A Nash equilibrium solution in an oligopoly market: The search for Nash equilibrium solutions with replicator equations derived from the gradient dynamics of a simplex algorithm

The present analysis applies continuous time replicator dynamics to the analysis of oligopoly markets. In the present paper, we discuss continuous game problems in which decision-making variables for each player are bounded on a simplex by equalities and non-negative constraints. Several types of problems are considered under conditions of normalized constraints and non-negative constraints. These problems can be classified into two types based on their constraints. For one type, the simplex constraint applies to the variables for each player independently, such as in a product allocation problem. For the other type, the simplex constraint applies to interference among all players, creating a market share problem. In the present paper, we consider a game problem under the constraints of allocation of product and market share simultaneously. We assume that a Nash equilibrium solution can be applied and derive the gradient system dynamics that attain the Nash equilibrium solution without violating the simplex constraints. Models assume that three or more firms exist in a market. Firms behave to maximize their profits, as defined by the difference between their sales and cost functions with conjectural variations. The effectiveness of the derived dynamics is demonstrated using simple data. The present approach facilitates understanding the process of attaining equilibrium in an oligopoly market.     

Modelling inter-supply chain competition with resource limitation and demand disruption

This paper proposes a comprehensive model for studying supply chain versus supply chain competition with resource limitation and demand disruption. We assume that there are supply chains with heterogeneous supply network structures that compete at multiple demand markets. Each supply chain is comprised of internal and external firms. The internal firms are coordinated in production and distribution and share some common but limited resources within the supply chain, whereas the external firms are independent and do not share the internal resources. The supply chain managers strive to develop optimal strategies in terms of production level and resource allocation in maximising their profit while facing competition at the end market. The Cournot–Nash equilibrium of this inter-supply chain competition is formulated as a variational inequality problem. We further study the case when there is demand disruption in the plan-execution phase. In such a case, the managers need to revise their planned strategy in order to maximise their profit with the new demand under disruption and minimise the cost of change. We present a bi-criteria decision-making model for supply chain managers and develop the optimal conditions in equilibrium, which again can be formulated by another variational inequality problem. Numerical examples are presented for illustrative purpose.     

具有不同决策规则的Cournot双寡头模型分析

以寡头市场中两家具有有限理性且以产量竞争的异质企业为背景,建立了一个离散时间的非线性动态古诺双寡头模型.讨论了模型的边界均衡点和Cournot-Nash均衡点的存在性和稳定性,给出了Cournot-Nash均衡点的稳定区域.通过数值模拟,利用单参数分岔图分析了随着参数的变化,系统的复杂动力学行为的变化情况.得出调整速度太大会导致Cournot-Nash均衡点失去稳定性,系统将产生混沌吸引子等复杂的动力学现象.此外初值极小的变化将导致系统产生巨大的波动.同时企业成本函数的差异性也会使得系统产生不同的分岔.     

The effect of firm number on equilibrium product positioning and pricing: a marketing–production perspective

We examine how product design and pricing are affected when the number of firms in the market increases by considering marketing and production variables. We employ a model that adopts marketing notions such as market shares and ideal points to dictate the attraction of the product to consumers. The model also incorporates production costs where the cost of the product increases linearly with increasing product attribute levels. The modeling framework facilitates the existence of a Nash equilibrium in prices and product positions. The number of firms in the market is exogenously defined. When the number of firms in the market increases, firms lower their prices and design their product with features closer to the market's ideal point. This results in lower profits.     

Revisiting Individual Evolutionary Learning in the Cobweb Model – An Illustration of the Virtual Spite-Effect

We examine the Cournot oligopoly model in the context of social and individual learning. In both models of learning, firms update their decisions about how much to produce via variants of the genetic algorithm updating procedure. Arifovic (1994) found that both models of social and individual learning converged to the Walrasian, competitive equilibrium. Vriend (2000) reports that the model of social learning converges to the Walrasian equilibrium outcome, while the model of individual learning converges to the Cournot–Nash equilibrium. We revisit the issue and conduct simulations varying elements of the updating algorithms, as well as of the underlying economic model. In the analysis of the outcomes of our simulations, we conclude that the convergence to the Cournot–Nash equilibrium is due to two things: the specific way in which production rules’ performance is evaluated coupled with a specific cost function specification.      

