking control method for modular robot manipulator (MRM) systems with input constraints is proposed using the adaptive dynamic programming (ADP) method based on integral reinforcement learning (IRL). First, a dynamic model of the MRM system is developed based on joint torque feedback technology, consisting of an n-joint subsystem related to interconnected dynamic coupling (IDC). Second, we design a robust compensation controller to handle the known model term and an optimal compensation controller to deal with the uncertainty term caused by the IDC and friction, respectively. In addition, a nonlinear disturbance observer is established to dispose of the negative effects caused by the uncertain sensor output disturbance. Third, based on differential game theory, we transform the optimal tracking control problem of the MRM system into an n-player NZS game problem. Then, the IRL-based ADP method is adopted, which relaxes the need for system partial unknown dynamic information, and only a critic neural network is used to solve the coupled Hamilton–Jacobi equation, so as to obtain the optimal control policy. Then, using Lyapunov theory, the tracking error of the MRM system is demonstrated to be uniformly ultimately bounded. Finally, the effectiveness and superiority of the proposed algorithm are verified through experiments.

Game-Theoretic Unlearnable Example Generator

Unlearnable example attacks are data poisoning attacks aiming to degrade the clean test accuracy of deep learning by adding imperceptible perturbations to the training samples, which can be formulated as a bi-level optimization problem. However, directly solving this optimization problem is intractable for deep neural networks. In this paper, we investigate unlearnable example attacks from a game-theoretic perspective, by formulating the attack as a nonzero sum Stackelberg game. First, the existence of game equilibria is proved under the normal setting and the adversarial training setting. It is shown that the game equilibrium gives the most powerful poison attack in that the victim has the lowest test accuracy among all networks within the same hypothesis space when certain loss functions are used. Second, we propose a novel attack method, called the Game Unlearnable Example (GUE), which has three main gradients. (1) The poisons are obtained by directly solving the equilibrium of the Stackelberg game with a first-order algorithm. (2) We employ an autoencoder-like generative network model as the poison attacker. (3) A novel payoff function is introduced to evaluate the performance of the poison. Comprehensive experiments demonstrate that GUE can effectively poison the model in various scenarios. Furthermore, the GUE still works by using a relatively small percentage of the training data to train the generator, and the poison generator can generalize to unseen data well. Our implementation code can be found at https://github.com/hong-xian/gue.

Rapid changes in synchronizability in conductance-based neuronal networks with conductance-based coupling.

Real neurons connect to each other non-randomly. These connectivity graphs can potentially impact the ability of networks to synchronize, along with the dynamics of neurons and the dynamics of their connections. How the connectivity of networks of conductance-based neuron models like the classical Hodgkin-Huxley model or the Morris-Lecar model impacts synchronizability remains unknown. One powerful tool to resolve the synchronizability of these networks is the master stability function (MSF). Here, we apply and extend the MSF approach to networks of Morris-Lecar neurons with conductance-based coupling to determine under which parameters and for which graphs the synchronous solutions are stable. We consider connectivity graphs with a constant non-zero row sum, where the MSF approach can be readily extended to conductance-based synapses rather than the more well-studied diffusive connectivity case, which primarily applies to gap junction connectivity. In this formulation, the synchronous solution is a single, self-coupled, or "autaptic" neuron. We find that the primary determining parameter for the stability of the synchronous solution is, unsurprisingly, the reversal potential, as it largely dictates the excitatory/inhibitory potential of a synaptic connection. However, the change between "excitatory" and "inhibitory" synapses is rapid, with only a few millivolts separating stability and instability of the synchronous state for most graphs. We also find that for specific coupling strengths (as measured by the global synaptic conductance), islands of synchronizability in the MSF can emerge for inhibitory connectivity. We verified the stability of these islands by direct simulation of pairs of neurons coupled with eigenvalues in the matching spectrum.

QESRL: Exploring Selfish Reinforcement Learning for Repeated Quantum Games

A novel unified learning algorithm that seamlessly applies to both classical and quantum non-zero sum games is presented. Building upon the exploring selfish reinforcement learning (ESRL) framework previously proposed in the context of classical games, we extend this approach to handle quantum games with imperfect information. A comparison is made between performance and fairness among agents learning using plain QRL vs. QESRL. The latter enables agents to explore and learn periodic policy strategies in quantum games, leveraging the quantization of games to uncover fairer results. By addressing the challenges posed by the expanded strategy space in quantum games, we test the algorithm’s scalability by increasing the number of agents. Empirical evidence is provided to showcase its performance and to compare classical and quantum game scenarios. The proposed learning algorithm represents a significant step towards understanding the convergence and optimality of strategies in non-zero sum games across classical and quantum settings, bringing us closer to harnessing the potential of independent reinforcement learning and quantum computing in game theory applications.

