## Reviews of Isaacs' Differential Games

 In 1965 Rufus Isaacs published the book Differential games. A mathematical theory with applications to warfare and pursuit, control and optimization. We list below short extracts from five reviews of the book, an extract from the Introduction and the publisher's description of the book.

1. Publisher's description:

One of the definitive works in game theory, this fascinating volume offers an original look at methods of obtaining solutions for conflict situations. Combining the principles of game theory, the calculus of variations, and control theory, the author considers and solves an amazing array of problems: military, pursuit and evasion, games of firing and maneuver, athletic contests, and many other problems of conflict. Beginning with general definitions and the basic mathematics behind differential game theory, the author proceeds to examinations of increasingly specific techniques and applications: dispersal, universal, and equivocal surfaces; the role of game theory in warfare; development of an effective theory despite incomplete information; and more. All problems and solutions receive clearly worded, illuminating discussions, including detailed examples and numerous formal calculations. The product of fifteen years of research by a highly experienced mathematician and engineer, this volume will acquaint students of game theory with practical solutions to an extraordinary range of intriguing problems.

2. From the Preface:

Although combat problems were its original motive, this book has turned out to be far from a manual of military techniques. Rather the result is a mathematical entity which fuses game theory, the calculus of variations, and control theory, and, because of its subsuming character, often transcends all three. As ideas burgeoned, they dictated their own course. Combat problems treated as genuine two-player games can be extremely difficult. (This matter is discussed at length in Chapter 11.) Their resolution demanded first a theory and it, one may logically say, is this book's real contribution. But it grew from solving problems. Each seemed accompanied by strange, new phenomena and, as each conception was mastered, still newer ones proliferated. The unanticipated was and still is a fascinating aspect of differential games. Baffling novelties never seem to cease appearing, and it is therefore yet hard to say how complete the theory is. To the reader seeking an introduction or a superficial acquaintance with the subject, I suggest the following program : Chapter 1 depicts the nature and scope of differential games. It presents typical problems, but says nothing about the mathematics or techniques needed for their solution. The latter can be sensed from Chapter 3, which is devoted to discrete models, some of them being quantized versions of problems later treated continuously. Because these discrete problems can be solved step by the reader can glimpse the concepts here without most of the formal mathematical tools. Chapter 2 essentially casts material such as appears in Chapter 1 in a formal mathematical mold, but the real theory does not begin until Chapter 4. Thus 1, 3, and a sketchy reading of 2 might be a good chapter order for casual acquaintance. The reader interested in military applications can turn to Chapter 11, either at once or after the foregoing prelude. The questions of what can be, should be, and may be done to attain military utility fill the early pages. The later ones, which contain specific illustrations, require the technology of the text for their understanding ; the reader can stop when he gets there. Possibly the lack of existence and uniqueness theorems will seem a heresy to some. The emphasis on the specific problem, although counter to current mathematical trends, I feel is good and, in this case, fitting. Without it, it is hard to see how the innovations of the theory could have come to light. Besides, the very opulence of their diversity would seem to preclude the above types of theorems, for such would run to unwieldy lengths were they to cover all cases. The applications themselves have become more diverse than I had at first dreamed, as the reader can discover by skimming the pages. Between its first publication - the Rand Reports, revised versions of which now constitute Chapters 1,2, 3, and 4 - there was a lapse of several years when I had no opportunity of giving differential games the concerted attention it required. An exceptional resumption occurred when I was on the staff of the Hughes Aircraft Company. The theory of collision avoidance between aircraft and ships is much more recondite than the uninitiated might suspect. An investigation, spurred by a series of headlined catastrophes, revealed an unexpected and elegant liaison with differential games. With cooperation rather than conflict between the two players, collision avoidance problems cohere to the same mathematical principals as games, providing "maximax" replaces minimax. Lack of space precludes the subject from the ensuing text; a separate publication will follow. As far as I was aware when writing it, this work was an original conception. But unavoidable delays in publication have possibly dimmed some of the sheen of its novelty. As has happened often before in the history of science, at the proper era the same concepts arise simultaneously and independently from widespread investigators. This work was largely a solitary task, and I was unaware of contemporary developments by others. In fact, it was just a few days after the completion of this manuscript (in March 1963) that I first saw the book by Pontryagin and others, which deals with minimising problems through the same basic devices as presented herein. The technique could be classed primarily as that of one-player games. In his dissertation of 1961, D L Kelendzeridze extended this technique to two players, and so to some extent his work tallies with mine. Besides these Soviet authors, the American contributions deal largely with the logical foundations of the subject. Berkovitz applies the calculus of variations to the strategy of one player, the opponent's being temporarily fixed. Fleming conceives a continuous strategy as lying between two discrete ones. These interesting devices of mathematical rigor appeared too late for their due incorporation in the present work. However, another native theory, roughly contemporaneous with this one, sounds so distinct that, while writing this book, I did not suspect an affinity. As developed by LaSalle and others, control theory is tantamount to that of one-player differential games and thus is largely a special case of the latter. I altered two terms of my Rand Reports to the present state and control variables in accord with control usage. The switching surfaces of this theory are similar to the singular surfaces of differential games. The question of controllability - what states can be reached from a given starting one - is essentially a specialization of differential games of kind (Chapters 8 and 9). Thus each science may enrich the other: control problems can be extended to games by adjoining an opposing player; the ideas of this book are applicable to control material by suppressing one player. However, I have retained, from the early genetic problems which suggested them, the names Pursuer and Evader for the two players, whatever be the nature of the game. An ensuing drawback has been the occasional impression of readers that the subject entails pursuit games exclusively. Without the allocation of time for the task by my then employers, the Institute for Defense Analyses, this volume would not have been completed. A vital part of my huge debt to this organization is owed in particular to Professor Bernard Koopman, now Director of Research, for his recognition of whatever value this work may have, both as a military tool and a mathematical theory. His willingness to accept a new idea, despite its unconventionality, is a virtue essential to our nation today. To my present employer, the Center for Naval Analyses, my gratitude is due for first bringing the work to published status as 'Differential Games', Research Contribution No. 1, on 3 December 1963 and for their tolerance in granting the time for final revisions.

