Finding Nash Equilibria Research Articles

Nash equilibria for relative investors with (non)linear price impact

We consider the strategic interaction of n investors who are able to influence a stock price process and at the same time measure their utilities relative to the other investors. Our main aim is to find Nash equilibrium investment strategies in this setting in a financial market driven by a Brownian motion and investigate the influence the price impact has on the equilibrium. We consider both CRRA and CARA utility functions. Our findings show that the problem is well-posed as long as the price impact is at most linear. Moreover, numerical results reveal that the investors behave very aggressively when the price impact is close to a critical parameter.

СРАВНЕНИЕ МЕТОДОВ НАХОЖДЕНИЯ РАВНОВЕСИЯ В ТЕОРИИ ИГР В БИБЛИОТЕКЕ NashPу

The article compares the performance and reliability of different methods for finding Nash equilibrium situations in antagonistic and bimatrix games. The NashPy library is used, which implements a convenient interface for searching for equilibrium situations. All methods are implemented in pure Python language. The comparison showed that the support_enumeration method finds all equilibrium situations, but its running time grows exponentially. Therefore, when the dimension is greater than 15, it is recommended to use the vertex_enumeration method, which also finds all equilibrium situations, but does it faster. If you need to find any one equilibrium, it is better to use the lemke_howson_enumeration method, which works very quickly, but returns only one solution. The article clearly presents graphs of the time spent searching for equilibrium depending on the dimension of the problem.

Finding Nash equilibrium based on reinforcement learning for bidding strategy and distributed algorithm for ISO in imperfect electricity market

Finding Nash equilibrium in an imperfect market with active and strategic resources is key to the successful operation of the electricity market. A hierarchical Nash distributed reinforcement learning (HNDRL) framework is proposed to realize the interaction between the market participants and independent system operator (ISO). In the bidding stage, multi-agent Nash policy reinforcement learning is utilized to update the bidding strategies for market participants, i.e., generation company (GenCo) and distributed company (DisCo), which maximizes their own profit with imperfect information. The bidding action is updated by the probability iteration to deal with the continuous bidding strategy. And in the market clearing stage, the distributed dynamic-average consensus optimization algorithm is proposed at the ISO level to obtain the locational marginal price (LMP) and transaction quantities. With the participation of the demand-side in the electricity market, the proposed HNDRL framework can reach a Nash equilibrium with the optimal bidding strategy and handle global resource constraints in a distributed way. In addition, theoretical analysis is provided to ensure that the participants’ strategy exponentially converges to the Nash equilibrium for both convex and non-convex problems. Finally, the IEEE 30-bus system is employed to illustrate the efficiency. Compared with the existing method, the simulation results show that the proposed HNDRL framework has a faster convergence rate with the range of 2.667–3.333 times and can obtain higher profit with the range of 1.4286%–7.1429% at the Nash equilibrium point.

Fuzzy approximation-based optimal consensus control for nonlinear multiagent systems via adaptive dynamic programming

This paper investigates the fuzzy approximation-based optimal consensus control problem for nonlinear multiagent systems with unknown perturbations. By constructing local error dynamics, the considered optimal consensus problem is reformulated as finding Nash-equilibrium solutions to zero-sum games. Then, by using sliding mode control technology and the concept of hierarchical design, a series of control signals are sequentially designed to regulate the consensus error and minimize the local value function. In addition, an identifier-critic architecture is developed by using generalized fuzzy hyperbolic models, where the identifier is employed to relax the requirement for complete system dynamics information, and the hierarchical sliding mode surface-based critic network is applied to approximate optimal control inputs. Finally, A simulation example is presented to illustrate the validity of the proposed approach.

Mean Field Games for Diel Vertical Migration with Diffusion

We present a theoretical framework, based on differential mean field games, for expressing diel vertical migration in the ocean as a game with a continuum of players. In such a game, each agent partially controls its own state by adjusting its vertical velocity but the vertical position in a water column is also subject to random fluctuations. A representative player has to make decisions based on aggregated information about the states of the other players. For this vertical differential game, we derive a mean field system of partial differential equations for finding a Nash equilibrium for the whole population. It turns out that finding Nash equilibria in the game is equivalent to solving a PDE-constrained optimization problem. We detail this equivalence when the expected fitness of the representative player can be approximated with a constant and solve both formulations numerically. We illustrate the results on simple numerical examples and construct several test cases to compare the two analytical approaches.

