Finding all Nash equilibria of a finite game using polynomial algebra

The set of Nash equilibria of a finite game is the set of nonnegative solutions to a system of polynomial equations. In this survey article, we describe how to construct certain special games and explain how to find all the complex roots of the corresponding polynomial systems, including all the Nash equilibria. We then explain how to find all the complex roots of the polynomial systems for arbitrary generic games, by polyhedral homotopy continuation starting from the solutions to the specially constructed games. We describe the use of Gröbner bases to solve these polynomial systems and to learn geometric information about how the solution set varies with the payoff functions. Finally, we review the use of the Gambit software package to find all Nash equilibria of a finite game.     

The company you keep: Qualitative uncertainty in providing a club good

Clubs are typically experience goods. Potential members cannot ascertain precisely beforehand their quality (dependent endogenously on the club's facility investment and number of users, itself dependent on its prices). Members with unsatisfactory initial experiences discontinue visits. We show that a monopoly profit maximizer never offers a free trial period for such goods. For quality functions homogeneous of degree of at least minus one, a welfare maximizer, motivated by distributional concerns to mitigate disappointed consumers' losses, always does. We demonstrate the robustness of this finding by showing that (i) without qualitative uncertainty (thus, no disappointed customers), neither welfarist nor monopolist offers free trials; and (ii) if the planner pursues an objective mixing welfare maximization with profit maximization, the likelihood of free trials increases with the weight put on welfare maximization. Regarding club quality and usage, the monopolist provides a socially excessive level of quality to repeat buyers when the quality function is homogeneous of degree zero. With nonhomogeneous quality functions, the monopolist permits too little club usage; quality may or may not be socially excessive.     

Dynamic Externalities and Policy Coordination

The paper introduces trade into dynamic models with externalities and capital accumulation, and evaluates the efficiency of the Cournot–Nash equilibrium. It considers mixed economies characterized by a blend of strategic and nonstrategic sectors. Also, there are two sources of interdependence: the existence of production externalities and the endogenous determination of market prices. It is shown that policy coordination is not needed when preferences are the same. In this case, the production externalities are internalized, so that an inefficient solution becomes the efficient integrated world equilibrium due to trade.     

东亚垂直分工网络与技术梯度研究——基于零部件贸易的视角

东亚因其迅速发展的国际生产网络而得到经济学界的广泛瞩目。本文在整理相关理论和实证文献的基础上,通过对人均收入水平的分析提出了东亚区域内广泛存在技术梯度差异的现实。作者根据Jones等人提出的国际垂直分割理论,建立增广引力模型,在区分零部件贸易和传统贸易的情况下,利用联合国贸易统计面板数据,对东亚垂直分工网络的技术结构进行了细致梳理。结果显示,近年来日本在核心部件上的作用有所下降,而四小龙实现了技术升级,作用得到较大的提升。作为东亚生产共享网络中的重要一极,中国同四小虎之间呈现出较强的互补性而非替代性。     

Production Costs,Scale Economies,and Technical Change in U.S. Textile and Apparel Industries

The elimination of quotas in textiles and apparel poses new threats from import competition. To survive, the sectors need to find least-cost methods of production. The production–cost structure of the U.S. textile and apparel industries is examined using a dual cost framework. A translog cost function is used to measure substitution elasticities between inputs, scale economies, and the nature of technical change. The scope for factor substitution in textiles remains limited with all substitution elasticities being less than unity. Labor and materials are complements in apparel production, but there is evidence of substitution between capital and labor. The rate of technical change is higher in textiles than in apparel. Given the intense import competition from low wage countries, in both industries, technical progress is labor saving. Overall, economies of scale are larger in apparel; however, scale economies have continued to increase in textiles.This research was supported by a grant from the National Textile Center.     

Tax efficiency in selected Indian states

This paper attempts to measure pure tax efficiency of fifteen major Indian states (Andhra Pradesh, Assam, Bihar, Haryana, Gujarat, Karnataka, Kerala, Madhya Pradesh, Maharashtra, Orissa, Punjab, Rajasthan, Tamilnadu, Uttar Pradesh and West Bengal) for the period 1980–81 to 1992–93 in a manner that allows this efficiency to vary both across time as well as across states. It is discovered that there is a moral hazard problem in the design of central grants in that higher grants by the central government to the state governments reduce efficiency of tax collection by these states. The less poor states are more efficient in tax collection. The rankings of states by tax efficiency for the various years do not converge. An index of aggregate tax efficiency is calculated and it appears that this index has been stagnating. It is argued that the weight placed on tax effort in the formula determining central grants to state governments should be increased to improve tax efficiency of state governments. First version received: November 1997/final version received: November 1998