Stability Of Nash Equilibrium Research Articles

Incentive-based fault-tolerant cooperative payoff design in weakly acyclic games

This paper proposes an incentive-based cooperative payoff design method to accommodate faults in weakly acyclic games, where the payoff functions of normal players are designed such that all players’ optimal payoffs are recovered into an admissible level. Besides, the stochastically stable Nash equilibrium (SSNE) is regarded as the optimal state to avoid some inefficient Nash equilibria. Firstly, without cooperation among players, the effect of faults on SSNE and the optimal payoff is analyzed deeply. Further, the conditions of fault accommodation are discussed under the proposed cooperative fault tolerance method, based on which an algorithm is designed to calculate the feasible incentive. Secondly, with cooperation among players, the effect of faults is analyzed again in special weakly acyclic games, namely potential games, and the simpler conditions of fault accommodation are provided. Finally, two examples of frequency assignment in distributed systems and power management in sensor networks illustrate the effectiveness of the proposed methods.

Coalition Formation Game Based Information-Energy Collaboration in Vehicle Edge Computing Networks

In vehicle edge computing network (VECN), a key issue for the roadside unit (RSU) is how to handle computationally intensive tasks timely and use energy efficiently. When an RSU cannot finish the task on time, computation offloading will be an effective approach. However, when offloading tasks most RSUs neglect the energy supply and do not consider computation requirement and energy supply jointly. In this paper, we consider a VECN where there are multiple RSUs. When an RSU has a computational task with latency constraint, it will offload the task to the nearby RSUs and/or cloud servers, and purchase electricity from the neighbors and/or power grid if it is also short of power. To conserve energy and reduce cost, not only the collaboration between the VECN and the cloud/grid, but also the collaboration of RSUs within the VECN is investigated. Specifically, we first formulate the information-energy collaboration problem, and study how the strategies of computation offloading and energy sharing interact with each other, where the RSU's individual revenue is considered. Next, since the problem is a mixed nonlinear programming problem that is hard to be solved, we reformulate it into a double-level coalition formation game, based on which an algorithm with low complexity is proposed. Finally, we prove theoretically that through finite iterations, the game-based algorithm can converge to a stable Nash equilibrium which is the solution to the original problem. Simulation results show that our scheme is superior to the existing ones in reducing edge network expenses while improving individual revenue.

Distributed Nash equilibrium seeking over strongly connected switching networks

The Nash equilibrium seeking over networks of N players has been studied under the assumption that the network is static and strongly connected or switching and every time strongly connected. In this paper, we further consider the case where the network is jointly strongly connected. Since a jointly strongly connected network can be disconnected at any time instant, the existing approach cannot handle such a case. Like the literature, assuming the pseudogradient dynamics has a globally exponentially stable Nash equilibrium, we first establish a distributed estimator for actions of all players over jointly strongly connected networks. Then we compose the pseudogradient dynamics with the distributed estimator to obtain an extended gradient system with N+1 subsystems. Under the assumption that the pseudogradient of the game is strongly monotone and Lipschitz continuous, we show that, starting from any initial condition, the state of every subsystem of the extended gradient system will exponentially converge to the Nash equilibrium of the game concerned.

Variable-sample method for the computation of stochastic Nash equilibrium

This article proposes a variable-sample method for the computation of stochastic stable Nash equilibrium, in which the objective functions are approximated, in each iteration, by the sample average approximation with different sample sizes. We start by investigating the contraction mapping properties under the variable-sample framework. Under some moderate conditions, it is shown that the accumulation points attained from the algorithm satisfy the first-order equilibrium conditions with probability one. Moreover, we use the asymptotic unbiasedness condition to prove the convergence of the accumulation points of the algorithm into the set of fixed points and prove the finite termination property of the algorithm. We also verify that the algorithm converges to the equilibrium even if the optimization problems in each iteration are solved inexactly. In the numerical tests, we comparatively analyze the accuracy error and the precision error of the estimators with different sample size schedules with respect to the sampling loads and the computational times. The results validate the effectiveness of the algorithm.

COVID-19 vaccine incentive scheduling using an optimally controlled reinforcement learning model

We model Covid-19 vaccine uptake as a reinforcement learning dynamic between two populations: the vaccine adopters, and the vaccine hesitant. Using data available from the Center for Disease Control (CDC), we estimate the payoff matrix governing the interaction between these two groups over time and show they are playing a Hawk–Dove evolutionary game with an internal evolutionarily stable Nash equilibrium (the asymptotic percentage of vaccinated in the population). We then ask whether vaccine adoption can be improved by implementing dynamic incentive schedules that reward/punish the vaccine hesitant, and if so, what schedules are optimal and how effective are they likely to be? When is the optimal time to start an incentive program, how large should the incentives be, and is there a point of diminishing returns? By using a tailored replicator dynamic reinforcement learning model together with optimal control theory, we show that well designed and timed incentive programs can improve vaccine uptake by shifting the Nash equilibrium upward in large populations, but only so much, and incentive sizes above a certain threshold show diminishing returns.

