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In this paper, we propose a game theoretic framework for secure physical layer authentication (PLA) by applying the spatial correlations of received signal strength (RSS) in the air-to-ground channels between an unmanned aerial vehicle (UAV), a legitimate transmitter, and a spoofer. Based on our hypothesis model, we first derive the false alarm rate and miss detection rate for the UAV receiver. We then formulate a zero-sum PLA game where the UAV receiver selects its RSS detection threshold while the spoofer selects its attack probability to maximize their respective utilities. We derive the Nash equilibrium (NE) and its existence condition for the proposed PLA game. Monte Carlo simulation results accurately verify our analytical expressions for the false alarm rate and miss detection rate. Numerical results highlight the complex interplay between the detection threshold, spoofing attack probability and UAV altitude in the derived NE. [ABSTRACT FROM AUTHOR]

The digital curling game is a two-player zero-sum extensive game in a continuous action space. There are some challenging problems that are still not solved well, such as the uncertainty of strategy, the large game tree searching, and the use of large amounts of supervised data, etc. In this work, we combine NFSP and KR-UCT for digital curling games, where NFSP uses two adversary learning networks and can automatically produce supervised data, and KR-UCT can be used for large game tree searching in continuous action space. We propose two reward mechanisms to make reinforcement learning converge quickly. Experimental results validate the proposed method, and show the strategy model can reach the Nash equilibrium. [ABSTRACT FROM AUTHOR]

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In this paper, we consider the two-player state and control path-dependent stochastic zero-sum differential game. In our problem setup, the state process, which is controlled by the players, is dependent on (current and past) paths of state and control processes of the players. Furthermore, the running cost of the objective functional depends on both state and control paths of the players. We use the notion of non-anticipative strategies to define lower and upper value functionals of the game, where unlike the existing literature, these value functions are dependent on the initial states and control paths of the players. In the first main result of this paper, we prove that the (lower and upper) value functionals satisfy the dynamic programming principle (DPP), for which unlike the existing literature, the Skorohod metric is necessary to maintain the separability of càdlàg (state and control) spaces. We introduce the lower and upper Hamilton–Jacobi–Isaacs (HJI) equations from the DPP, which correspond to the state and control path-dependent nonlinear second-order partial differential equations. In the second main result of this paper, we show that by using the functional Itô calculus, the lower and upper value functionals are viscosity solutions of (lower and upper) state and control path-dependent HJI equations, where the notion of viscosity solutions is defined on a compact κ -Hölder space to use several important estimates and to guarantee the existence of minimum and maximum points between the (lower and upper) value functionals and the test functions. Based on these two main results, we also show that the Isaacs condition and the uniqueness of viscosity solutions imply the existence of the game value. Finally, we prove the uniqueness of classical solutions for the (state path-dependent) HJI equations in the state path-dependent case, where its proof requires establishing an equivalent classical solution structure as well as an appropriate contradiction argument. [ABSTRACT FROM AUTHOR]

We apply dynamic programming principle to discuss two optimal investment problems by using zero-sum and nonzero-sum stochastic game approaches in a continuous-time Markov regime-switching environment within the frame work of behavioral finance. We represent different states of an economy and, consequently, investors' floating levels of psychological reactions by a D-state Markov chain. The first application is a zero-sum game between an investor and the market, and the second one formulates a nonzero-sum stochastic differential portfolio game as the sensitivity of two investors' terminal gains. We derive regime-switching Hamilton–Jacobi–Bellman–Isaacs equations and obtain explicit optimal portfolio strategies with Feynman–Kac representations of value functions. We illustrate our results in a two-state special case and observe the impact of regime switches by comparative results. [ABSTRACT FROM AUTHOR]

