Complex dynamic properties of Cournot duopoly games with convex and log-concave demand function

A Cournot duopoly game is proposed where the interdependence between firms depends on convex and log-concave demand function. In this paper, a model of two rational firms that are in competition and produce homogeneous commodities is introduced. The equilibrium points of this model are obtained and their dynamical characteristics such as stability, bifurcation and chaos are investigated. Furthermore, a multi-team Cournot game is introduced. Through simulation the dynamical characteristics of the equilibrium points of this game are illustrated.     

有限理性与一类多主从博弈问题的良定性

利用Ky Fan不等式证明了一类多主从博弈平衡点的存在性,并且定义了此类多主从博弈的有限理性函数.在非线性问题的良定性的框架下,使用有限理性证明了此类多主从博弈问题是广义Had-amard良定的和广义Tykhonov良定的.     

On modifications of Puu's dynamical duopoly

Puu's economic dynamical model is extended to the following cases: Bounded rationality, differentiated goods and bounded rationality with delay. It is shown that delay increases stability. Hence firms using bounded rationality with delay has a higher chance of reaching Nash equilibrium. Stability and instability (cycles and chaos) conditions for all these cases are determined.     

The dynamics of Bowley's model with bounded rationality

A nonlinear dynamical system which describe the time evolution of n-competitors in a Cournot game (Bowley's model) with bounded rationality is analyzed. The existence and stability of the equilibria of this system is studied. The stability conditions of the steady states for two and three players are explicitly computed. Complex behavior such as cycles and chaotic behavior are observed by numerical simulation. Delayed Bowley's with bounded rationality in monopoly is studied. We show that firms using bounded rationality with delay has a higher chance of reaching Nash equilibrium.     

Redefinition of Belief Distorted Nash Equilibria for the Environment of Dynamic Games with Probabilistic Beliefs

In this paper, a new concept of equilibrium in dynamic games with incomplete or distorted information is introduced. In the games considered, players have incomplete information about crucial aspects of the game and formulate beliefs about the probabilities of various future scenarios. The concept of belief distorted Nash equilibrium combines optimization based on given beliefs and self-verification of those beliefs. Existence and equivalence theorems are proven, and this concept is compared to existing ones. Theoretical results are illustrated using several examples: extracting a common renewable resource, a large minority game, and a repeated Prisoner’s Dilemma.     

Analysis of the dynamics of Cournot team-game with heterogeneous players

In this paper, we study a dynamical system of a two-team Cournot game played by a team consisting of two firms with bounded rationality and a team consisting of one firm with naive expectation. The equilibrium solutions and the conditions of their locally asymptotic stability are studied. It is demonstrated that, as some parameters in the model are varied, the stability of the equilibrium will get lost and many such complex behaviors as the period bifurcation, chaotic phenomenon, periodic windows, strange attractor and unpredictable trajectories will occur. The great influence of the model parameters on the speed of convergence to the equilibrium is also shown with numerical analysis.     

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.     

不完备偏好扩展式博弈的序贯均衡

将Kreps和Wilson提出的序贯均衡解概念推广到了存在不完备偏好的情形.首先给出了一个修正的颤抖手完美均衡的概念,然后应用它去证明不完备偏好扩展式博弈序贯均衡的存在性.     

On Puu’s incomplete information formulation for the standard and multi-team Bertrand game

Puu’s incomplete information dynamical system is introduced and applied for Bertrand Duopoly. Multi-team Bertrand game is formulated. It is a generalization of Liu’s work to dynamical non-convex multi-team games.     

On some variants of dynamical systems

A dynamical system motivated by discrete physics is studied. Fuzzy dynamical systems are used to study fuzzy discrete replicator dynamics of hawk–dove (HD) and prisoner's dilemma (PD) games. New solutions are obtained. Finally a preliminary study for fuzzy predator–prey model is studied and a new equilibrium is found.     

On Cournot–Bertrand competition with differentiated products

In this paper, we present a game theoretic framework for Cournot–Bertrand competition based on a nonlinear price function. The competition is between two firms and is assumed to take place in terms of pricing decision and quantity produced. However, the proposed objective function has not been used in literature before, yet the throughput obtained in this paper generalizes some of the existing results in literature. The competitive interaction between firms is described and analyzed using best-reply reaction, proposed adaptive adjustment and bounded rationality approach. The condition of stability of Nash equilibrium (NE) is induced by these approaches. Interestingly, we prove that there exists exactly a unique NE. Furthermore, it is noticed that when firms adopt best-reply and the proposed adaptive adjustment, the firms’ strategic variables become asymptotically stable. On the contrary, when the bounded rationality is used both quantity and price behave chaotically due to bifurcation occurred.     

Explaining cooperation in the finitely repeated simultaneous and sequential prisoner’s dilemma game under incomplete and complete information

Explaining cooperation in social dilemmas is a central issue in behavioral science, and the prisoner’s dilemma (PD) is the most frequently employed model. Theories assuming rationality and selfishness predict no cooperation in PDs of finite duration, but cooperation is frequently observed. We therefore build a model of how individuals in a finitely repeated PD with incomplete information about their partner’s preference for mutual cooperation decide about cooperation. We study cooperation in simultaneous and sequential PDs. Our model explains three behavioral regularities found in the literature: (i) the frequent cooperation in one-shot and finitely repeated N-shot games, (ii) cooperation rates declining over the course of the game, and (iii) cooperation being more frequent in the sequential PD than in the simultaneous PD.     