A class of balanced binary sequences with two-valued non-zero autocorrelation sum and good crosscorrelation sum

A class of balanced binary sequences with two-valued non-zero autocorrelation sum and good crosscorrelation sum

Nonzero-Sum Game-Based Voltage Recovery Consensus Optimal Control for Nonlinear Microgrids System.

Since most of the existing models based on the microgrids (MGs) are nonlinear, which could cause the controller oscillate, resulting in the excessive line loss, and the nonlinear could also lead to the controller design difficulty of MGs system. Therefore, this article researches the distributed voltage recovery consensus optimal control problem for the nonlinear MGs system with N -distributed generations (DGs), in the case of providing stringent real power sharing. First, based on the distributed cooperative control concept of multiagent systems and the critic neural networks (NNs), a novel distributed secondary voltage recovery consensus optimal control protocol is constructed via applying the backstepping technique and nonzero-sum (NZS) differential game strategy to realize the voltage recovery of island MGs. Meanwhile, the model identifier is established to reconstruct the unknown NZS games systems based on a three-layer NN. Then, a critic NN weight adaptive adjustment tuning law is proposed to ensure the convergence of the cost functions and the stability of the closed-loop system. Furthermore, according to Lyapunov stability theory, it is proven that all signals are uniform ultimate boundedness in the closed loop system and the voltage recovery synchronization error converges to an arbitrarily small neighborhood of the origin near. Finally, some simulation results in MATLAB illustrate the validity of the proposed control strategy.

Thermodynamic relationships between time, capital, and labour

In the theoretical achievements of economics, the past has left many insinuations, not always accurate terms and also controversies. This does not burden the scholars whose diligent efforts fell on the immature ground of general knowledge, in particular the lack of thermodynamics. The categories of capital, profit, labour are basic examples. There was also a lack of understanding of the nature of time. This issue was as if left to physicists, despite the fact that the use of present and future value was and is a significant achievement in economics and accounting. On the other hand, among those engaged in physics, there are divergences about time travel. The great idea of the economy as a game with nature with a non-zero positive sum has also not spread in economic theory, reinforcing the aspiration to belong to the natural sciences. Therefore, we take up the consideration of the capital-labour-time triad as an important basic element of economic knowledge. In our view, this correctly understood triad will correctly illuminate many economic issues and can form the hard core of a scientific program for improving economic and accounting theory. Thus, the epistemology of I. Lakatos defines the methodology of ongoing research, in which positive heuristics and empirical verification are the engine of scientific action.

Non-Zero Sum Nash Game for Discrete-Time Infinite Markov Jump Stochastic Systems with Applications

This paper is to study finite horizon linear quadratic (LQ) non-zero sum Nash game for discrete-time infinite Markov jump stochastic systems (IMJSSs). Based on the theory of stochastic analysis, a countably infinite set of coupled generalized algebraic Riccati equations are solved and a necessary and sufficient condition for the existence of Nash equilibrium points is obtained. From a new perspective, the finite horizon mixed robust H2/H∞ control is investigated, and summarize the relationship between Nash game and H2/H∞ control problem. Moreover, the feasibility and validity of the proposed method has been proved by applying it to a numerical example.

Fully cooperative games with state and input constraints using reinforcement learning based on control barrier functions

AbstractThis paper provides a novel safe reinforcement learning (RL) control algorithm to solve safe optimal problems for fully cooperative (FC) games of discrete‐time multiplayer nonlinear systems with state and input constraints. The FC game is a special case of nonzero‐sum (NZS) games, where all players cooperate to accomplish a common task. The algorithm is proposed based on the policy iteration (PI) framework utilizing only the measured data along the system trajectories in the environment. Different from most works about PI, an effective method of obtaining initial safe and stable control policies is given here. In addition, control barrier functions (CBFs) and an input constraint function are introduced to augment reward functions. And the monotonically nonincreasing property of the iterative value function in the PI algorithm maintains the safe set forward invariant. Then, the neural networks are employed to approximate the system dynamics, the iterative control policies, and the iterative value function, respectively. Furthermore, the proposed algorithm is supported by theoretical proofs that guarantee both safety and convergence. Finally, the effectiveness and safety of the algorithm are illustrated by the results of the simulation.