Network densification is recognized as the key technology to meet the ever growing demand of data traffic in next generation wireless networks. However the proliferation of small cell base stations (SCBSs) introduces vulnerabilities to information security, as they are prone to eavesdropping attacks. In this paper, it is studied how user association strategies can be specifically tailored to meet this security challenge in a ultra dense heterogeneous cellular network(UDHCN) with a group of colluding eavesdroppers. In particular, the situation is considered where eavesdroppers may have limited capabilities in intercepting and decoding data traffic. In this setting each eavesdropper has to make its own decision on eavesdropping targets, and they are able to take concerted actions to sabotage information security. To address this challenge, a zero sum game framework to reflect the conflicting interests of users and eavesdroppers is proposed and a user association scheme is devised that aims to maximize system sum secrecy rate against these adversaries whose actions intend to minimize sum secrecy rate. The case that all adversaries have limited eavesdropping capabilities is first considered, and it is shown that the corresponding zero sum game is indeed a bilinear game, and its Nash equilibrium solution can be readily approximated using a saddle point Frank Wolfe(SP‐FW)algorithm. Then the framework is extended to the case where adversaries with a diverse configuration of eavesdropping capabilities exist. In this case, it is shown that the underlying minimax optimization problem is indeed a nonsmooth convex‐nonconcave one. A two stage method is proposed by first smoothing the objective function with a Hermite cubic polynomial approximation, and then obtaining a nearly stationary solution via a recently proposed proximal point gradient descent method. Simulation results show that the proposed framework leads to significant increases in sum secrecy rate and secrecy probability against both the capability‐limited adversaries and a hybrid type of adversaries. [ABSTRACT FROM AUTHOR]

In this paper, we consider a zero-sum differential game that can be stopped by any of the participants at one of time points known in advance. The cost functional may depend both on the time when the game ends and the position of the system at that time and on the player who initiates the game stopping. When optimizing the expectation of this functional, each player, based on the information the player has about the realized trajectory of the system, makes decisions both about his/her probability of stopping the game and about his/her own control of the conflict-controlled system; however, nondeterministic distribution rules are also allowed. The existence of the game value is shown under the assumption of a saddle point condition in the small game (the Isaacs condition). The construction of a stochastic guide whose model is a continuous-time Markov chain was used to construct approximately optimal strategies for the players. [ABSTRACT FROM AUTHOR]

There is an abundance of perspectives and interpretations about the practices and actions of the Russian Federation on states and individuals, especially at the information level. However, in some respects, Western perspectives tend to simplify Russian actions and categorise them as zero-sum games. In this respect, the paper tries to interpret the actions of the Russian Federation, especially at the informational level, by defining key concepts, as well as to provide a perspective on how Russian military theorists use the notion of "hostile actions" in the context of designing peacetime operations. Moreover, the complexity of the Russian military theoretical background is highlighted, with brief comparisons between the theoretical forms of some concepts from the Soviet period and the present. The paper also reviews some conceptions of the hybrid form of Russian action, attempts to synthesise certain views of Russian military theorists and, based on Russian strategic culture, to show the levels of action in the context of a series of operations. Finally, the paper proposes an orientation model that addresses a holistic perspective on how the subversive and informational actions of the Russian Federation can influence the policy and security of a state targeting the individual, society and the political system. [ABSTRACT FROM AUTHOR]

This study proposes a novel way to deal with uncertainty in a two-person zero-sum matrix game with payoffs expressed as fuzzy rough numbers. Complete and reasonable solutions to these types of games are obtained. In this research we develop two linear programming models with upper and lower approximation intervals of fuzzy rough numbers and handle multi-objective crisp linear programming models by incorporating trapezoidal fuzzy rough numbers as payoffs. To provide each opponent with the optimal strategy and value of the game, the usual simplex approach is applied. Finally, two numerical examples demonstrate the matrix game outcomes using Wolfram Cloud. [ABSTRACT FROM AUTHOR]

Editorials, ischemia, mortality, myocardial ischemia, coronary angiography Keywords: Editorials; coronary angiography; ischemia; mortality; myocardial ischemia EN Editorials coronary angiography ischemia mortality myocardial ischemia 20 22 3 12/29/22 20230103 NES 230103 B Article, see p 8 b According to the ancient Greeks, the lives of the gods and mortals were immutably determined by 3 Fates: sisters Clotho, who spins the thread of life; Lachesis, who assigns each their destiny; and Atropos, whose scissors snip the thread of life at its end. Analyses from ISCHEMIA,[10] and COURAGE before it,[11] have demonstrated a time-dependent effect of an invasive strategy on patient-reported health status, most notably symptom burden among patients with more severe anginal symptoms at baseline. [Extracted from the article]

