Strategy Nash Equilibrium Research Articles

Kleptoparasitic interactions modeling varying owner and intruder hunger awareness

We consider a game theoretical model of kleptoparasitic interaction between two individuals, the Owner and the Intruder. The Owner is in possession of a resource and must decide whether to defend the resource against the Intruder or flee. If the Owner defends, the Intruder must decide whether to fight with the Owner or flee. The outcome of the fight depends on the hunger of the individuals, the hungrier the individual is, the more likely they are to win the fight. We consider three scenarios: (a) both individuals know their own and their opponent’s hunger, (b) individuals only know their own hunger but not that of their opponent, and (c) individuals do not know their own nor the opponent’s hunger levels. We determine Nash equilibrium strategies in each scenario. We conclude that Owner is generally willing to defend more often than the Intruder is willing to attack. Also, the Intruder’s payoff is largest in the full information case; but the Owner may benefit in the no information or partial information cases when the cost of the fight is neither too large nor too small.

Stochastic differential investment and reinsurance games with nonlinear risk processes and VaR constraints

This paper investigates a class of non-zero-sum stochastic differential investment and reinsurance games between two insurance companies. We allow both insurers to purchase a proportional reinsurance contract and invest in risky and risk-free assets. When applying the generalized mean–variance premium principle in determining reinsurance premium, the surplus process becomes quadratic in the retained proportion of the claims. The optimization criterion of each insurer is to maximize the expected utility of the insurer’s terminal performance relative to that of his competitor. In addition, we incorporate dynamic Value-at-Risk (VaR) constraints in the optimization problems of both insurers to satisfy the capital requirements from regulators. The results show that this game problem can be converted to solving a system of nonlinear equations by means of dynamic programming principle and Karush–Kuhn–Tucker (KKT) conditions. Specifically, when both insurers are constant absolute risk aversion (CARA) institutions and the reinsurance premium principle reduces to the expected value principle, we derive the simplified expressions for the Nash equilibrium strategies. Finally, we use some numerical examples to illustrate the effects of several model parameters on the Nash equilibrium strategies under three different scenarios.

Lanchester duopoly model revisited: Advertising competition under time inconsistent preferences

In this paper, we study a finite time horizon advertising dynamic game under the assumption of time-inconsistent preferences. Specifically, we consider two types of discounting, heterogeneous discounting and hyperbolic discounting. In the case of heterogeneous discounting, the relative importance of the final function will increase/decrease as the end of the planning horizon approaches compared with current payoffs. Whereas when agents discount future payoffs hyperbolically, their discount rates diminish rapidly in earlier stages and then slowly in the long term. We compute time-inconsistent and time-consistent feedback Nash equilibrium strategies, and compare them with those of the standard discounting case. Our results reveal that heterogeneous discounting would lead to some last-minute changes, i.e., some adapting behaviours in the last years in accordance with their increasing/decreasing valuations of the final state. Under some circumstances, the change can be so radical that the pre-commitment solution takes the contrary path of time-consistent strategies. Concerning the competition under hyperbolic discounting, the temporal evolution of advertising efforts show a quite different nature. Different strategies exhibit disparity in the beginning, and encounter in the neighbourhood in the end, which is contrary to the heterogeneous discounting. Besides, a strong commitment power might induce over investment.

Particle Swarm Optimization Algorithm With Self-Organizing Mapping for Nash Equilibrium Strategy in Application of Multiobjective Optimization.

In this article, the Nash equilibrium strategy is used to solve the multiobjective optimization problems (MOPs) with the aid of an integrated algorithm combining the particle swarm optimization (PSO) algorithm and the self-organizing mapping (SOM) neural network. The Nash equilibrium strategy addresses the MOPs by comparing decision variables one by one under different objectives. The randomness of the PSO algorithm gives full play to the advantages of parallel computing and improves the rate of comparison calculation. In order to avoid falling into local optimal solutions and increase the diversity of particles, a nonlinear recursive function is introduced to adjust the inertia weight, which is called the adaptive particle swarm optimization (APSO). In addition, the neighborhood relations of current particles are constructed by SOM, and the leading particles are selected from the neighborhood to guide the local and global search, so as to achieve convergence. Compared with several advanced algorithms based on the eight multiobjective standard test functions with different Pareto solution sets and Pareto front characteristics in examples, the proposed algorithm has a better performance.

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.

