Optimal Mixed Strategy Research Articles

Formulation of Game Model as a Linear Programming Problem using Various Models

Game theory is the examination of strategic interactions between two or more individuals known as players who act based on their individual self-interest within a framework known as the game. Every player possesses a set of possible actions referred to as strategies, from which they make selections. In a two-person zero sum game, each of the two players has at least two strategies. In such a game problem where both players have no inferior strategies, we can determine the optimal mixed strategies of the game problem by converting it to a linear programming problem and solving it using the simplex method or variations of it. In this paper, consideration of some existing models along with our proposed model on the conversion of game problem to LPP was made. We compared the results across the various models considered. The results obtained revealed that our proposed model on the conversion of game problem to LPP produced a higher value of the game compared to the others considered; and thus, produced better performance.

DECISION THEORETIC BOOTSTRAPPING

The design and testing of supervised machine learning models combine two fundamental distributions: (1) the training data distribution and (2) the testing data distribution. Although these two distributions are identical and identifiable when the data set is infinite, they are imperfectly known when the data are finite (and possibly corrupted), and this uncertainty must be taken into account for robust uncertainty quantification (UQ). An important case is when the test distribution is coming from a modal or localized area of the finite sample distribution. We present a general decision theoretic bootstrapping solution to this problem: (1) partition the available data into a training subset and a UQ subset; (2) take m subsampled subsets of the training set and train m models; (3) partition the UQ set into n sorted subsets and take a random fraction of them to define <i>n</i> corresponding empirical distributions μ<sub>j</sub>; (4) consider the adversarial game where Player I selects a model i ∈ {1,.....,m}, Player II selects the UQ distribution μ<sub>j</sub>, and Player I receives a loss defined by evaluating the model <i>i</i> against data points sampled from μ<sub>j</sub>; (5) identify optimal mixed strategies (probability distributions over models and UQ distributions) for both players. These randomized optimal mixed strategies provide optimal model mixtures, and UQ estimates given the adversarial uncertainty of the training and testing distributions represented by the game. The proposed approach provides (1) some degree of robustness to in-sample distribution localization/concentration and (2) conditional probability distributions on the output space forming aleatory representations of the uncertainty on the output as a function of the input variable.

On the Existence of Stationary Nash Equilibria for Mean Payoff Games on Graphs

In this paper we extend the classical concept of positional strategies for a mean payoff game to a general mixed stationary strategy approach, and prove the existence of mixed stationary Nash equilibria for an arbitrary m-player mean payoff game on graphs. Traditionally, a positional strategy represents a pure stationary strategy in a classical mean payoff game, where a Nash equilibrium in pure stationary strategies in general may not exist. Based on a constructive proof of the existence of specific equilibria for an m-player mean payoff game we propose a new approach for determining the optimal mixed stationary strategies. Additionally we characterize and extend the general problem of the existence of pure stationary Nash equilibria for some special classes of mean payoff games.

Learning-Based Multi-Domain Anti-Jamming Communication with Unknown Information

Due to the open nature of the wireless channel, wireless networks are vulnerable to jamming attacks. In this paper, we try to solve the anti-jamming problem caused by smart jammers, which can adaptively adjust the jamming channel and the jamming power. The interaction between the legitimate transmitter and the jammers is modeled as a non-zero-sum game. Considering that it is challenging for the transmitter and the jammers to acquire each other’s information, we propose two anti-jamming communication schemes based on the Deep Q-Network (DQN) algorithm and hierarchical learning (HL) algorithm to solve the non-zero-sum game. Specifically, the DQN-based scheme aims to solve the anti-jamming strategies in the frequency domain and the power domain directly, while the HL-based scheme tries to find the optimal mixed strategies for the Nash equilibrium. Simulation results are presented to validate the effectiveness of the proposed schemes. It is shown that the HL-based scheme has a better convergence performance and the DQN-based scheme has a higher converged utility of the transmitter. In the case of a single jammer, the DQN-based scheme achieves 80% of the transmitter’s utility of the no-jamming case, while the HL-based scheme achieves 63%.