Electric Vehicles Charging Dispatch and Optimal Bidding for Frequency Regulation Based on Intuitionistic Fuzzy Decision Making

The spread of electric vehicles (EVs) could reduce greenhouse gas emissions and achieve sustainable travel patterns. However, the rapidly increasing charging demand will bring challenges to the operation of charging stations and power systems. Therefore, a two-stage EV management scheme is introduced in this article to overcome these challenges and promote sustainable transport. A charging dispatch model based on fuzzy multicriteria decision making is proposed in the first stage, where users' preferences are in the form of intuitionistic fuzzy sets to address the fuzziness and uncertainty of subjective factors and human judgment. A <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\sigma$</tex-math></inline-formula> -cut similarity matrix is proposed to increase the users' satisfaction by excluding options with lower similarity. In the second stage, a noncooperative game model is proposed to incentive EVs to participate in supplementary frequency regulation (SFR). A fuzzy set is employed to reflect users' willingness to adjust charging power. The existence and uniqueness of the Nash equilibrium are investigated. Moreover, a distributed proximal best response algorithm with linear convergence is employed to find Nash equilibrium. Numerical simulations indicate that the proposed method can reduce charging costs while meeting users' preferences and facilitate EVs to participate in SFR.

Neural Fictitious Self-Play for Radar Antijamming Dynamic Game With Imperfect Information

One emerging issue in modern electronic warfare is the competition between the radar and jammer, which in principle can be viewed as a noncooperative game with two players. In practice, the interaction between the radar and jammer involves multiple rounds as well as partial observation. This makes the competition become a dynamic game with imperfect information. Antijamming strategy design for such kind of a game is still unclear. In this work, the competition between a frequency agile radar and a transmit/receive time-sharing jammer is considered. We utilize an extensive-form game (EFG) with imperfect information to model the multiple rounds interaction between the radar and jammer. For the established EFG, finding Nash equilibrium (NE) strategies is a computationally-intractable task since the number of information states grows exponentially with respect to game stages. Instead, a sampled-based learning method called neural fictitious self play algorithm is used to find approximate NE strategies (ANES). Simulation results show that ANES can be obtained and outperform the common elementary and advanced strategies from the perspective of detection performance.

행위자-기반 기법을 이용한 불완비정보 전력시장에서의 균형상태 해석

This paper presents a methodology of finding Nash Equilibrium (NE) by using an agent-based approach. A game theoretic method to solve the NE is applicable under the assumption of complete information on the electricity market and the participants. In the real market, however, Generation Companies (Gencos) submit supply bids to the Market Operator (MO) with incomplete information. Agent-based approaches are more suitable for the electricity market with incomplete information than the microeconomic method. In this paper, the payoff matrices are used as key information for deciding strategic bids. In the process of repeated auction with incomplete information, each agent can calculate efficient partial payoff matrix by using the results after each auction. The Best Response (BR) technique is applied to the partially constructed payoff matrices for finding the NE. The simulation process is divided into three modes, first is a preliminary step of constructing the payoff matrices, second is a step of tracking the NEs, and the third is a step of applying to the varying load pattern. The simulation results show that this methodology is applicable to the electricity market with incomplete information.

Jamming Attack Mitigation in CR-IoT Using Game Theory

Jamming attack is Cognitive radio Internet of thing network disables the spectrum sharing and reduces the spectrum utilization .detection and mitigation of such attacks is the main component in realizing cognitive radio-based spectrum sharing in cognitive radio-based internet of thing network this work proposes a game theory-based jamming attack mitigation strategy. The problem of jamming attack mitigation is modeled as a zero-sum game and solved by finding Nash equilibrium. The cognitive node which tries to share the spectrum pays the zero-sum game with the jamming attacker and selects the best strategy of selecting the best frequency without getting into the jamming attack. The result of the proposed mechanism proves that the gaming mechanism can tackle the jamming attack.