Comparative dynamics in an overlapping-generations model: the effects of quasi-rational discrete choice on finding and maintaining Nash equilibrium

Many models of Nash Equilibrium are complex enough that it becomes difficult to ascertain if and under what conditions the economic players can find and maintain this equilibrium. Using an analytical overlapping- generations model of goods, labor, and banking markets and quasi-rational discrete choice decision making, we find through agent-based simulations that Nash Equilibrium in goods market prices is stable when firms are sufficiently sensitive to changes in profits. In addition to verifying the analytical Nash outcome, the simulations verify that their economic agents, decision rules, and other protocols correspond to and maintain consistency with the analytical theory and identify important bounds of the analytical model.      

Two-node market under imperfect competition

In this paper, we consider a Cournot auction with uniform nodal prices for a two-node market. The structure of each local market is an oligopoly. We demonstrate how the type of Nash equilibrium depends on the throughput. Finally, we investigate the optimum throughput problem under an imperfect competition in the market.     

Modeling Multileader–Follower Noncooperative Stackelberg Games

This paper presents a Stackelberg–Nash game for modeling multiple leaders and followers. The model involves two Nash games restricted by a Stackelberg game. We propose a computational approach to find the equilibrium point based on the extraproximal method for ergodic controlled finite Markov chains. The extraproximal method consists of a two-step iterated procedure: the first step is a prediction and the second is a basic adjustment of the previous step. We formulate the game as coupled nonlinear programming problems using the Lagrange principle. The Tikhonov’s regularization method is used to guarantee the convergence to a unique equilibrium point. Validity of the method is demonstrated applying this framework to model an oligopoly competition.     

A newsvendor pricing game

This paper considers a horizontal market of multiple firms that face stochastic price-dependent demand. The firms make joint pricing/inventory decisions and use price to compete for market demand. With fairly general demand models that are price-dependent, stochastic, and substitutable among firms, we prove the existence and uniqueness of the pure-strategy Nash equilibrium. The market at the equilibrium exhibits a bias toward under-pricing caused by competition; specifically, raising prices at any equilibrium of the game increases the total system profit, and at any joint-optimal set of pricing levels each self-interested firm has an incentive to lower its price. This result closely parallels that obtained in the inventory competition games in which prices are fixed and the bias is toward overstocking.     

Information sharing vs. privacy: A game theoretic analysis

Sharing cyber security information helps firms to decrease cyber security risks, prevent attacks, and increase their overall resilience. Hence it affects reducing the social security cost. Although previously cyber security information sharing was being performed in an informal and ad hoc manner, nowadays through development of information sharing and analysis centers (ISACs), cyber security information sharing has become more structured, regular, and frequent. This is while, the privacy risk and information disclosure concerns are still major challenges faced by ISACs that act as barriers in activating the potential impacts of ISACs.This paper provides insights on decisions about security investments and information sharing in consideration of privacy risk and security knowledge growth. By the latest concept i.e. security knowledge growth, we mean fusing the collected security information, adding prior knowledge, and performing extra analyses to enrich the shared information. The impact of this concept on increasing the motivation of firms for voluntarily sharing their sensitive information to authorities such as ISACs has been analytically studied for the first time in this paper. We propose a differential game model in which a linear fusion model for characterizing the process of knowledge growth via the ISAC is employed. The Nash equilibrium of the proposed game including the optimized values of security investment, and the thresholds of data sharing with the price of privacy are highlighted. We analytically find the threshold in which the gain achieved by sharing sensitive information outweighs the privacy risks and hence the firms have natural incentive to share their security information. Moreover, since in this case the threshold of data sharing and the security investment levels chosen in Nash equilibrium may be lower than social optimum, accordingly we design mechanisms which would encourage the firms and lead to a socially optimal outcome. The direct impact of the achieved results is on analyzing the way ISACs can convince firms to share their security information with them.     