Optimal pure strategies for a discrete search game

Consider a two-person zero-sum search game between a Hider and a Searcher. The Hider chooses to hide in one of n discrete locations (or “boxes”) and the Searcher chooses a search sequence specifying which order to look in these boxes until finding the Hider. A search at box i takes ti time units and finds the Hider—if hidden there—independently with probability qi, for i=1,…,n. The Searcher wants to minimize the expected total time needed to find the Hider, while the Hider wants to maximize it. It is shown in the literature that the Searcher has an optimal search strategy that mixes up to n distinct search sequences with appropriate probabilities. This paper investigates the existence of optimal pure strategies for the Searcher—a single deterministic search sequence that achieves the optimal expected total search time regardless of where the Hider hides. We identify several cases in which the Searcher has an optimal pure strategy, and several cases in which such optimal pure strategy does not exist. An optimal pure search strategy has significant practical value because the Searcher does not need to randomize their actions and will avoid second guessing themselves if the chosen search sequence from an optimal mixed strategy does not turn out well.

Learning-Based Secure Control for Multichannel Networked Systems Under Smart Attacks

This paper is concerned with the security control of a class of discrete-time linear cyber-physical systems (CPSs) subject to denial-of-service (DoS) attacks. To enhance the inherent resistance of the CPS against damage and attacks, a multi-channel network is employed for remote information interaction between the ingredients of the system. In this way, a complex interaction process will be formed between the signal sender and the smart malicious adversary. Specifically, in the context of using the multi-channel network, the malicious adversary has to maximize the attack success probability within its energy constraint. As a counterpart, the system tries to mitigate the negative impact of such attacks on CPS control performance. For the purpose of designing a control strategy to cope with the attacks, this interaction process is formulated as a zero-sum stochastic game, while the Nash equilibrium solution of this problem is found with the help of the proposed learning algorithm, and the optimal mixed strategies for both attackers and defenders are derived. Further, for the CPS driven by the obtained decision-making strategies, a Kalman filter-based active dynamic output feedback resilient controller is proposed. Finally, the effectiveness of the developed optimal defense strategies and the resilient controller is demonstrated by extensive case studies on the servo motor experimental platform.

Autonomous Maneuver Decision for Unmanned Aerial Vehicle via Improved Pigeon-Inspired Optimization

As a crucial technology of air-to-air confrontation, autonomous maneuver decision has attracted wide attention in recent years. This paper proposes an improved pigeon-inspired optimization method to realize autonomous maneuver decision for unmanned aerial vehicles (UAVs) rapidly and accurately in an aerial combat engagement. The maneuver library is designed, including some advanced offensive and defensive maneuvers. A dependent set of trial maneuvers is generated to help UAVs make decisions in any tactical situation, and a future engagement state of the opponent UAV is predicted for each trial maneuver. The core of the decision-making process is that the objective function to be optimized is designed using the game mixed strategy, and the optimal mixed strategy is obtained by the improved pigeon-inspired optimization. A comparative analysis with other classical optimization algorithms highlights the advantage of the proposed algorithm. The simulation tests are conducted under four different initial conditions, namely, neutral, offensive, opposite, and defensive conditions. The simulation results verify the effectiveness of the proposed autonomous maneuver decision method.

A Note on Linear Complementarity via Two-Person Zero-Sum Games

The matrix [Formula: see text] of a linear complementarity problem can be viewed as a payoff matrix of a two-person zero-sum game. Lemke’s algorithm can be successfully applied to reach a complementary solution or infeasibility when the game satisfies the following conditions: (i) Value of [Formula: see text] is equal to zero. (ii) For all principal minors of [Formula: see text] (transpose of [Formula: see text]) value is non-negative. (iii) For any optimal mixed strategy [Formula: see text] of the maximizer either [Formula: see text] or [Formula: see text] for each coordinate [Formula: see text].

A Classical Search Game in Discrete Locations

Consider a two-person zero-sum search game between a hider and a searcher. The hider hides among n discrete locations, and the searcher successively visits individual locations until finding the hider. Known to both players, a search at location i takes ti time units and detects the hider—if hidden there—independently with probability [Formula: see text] for [Formula: see text]. The hider aims to maximize the expected time until detection, whereas the searcher aims to minimize it. We prove the existence of an optimal strategy for each player. In particular, any optimal mixed hiding strategy hides in each location with a nonzero probability, and there exists an optimal mixed search strategy that can be constructed with up to n simple search sequences. Funding: This work was supported by the Engineering and Physical Sciences Research Council [Grant EP/L015692/1] STOR-i Centre for Doctoral Training.

Dynamical intervention planning against COVID-19-like epidemics.