We introduce and study the class of semidefinite games, which generalizes bimatrix games and finite N-person games, by replacing the simplex of the mixed strategies for each player by a slice of the positive semidefinite cone in the space of real symmetric matrices. For semidefinite two-player zero-sum games, we show that the optimal strategies can be computed by semidefinite programming. Furthermore, we show that two-player semidefinite zero-sum games are almost equivalent to semidefinite programming, generalizing Dantzig’s result on the almost equivalence of bimatrix games and linear programming. For general two-player semidefinite games, we prove a spectrahedral characterization of the Nash equilibria. Moreover, we give constructions of semidefinite games with many Nash equilibria. In particular, we give a construction of semidefinite games whose number of connected components of Nash equilibria exceeds the long standing best known construction for many Nash equilibria in bimatrix games, which was presented by von Stengel in 1999. [ABSTRACT FROM AUTHOR]

The infinite-horizon zero-sum game of a linear system can be resorted to solve a Game algebraic Riccati equation (GARE) with indefinite quadratic term. Double-loop policy iteration algorithm is often used to find the solution of such GARE, but its calculation is usually time-consuming. In this work, we propose a novel model-based single-loop policy iteration algorithm to solve GARE and the convergence of the algorithm is guaranteed by the boundness of the iterative sequence and the comparison result. Furthermore, we devise a data-driven single-loop policy iteration algorithm for solving linear zero-sum games, without requiring the knowledge of system dynamics. Compared to the existing Newton's single-loop methods, the initialization of our algorithms is significantly relaxed and easier to implement. Two numerical examples are included to illustrate the proposed algorithms. [ABSTRACT FROM AUTHOR]

• A data-driven waterlogging risk mapping method was proposed based on long-tail learning and game theory. • A novel F (α , β) score was defined to both consider precision and recall and trade-off equity and efficiency in public emergency resource allocation. • A framework for evaluating and enhancing urban waterlogging risk mitigation capabilities was proposed, targeting nonprofit operations and government bailouts. Because disaster reduction resources are always constrained, risk mitigation measures and policies need to be weighted toward efficiency or equity. Greater efficiency focuses on high-risk areas, and greater equity focuses on identifying additional potential risk areas. To address this efficiency-equity tradeoff decision, a data-driven urban waterlogging risk mitigation capability evaluation and improving framework was proposed for nonprofit operations and government bailouts. First, a novel evaluation index was defined, the F (α , β) score, to consider both precision and recall in the binary classification tasks and to seek an effective efficiency-equity trade-off, after which a two-person, zero-sum matrix game model was developed to calculate the α and β parameters and select the highest valued optimal model. Then, using this model, a risk probability map was generated for Wuhan, China, and data envelopment analysis was used to conduct a risk mitigation capability evaluation and identify the needed policy improvements. It was found that: (1) vulnerability was the primary risk-related factor, followed by adaptability and restorability; (2) more importance should be attached to equity than efficiency when implementing risk reduction capability evaluations, that is, authorities should reduce resource investments in extremely high-risk DEA-inefficient areas and identify a greater number of potential risk areas; (3) there were around 1.55% of extremely high-risk areas, which were mainly concentrated in the Jianghan and Wuchang districts and around 9% of high-risk areas, mainly located in the Hongshan district; and (4) northeastern and southwestern Jianghan district could be a valuable reference for urban waterlogging management and disaster reduction. However, excessive investment in transportation infrastructure and construction in DEA inefficient areas should be appropriately reduced. [ABSTRACT FROM AUTHOR]

In this paper, the zero-sum game problem for linear discrete-time Markov jump systems is solved by two novel model-free reinforcement Q -learning algorithms, on-policy Q -learning and off-policy Q -learning. Firstly, under the framework of the zero-sum game, the game-coupled algebraic Riccati equation is derived. On this basis, subsystem transformation technology is employed to decouple the jumping modes. Then, a model-free on-policy Q -learning algorithm is introduced in the zero-sum game architecture to obtain the optimal control gain by measured system data. However, the probing noise will produce biases in on-policy algorithm. Thus, an off-policy Q -learning algorithm is proposed to eliminate the effect of probing noise. Subsequently, convergence is discussed for the proposed methods. Finally, an inverted pendulum system is employed to verify the validity of the proposed methods. • The zero-sum game problem for linear discrete-time Markov jump systems is addressed with the reinforcement learning method. • A novel model-free parallel off-policy Q-learning method is developed, which is not affected by probing noise. • The subsystem transformation technology is employed decoupling the jumping modes to solve the game-coupled algebraic Riccati equation. • The convergence of the proposed methods, on-policy and off-policy, for Markov jump systems are both discussed in this paper. [ABSTRACT FROM AUTHOR]

