Competition analysis of a triopoly game with bounded rationality

A dynamic Cournot game characterized by three boundedly rational players is modeled by three nonlinear difference equations. The stability of the equilibria of the discrete dynamical system is analyzed. As some parameters of the model are varied, the stability of Nash equilibrium is lost and a complex chaotic behavior occurs. Numerical simulation results show that complex dynamics, such as, bifurcations and chaos are displayed when the value of speed of adjustment is high. The global complexity analysis can help players to take some measures and avoid the collapse of the output dynamic competition game.     

Application of duopoly multi-periodical game with bounded rationality in power supply market based on information asymmetry

In the power market, each entity is not completely rational when generate strategy, and the market information held by each entity is not exactly the same. In this paper, duopoly power providers with different selling adjustment structures are simplified from the actual grid background, where one provider can sell part of its power to another at contract price to store the power. Each provider is trying to maximize its profit by adjusting its power selling strategy. The process of evolutionary game with multi-periods bounded rational is established. One provider adjusts its selling strategy through the multi-periods market price and another through its multi-periods marginal profit. The quantity of power sold by each provider will tend to Nash equilibrium and how information asymmetry affects the stability of Nash equilibrium is analyzed through comparing dynamic power selling with and without information asymmetry. Information asymmetry has a great impact on one provider but not another. The numerical simulations also show that the information asymmetry will increase the stability region of the system. Different adjustment suggestions are proposed for different providers when information asymmetry occurs.     

COMPETITION AND COOPERATION IN A DYNAMICAL MODEL OF NATURAL RESOURCES

Abstract In this paper, we propose a model describing the commercial exploitation of a common renewable resource by a population of strategically interacting agents. Players can cooperate or compete; cooperators maximize the payoff of their group while defectors maximize their own profit. The partition of the players into two groups, defectors and cooperators, results from the players' choices, so it is not predetermined. This partition is decided as a Nash equilibrium of a static game. It is shown that different types of players can exist in an equilibrium; more precisely, depending on the parameter values such as resource stock, cost, and so on, there might be equilibria only with defectors, cooperators, or with a combination of cooperators and defectors. In any case the total harvest depends on the renewable resource stock, so it influences agents' positions. It is assumed that at each time period the agents harvest according to Nash equilibrium, which can be combined with a dynamic model describing the evolution of fish population. A complete analysis of the equilibria is presented and their stability is analysed. The effect of the different Nash equilibria on the stability of the fish stock, showing that full cooperation is the most stable case, is examined.     

Dynamic modeling of the power market and analysis of its complex behavior based on a nonlinear complementarity function

In order to accurately simulate the dynamic decision-making behaviors of market participants, a new dynamic model of power markets that considers the constraints of realistic power networks is proposed in this paper. This model is represented by discrete difference equations embedded within the optimization problem of market clearing. Compared with existing dynamic models, the remarkable characteristic of the proposed model is twofold: it accurately reflects the process of market clearing by the Independent System Operator (ISO) while considering the inherent physical characteristics of power networks, i.e., the complex network constraints; and it describes the market condition that the generation and demand sides bid simultaneously. Using a nonlinear complementary function, the complex discrete difference dynamic model is transformed into a set of familiar discrete difference algebraic equations. Then, the complex dynamic behaviors of power markets are quantitatively analyzed. Corresponding to different operating conditions of power network, such as congestion or non-congestion, the Nash equilibrium of power markets and its stability are calculated, and the periodic and even chaotic dynamic behaviors are exhibited when the market parameters are beyond the stability region of the Nash equilibrium.     

Complex dynamics and chaos control of duopoly Bertrand model in Chinese air-conditioning market

A dynamic duopoly Bertrand model with quadratic cost function which is closer to reality and different from previous researches is discussed. The model is applied into air-conditioning market where the boundary equilibrium point is locally stable. Numerical simulations illustrate that the stability of Nash equilibrium strongly depends on the speed of adjustment of bounded rational player. The adjustment speeds and the degree of substitutability may undermine the stability of the equilibrium and cause a market structure to behave chaotically. The Lyapunov dimension of the chaos attractor is 1.9585 under some conditions. The stabilization of the chaotic behavior can be obtained by reducing the degree of substitutability. The results have an important theoretical and practical significance to Chinese air-conditioning market.     

Persistent equilibria in strategic games

A perfect equilibrium [Selten] can be viewed as a Nash equilibrium with certain properties of local stability. Simple examples show that a stronger notion of local stability is needed to eliminate unreasonable Nash equilibria. The persistent equilibrium is such a notion. Properties of this solution are studied. In particular, it is shown that in each strategic game there exists a pesistent equilibrium which is perfect and proper.     