COVID-19 has got us to face a new situation where, for the lack of ready-to-use vaccines, it is necessary to support vaccination with complex non-pharmaceutical strategies. In this paper, we provide a novel Mixed Integer Nonlinear Programming formulation for fine-grained optimal intervention planning (i.e., at the level of the single day) against newborn epidemics like COVID-19, where a modified SIR model accounting for heterogeneous population classes, social distancing and several types of vaccines (each with its efficacy and delayed effects), allows us to plan an optimal mixed strategy (both pharmaceutical and non-pharmaceutical) that takes into account both the vaccine availability in limited batches at selected time instants and the need for second doses while keeping hospitalizations and intensive care occupancy below a threshold and requiring that new infections die out at the end of the planning horizon. In order to show the effectiveness of the proposed formulation, we analyze a case study for Italy with realistic parameters.

Decision-making method based on game theory and grey theory in a single-value neutrosophic environment and its application to typhoon disaster assessment

PurposeThe Chinese believe that “man will conquer the sky” and “fighting with the sky brings endless joy”. Considering that disaster assessment can be regarded as a two-person, zero-sum game problem between nature and human beings, this paper proposes a multi-attribute decision-making method based on game theory and grey theory in a single-value neutrosophic set environment. Due to the complexity and uncertainty of the decision-making environment, the method builds a decision matrix based on single-valued neutrosophic numbers.Design/methodology/approachFirst, the authors use the single-value neutrosophic information entropy to calculate the attribute weights and the weighted decision matrix. Second, the optimal mixed strategy method based on linear programming solves the optimal mixed strategy for both sides of the game so that the expected payoff matrix can be obtained. Finally, grey correlation analysis is used to obtain the closeness coefficient of each alternative based on the expectation payoff matrix to identify the ranking result of the alternative.FindingsAn example is used to verify the effectiveness of the proposed method, and its rationality is verified through a comprehensive comparison and analysis of the various aspects.Practical implicationsThe proposed decision-making method can be applied to typhoon disaster assessment. Such assessment results can provide intelligent decision support to the relevant disaster management departments, thereby reducing the negative impact of typhoon disasters on society, stabilizing society and improving people's happiness. Further, the method can be used for decision-making, recommendation and evaluation in other fields.Originality/valueThe proposed method uses single-value neutrosophic numbers to solve the information representation problem of decision-making in a complex environment. Under a new perspective, game theory is used to handle the decision matrix, while grey relational analysis converts inexact numbers to exact numbers for comparison and sorting. Thus, the proposed method can be used to make reasonable decisions while preserving information to the extent possible.

A q-rung orthopair fuzzy non-cooperative game method for competitive strategy group decision-making problems based on a hybrid dynamic experts\u2019 weight determining model

How to select the optimal strategy to compete with rivals is one of the hottest issues in the multi-attribute decision-making (MADM) field. However, most of MADM methods not only neglect the characteristics of competitors’ behaviors but also just obtain a simple strategy ranking result cannot reflect the feasibility of each strategy. To overcome these drawbacks, a two-person non-cooperative matrix game method based on a hybrid dynamic expert weight determination model is proposed for coping with intricate competitive strategy group decision-making problems within q-rung orthopair fuzzy environment. At the beginning, a novel dynamic expert weight calculation model, considering objective individual and subjective evaluation information simultaneously, is devised by integrating the superiorities of a credibility analysis scale and a Hausdorff distance measure for q-rung orthopair fuzzy sets (q-ROFSs). The expert weights obtained by the above model can vary with subjective evaluation information provided by experts, which are closer to the actual practices. Subsequently, a two-person non-cooperative fuzzy matrix game is formulated to determine the optimal mixed strategies for competitors, which can present the specific feasibility and divergence degree of each competitive strategy and be less impacted by the number of strategies. Finally, an illustrative example, several comparative analyses and sensitivity analyses are conducted to validate the reasonability and effectiveness of the proposed approach. The experimental results demonstrate that the proposed approach as a CSGDM method with high efficiency, low computation complexity and little calculation burden.

Dynamic Information Design: A Simple Problem on Optimal Sequential Information Disclosure

We study a dynamic information design problem in a finite-horizon setting consisting of two strategic and long-term optimizing agents, namely a principal (he) and a detector (she). The principal observes the evolution of a Markov chain that has two states, one “good” and one “bad” absorbing state, and has to decide how to sequentially disclose information to the detector. The detector’s only information consists of the messages she receives from the principal. The detector’s objective is to detect as accurately as possible the time of the jump from the good to the bad state. The principal’s objective is to delay the detector as much as possible from detecting the jump to the bad state. For this setting, we determine the optimal strategies of the principal and the detector. The detector’s optimal strategy is described by time-varying thresholds on her posterior belief of the good state. We prove that it is optimal for the principal to give no information to the detector before a time threshold, run a mixed strategy to confuse the detector at the threshold time, and reveal the true state afterward. We present an algorithm that determines both the optimal time threshold and the optimal mixed strategy that could be employed by the principal. We show, through numerical experiments, that this optimal sequential mechanism outperforms any other information disclosure strategy presented in the literature. We also show that our results can be extended to the infinite-horizon problem, to the problem where the matrix of transition probabilities of the Markov chain is time-varying, and to the case where the Markov chain has more than two states and one of the states is absorbing.