Stochastic differential reinsurance games in diffusion approximation models

In this paper, we study some noncooperative and cooperative stochastic differential games between two insurers with reinsurance controls. The surplus processes are modeled by diffusion approximation processes and the insurers can purchase reinsurance dynamically (i.e. control the drift and diffusion terms continuously over time). We consider two types of reinsurance: quota-share (QS) reinsurance and excess-of-loss (XL) reinsurance. In the noncooperative game, one insurer tries to minimize the probability that the surplus difference of the two insurers reaches a low target before it hits a high target, while the other aims to maximize the probability. We consider two cases of the game: in the first case, both insurers purchase XL reinsurance; and in the second case, one insurer purchases QS reinsurance while the other purchases XL reinsurance. In some parameter cases, we solve the game by finding the value function and Nash equilibrium strategy explicitly. In some parameter cases, the value function and Nash equilibrium strategy do not exist and we find the sup-value and sub-value functions of the game. We also establish and solve a cooperative game in which both insurers make joint efforts to minimize the probability that the sum of surplus processes reaches a low target before it hits a high target. Numerical examples and economic implications are given to illustrate the results.

Regular potential games

A fundamental problem with the Nash equilibrium concept is the existence of certain “structurally deficient” equilibria that (i) lack fundamental robustness properties, and (ii) are difficult to analyze. The notion of a “regular” Nash equilibrium was introduced by Harsanyi. Such equilibria are isolated, highly robust, and relatively simple to analyze. A game is said to be regular if all equilibria in the game are regular. In this paper it is shown that almost all potential games are regular. That is, except for a closed subset with Lebesgue measure zero, all potential games are regular. As an immediate consequence of this, the paper also proves an oddness result for potential games: In almost all potential games, the number of Nash equilibrium strategies is finite and odd. Specialized results are given for weighted potential games, exact potential games, and games with identical payoffs. Applications of the results to game-theoretic learning are discussed.

Optimal Energy Sharing for Cooperative Relaying in a Random Access Network With Energy Harvesting Nodes

In this article, we consider an energy harvesting cooperative network comprised of two data sources and one relay. All nodes transmit based on a predefined slotted Aloha protocol in a random access environment. The main challenge we study is to find how the relay node must share its energy between two sources to optimize their QoS metrics, i.e., throughput and average transmission delay (ATD), separately. Thus, according to our multi-source scenario, we encounter a multi-objective optimization problem (MOOP). Since the relay does not prioritize between two sources, in order to solve the MOOP, we optimize equal-weight linear aggregation of the QoS metrics of two sources. Interestingly, we show that our problem can be modeled as an exact potential game in which the Nash equilibrium strategy is an optimal and stable solution of the related MOOP. The cooperative relaying strategy proposed in this article assures that the throughput and ATD of each source is better than the corresponding metrics without relay. Finally, we present numerical results to show how much the derived cooperative relaying strategies improve the QoS of each source.

Subcontracting New Product Development Projects: The Role of Competition and Commitment

ABSTRACTShould a firm, which seeks to subcontract a new product development project, leverage competition among potential suppliers and ask all of them to engage in research and development in parallel? Or should it first invite offers and commit to the supplier with the best offer, before only this supplier engages in development? Building on analytical literature on both formats, we apply game theory to answer these questions. We identify Bayesian Nash equilibrium strategies and characterize advantages of both formats. We find that having multiple suppliers engage in new product development in parallel is favored only if enough suppliers can be attracted, which is the case when development uncertainty and learning benefits are high. The participation decision also depends on the specific structure of the project's development costs. If administrative overhead and material costs are substantial, while engaging in development and exerting effort is relatively cheap but does not offer many learning opportunities, the number of suppliers who would be willing to engage in parallel development is limited. First inviting offers and selecting the best supplier to exclusively engage in new product development then becomes more attractive for the buyer. We discuss further implications and characterize environments that may foster more innovativeness in this context.

A Survey of Nash Equilibrium Strategy Solving Based on CFR

Recently, with the rapid development of artificial intelligence technology, there are growing researchers drawing their attention on the field of computer game. In two-player zero-sum extensive games with imperfect information, counterfactual regret (CFR) method is one of the most popular method to solve Nash equilibrium strategy. Therefore, we have carried on a wide range of research and analysis on the CFR and its related improved methods in recent years. In this paper, we firstly introduce the process of solving the strategy of Nash equilibrium based on CFR method. Then, some related improved methods of CFR are reviewed. The bare extended experiments are carried out to help researchers to understand these methods more conveniently. Further, some successful applications and common test platforms are described. Finally, the paper is ended with a conclusion of CFR-based methods and a prediction of future development.