Model-free learning for multi-agent stochastic games is an active area of research. Existing reinforcement learning algorithms, however, are often restricted to zero-sum games and are applicable only in small state-action spaces or other simplified settings. Here, we develop a new data-efficient Deep-Q-learning methodology for model-free learning of Nash equilibria for general-sum stochastic games. The algorithm uses a locally linear-quadratic expansion of the stochastic game, which leads to analytically solvable optimal actions. The expansion is parametrized by deep neural networks to give it sufficient flexibility to learn the environment without the need to experience all state-action pairs. We study symmetry properties of the algorithm stemming from label-invariant stochastic games and as a proof of concept, apply our algorithm to learning optimal trading strategies in competitive electronic markets. [ABSTRACT FROM AUTHOR]

We consider a model of robust learning in an adversarial environment. The learner gets uncorrupted training data with access to possible corruptions that may be affected by the adversary during testing. The learner's goal is to build a robust classifier, which will be tested on future adversarial examples. The adversary is limited to k possible corruptions for each input. We model the learner-adversary interaction as a zero-sum game. This model is closely related to the adversarial examples model of Schmidt et al. (2018); Madry et al. (2017). Our main results consist of generalization bounds for the binary and multiclass classification, as well as the real-valued case (regression). For the binary classification setting, we both tighten the generalization bound of Feige et al. (2015), and are also able to handle infinite hypothesis classes. The sample complexity is improved from ... for any α>0. Additionally, we extend the algorithm and generalization bound from the binary to the multiclass and real-valued cases. Along the way, we obtain results on fat-shattering dimension and Rademacher complexity of k-fold maxima over function classes; these may be of independent interest. For binary classification, the algorithm of Feige et al. (2015) uses a regret minimization algorithm and an ERM oracle as a black box; we adapt it for the multiclass and regression settings. The algorithm provides us with near-optimal policies for the players on a given training sample. [ABSTRACT FROM AUTHOR]

Consider a scenario where a player chooses an action in each round t out of T rounds and observes the incurred cost after a delay of dt rounds. The cost functions and the delay sequence are chosen by an adversary. We show that in a non-cooperative game, the expected weighted ergodic distribution of play converges to the set of coarse correlated equilibria if players use algorithms that have "no weighted-regret" in the above scenario, even if they have linear regret due to too large delays. For a two-player zero-sum game, we show that no weighted-regret is sufficient for the weighted ergodic average of play to converge to the set of Nash equilibria. We prove that the FKM algorithm with n dimensions achieves an expected regret of O - nT 3 4 + p nT 1 3D 13 - and the EXP3 algorithm with K arms achieves an expected regret of O -p logK (KT + D) - even when D = PT t=1 dt and T are unknown. These bounds use a novel doubling trick that, under mild assumptions, provably retains the regret bound for when D and T are known. Using these bounds, we show that FKM and EXP3 have no weighted-regret even for dt = O (t log t). Therefore, algorithms with no weighted-regret can be used to approximate a CCE of a finite or convex unknown game that can only be simulated with bandit feedback, even if the simulation involves significant delays. [ABSTRACT FROM AUTHOR]

The problem of the existence and determining equilibria in pure stationary strategies for a two-player zero-sum average stochastic positional game is considered. We show that for such a game there exists the value and players may achieve the value by applying pure stationary strategies of choosing the actions in their positions. Based on a constructive proof of these results we propose an algorithmic approach for determining the optimal pure stationary strategies of the players. [ABSTRACT FROM AUTHOR]

Purpose: This paper aims to present a simple behavioural explanation of the prohibition of speculation in Islamic finance. Design/methodology/approach: This paper proposes a theoretical model that describes how investors from low income strata of the society may be prone to make sub-optimal decisions when they compare their outcome from a speculative trading activity to that of the counterparty to the trade and perceive inequity to exist. Findings: When individuals from low income strata of the society compare their current situation with the average income of the society, they perceive themselves to be in a loss. This creates a loss frame within which they then evaluate all future outcomes. When such individuals invest in speculative trading activities and incur a loss, they compare their outcome from the trade to that of the counterparty to the trade. As speculative trades are a zero sum game, the counterparty makes an equivalent gain from the trade. Thus, the comparison leads to a perception of inequity. This perception of inequity is aggravated by the loss frame within which the investor is operating. The aggravated inequity aversion may then motivate the investor to make further sub-optimal decisions like repeated speculative trading activities. The Islamic prohibition on speculative trading activities may serve to protect low income investors from entering into such cycles of sub-optimal decisions. Originality/value: This paper offers a unique explanation of why day trading and short selling may be prohibited in Islamic capital markets. [ABSTRACT FROM AUTHOR]

