×

A survey of dynamical games in economics. (English) Zbl 1237.91002

Surveys on Theories in Economics and Business Administration 1. Hackensack, NJ: World Scientific (ISBN 978-981-4293-03-7/hbk; 978-981-4293-04-4/ebook). xiv, 275 p. (2010).
Game theory has come into view of the growing need for scientists and economists, and has dealt with tactical interactions among multiple decision makers. Antonine Augustin Cournot’s (1838) work is the first important study in game theory (a concept close to Nash equilibrium for a duoploly model). In the 20th century, people like Zermelo and Borel became interested in two-player zero-sum and parlor games. The seminal work of game theory is the famous book by [J. von Neumann and O. Morgenstern, Theory of games and economic behavior. Princeton, New Jersey: Princeton University Press. (1944; Zbl 0063.05930)]. The mathematical formulation and study of the dynamic games (differential games) was initiated by [R. Isaacs, Differential games. A mathematical theory with applications to warfare and pursuit, control and optimization. (The SIAM Series in Applied Mathematics) New York etc.: John Wiley and Sons, Inc. (1965; Zbl 0125.38001)]. Dynamic games are models of interactive players and the surrounding world evolving over time. There are two types of differential games: (i) deterministic ones and (ii) stochastic ones.
The first important theorem in game theory is the minimax theorem proved by John von Neumann (1928). The second important is the equilibrium concept introduced by John Nash in 1950. John Nash, John Harsanyi and Reinhard Selten were awarded the Nobel Prize in Economics for their pioneering analysis of equilibria in the theory of noncooperative games. Nash equilibrium theory is now recognized as one of the outstanding intellectual advances of the twentieth century. The Oscar winning film “A Beautiful Mind” about John Nash was released in 2001. A great majority of people started to pay attention to game theory and its usefulness in social science and economy. More recently, Robert Aumann and Thomas Schelling were awarded the Nobel Prize in 2005 for having enhanced our understanding of conflict and cooperation through game-theory analysis, and the Nobel Prize in 2007 went to L. Hurwicz, E. Maskin and R. Myerson for having laid the foundations of mechanism design theory through game theory.
The close relation between differential games and optimal control theory was a fusion to motivate interest in differential games. The earlier work on differential games was based on the Hamiltonian-Jacobi-Isaacs (HJI) dynamic programming. Berkovitz, Fleming, Elliott, Friedman, Kalton, Krasovskii, Subbotin, Uchida among others showed that smooth solutions for HJI do not exist in general and nonsmooth solutions are highly nonunique. Crandall and Lions, Fleming and Soner introduced the viscosity solutions for HJI to characterize the value function as the unique solution satisfying certain boundary conditions. Swiech later gave a rigorous study on the viscosity solution of the Hamilton-Jacobi-Bellman-Isaacs equation in infinite dimensions. The dynamic games as pursuit-evasion games, viscosity solutions, zero-sum games, cooperative and noncooperative games can be adapted to social sciences and economics.
C. F. Roos [“A mathematical theory of competition”, Amer. J. 47, 163–175 (1925; JFM 51.0304.04); “A dynamic theory of economics”, J. Polit. Econ. 35, 632–656 (1927)], first introduced the game theory to economics. S. Clemhout and H. Y. Wan jun. [“Differential games: economic applications”, in: Handbook of game theory with economic applications. Vol. 2. Amsterdam: Elsevier. 801–825 (1994; Zbl 0925.90087)] gave an excellent survey on dynamic games with special emphasis on economic applications. There are other excellent books, for example [T. Basar and G. J. Olsder, Dynamic noncooperative game theory. London: Academic Press. (1995; Zbl 0828.90142)], [E. Dockner, S. Jørgensen, Ngo Van Long and G. Sorger, Differential games in economics and management science. Cambridge: Cambridge University Press. (2000; Zbl 0996.91001)] and [S. Jørgensen and G. Zaccour, Differential games in marketing. Boston: Kluwer Academic Publishers. (2004)]. For the political economy, a central result in macroeconomic policy-making is that the time-consistent equilibrium is typically inferior to the first-best if the private sector believes that a benevolent government is unable to stick to a time path of its policy. The reaction of this finding, economists argued that a welfare improvement can be obtained if some partial commitments can be institutionalized, such as central bank to target the inflation rate or nominal money supply. Barro and Gordan (1983) model the government to boost employment by creating unanticipated inflation, Rogoff (1985) shows that the appointment of a conservative central banker in this model will lead to a low time-consistent rate of inflation. Ploeg (1995) models that the government has incentive to use temporary bouts of inflation and taxation to financing spending, and commit to a time path of government spending though it cannot commit to a time path of real taxes and seigniorage.
For the dynamic games in international economics and development economics, the sticky-price model assumes two firms that produce a homogeneous product and that use output as their control variable. Fershtman and Kamien (1987) show that the equilibrium feedback strategy of each firm in the sticky-price model has the property that its output is an increasing function of the current price, and the industry output in the steady state is higher than the static Nash-Cournot industry output. The tariff and quota in this model are not equivalent under dynamic duopolistic competition. Dockner and Haug (1990) give two equilibrium feedback output rules provided the tariff rate is constant. The equilibrium feedback rules illustrate that outputs respond positively to increases in the market price, and the long-run market prices under tariff and quota are the same when the number of firms goes to infinity. Fujiwara (2009) studies the welfare effect of trade involving a natural resource and reciprocal dumping via dynamic games. A surprising result is that trade liberalization reduces welfare. Fujiwara and Long (2000) model a dynamic contest between a home firm and a foreign firm that compete for a government procurement contract at each point of time, and focus on the open-loop Nash equilibrium. They find that trade liberalization, in the form of reducing bias against the foreign firm, improves both domestic and global welfare if the foreign firm’s profit is sufficiently large or the initial degree of home bias is sufficiently small.
The book under review gives an excellent survey on dynamic games in economics. The first chapter introduces basic concepts and examples to illustrate the ideas in dynamic game theory. Then the author lists six subfields in each chapter to survey the information about dynamic games in environment economics, natural resources economics, trade an development economics, industrial organization, public economics and macroeconomics. The layout of each chapter is a balance between model developments and research results in this direction as well as the summary of some related works. The exposition in each chapter is novel, important concepts are emphasized. Nevertheless, the leading conjectures or main problems in the subject are not presented and some technical issues are not addressed. It would also be helpful to have future research topics or directions. Other interesting applications of dynamic games in economics can also be found in [M. L. Petit, Control theory and dynamic games in economic policy analysis. Cambridge etc.: Cambridge University Press. (1990; Zbl 0726.90002)]. Overall, this is an outstanding book for any graduate student or researcher interested in dynamics games with economic applications.

MSC:

91-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to game theory, economics, and finance
91A25 Dynamic games
91A80 Applications of game theory
PDFBibTeX XMLCite
Full Text: Link