Study on a 3-dimensional game model with delayed bounded rationality

A nonlinear dynamic triopoly game model is studied based on the theory of nonlinear dynamics and previous researches in this paper. A lagged structure is introduced to the model to study stability conditions of the Nash equilibrium under a local adjustment process when players price their products with delayed bounded rationality. Numerical simulations are provided to demonstrate the complexity of system evolvement and influence of the strategy of delayed bounded rationality on system stability. We find that besides the lagged structure, suitable delayed parameters are also important factors to eliminate chaos or expand the stable region of the system, and various players’ adjustment parameters have different effect on stability of the system.     

Approximating Nash equilibrium for production control with sticky price

We study a mean field game problem arising from the production control for multiple firms with price stickiness in the commodity market. The price dynamics for each firm is described as a (controlled) jump-diffusion process with mean-field interaction. Each firm aims to maximize her expectation of cumulative net profit coupled with each other through price processes. By solving the limiting control problem and a fixed-point problem, we construct an explicit approximating Nash equilibrium when the number of firms grows large.     

多用户类多准则交通分配的势博弈与拥挤定价

交通管理者在解决路网拥挤问题时,并不知道出行者的出行效用,同时管理者难以对出行者的路径选择行为做出准确的观测.运用势博弈理论分析多用户类多准则交通行为的演化过程,得到了固定需求和弹性需求情形下的可容许动态(一种刻画出行者通过转换路径增加当前效用的近似调整行为的演化动态),证明当路段时间函数和逆需求函数为严格单调、连续、可微时,所对应的交通分配是势博弈问题的惟一Nash均衡点.进一步研究了固定需求下的可变拥挤道路收费问题,得到了在当前系统状态下实现系统最优交通分配的拥挤收费水平.     

三群体2×2×2非对称演化博弈稳定性分析

针对目前三方演化博弈的稳定性研究不足这一问题,利用复制动态方程构建了一般化的三维动力系统,首先讨论了单群体策略演化趋势,接着根据李雅普诺夫稳定性理论分析了系统的渐进稳定性,并结合单群体策略的演化趋势对系统稳定性作了深入研究。研究表明:严格纯策略纳什均衡是ESS,不严格纯策略纳什均衡是线性策略收敛(自定义概念),所有类型的混合策略纳什均衡均为鞍点,共同划分了ESS的吸引域,并证明了零特征值非ESS定理,以及ESS不共边定理,在此基础上给出了N维双策略系统中ESS的最多个数。最后,设计了六组经典算例,首先结合研究结果分析了算例,接着对算例进行系统仿真,仿真结果与理论分析一致,为演化博弈的进一步研究提供借鉴与启发。     

考虑相对业绩的最优投资选择博弈问题研究

研究了具有相互作用的两个竞争机构投资者之间的离散时间最优投资选择博弈问题,每个机构投资者都考虑其竞争对手的相对业绩.机构投资者可以投资于相同的无风险资产和不同的具有相关关系的风险股票,以反映投资的资产专门化.机构投资者选择投资组合策略使得期望终端绝对财富和相对财富的效用最大.首先,定义了Nash均衡投资组合选择策略.然后,在机构投资者具有指数效用函数的假设下,得到了Nash均衡投资组合选择策略和值函数的显示表达式,分析了机构投资者之间的竞争对Nash均衡投资组合选择策略的影响.最后,通过数值计算给出了各种情况下Nash均衡投资组合选择策略和值函数与模型主要参数之间的关系.结果表明:机构投资者之间的竞争会影响其对风险的承担,投资机会集对机构投资者的Nash均衡投资组合选择策略和值函数与模型主要参数之间的关系会产生很大的影响.     

A class of games having Pareto optimal Nash equilibria

A simple two-person exchange model is considered first as a cooperative game without side payments, then as a non-cooperative game. It is shown that the Nash equilibrium solution of the latter agrees with the von Neumann-Morgenstern and core solutions of the former. This is made possible by the Pareto optimality of the Nash equilibrium, which is the main theoretical result presented.     

On the complexity of simultaneous price-quantity adjustment processes

While the Walrasian price tâtonnement represents the traditional dynamic process in the general equilibrium context with and without production, Walras and other classics designed the process exclusively for pure exchange economies. In productive economies, the short-run output adjustment of existing firms and the entry/exit of firms should be modeled as well. So-called cross-dual processes which represent the classical approach to the dynamics of productive economies are discussed and extended. Complex motion can emerge in a discrete-time version of the original two-dimensional system when the aggregate demand function has a non-standard shape. A simultaneous process of price and short-run quantity adjustment with free entry and exit of competitive firms in a single market with a continuum of firms can generate closed orbits via a Hopf bifurcation when the slope of the demand function is positive at equilibrium. When the continuum economy is replaced by an economy with a finite number of firms, noisy limit cycles and complicated behavior can be observed.     