Safe Subgame Resolving for Extensive Form Correlated Equilibrium

Correlated Equilibrium is a solution concept that is more general than Nash Equilibrium (NE) and can lead to outcomes with better social welfare. However, its natural extension to the sequential setting, the Extensive Form Correlated Equilibrium (EFCE), requires a quadratic amount of space to solve, even in restricted settings without randomness in nature. To alleviate these concerns, we apply subgame resolving, a technique extremely successful in finding NE in zero-sum games to solving general-sum EFCEs. Subgame resolving refines a correlation plan in an online manner: instead of solving for the full game upfront, it only solves for strategies in subgames that are reached in actual play, resulting in significant computational gains. In this paper, we (i) lay out the foundations to quantify the quality of a refined strategy, in terms of the social welfare and exploitability of correlation plans, (ii) show that EFCEs possess a sufficient amount of independence between subgames to perform resolving efficiently, and (iii) provide two algorithms for resolving, one using linear programming and the other based on regret minimization. Both methods guarantee safety, i.e., they will never be counterproductive. Our methods are the first time an online method has been applied to the correlated, general-sum setting.

Research on the Development of Electricity Market Based on Performance Guarantee

China’s electricity market is currently in the development stage, and the related system is not comprehensive enough. Therefore, it is a key issue to standardize the trading behavior of the market entities and improve the operation efficiency of the electricity market. To solve this problem, this paper takes advantage of Cournot duopoly model to find Nash equilibrium on the electricity supply side and constructs a bank performance guarantee model based on the profit function on the electricity sales side. This study also sorts out the function of performance guarantee, its mechanism for risk prevention, and the feasibility of promoting the performance guarantee mechanism and other supporting systems. Finally, the conclusion is drawn that the performance guarantee mechanism can effectively prevent the risks in the electricity market, and it can be promoted with other supporting systems.

Finding nash equilibrium for imperfect information games via fictitious play based on local regret minimization

Finding Nash equilibrium in the domain of imperfect information games as a challenging problem has received much attention. Neural Fictitious Self-Play (NFSP) is a popular model-free machine learning algorithm and has computed approximate Nash equilibrium on such games. However, the deep reinforcement learning method used to approximate the best response in NFSP requires reaching a fully observable Markov state, while the states in imperfect information games are partially observable and non-Markovian, which results in a poor approximation of the best response. Thus, NFSP needs more iterations to converge. In this study, we present a new reinforcement learning method that is inspired by counterfactual regret minimization to relax the Markov requirement by iteratively updating policy according to the regret matching process. Combining this new reinforcement learning algorithm with fictitious play, we further present a novel algorithm to find approximate Nash equilibrium in zero-sum imperfect information games. Experimental results in three benchmark games show that this new algorithm can find approximate Nash equilibrium effectively and converge much faster compared with baseline.

A Decentralized Game Theoretic Approach for Virtual Storage System Aggregation in a Residential Community

The current electricity market is better described as an oligopoly than a market of perfect competition from which, in fact, it may be rather far. The increasing penetration of residential distributed energy resources has led to a significant number of prosumers in the electricity market. Microgrid community, a group of single controllable entity prosumers, is a promising component of the smart grid which will potentially yield a free electricity market. In this paper, we present a novel formation for a residential community microgrid that includes a coalition of prosumer households with solar photovoltaic systems. These households are connected through a virtual power bank that consists of households’ storage batteries and that mediates the communications between the households and the main grid. Using an application of mean field game theory, we find Nash Equilibrium strategies under which such sharing could minimize a linear combination of the households’ energy generation cost, energy consumption cost and revenue of sold energy. The resulting approach is tested on a case study of a constructed community micro-grid in Montreal, Quebec, Canada. The proposed mean field game approach can help decrease the aggregated cost and the individual energy cost. A comparative analytical study on the benefit of sharing was also performed, demonstrating that each prosumer is expected to have at least a 40 percent reduction on their individual cost if belonging to a microgrid community of 100 prosumers located in Montreal city.

Using BTA Algorithm for finding Nash equilibrium problem aiming the extraction of rules in rule learning

It is crystal clear that discovering the rules for finding a specific pattern among given data for extraction of association rules in rule-based learning systems has been defined in previous researches. Making use of game theory for the processes contributing to discovery of rules can be seen in numerous researches. In recent years, modeling based on game theory in rule learning sphere has gained much more attention for computer scientists. When two or more players use different strategies independently, the strategy game modeling could be used. In this view, strategic play is a desirable model for situations with no permanent strategic relationship among interactions. In addition, Nash equilibrium is the most widely used solution concept in game theory. This concept is a state-of-the-art interpretation of a strategy game. Each player has an accurate prediction of other players’ behavior and acts according to such a rational prediction. In the present study, by extracting rules from frequent patterns we have presented a model that can extract learning rules by abstraction based on game theory, which can be used not only for association rules but also for rule-based learning systems. Also, the introduced method can be easily generalized to fuzzy data. To find Nash equilibrium (FNE) in the proposed method, we used meta-heuristic bus transportation algorithm. The results indicated that the method reduces computational complexity in the associate rule discovery process and rule learning, provided that FNE is solved.