A Coalition Formation Game-Based Multi-User Grouping Approach in the Jamming Environment

Due to the broadcast nature of wireless communication, it is vulnerable to malicious jamming attacks. Meanwhile, multiple users need to share the limited spectrum resources to enhance the spectrum utilization efficiency because of the scarcity of wireless spectrum resources. Therefore, wireless communication systems need to deal with the malicious jamming attacks and mutual interference among different users. In this paper, a multi-user anti-jamming grouping approach based on a coalition formation game is studied. The proposed approach can maximize users’ transmission rate while avoiding channel jamming through optimization of the coalition formation strategy under the influence of malicious jamming. With the help of the exact potential energy game, we demonstrate that the proposed game model can obtain stable Nash equilibrium solutions. Lastly, the effectiveness of the proposed algorithm is demonstrated through numerical.

Multi-oligarch dynamic game model for regional power market with renewable portfolio standard policies

Renewable portfolio standard and tradable green certification have become key policies for China to achieve carbon neutralization, which pose new challenges to strategies of players in the actual power market. A multi-oligarch dynamic electricity sales game model in regional power supply market is established, which consists of a renewable energy power enterprise, a traditional enterprise and an energy storage enterprise. The stability conditions of the system together with the effects of the macro-policies and information asymmetry on the game are further analyzed. According to the strategic interaction of stakeholders, the results demonstrate that, compared with the market transaction price weight, the marginal profit weight of energy storage enterprise has a significant impact on strategies of players, while information asymmetry has a depraved impact on the stability. Further, through the proposition of a partially shared signal transmission mitigation mechanism under macro-control, it is evidenced to effectively cushion the bad impact of information asymmetry on the stability of Nash equilibrium. Finally, the implementation of renewable portfolio standard and tradable green certification in the long run, will improve the stability of the system.

Global supply chains, trade agreements and rules of origin

AbstractFree Trade Agreements (FTAs) usually come with restrictions on the use of intermediate inputs in order for final goods to qualify for free trade. I focus on Rules of Origin (RoO), which limit expenses on nonmember country's intermediate inputs. In a three‐country FTA formation game, I introduce international trade in intermediate inputs and RoO restrictions. In the case of symmetric countries, I show that as countries become more involved in global supply chains, measured by their input shares in foreign final goods production, global free trade is less likely to be a stable equilibrium outcome. Free riding is the main problem preventing countries from liberalising trade. Countries are better off being nonmembers of FTAs between the other two countries relative to global free trade. Rules of Origin can solve this problem by limiting the benefits countries get from other countries' free trade agreements. In the case of asymmetric countries, an additional incentive exists for the smaller country not to join: such a country gives up more than it gains from joining an FTA for a sufficiently high degree of asymmetry in country sizes. I show that global free trade is a stable Nash equilibrium under a larger region of asymmetric country parameter space in the case of RoO than without it. Therefore, it is shown that RoO is essential in order to attain global free trade.

Large Games: Robustness and Stability

This review focuses on properties related to the robustness and stability of Nash equilibria in games with a large number of players. Somewhat surprisingly, these equilibria become substantially more robust and stable as the number of players increases. We illustrate the relevant phenomena through a binary-action game with strategic substitutes, framed as a game of social isolation in a pandemic environment.

A Brief Explanation of the Logic behind the Location Choice of Fast Food Restaurants in China

Chinese market is relative new for fast food chains. In this article, we will explore the competition between the McDonald’s and KFC. We will review two papers, each one focusing on one different aspect of the competition mode we are studying. For the Mayer’s model, we can apply the profit’s concept into our Hotelling model. For the electoral vote model, we can incorporate the concept of market share. We will try to quantify the variables involved in the real world fast food market. We will assume an imaginary line of positions for the firms to choose. Then, in the section of Theoretical Nash Equilibrium, we will try to solve the strategic setting and apply the principles further in the extensions. Finally we will find empirical evidence, which conforms perfectly to our propositions. To sum up, we propose that KFC and McDonald’s clustering at the center are of reason, generating maximum profit and market share, creating a stable Nash Equilibrium.