Malware propagation model in wireless sensor networks under attack–defense confrontation

An important security problem faced by wireless sensor networks (WSNs) is malware propagation. Traditional malware propagation models take less consideration of the impact of the attack and defense processes on malware propagation results. We analyzed the micro-mechanism of malware propagation in WSNs from the perspective of the game theory, established the attack–defense game model of WSNs, derived the mixed Nash equilibrium solution of the game model, and derived the contagion probability of malwares according to the mixed Nash equilibrium strategy of both sides in the game, thus establishing the malware propagation model in WSNs. From the analysis of the theoretical model, we deduced the relationship between the steady-state infection ratio and the game parameters and obtained the conditions for the long-term existence of malwares. In addition, we simulated the propagation process of malwares in WSNs with cellular automaton and the simulation results verified the correctness of the theoretical model. The results have theoretical guiding significance for suppressing the propagation of malwares in WSNs.

A Differential Game of Ecological Compensation Criterion for Transboundary Pollution Abatement under Learning by Doing

This paper studies a stochastic differential game of transboundary pollution abatement between two kinds of ecological compensation and the abatement policy, in which the learning by doing is taken into account. Emission and pollution abatement between upstream and downstream region in the same basin is a Stackelberg game, and the downstream regions provide economic compensation for pollution abatement in the upstream region. We discuss the feedback Nash equilibrium strategies of proportional compensation and investment compensation, and it is found that an appropriate ecological compensation ratio can improve the investment level of pollution abatement in the two regions by accumulating experience in the process of learning by doing. In the long term, the investment compensation mechanism is an effective transboundary pollution abatement measure that can continuously reduce the water pollution stock in the upstream and downstream.

Non-zero-sum reinsurance and investment game between two mean-variance insurers under the CEV model

This paper considers a non-zero-sum stochastic differential game between two competitive mean-variance insurers, who aim to seek the time-consistent reinsurance and investment strategies. These two insurers are allowed to purchase proportional reinsurance to mitigate individual claim risks; and can invest in one risk-free asset and one risky asset whose price dynamics follows the constant elasticity of variance model. The main objective of each insurer is to maximizing the mean-variance utility of his relative terminal wealth with respect to that of his competitor. Applying the techniques of stochastic control theory, we derive the Nash equilibrium reinsurance and investment strategies explicitly and present the corresponding verification theorem. Furthermore, Nash equilibrium strategies and value functions are also provided under the diffusion approximation model. Finally, we perform some numerical examples to illustrate the influence of model parameters on the equilibrium reinsurance and investment strategies and draw some economic interpretations from these results.

A Trust-Game-Based Access Control Model for Cloud Service

In order to promote mutual trust and win-win cooperation between the users and the providers, we propose a trust-game-based access control model for cloud service in the paper. First, we construct a trust evaluation model based on multiple factors, such as the direct trust, feedback trust, reward punishment, and trust risk and further propose a weight method by maximum discrete degree and information entropy theory; second, we combine trust evaluation with the payoff matrix for game analysis and calculate the mixed Nash equilibrium strategy for the users and service providers; third, we give the game control condition based on trust level prediction and payment matrix to encourage participants to make honest strategy. Experimental results show that our research has good effect in terms of acceptance probability, deception probability, accuracy of trust evaluation, and cooperation rate in the cloud service.

Reinsurance–investment game between two mean–variance insurers under model uncertainty

This paper investigates a class of robust non-zero-sum reinsurance–investment stochastic differential games between two competing insurers under the time-consistent mean–variance criterion. We allow each insurer to purchase a proportional reinsurance treaty and invest his surplus into a financial market consisting of one risk-free asset and one risky asset to manage his insurance risk. The surplus processes of both insurers are governed by the classical Cramér–Lundberg model and each insurer is an ambiguity-averse insurer (AAI) who concerns about model uncertainty. The objective of each insurer is to maximize the expected terminal surplus relative to that of his competitor and minimize the variance of this relative terminal surplus under the worst-case scenario of alternative measures. Applying techniques in stochastic control theory, we obtain the extended Hamilton–Jacobi–Bellman (HJB) equations for both insurers. We establish the robust equilibrium reinsurance–investment strategies and the corresponding equilibrium value functions of both insurers by solving the extended HJB equations under both the compound Poisson risk model and its diffusion-approximated model. Finally, we conduct some numerical examples to illustrate the effects of several model parameters on the Nash equilibrium strategies.