The article focuses on the complexity of the concept of color, which has been studied by artists, philosophers, and scholars for centuries. It mentions the despite efforts to categorize color, it remains a challenging and subjective phenomenon that requires a holistic understanding. It suggests exploring the Congo and the African continent from the perspective of color and analysis as a way to challenge stereotypes and broaden knowledge.

Unmanned combat has stepped into the stage of modern war, and multi unmanned ground vehicle (MUGV) cooperative combat will become the main style of land combat in the future. The evaluation of system effectiveness is the core problem of equipment demonstration and combat method research. A set of exploratory simulation analysis method based on multi agent deep reinforcement learning is established to evaluate the combat effectiveness of MUGV system. The MUGV confrontation process is modeled as a zero sum stochastic game (ZSG) model. By using multi-agent deep reinforcement learning MADRL to explore the Nash equilibrium solution of ZSG model under the condition of different scale of enemy unmanned ground vehicle, and analyzing the constraints such as the winning rate and the duration of combat under the condition of Nash equilibrium, the combat effectiveness evaluation of MUGV system is completed. Finally, the application example of operational effectiveness evaluation of MUGV system is given. Thus, a more reliable and available system performance evaluation method is established. [ABSTRACT FROM AUTHOR]

Through the theoretical lenses of cultural gaming and paradigmatic assumptions of Eastern and Western cultures, this article looks into the interrelationships between game intentions and communication strategies. Through a thematic analysis of the Chinese traditional cultural classics from the pre-Qin (221 BCE–206 BCE) to the Republic of China (1912–1949) from the Chinese Text Project, three featured themes of water including the best and the softest, carrying boats and capsizing boats, and the most violent and the strongest have emerged in correspondence to the positive zero sum, zero sum, and negative zero sum game intentions. Thus, a water and game theory for intercultural communication has been constructed. The theory is applicable to managing intercultural communication barriers, esp. systematic misunderstanding between individuals from China and the West, between Chinese enterprises and their counterparts in other countries, and between China and other countries on various intercultural communication occasions. [ABSTRACT FROM AUTHOR]

The European Union is bound to respect the national identities of member states. States might, on occasion, define their national identities in ways that breach inclusive constitutional values (e.g. equality) protected under TEU Article 2. The assumption behind the recognition of diverse constitutional identities is the peaceful coexistence of both, which is challenged by illiberal national developments. We see the relationship between the constitutional recognition of exclusive values (e.g. dominant ethnicity or religion) and inclusive constitutional values as a zero-sum game; every gain by the proponents of emergent authoritarianism translates to a loss for constitutional democracy. While exclusive norms appear in virtually every constitutional system, a critical mass of exclusive values can lead to the hollowing out of a democratic order, both at the national and supranational levels. To try to identify the line where this shift happens, we rely on the limits of toleration and recognition of exclusive norms and identity elements of minority communities in liberal theories of multiculturalism (e.g. Raz, Taylor, Kymlicka). We think that the case of illiberal minorities raises structurally similar theoretical questions, insights and experiences as the dilemma defined above, namely the challenge of illiberal states undermining fundamental EU values. [ABSTRACT FROM AUTHOR]

Rough interval (RI) is an appropriate generalization of the crisp interval and is very much useful to express uncertain parameters or partially unknown variables when they contain dual‐layer information. In real‐life problems, there are many situations where players of a matrix game cannot assess their pay‐offs by using fuzzy sets/intuitionistic fuzzy sets or ordinary intervals. RIs are used as an excellent tool to handle such situations. This paper explores matrix games with RI pay‐offs and investigates two different solution methodologies to solve such a game. In the first approach, a pair of auxiliary linear programming problems with RI coefficients for the players has been constructed. Then each of the two auxiliary programming problems is converted into two linear programming problems with interval coefficients (LPPICs) using lower and upper approximation intervals of the RI. Finally from each LPPIC, two classical linear programming problems (LPPs) are constructed. In the second approach, the expected value technique for RI is used for transforming auxiliary mathematical programming models under the RI environment to crisp LPP. A case study on the telecom market share problem is considered to show the applicability of the proposed approaches and results are compared and analyzed with an existing method. [ABSTRACT FROM AUTHOR]