考虑相互依赖性的信息系统安全投资及协调机制

考虑信息系统安全相互依赖情形下最优化信息系统连续时间安全投资水平是一个值得研究的问题。首先讨论了非合作博弈下信息系统安全投资的最优策略选择,在此基础上讨论了安全投资效率参数、黑客学习能力、传染风险对信息系统脆弱性及信息系统安全投资率的影响。其次,在推导出两企业在合作博弈情形下最优策略选择的基础上,对比两种情形下的博弈均衡结果,得出合作博弈下的投资水平高于非合作博弈下的投资水平。原因是两个企业的相互依赖关系隐含着企业投资的负外部性,从而导致企业投资不足。最后,构建一种双边支付激励机制消除企业投资不足问题,从而使企业达到合作博弈下的最优投资水平,提高两个企业的收益。     

一类输出耦合混沌复杂动态网络不稳定平衡点的控制

研究节点输出耦合混沌复杂动态网络不稳定平衡点的控制问题,基于输出控制思想,提出网络节点不稳定平衡点的全局控制方法以及牵制控制方法,将混沌复杂动态网络的所有节点镇定到其平衡点.利用李稚普诺夫稳定性理论,得到控制器参数选择条件,以蔡氏混沌电路作为网络节点动态进行仿真研究,证明该方法的有效性.     

Modeling international investment decisions for financial holding companies

This research analyzes the internationalization process model developed by Johanson and Vahlne and derives two integer programming investment decision models that consider the risk attitudes of investment firms. Johanson and Vahlne’s model provides a starting point for building a model that suits the investment approach and decision making process of financial holding companies. In practice, when firms make an international investment decision, there is a need for a model that can generate outputs based on financial measures such as profit, investment returns, and tolerable levels of risk. Thus, in this paper, Johanson and Vahlne’s concepts are studied and financial managers are interviewed to derive models that match the investment decision procedures of the firms. The model helps firms manage the risks of their investments and derive accurate investment strategies based on investment objectives and constraints.     

不完全信息动态二维价格博弈模型及其分析

单位生产成本为不完全信息条件下,本文首先讨论了两个企业关于具有一定替代性的两种产品价格的动态二维博弈模型,并求得其精练Bayes均衡.然后分析了当两种产品不存在替代关系时,企业对这两种产品价格进行动态博弈的精练Bayes均衡相当于对这两种产品单独进行博弈的精练Bayes均衡的简单组合.     

Bounded rationality and correlated equilibria

We study an interactive framework that explicitly allows for nonrational behavior. We do not place any restrictions on how players’ behavior deviates from rationality, but rather, on players’ higher-order beliefs about the frequency of such deviations. We assume that there exists a probability p such that all players believe, with at least probability p, that their opponents play rationally. This, together with the assumption of a common prior, leads to what we call the set of p-rational outcomes, which we define and characterize for arbitrary probability p. We then show that this set varies continuously in p and converges to the set of correlated equilibria as p approaches 1, thus establishing robustness of the correlated equilibrium concept to relaxing rationality and common knowledge of rationality. The p-rational outcomes are easy to compute, also for games of incomplete information. Importantly, they can be applied to observed frequencies of play for arbitrary normal-form games to derive a measure of rationality \(\overline{p}\) that bounds from below the probability with which any given player chooses actions consistent with payoff maximization and common knowledge of payoff maximization.     

An attrition game on an acyclic network

This paper deals with noncooperative games in which two players conflict on a network through an attrition phenomenon. The associated problem has a variety of applications, but we model the problem as a military conflict between an attacker and a defender on an acyclic network. The attacker marches from a starting node to a destination node, expecting to keep his initial members untouched during the march. The defender deploys his forces on arcs to intercept the attacker. If the attacker goes through an arc with deployed defenders, the attacker incurs casualties according to Lanchester’s linear law. In this paper, we discuss two games having the number of remaining attackers as the payoff and propose systems of linear programming formulations to derive their equilibrium points. One game is a two-person zero-sum (TPZS) one-shot game with no information and the other is a TPZS game with two stages separated by information acquisition about players’ opponents.     

Robust Multiple Objective Game Theory

In this paper, we propose a distribution-free model instead of considering a particular distribution for multiple objective games with incomplete information. We assume that each player does not know the exact value of the uncertain payoff parameters, but only knows that they belong to an uncertainty set. In our model, the players use a robust optimization approach for each of their objective to contend with payoff uncertainty. To formulate such a game, named “robust multiple objective games” here, we introduce three kinds of robust equilibrium under different preference structures. Then, by using a scalarization method and an existing result on the solutions for the generalized quasi-vector equilibrium problems, we obtain the existence of these robust equilibria. Finally, we give an example to illustrate our model and the existence theorems. Our results are new and fill the gap in the game theory literature.