Strategies for Vaccine Prioritization and Mass Dispensing.

We propose a system that helps decision makers during a pandemic find, in real time, the mass vaccination strategies that best utilize limited medical resources to achieve fast containments and population protection. Our general-purpose framework integrates into a single computational platform a multi-purpose compartmental disease propagation model, a human behavior network, a resource logistics model, and a stochastic queueing model for vaccination operations. We apply the modeling framework to the current COVID-19 pandemic and derive an optimal trigger for switching from a prioritized vaccination strategy to a non-prioritized strategy so as to minimize the overall attack rate and mortality rate. When vaccine supply is limited, such a mixed vaccination strategy is broadly effective. Our analysis suggests that delays in vaccine supply and inefficiencies in vaccination delivery can substantially impede the containment effort. Employing an optimal mixed strategy can significantly reduce the attack and mortality rates. The more infectious the virus, the earlier it helps to open the vaccine to the public. As vaccine efficacy decreases, the attack and mortality rates rapidly increase by multiples; this highlights the importance of early vaccination to reduce spreading as quickly as possible to lower the chances for further mutations to evolve and to reduce the excessive healthcare burden. To maximize the protective effect of available vaccines, of equal importance are determining the optimal mixed strategy and implementing effective on-the-ground dispensing. The optimal mixed strategy is quite robust against variations in model parameters and can be implemented readily in practice. Studies with our holistic modeling framework strongly support the urgent need for early vaccination in combating the COVID-19 pandemic. Our framework permits rapid custom modeling in practice. Additionally, it is generalizable for different types of infectious disease outbreaks, whereby a user may determine for a given type the effects of different interventions including the optimal switch trigger.

Random Actions in Experimental Zero-Sum Games

A mixed strategy, a strategy of unpredictable actions, is applicable to business, politics, and sports. Playing mixed strategies, however, poses a challenge, as the game theory involves calculating probabilities and executing random actions. I test i.i.d. hypotheses of the mixed strategy Nash equilibrium with the simplest experiments in which student participants play zero-sum games in multiple iterations and possibly figure out the optimal mixed strategy (equilibrium) through the games. My results confirm that most players behave differently from the Nash equilibrium prediction for the simplest 2x2 zero-sum game (matching-pennies) and 3x3 zero-sum game (e.g., the rock-paper-scissors game). The results indicate the need to further develop theoretical models that explain a non-Nash equilibrium behavior.

A Class of Optimal Switching Mixed Data Injection Attack in Cyber-Physical Systems

This letter considers a class of switching mixed data injection attacks using input derivatives in cyber-physical systems with linear quadratic(LQ) cost from the perspective of the attacker. The attacker injects data mixed with false data and its derivative into a healthy system, which destroys the performance of the original system. For this situations, the optimal mixed data injection attack strategy is designed to minimize the quadratic cost and most damage the performance of the system. After that, in order to increase the complexity, concealment and energy saving of the attack, we designed the optimal switching mixed data injection attack strategy. Finally, numerical results and comparative experiments are provided to illustrate the effectiveness of the proposed method.

Protecting the Internet of Vehicles Against Advanced Persistent Threats: A Bayesian Stackelberg Game

Connected vehicles are essential for the deployment of intelligent transportation services. However, the high level of connectivity in today's Internet of vehicles (IoV) and the extreme reliance on the data collected from the smart transportation infrastructure widen the space of security vulnerabilities, making the IoV a potential target for cyberattacks. This article investigates novel sophisticated ways to exploit the IoV and launch intelligent attacks on road traffic services by creating persistent impact and reducing detection chances. This article models the processes of attack and defense as a cybersecurity Stackelberg game leading to optimal mixed strategies for both the attackers and the IoV defense system, where the latter optimally deploys the available security resources within the transportation infrastructure to minimize the impact of attacks and improve their detection. The game is of Bayesian type and considers several types of data corruption attacks that occur according to a probability distribution that we determine based on a rigorous risk assessment approach. The results show that our game model and solution allow us to reduce the impact of advanced persistent threats compared to a uniform defense design that is indifferent to attackers' strategies and types. The solution could be integrated into the design of IoV intrusion detection systems to increase their robustness.