Best-response dynamics in zero-sum stochastic games

We define and analyse three learning dynamics for two-player zero-sum discounted-payoff stochastic games. A continuous-time best-response dynamic in mixed strategies is proved to converge to the set of Nash equilibrium stationary strategies. Extending this, we introduce a fictitious-play-like process in a continuous-time embedding of a stochastic zero-sum game, which is again shown to converge to the set of Nash equilibrium strategies. Finally, we present a modified δ-converging best-response dynamic, in which the discount rate converges to 1, and the learned value converges to the asymptotic value of the zero-sum stochastic game. The critical feature of all the dynamic processes is a separation of adaption rates: beliefs about the value of states adapt more slowly than the strategies adapt, and in the case of the δ-converging dynamic the discount rate adapts more slowly than everything else.

Stochastic Congestion Game for Load Balancing in Mobile-Edge Computing

Mobile-edge computing can reduce task execution delay and improve the Quality of Experience (QoE) for the network edge users. However, when there are multiple independent cloudlets in the network with the mobile users offloading tasks randomly, how to maintain the load balancing of the independent cloudlets, how to improve the quality of service and users' QoE are still issues need to be solved. To this end, we study these issues from the perspective of game theory and propose decentralized learning algorithms. First, we turn the cloudlets load-balancing issue into a competition that each user minimizes its task execution time, and then a stochastic congestion game with incomplete information is proposed. Second, based on the existence proof of the Nash equilibria by using potential game theory, we propose a multiuser decentralized learning algorithm to obtain the pure Nash equilibrium strategy of each user. Then, an ordinary differential equation is derived to prove the convergence of the algorithm. Finally, we propose two application scenarios, one is for static users and the other is for dynamic users, and then the performances of the algorithm in a static scenario is tested. In order to adapt to dynamic scenarios, and further improve the performance and reduce communication costs, we propose a decentralized learning algorithm with termination condition. The experiments show that this algorithm can improve the load balancing of the multicloudlet system, and enhance the quality of service.

Automated temporal equilibrium analysis: Verification and synthesis of multi-player games

In the context of multi-agent systems, the rational verification problem is concerned with checking which temporal logic properties will hold in a system when its constituent agents are assumed to behave rationally and strategically in pursuit of individual objectives. Typically, those objectives are expressed as temporal logic formulae which the relevant agent desires to see satisfied. Unfortunately, rational verification is computationally complex, and requires specialised techniques in order to obtain practically useable implementations. In this paper, we present such a technique. This technique relies on a reduction of the rational verification problem to the solution of a collection of parity games. Our approach has been implemented in the Equilibrium Verification Environment (EVE) system. The EVE system takes as input a model of a concurrent/multi-agent system represented using the Simple Reactive Modules Language (SRML), where agent goals are represented as Linear Temporal Logic (▪) formulae, together with a claim about the equilibrium behaviour of the system, also expressed as an ▪ formula. EVE can then check whether the ▪ claim holds on some (or every) computation of the system that could arise through agents choosing Nash equilibrium strategies; it can also check whether a system has a Nash equilibrium, and synthesise individual strategies for players in the multi-player game. After presenting our basic framework, we describe our new technique and prove its correctness. We then describe our implementation in the EVE system, and present experimental results which show that EVE performs favourably in comparison to other existing tools that support rational verification.

NASH EQUILIBRIUM STRATEGIES REVISITED IN SOFTWARE RELEASE GAMES

A software release game was formulated by Zeephongsekul and Chiera [Zeephongsekul, P. & Chiera, C. (1995). Optimal software release policy based on a two-person game of timing. Journal of Applied Probability 32: 470–481] and was reconsidered by Dohi et al. [Dohi, T., Teraoka, Y., & Osaki, S. (2000). Software release games. Journal of Optimization Theory and Applications 105(2): 325–346] in a framework of two-person nonzero-sum games. In this paper, we further point out the faults in the above literature and revisit the Nash equilibrium strategies in the software release games from the viewpoints of both silent and noisy type of games. It is shown that the Nash equilibrium strategies in the silent and noisy of software release games exist under some parametric conditions.

A state-feedback Nash equilibrium for the general Target–Attacker–Defender differential game of degree in arbitrary dimensions

In this manuscript we formulate the general Target–Attacker–Defender differential game of degree in both its continuous-time and discrete-time turn-based variants in n-dimensional euclidean space. In this three-agent engagement, the Attackers goal is to get as close as possible to the Target before collision with the Defender, whilst the Target and Defender coordinate to achieve the opposite. The most general setting for this zero-sum differential game is considered, where the agents move at speeds not necessarily equal, and the Nash equilibrium strategies are proven in the discrete-time turn-based variant.