Games and Economic Behavior 46 (2004) 215–218 www.elsevier.com/locate/geb Handbook of Game Theory, Vol. 3 Edited by Robert Aumann and Sergiu Hart, Elsevier, New York, 2002. This is the final volume of three that organize and summarize the game theory literature, a task that becomes more necessary as the field grows. In the 1960s the main actors could have met in a large room, but today there are several national conferences with concurrent sessions, and the triennial World Congress. MathSciNet lists almost 700 pieces per year with game theory as their primary or secondary classification. The problem is not quite that the yearly output is growing—this number is only a bit higher than two decades ago— but simply that it is accumulating. No one can keep up with the whole literature, so we have come to recognize these volumes as immensely valuable. They are on almost everyone’s shelf. The initial four chapters discuss the basics of the Nash equilibrium. Van Damme focuses on the mathematical properties of refinements, and Hillas and Kohlberg work up to the Nash equilibrium from below, discussing its rationale with respect to dominance concepts, rationalizability, and correlated equilibria. Aumann and Heifetz treat the foundations of incomplete information, and Raghavan looks at the properties of two-person games that do not generalize. There is less overlap among the four chapters than one would expect, and it is valuable to get these parties’ views on the basic questions. It strikes me as lively and healthy that after many years the field continues to question and sometimes modify its foundations. A difficulty is that some chapters are several years old. Van Damme states that he wrote his piece on refinements in 1994, and adds a 1996 postscript noting a result that seems to challenge the whole project. For a game g, if we choose a strategy profile s and subset C of the players, we can define the reduced game with respect to s and C as the one where the players outside of C automatically play the strategies assigned to them in s, while the players in C have the same choices as in g. Consistency of a solution concept, by a definition of Peleg and Tijs (1996), requires that if s is a solution of the original g, then the players in C are at a solution of the reduced game when they use the strategies assigned by s. However, Norde et al. (1996) show that the only solution concept that is non- empty, individually rational, and consistent is the Nash equilibrium. Thus any reasonable refinement of Nash will run afoul of consistency. It seems hard for refinement advocates to dismiss consistency, since it is so close to the basic rationale for the Nash equilibrium. Given van Damme’s role as an integrator and expounder of different solution concepts, this reviewer would like to hear his views on what this surprising result means for refinement theory. 0899-8256/2003 Published by Elsevier Inc. doi:10.1016/S0899-8256(03)00172-6 216 Book review / Games and Economic Behavior 46 (2004) 215–218 Oliver Wendell Holmes wrote about a deacon who made his one-horse shay out of parts of exactly equal sturdiness, so that they all fell apart at once—“all at once and nothing first, as bubbles do when they burst.” The problem for a review volume is to bring all the parts together at once so the surveys will be equally current, and that seems to be harder. There are perverse incentives here—the chapter submitted first will be the most out of date. The survey of stochastic games has a good answer: Mertens’ article stops at 1994 and Vielle brings it up to date. Both papers make this difficult subject more accessible with clear examples. This volume of the Handbook includes some less standard topics. Ali Khan and Sun discuss many-player non-cooperative games, and von Stengel writes on the computation of equilibria for two-person games, as used in the McKelvey, McLennan, and Turocy’s GAMBIT and Pfeffer and Koller’s GALA, computer programs to find numerical equilibria. Downloadable flexible programs that find equilibria or refinements are a new development made possible by fast personal computers and the internet. They are improving and it will be interesting to see how they interact with theoretical research. A group of articles surveys non-cooperative applications to specific contexts. Banks writes on political institutions, and Benoit and Kornhauser on legal applications. Ausubel, Cramton, and Deneckere integrate the literature on bargaining with incomplete informa- tion, emphasizing that element as a source of inefficiency. Palfrey surveys one part of implementation theory, those social choice rules for which there exists some implementing mechanism yielding the rule as an equilibrium (as opposed, for example, to rules imple- mentable by strategies that are dominant or resistant to coalitions). An important milestone was Tirole’s 1988 text on industrial organization, which showed that an entire field of economics could be treated in a unified way by a game-theoretic approach. Bagwell and Wolinsky survey industrial organization, focusing on those results for which game theory added new content (as opposed to it formalizing existing ideas). They translate the theories into a basic model in which two firms compete over two periods, then modify the model to discuss international trade, entry deterrence, predation, collusion, and promotional “sales.” Avenhaus and von Stengel survey arms control inspection games, including the problem of allocating a limited quota of inspections among suspicious events, as well as “material accountancy,” meaning monitoring reactors under the Non-Proliferation Treaty, to detect any nuclear material that is being diverted to weapons production. There will be some noise in the data, and possibly an adversary trying to escape detection. The literature is part of operations research in that it uses readily measured variables to generate specific advice. The authors stop short of saying that the authorities have used these techniques, and it would be interesting to know why not. These are momentous issues that have just been used to justify a war. The volumes do not include game theoretical applications to business subjects, such as finance, operations management, or financial and managerial accounting, but this gap is filled by the recent book edited by Chatterjee and Samuelson (2001). As a researcher in political science I sometimes find the game-theoretical literature frus- trating, since it talks as if it were describing social conflict but pays so little systematic attention to empirical facts. Only two articles in the volume show a sustained interest in evidence, and only one of these involves naturally occurring social phenomena. Shubik Book review / Games and Economic Behavior 46 (2004) 215–218 217 surveys experimentation directed at testing game-theoretical ideas, excluding, reasonably, the thousands of experiments where subjects played matrix games to investigate person- ality, culture, communication or other contextual variables. The other chapter is Grief’s summary of economic history applications. It is worth reading to see how game theory can be more than a mathematical structure, how it can explain phenomena in the world. The literature focuses on the design of rules that maintain stable economic relationships and coalitions—Jewish Maghribi traders in the eleventh century Mediterranean area, or Amer- ican merchants operating in Mexican California, or the credit system in pre-revolutionary France. The contexts change over time and place, and this generates a natural laboratory, as players choose different institutions to fit their situations. The quest to reduce equilibria described in van Damme’s chapter is less pressing for these writers, who want a degree of flexibility for their theories. Movement from one equilibria to another is something to be explained from the empirical context rather than a general criterion of rationality. Surveys by Winter, McLean, Neyman, Monderer and Samet, and Hart discuss the Shapley value and its generalizations: extensions to NTU games, games with an infinity of players, and games representing competitive economies. Mertens treats applications to taxation and public goods. Derived from the work of Aumann and Kurz, this area strikes me as especially interesting, but from the dates of the references it seems to have halted in the 1980s. Around the time of the founding, strategic form and characteristic function theory were conceptually united, the latter seen as the natural extension of the former using a definition of the characteristic functions based on the strategic form. The year 1952 saw the first mathematical textbook in which J.C.C. McKinsey explained n-person games using characteristic functions and added a footnote that “another approach to this subject” had been recently developed by Nash (McKinsey, 1952). Of course, this other approach took over as the natural way to generalize two-person games, while characteristic function theory came to be seen as a separate topic. Some results have appeared uniting the two approaches, but the articles in this section suggest that this has not been a major research theme. The Handbook reflects the field, but it also has had a role in defining it by including or omitting a particular literature. Differential game theory is underrepresented compared to the volume of work, and combinatorial games (surveyed by Guy, 1996) are not there at all. This makes sense in that these parts interact very little with the rest of the field. However, there is one aspect of the Handbook’s chosen boundary that I think is unfortunate. By its title and emphasis it is a book of theory with a focus on economics—empirical applications are there but treated more briefly. The effect on both theoreticians and appliers is for the worse, since the former go off on flights and miss some important applications, and applied users stay less sophisticated.