Using Statistical Estimates in the Game with Nature as an Investment Model

Purpose of the study. The aim of the research is to develop new principles of decision making (principles of optimality) in games with nature and their application to analyze statistical data and choose strategies for stock investment.Materials and methods. We analyze Russian and foreign bibliography on the research problem. A model of decision making in a game with nature with known state probabilities is proposed. The mathematical expectation of the player's payoff is taken as an assessment of efficiency, and the standard deviation or variance is taken as a risk assessment. This two-criterion task is formalized by transferring the efficiency assessment into a constraint. As a result, for the case of mixed strategies, a nonlinear (quadratic) task of mathematical programming arises. To solve it, an approach based on the Lagrange function and the Karush-Kuhn-Tucker optimality conditions is used. As an application of the methods obtained, the problems of stock investment are considered.Results. Analytical methods for solving the indicated optimization problem and an algorithm for finding optimal mixed strategies are obtained. Practical examples of application of the proposed approach on real statistical data are given. As the initial data in this study, we used stock quotes of Russian companies in the electric power industry for the period from 01.07.2020 to 01.10.2020, taken from the website of the FINAM Investment Company. The developed method allows one to find the optimal strategy and the corresponding values of profitability and risk based on only the initial data (statistical characteristics of financial instruments and the threshold value of profitability), i.e. provides, in our opinion, a convenient analysis tool for the investor.Conclusion. The concept of the principle of optimality in decision making problems under conditions of incomplete information is very ambiguous. The decision maker should be able to choose from a range of decision making models that reflect the dependence of the type of rational behavior on the available information and the attitude to risk. The paper proposes a model of this type for the case of probabilistic uncertainty, which leads to the problem of minimizing variance as a risk assessment with a lower bound on the mathematical expectation as an assessment of efficiency.

Renewable Energy Investment under Carbon Emission Regulations

This paper considers a risk-neutral energy supplier who operates an electricity plant in an uncertain demand market. To characterize the impact of carbon emission quota and emergency supply cost of coal electricity, we first consider four different cases, (1) traditional energy without carbon emission restriction; (2) traditional energy with carbon emission quota; (3) mixed energy (both traditional and renewable energy) without carbon emission restriction, and (4) mixed energy with carbon emission quota, to find the optimal renewable energy investment level and coal inventory mechanism for an energy supplier. Then, through the analysis we derive the resulting equilibriums: coal inventory for electricity generation and the investment of renewable energy capacity. By comparing the performances under different scenarios, we find that (a) renewable energy establishment can mitigate the depression of carbon emission constraint, (b) the energy supplier can obtain positive benefits from optimal mixed energy strategy if the additional emergency cost of traditional energy is not too high, and (c) the optimal renewable energy capacity level is decreasing in the carbon emission quota. Our numerical simulations imply that the impact of additional emergency cost to the profit difference between mixed energy strategy and traditional energy strategy is mediated by carbon emission quota.

A Cybersecurity Insurance Model for Power System Reliability Considering Optimal Defense Resource Allocation

With the increasing application of Information and Communication Technologies (ICTs), cyberattacks have become more prevalent against Cyber-Physical Systems (CPSs) such as the modern power grids. Various methods have been proposed to model the cybersecurity threats, but so far limited studies have been focused on the defensive strategies subject to the limited security budget. In this paper, the power supply reliability is evaluated considering the strategic allocation of defense resources. Specifically, the optimal mixed strategies are formulated by the Stackelberg Security Game (SSG) to allocate the defense resources on multiple targets subject to cyberattacks. The cyberattacks against the intrusion-tolerant Supervisory Control and Data Acquisition (SCADA) system are mathematically modeled by Semi-Markov Process (SMP) kernel. The intrusion tolerance capability of the SCADA system provides buffered residence time before the substation failure to enhance the network robustness against cyberattacks. Case studies of the cyberattack scenarios are carried out to demonstrate the intrusion tolerance capability. Depending on the defense resource allocation scheme, the intrusion-tolerant SCADA system possesses varying degrees of self-healing capability to restore to the good state and prevent the substations from failure. If more defense resources are invested on the substations, the intrusion tolerant capability can be further enhanced for protecting the substations. Finally, the actuarial insurance principle is designed to estimate transmission companies’ individual premiums considering correlated cybersecurity risks. The proposed insurance premium principle is designed to provide incentive for investments on enhancing the intrusion tolerance capability, which is verified by the results of case studies.