An eavesdropping and jamming dilemma with sophisticated players

Wireless networks are susceptible to malicious attacks, especially those involving eavesdropping and jamming. In this paper, we consider a communication scenario involving a transmitter who wishes to communicate secretly and reliably with a receiver, while an adversary wants to obstruct this communication by means of either eavesdropping or jamming. The transmitter as well as the adversary wants to achieve its own goal in a manner that is as unpredictable as possible to its rival. We model this problem by a non-zero sum game. The expected throughput that is delivered secretly and non-jammed to the receiver is considered as the metric that reflects communication secrecy and reliability. The entropy of the player’s strategies is considered as the metric to reflect the player’s unpredictability. The equilibrium is found in closed form, and parameters of the transmitter’s utility supporting both goals are optimized via a proportional fairness approach.

Compact quantum group structures on type-I $\mathrm{C}^*$-algebras

We prove a number of results having to do with equipping type-I \mathrm{C}^* -algebras with compact quantum group structures, the two main ones being that such a compact quantum group is necessarily co-amenable, and that if the \mathrm{C}^* -algebra in question is an extension of a non-zero finite direct sum of elementary \mathrm{C}^* -algebras by a commutative unital \mathrm{C}^* -algebra then it must be finite-dimensional.

Building Bridges Instead of Putting Up Walls: Connecting the “Teams” to Improve Soccer Players’ Support

The increase in the economic value of soccer occurred in parallel with an increase in competing demands. Therefore, clubs and federations evolved to greater specialization (e.g., state-of-the-art facilities and high-profile expertise staff) to support players’ performance and health. Currently, player preparation is far from exclusively club or national team centered, and the lack of control in each player’s environment can be more prevalent than expected. For example, an elite group of professional players faces disruptions in the season club-oriented planification due to involvement in national teams. Moreover, as elite players’ financial resources grow, it is common for them to employ specialized personal staff (e.g., strength and conditioning, nutritionist, and sports psychologist) to assist in their preparation, resulting in complex three-fold relationships (i.e., club, player’s staff, national team). Although efforts have been made to improve communication with and transition from the club to the national team supervision, this new reality (club-players’ staff) may generate serious compound role-related problems and difficulties in monitoring load and training adaptation and having a unified message. Therefore, efforts must be implemented to ensure a more informed management of the players’ performance environment, where the existence and impact of these various personal staff are considered to avoid a long-term non-zero sum for all intervening parties. If left unchecked, current professional thinking may collide or overlap, potentially triggering conflict escalation and impairing athletic performance or health, especially if effective communication routes are not adequately established. Moreover, diluted personal responsibility regarding performance may ensue, resulting in decreased productivity from all involved, which may cause more harm than benefits for the player’s overall health and performance. This emerging reality calls for developing a joint working framework (i.e., between the player’s personalized support team and the clubs’ team) and better managing of a player-centered process.

Best-Response Dynamics for Evolutionary Stochastic Games

We introduce a version of best-response (BR) dynamics for stochastic games using the framework of evolutionary stochastic games proposed in [Flesch, J., Parthasarathy, T., Thuijsman, F. and Uyttendaele, P. [2013] Evolutionary stochastic games, Dyn. Games Appl. 3, 207–219]. This dynamics is defined on the simplex of all population distributions, as in classical evolutionary games. In this BR dynamics, it is shown that the set of all population distributions inducing Nash equilibria is globally asymptotically stable, for stable stochastic games. Also, if we transform the BR dynamics into the space of stationary strategies, the set of all Nash Equilibria is asymptotically stable. Another version of BR dynamics is defined in the space of stationary strategies for zero-sum stochastic games by [Leslie, D. S., Perkins, S. and Xu, Z. [2020] Best-response dynamics in zero-sum stochastic games, J. Econ. Theory 189, 105095]. To compare both versions of BR dynamics, we extend the evolutionary stochastic games model to asymmetric stochastic games and employ an equivalent definition of Nash equilibria. This comparison is illustrated using examples and it is observed that the trajectories of both versions of BR dynamics do converge to the same Nash equilibrium. The BR dynamics proposed in this paper has the advantage of being valid for nonzero sum stochastic games and conforms to both classical game dynamics and the dynamics of stationary strategies in stochastic games.