Evolutionary Random Access Game With Objective and Subjective Players

We consider an evolutionary game on a graph under the condition that players perception of uncertainty and decision making can follow either an objective expected utility theory (EUT) model or a subjective prospect theory (PT) model. Each player chooses one of two strategies with probabilities $x$ and $1-x$ , where the subjective players bias their choices of the probabilities to be $w(x)$ and $w(1-x)$ to reflect the probability weighting effect of PT. We derive the conditions to achieve a locally asymptotically stable Nash equilibrium (NE) and the necessary conditions to achieve an evolutionary stable strategy (ESS). We illustrate these results through a simple random access channel game. We find that players’ behavior is affected by the total number of players as well as the number of each type of players (objective or subjective). We show that increasing the number of interacting players induces an aggressive transmission strategy and the behavior becomes similar for both types of players. Moreover, we let the objective and subjective players adjust their transmission under throughput rewards, delay penalties, and energy costs. The EUT players’ throughput is higher than the PT players’ throughput for a small number of interacting players and equal to if the number of interacting players increases.

Grassmannian reduction of cucker-smale systems and dynamical opinion games

<p style='text-indent:20px;'>In this note we study a new class of alignment models with self-propulsion and Rayleigh-type friction forces, which describes the collective behavior of agents with individual characteristic parameters. We describe the long time dynamics via a new method which allows us to reduce analysis from the multidimensional system to a simpler family of two-dimensional systems parametrized by a proper Grassmannian. With this method we demonstrate exponential alignment for a large (and sharp) class of initial velocity configurations confined to a sector of opening less than <inline-formula><tex-math id="M1">\begin{document}$ \pi $\end{document}</tex-math></inline-formula>.</p><p style='text-indent:20px;'>In the case when characteristic parameters remain frozen, the system governs dynamics of opinions for a set of players with constant convictions. Viewed as a dynamical non-cooperative game, the system is shown to possess a unique stable Nash equilibrium, which represents a settlement of opinions most agreeable to all agents. Such an agreement is furthermore shown to be a global attractor for any set of initial opinions.</p>

Dynamics of a Bertrand Duopoly Game of the Greek Oil Market and Application of the d-Backtest Method

This work presents the complexity that emerges in a Bertrand duopoly between two companies in the Greek oil market, one of which is semi-public and the other is private. The game uses linear demand functions for differentiated products from the existing literature and asymmetric cost functions that arose after approaches using the published financial reports of the two oil companies (Hellenic Petroleum and Motor Oil). The game is based on the assumption of homogeneous players who are characterized by bounded rationality and follow an adjustment mechanism. The players’ decisions for each time period are expressed by two difference equations. A dynamical analysis of the game’s discrete dynamical system is made by finding the equilibrium positions and studying their stability. Numerical simulations include bifurcation diagrams and strange attractors. Lyapunov numbers’ graphs and sensitivity analysis in initial conditions prove the algebraic results and reveal the complexity and chaotic behavior of the system focusing on the two parameters k1 and k2 (speed of adjustment for each player). The d-Backtest method is applied through which an attempt is made to control the chaos that appears outside the stability space in order to return to the locally asymptotically stable Nash equilibrium for the system.

Chaotic Behaviors in a Nonlinear Game of Two-Level Green Supply Chain with Government Subsidies

In this paper, a two-level green supply chain composed of a manufacturer and a retailer is taken as the background. Considering the consumer’s double consumption preference and the manufacturer’s green product R&D investment, a differential game model of the green supply chain under the government cost subsidy strategy is constructed. Firstly, the equilibrium points of the system are solved and their stability is discussed and analyzed. Secondly, the dynamic evolution process of Nash equilibrium under the parameters of green degree, green preference coefficient, retail channel preference coefficient, coefficient of the sensitivity of price, and adjustment speed are described by numerical simulation. The results show that the two ways of a system entering chaos are Flip bifurcation and N-S bifurcation, respectively, by 2D bifurcation graph, and it is also verified in 1D bifurcation diagram. When the bifurcation parameters are small, the system maintains Nash equilibrium stability. If the green degree of products is increased, the green preference coefficient will also increase; on the contrary, the retail preference coefficient will decrease. Research and development cost subsidy policy can effectively improve the green degree of products and increase the sales volume of products, so as to improve the profit of supply chain members.