Solving leaderless multi-cluster games over directed graphs

We are concerned with finding Nash Equilibria in agent-based multi-cluster games, where agents are separated into distinct clusters. While the agents inside each cluster collaborate to achieve a common goal, the clusters themselves are considered to be virtual players that compete against each other in a non-cooperative game with respect to a coupled cost function. In such scenarios, the intra-cluster problem and the game between the clusters need to be solved simultaneously. Therefore, the resulting inter-cluster Nash Equilibrium should also be a minimizer of the social welfare problem inside the clusters. In this work, this setup is cast as a distributed optimization problem with sparse state information. Hence, critical information, such as the agent’s cost functions, remains private. We present a distributed algorithm that converges with a linear rate to the optimal solution. Furthermore, we apply our algorithm to an extended Cournot game to verify our theoretical results.

Linear model for coputation of equilibrium point in bimatrix game

The methods for finding Nash equilibrium in bimatrix game are introduced in this paper. Linear optimization model with binary variables for coputation of all equilibrium points in nongenerate bimatrix game is proposed.

Nash-equilibrium algorithm and incentive protocol for a decentralized decision and scheduling problem in sustainable electroplating plants

This paper addresses a decentralized production decision and scheduling problem in smart electroplating plants considering environmental sustainability. Due to the characteristics of the electroplating process, part processing times are confined to time-window constraints and controlled by independent autonomous agents with the support of multi-agent and distribution manufacturing technologies. It is shown that part processing times in chemical/physical tanks not only determine the productivity but also affect the environmental cost. Hence, two essential issues should be addressed: (a) how to find equilibrium strategies of agents (i.e., part processing times) in such a decentralized manufacturing setting; and (b) how to design an incentive protocol inducing the equilibrium strategies towards the optimal ones concerning the productivity and the environmental cost simultaneously. To deal with the first issue, this paper establishes a non-cooperative sequential game model, which simulates the evolution process of the strategies from an unstable state to an equilibrium one. Then, an iterative algorithm is developed to find Nash equilibrium strategies based on a cyclic bi-value graph. To address the second issue, a novel heuristic rule is proposed for the incentive protocol by exploring the characteristics of time-window constraints and environmental cost functions. The results of numerical experiments validate the performance of the Nash-equilibrium algorithm and the effectiveness of the heuristic rule.

A Column-and-Constraint Generation Algorithm to Find Nash Equilibrium in Pool-Based Electricity Markets

Equilibrium analysis is crucial in electricity market designs, with Nash equilibrium recognized as the most powerful one. Its most prominent hindrance, however, is an efficient methodology to compute an equilibrium point in large-scale systems. In this work, a Column-and-Constraint Generation (CCG) algorithm is proposed to tackle this challenge. More precisely, the master problem finds a candidate for Nash equilibrium and the oracle identifies whether this candidate point is indeed an equilibrium. A set of numerical experiments was conducted, comparing its computational performance with the solution of an Equilibrium Problem with Equilibrium Constraint (EPEC). We identify that the proposed algorithm overcomes the benchmark in the magnitude of 20 times on average and more than 30 times in the most demanding instances. Furthermore, the scalability of the EPEC formulation is challenged even for medium-scale instances, whilst the proposed algorithm was able to handle all tested instances in a reasonable computational time.

Численный метод нахождения равновесий Нэша и Штакельберга в моделях контроля качества речных вод

Численный метод нахождения равновесий Нэша и Штакельберга в моделях контроля качества речных вод

탄소배출권 시장과 연계된 경쟁형 전력시장 연구

This study is about power generation operation of fossil fuel generators considering the greenhouse gas emissions trading market. The determination of generating power is related to fuel costs as well as consumption of the emission allowance. Therefore, the optimal generation considers both electricity market and emissions trading market at the same time. Consumption in excess of the allocated emission allowance increases sales in the electricity market, but the cost of purchasing the allowance increases. The opposite occurs when the allowance remains. The relationship between these two markets is modeled in the form of oligopolistic competition to find Nash Equilibrium(NE). The effect of marginal cost and load demand on NE is studied by analyzing the fair price for the emissions trading.