A methodology for solving bimatrix games under 2-tuple linguistic environment

In a game problem, when the situation is formulated as the gain of one equals the loss of other, the game is called zero-sum. But actually, one player's win is not always the other's loss. Such a game is referred to as non-zero sum or bimatrix game. The work provided in this research is primarily elaborated on linguistic bimatrix games under 2-tuple linguistic environment. This class of bimatrix games consists of payoffs having entries as linguistic 2-tuples, that are based on a finite subscript-symmetric additive linguistic set defined in the domain , where g is a positive integer. In this direction, a novel linguistic equilibrium concept is defined to solve such games, and a methodology based on a nonlinear linguistic programme is proposed to compute mixed Nash equilibrium. The proposed methodology is sufficient to evaluate the Nash equilibrium and the maximum expected payoffs corresponding to each player simultaneously in a single programme. The necessary and sufficient conditions for the existence of Nash equilibrium and a computation procedure are also provided. Finally, some hypothetical working examples are given to show the practical applicability of the defined methodology. The proposed methodology is also applied to solve two-player constant sum linguistic games.

The [formula omitted]-vortex problem in a doubly periodic rectangular domain with constant background vorticity

We formulate the N point vortex problem in a doubly-periodic rectangle with a constant background vorticity using the hydrodynamic Green’s function. We derive an explicit formula for the hydrodynamic Green’s function and the velocities of the point vortices in terms of the Schottky–Klein prime function using a conformal mapping approach. The sum of vortex strengths can be arbitrary, and when it is zero we recover previous known results including the integrability of the two and three-vortex problems. A non-zero sum of vortex strengths leads to a constant background vorticity. We derive a Hamiltonian structure for the equations and show that the two-vortex problem is integrable, and classify all possible vortex motions. In addition, for general N, we obtain several fixed lattice equilibrium configurations including single-layered and double-layered equilibria. In the latter case, we also obtain lattice configurations with defects consisting of point vortices with inhomogeneous strengths. We find equilibria arranged in doubly-periodic rectangular and parallelogram lattices consisting of N=1,2,3,4 vortices per fundamental lattice.

Linear-Quadratic Non-Zero Sum Backward Stochastic Differential Game With Overlapping Information

This article studies a linear quadratic non-zero sum stochastic differential game with overlapping information, where the state dynamics are described by a backward stochastic differential equation and the information obtained by two players has a common part but no inclusion relation. The open-loop Nash equilibrium strategy is given by some conditional mean-field stochastic differential equations. In addition, coupled Riccati equations are introduced to express the state feedback form of the Nash equilibrium strategy.

Feature Selection Using Games with Imperfect Information (FSGIN)

Game Theory (GT) is the study of strategic decision making. By virtue of its importance, several GT based methodologies for Feature Selection (FS) are proposed in recent times. FS problem can be abstracted as a game by considering each feature as a player and their values as their strategies. Additionally, overall goal of the game is set to classify a data instance appropriately. Most of the existing GT based FS techniques are restricted to Zero Sum Games, Non-Zero Sum Games and Cooperative Games. The classical setting of assuming that all the details of players are known to all players cannot hold in many real-world problems. When the given features are independent, they cannot be treated alike and a characteristics based uncertainty persists among the features. This uncertainty is handled by none of the game forms used in the existing methods. Unlike the mentioned game techniques, Bayesian Games (BG) address the games with imperfect information. This paper investigate the FS problem in terms of BG and proposes a novel method to select the best features. The proposed BG based FS method is a filter type FS method and it starts with identifying Principle Features (PF) and proceeds to play global pairwise Bayesian games between those PF to obtain feature scores. Later, the features are ranked using these scores. In the final stage, a forward selection method with Support Vector Machine (SVM) is used to evaluate the classification performance of the ranked features and helps in the selection of the optimal set of features. Besides these, to improve the scalability of the proposed method MapReduce paradigm is exploited. In order to show the efficacy of the proposed method, experiments are carried out with seven real-world datasets from UCI and Statlog repositories. The promising results showed a significant improvement in the classification performance with fewer selected features than which is achieved using the existing methods.

Tight globally simple nonzero sum Heffter arrays and biembeddings

AbstractSquare relative nonzero sum Heffter arrays, denoted by , have been introduced as a variant of the classical concept of Heffter array. An is an partially filled array with elements in , where , whose rows and whose columns contain filled cells, such that the sum of the elements in every row and column is different from 0 (modulo ) and, for every not belonging to the subgroup of order , either or appears in the array. In this paper we give direct constructions of square nonzero sum Heffter arrays with no empty cells, , for every odd, when is a divisor of and when . The constructed arrays have also the very restrictive property of being “globally simple”; this allows us to get new orthogonal path decompositions and new biembeddings of complete multipartite graphs.