The editorial from Critical Criminology discusses the global events of the Middle East conflict and the social harms of imperialism and colonialism. It highlights the need to use a critical criminological lens to understand the socio-historical factors behind these conflicts. The editorial also mentions the consequences of structural violence in Ecuador and the formation of militarized parallel states by organized crime. The journal aims to encourage a new imagination shaped by radical social praxis and features articles on various topics such as queer victimology, female incarceration, green criminology, crime mapping, youth homelessness, and biological criminology. The editorial also pays tribute to John Hagedorn, a prominent figure in critical gang studies. The editorial concludes with a reminder to submit proposals for the next ASC Conference and expresses gratitude to the book editor and graduate researcher. [Extracted from the article]

A nonnegative tensor I A i is I hyperstochastic i if Graph HT ht for all I i i , I j i . Let Graph HT ht be a I k i -tensor. For Graph HT ht and Graph HT ht , we denote by Graph HT ht the sum of the I i i th slice in the I j i th direction. [Extracted from the article]

In this paper, zero-sum mean-field type games (ZSMFTG) with linear dynamics and quadratic cost are studied under infinite-horizon discounted utility function. ZSMFTG are a class of games in which two decision makers whose utilities sum to zero, compete to influence a large population of indistinguishable agents. In particular, the case in which the transition and utility functions depend on the state, the action of the controllers, and the mean of the state and the actions, is investigated. The optimality conditions of the game are analysed for both open-loop and closed-loop controls, and explicit expressions for the Nash equilibrium strategies are derived. Moreover, two policy optimization methods that rely on policy gradient are proposed for both model-based and sample-based frameworks. In the model-based case, the gradients are computed exactly using the model, whereas they are estimated using Monte-Carlo simulations in the sample-based case. Numerical experiments are conducted to show the convergence of the utility function as well as the two players' controls. [ABSTRACT FROM AUTHOR]

It is possible to endorse virtuous leadership and at the same time to accede that virtue and effectiveness will sometimes part ways; and if this is the case, then we must be open to the possibility that leadership guidance that seeks to do honor to the ideals of virtue while acknowledging the realistic constraints faced by fixed social leaders should try to find a balanced approach between virtuous and effective leadership. Emphasizing the importance of virtue for leadership is entirely appropriate: moral virtue is crucial for strong leadership. LEADERSHIP GUIDANCE BETWEEN MORAL VIRTUE AND EFFECTIVENESS The upshot of all of this is that purely virtue-based approaches to leadership will inevitably overlook, or give short shrift, to an ineliminable and crucial portion of what it means to be a leader in a fixed social position, and to that extent such approaches will be of limited practical value. An approach to leadership guidance that navigates between virtue and effectiveness must also give place to prudential dictates that, while not constituting egregious violations of virtue, would likely be frowned upon by defenders of traditional virtue. [Extracted from the article]

We study a two player zero sum game where the initial position is not communicated to any player. The initial position is a function of a couple where is communicated to player Ⅰ while is communicated to player Ⅱ. The couple is chosen according to a probability measure. We show that the game has a value and, under additional regularity assumptions, that the value is a solution of Hamilton Jacobi Isaacs equation in a dual sense. [ABSTRACT FROM AUTHOR]

In a deregulated power market, the economical-risk due to real-time pricing is critical as it is not permitted to alter the decisions taken once. The existing generation market needs improvement in terms of enhancing its effectiveness and reliability. This article presents a game-theoretic approach-based bidding strategy decision-making through a case study. In this case, three thermal generating units feed three different constant loads for base load demand, one at a time. The economic load dispatch has been obtained using MATLAB software applying the particle swarm optimization (PSO) method. Three different bidding strategies for individual generators have been chosen to create 27 combinations to create data where the zero-sum game theory is applied. The marginal costs are calculated for each of the 27 combinations to formulate a game theory matrix. The game theory dominance method is then applied to obtain the market clearing price (MCP). The proposed methodology can help the GENCOs in making a profit or reducing the risk of making a loss by making a judicious selection among the possible available strategies. [ABSTRACT FROM AUTHOR]