An insurance network: Nash equilibrium

We consider d insurance companies whose surplus processes are r.c.l.l. functions (like the sample paths of perturbed Levy processes). Suppose they have a treaty to diversify risk; accordingly, if one company needs a certain amount to prevent ruin, the other companies pitch in previously – agreed – upon fractions of the amount, and any shortfall is got from external sources. With each company trying to minimise its repayment liability, the situation is viewed upon as a d-person dynamic game with state space constraints and a Nash equilibrium is sought. Under certain natural conditions, it is shown that the Skorokhod problem of probability theory provides a (unique) Nash equilibrium. The thrust of the paper is entirely deterministic.     

Market Power Issues in Bid-Based Hydrothermal Dispatch

The objective of this work is to investigate market power issues in bid-based hydrothermal scheduling. Initially, market power was simulated with a single stage Cournot–Nash equilibrium model. In this static model the equilibrium was calculated analytically. It was shown that the total production of N strategic agents is smaller than the least-cost solution by a factor of (N/(N+1)). Market power analysis for multiple stages was then carried through a stochastic dynamic programming scheme, where the decision in each stage and state is the Cournot–Nash equilibrium of a multi-agent game. Case studies with data taken from the Brazilian system are presented.     

无限阶段网络博弈中合作解的策略稳定性

合作博弈的经典合作解不满足时间一致性, 并缺乏策略稳定性. 本文研究无限阶段网络博弈合作解的策略稳定性理论. 首先建立时间一致的分配补偿程序实现合作解的动态分配, 然后建立针对联盟的惩罚策略, 给出合作解能够被强Nash均衡策略支撑的充分性条件, 最后证明了博弈中的惩罚策略局势是强Nash均衡, 从而保证了合作解的策略稳定性. 作为应用, 考察了重复囚徒困境网络博弈中Shapley值的策略稳定性.     

Nonlinear duopoly games with positive cost externalities due to spillover effects

A Cournot duopoly game is proposed where the interdependence between the quantity-setting firms is not only related to the selling price, determined by the total production through a given demand function, but also on cost-reduction effects related to the presence of the competitor. Such cost reductions are introduced to model the effects of know-how spillovers, caused by the ability of a firm to take advantage, for free, of the results of competitors' Research and Development (R&D) results, due to the difficulties to protect intellectual properties or to avoid the movements of skilled workers among competing firms. These effects may be particularly important in the modeling of high-tech markets, where costs are mainly related to R&D and workers' training. The results of this paper concern the existence and uniqueness of the Cournot–Nash equilibrium, located at the intersection of non-monotonic reaction curves, and its stability under two different kinds of bounded rationality adjustment mechanisms. The effects of spillovers on the existence of the Nash equilibrium are discussed, as well as their influence on the kind of attractors arising when the Nash equilibrium is unstable. Methods for the global analysis of two-dimensional discrete dynamical systems are used to study the structure of the basins of attraction.     

Absolutely expedient algorithms for learning Nash equilibria

This paper considers a multi-person discrete game with random payoffs. The distribution of the random payoff is unknown to the players and further none of the players know the strategies or the actual moves of other players. A class of absolutely expedient learning algorithms for the game based on a decentralised team of Learning Automata is presented. These algorithms correspond, in some sense, to rational behaviour on the part of the players. All stable stationary points of the algorithm are shown to be Nash equilibria for the game. It is also shown that under some additional constraints on the game, the team will always converge to a Nash equilibrium. Dedicated to the memory of Professor K G Ramanathan     

Nash equilibria in competitive project scheduling

We consider the problem of scheduling activities of a project by a firm that competes with another firm that has to perform the same project. The profit that a firm gets from each activity depends on whether the firm finishes the activity before or after its competitor. It is required to find a Nash equilibrium solution or show that no such solutions exist. We present a structural characterization of Nash equilibrium solutions, and a low order polynomial algorithm for the problem.     

Hedonic coalition formation games: A new stability notion

This paper studies hedonic coalition formation games where each player’s preferences rely only upon the members of her coalition. A new stability notion under free exit-free entry membership rights, referred to as strong Nash stability, is introduced which is stronger than both core and Nash stabilities studied earlier in the literature. Strong Nash stability has an analogue in non-cooperative games and it is the strongest stability notion appropriate to the context of hedonic coalition formation games. The weak top-choice property is introduced and shown to be sufficient for the existence of a strongly Nash stable partition. It is also shown that descending separable preferences guarantee the existence of a strongly Nash stable partition. Strong Nash stability under different membership rights is also studied.