Competitive Delivered Spatial Pricing

This paper contributes to the literature by studying price and production competition between spatially distributed profit maximizing firms. Firms compete by setting delivered prices, planning production, and sending output to each market. Both elastic demand and non-linear production costs are assumed. A non-cooperative game is defined, and its properties are characterized. We find spatial pricing patterns similar to those found by Hoover (1936). Existence and general properties of the Nash price and production equilibrium are shown and sufficient conditions that guarantee the existence of an unique price-production-transportation equilibrium are presented. A convergent algorithm is shown to find the equilibrium and is demonstrated with an example.     

基于MAS市场机制的动态计算资源调度模型研究

针对动态计算网格资源调度问题,结合多Agent系统(multi agent syste,MAS)协同技术和市场竞价博弈机制,对计算网格资源分配技术进行了深入研究,提出了能够反映供求关系的基于市场经济的网格资源调度模型,该模型一方面能够充分利用消费者Agent的协商能力,另一方面能够充分考虑消费者的行为,使得消费者的资源申请和分配具有较高的合理性和有效性.同时,设计了消费者的效用函数,论证了资源分配博弈中Nash均衡点的存在性和惟一性以及Nash均衡解.基于所提资源调度模型,设计了一种网格资源调度算法.仿真实验表明,资源调度算法能够为消费者的资源数量提供参考,规范消费者竞价,从而使得整个资源的分配趋于合理.     

Game theory applied to dynamic duopoly problems with production constraints

In this paper an application of differential game theory in the area of microeconomics is presented. The problem considered is that of a dynamic duopoly where two firms each limited by a maximum capacity of production, share the same market, and try simultaneously but independently to maximize their profits over a certain planning horizon. While the static duopoly theory does not address itself to the question of the process by which changes in the price are brought about, but only compares the prices before and after the change takes place, the dynamic market theory, considered in this paper, allows for an analysis of how the price changes with time and what trajectory it follows. Necessary conditions for the existence of a Nash equilibrium solution in the general case are discussed and more specific results for the special case of linear demand and quadratic cost functions are developed.     

On oligopoly spectrum allocation game in cognitive radio networks with capacity constraints

Dynamic spectrum sharing is a promising technology to improve spectrum utilization in future wireless networks. The flexible spectrum management provides new opportunities for licensed primary user and unlicensed secondary users to reallocate the spectrum resource efficiently. In this paper, we present an oligopoly pricing framework for dynamic spectrum allocation in which the primary users sell excessive spectrum to the secondary users for monetary return. We present two approaches, the strict constraints (type-I) and the QoS penalty (type-II), to model the realistic situation that the primary users have limited capacities. In the oligopoly model with strict constraints, we propose a low-complexity searching method to obtain the Nash Equilibrium and prove its uniqueness. When reduced to a duopoly game, we analytically show the interesting gaps in the leader–follower pricing strategy. In the QoS penalty based oligopoly model, a novel variable transformation method is developed to derive the unique Nash Equilibrium. When the market information is limited, we provide three myopically optimal algorithms “StrictBEST”, “StrictBR” and “QoSBEST” that enable price adjustment for duopoly primary users based on the Best Response Function (BRF) and the bounded rationality (BR) principles. Numerical results validate the effectiveness of our analysis and demonstrate the convergence of “StrictBEST” as well as “QoSBEST” to the Nash Equilibrium. For the “StrictBR” algorithm, we reveal the chaotic behaviors of dynamic price adaptation in response to the learning rates.     