The Dynamic Rent-Seeking Games with Policymaker Cost and Competition Intensity

In this paper, a dynamic rent-seeking game incorporating policymaker cost and competition intensity is considered. On the basis of the political environment and rent-seekers with incomplete information set, the locally asymptotic stability of Nash equilibrium is proved. The competition intensity and policymaker cost could enlarge the stability region of Nash equilibrium. The higher the competition intensity is, the more the opponent’s expenditure reduces the player’s success probability, which is beneficial to the maintenance of Nash equilibrium. The higher the policymaker cost is, the less easily both players succeed and the more stable the rent-seeking market is. As the competition parameter decreases or the expenditure parameter increases, there will be chaos in a rent-seeking market. Chaos control is in order to stabilize the equilibrium of the rent-seeking game.

On the Existence of Stable Equilibria in Monotone Games

This paper shows that under very general conditions, there exists a locally stable Nash equilibrium in games of strategic complements (GSC), as well as in the more general case of games with non-decreasing best response correspondences. While it is well known that in such cases a unique equilibrium is globally stable, no equilibrium can be globally stable when multiple equilibria exist. However, the existence of a locally stable equilibrium remains an open question, as we give examples of GSC in which no stable equilibrium exists. One main advantage of our approach is that our results can be derived simply by exploiting the monotonicity properties of the game, and do not require any differentiability assumptions. Results on equilibrium refinement follow as a corollary under slightly stronger assumptions, in the sense that games with two equilibria possess exactly one locally stable equilibrium.

Distributed Nash Equilibrium Seeking for a Dynamic Micro-grid Energy Trading Game with Non-quadratic Payoffs

To model energy trading interactions among prosumers, a generic non-quadratic dynamic game model is proposed and solved for a stable Nash equilibrium. The framework is modeled as a non-cooperative, infinite strategy, multiplayer game where the participating players can possess a non-quadratic payoff function. Moreover, the market framework realizes a dynamic mapping from strategy set to payoffs of the participating players. It also carries an embedded notion of reliability and fairness by utilizing the concept of market reputation index. The existence of multiple Nash equilibria is discussed as well as the criteria for stability of Nash equilibrium. It is demonstrated that when an extremum seeking approach is used to model Nash seeking behavior of the players, the system converges to one of the stable Nash equilibriums depending on the initial condition. Simulation results are presented to verify the theoretical results, and a complete day with multiple prosumers is simulated to demonstrate effectiveness of the approach. The results are evaluated using several reliability matrices to show that the proposed framework results in an increased local generation, increased payoffs for prosumers, lower market clearing prices, and higher reliability.

A Smart Game for Data Transmission and Energy Consumption in the Internet of Things

The current trend in developing smart technology for the Internet of Things (IoT) has motivated a lot of research interest in optimizing data transmission or minimizing energy consumption, but with little evidence of proposals for achieving both objectives in a single model. Using the concept of game theory, we develop a new MAC protocol for IEEE 802.15.4 and IoT networks in which we formulate a novel expression for the players’ utility function and establish a stable Nash equilibrium (NE) for the game. The proposed IEEE 802.15.4 MAC protocol is modeled as a smart game in which analytical expressions are derived for channel access probability, data transmission probability, and energy used. These analytical expressions are used in formulating an optimization problem (OP) that maximizes data transmission and minimizes energy consumption by nodes. The analysis and simulation results suggest that the proposed scheme is scalable and achieves better performance in terms of data transmission, energy-efficiency, and longevity, when compared with the default IEEE 802.15.4 access mechanism.

Learning with minimal information in continuous games

While payoff‐based learning models are almost exclusively devised for finite action games, where players can test every action, it is harder to design such learning processes for continuous games. We construct a stochastic learning rule, designed for games with continuous action sets, which requires no sophistication from the players and is simple to implement: players update their actions according to variations in own payoff between current and previous action. We then analyze its behavior in several classes of continuous games and show that convergence to a stable Nash equilibrium is guaranteed in all games with strategic complements as well as in concave games, while convergence to Nash equilibrium occurs in all locally ordinal potential games as soon as Nash equilibria are isolated.

On the existence and stability of equilibria in N-firm Cournot–Bertrand oligopolies

This paper takes a novel approach to studying the existence and stability of Nash equilibria in N-firm Cournot–Bertrand oligopolies. First, we show that such games can be monotonically embedded into a game of strategic heterogeneity, so that each firm best responds to the choices of all other firms in a monotonic way. We then show that this monotonicity can be exploited to derive conditions which guarantee the existence of a unique, dominance solvable Nash equilibrium which is stable under all adaptive dynamics. These conditions constitute a strict improvement over existing results in the literature. Finally, we examine the effect on these conditions resulting from additional firms entering the market.