一种多智能体系统任务竞争模型及算法研究

致力于解决多智能体系统中的任务分配问题，基于社会生活中的竞争现象提出了一种多智能体竞争模型，同时提出了解决多智能体任务分配的详细算法．文章引入博弈论来研究存在相互外部约束条件下的个体选择问题．为了克服求解纳什均衡点的复杂性，本文采用了一步纳什均衡的方法．仿真结果证明了本模型的合理性和算法的有效性．     

RFID-enabled item-level product information revelation

We consider a homogeneous product market and the incentive for oligopolists to share item-level product information with their customers. Enabled by Radio Frequency Identification technology, each firm has the option to record and reveal item-level information of a proportion of its products. We consider a two-stage game where each firm first decides its production plan and then determines its level of information revelation. With a constant clearance discount rate, we derive pure strategy equilibria that are subgame perfect and demonstrate that complete information sharing is the unique Nash equilibrium for the game when the common demand is volatile and that no information revelation is the unique Nash equilibrium when demand is not volatile. Furthermore, we show that the Nash equilibrium is the same with a decreasing clearance discount rate and that neither complete information revelation nor zero information revelation is consistent with an equilibrium with an increasing discount rate. Results are similar in a duopoly non-homogeneous product market scenario.     

信息不对称条件下寡头集团转移定价决策分析

研究信息不对称条件下存在中间产品外部市场的寡头集团转移定价决策问题.在不完全竞争中间产品外部市场条件下,当集团之间信息不对称时,对中间产品内外部实行单一定价,转移价格将大于中间产品的边际成本;对中间产品内外部实行差别定价时,转移价格则等于中间产品的边际成本.通过对两种策略的比较可知,寡头集团实行差别定价还是单一定价,取决于中间产品外部市场和最终产品市场的需求价格弹性.     

A game theory-based model for product portfolio management in a competitive market

In today’s competitive markets, effective product portfolio is critical for manufacturers that offer several products. From manufacturers’ perspective, the diversity must be maintained in a level in which the engineering costs do not exceed the acquired advantages of increased market share. On the other hand, product portfolio diversity is prominent for customers. In addition, manufacturers should always be careful about competitors activity. Therefore, we consider the problem of product portfolio management (PPM) in a competitive environment. This paper constructs a game theory-based mathematical model to deal with this new PPM problem. In this presented mathematical model, the PPM problem is formulated as a 2-person non-cooperative game with complete information. Each player has a set of strategies which correspond to the feasible product portfolios. Every payoff is determined by the procedure that considers the customer–engineering interaction in product portfolio planning, which aims to optimize product portfolio for a target market segment, and proposed a maximizing surplus share model for it. Therefore, obtaining the optimal product portfolio is determined by the Nash equilibrium point of this game. Finally, a numerical example is presented to demonstrate the feasibility of the approach.     

Interacting oligopolistic and oligopsonistic Cournot markets

This paper considers the model of several interacting Cournot markets. Some of them are final goods markets, while the others are resource markets. The markets interact by sharing the same set of economic agents (producers), so that the latter are oligopsonists on resource markets and simultaneously oligopolists on goods markets. Each producer strategically chooses its supply volumes on each goods market and its purchase volume of resources in accordance with technology and expected supply effects on prices. We prove that in the case of linear demand and supply functions the model of interacting Cournot markets is reduced to a potential game; hence, the Nash equilibrium problem is equivalent to a mathematical programming problem. We also discuss the advantages and special features of such a representation of interacting oligopolistic and oligopsonistic markets.     

Adoption of new online services in the presence of network externalities and complementarities

A typical online user utilizes multiple services in the same service category concurrently due to the fact that many online service markets are characterized by the coexistence of multiple services offering complementary features. Along with an inherent network externality feature, this complementary nature among competing online services complicates predictions of adoption patterns in these markets. This paper extends the adoption function model of Arthur [W.B. Arthur, Competing technologies, increasing returns, and lock-in by historical events, The Economic Journal 99 (1989) 116–131] and applies it to the online service market in an attempt to explain various cases of adoption behavior. The proposed model predicts that there exists a first-mover’s advantage in this market. Specifically, when network externality is large enough, the follower is confined to a low market share, even though it provides the same level of service as that provided by the leader. However, this first-mover’s advantage can be overcome in cases where perturbations are caused by the heterogeneity of consumers or by service value uncertainty. In addition, a two-step punctuated equilibrium may exist: under specific conditions, market share distribution may be stable for a while at certain levels and then move into actual equilibrium.