3. From the Introduction:

It is only a few years since J von Neumann and O Morgenstern innovated the theory of games. Their work gave us such essential items as the concept of a strategy, the value of a game and, probably most important of all, a workable and sound delineation of an optimal strategy which might be either pure or mixed. The reader of this book should be conversant with these terms as they relate to zero-sum, two-player games, but further technical knowledge of game theory will be needed but rarely. These researches will be essentially self-contained; although they could not have existed without the above pioneering work, its ideas for us will be standards and models rather than tools. Game theory seemed to leap at once to the status of an accepted classic. The subject was innundated by a flood of papers and a wave of books. Optimism ran high at certain periods as to the revolution pending in certain domains of its applications, predominantly warfare and economics. In the former, which we are far better equipped to discuss, very little happened. What are the reasons for the failure of such high hopes? It seems there are two. One is the increased difficulty of the problems when - and such is the essence of game theory - there are two opponents with conflicting aims and each is to make the best possible decisions understanding and taking into account that his antagonist is doing the same. Such a situation is entirely different from what we might call the corresponding class of problems in classical analysis. As we shall discuss further in Chapter 3, it is possible to cast many of these problems in the form of one-player games. The added difficulty with two interacting players can be enormous. The second reason is the lack of methods for obtaining answers. Of the deluge of material written on game theory, most concerns general theorems and results, often of the highest calibre mathematics but very little of usable techniques for obtaining practical answers. Such, as the previous paragraph states, is not always easy, but even so it is seldom the goal. Mathematics today tends to favor more abstract and general ends.

4. Review by: Lamberto Cesari.
Mathematical Reviews, MR0210469 (35 #1362).

This book is an account of the main lines of 20 years of the author's pioneering work on the subject, which to a large extent appears here for the first time in print. The term "differential games" was coined by the author in 1951, and this term emphasizes the main point: games (generally two-player games) involving a system of ordinary differential equations. ... The book is a remarkable achievement as it presents a lifetime exploration into a class of difficult problems. It is the variety of phenomena in the large and the many types of singularities which, by lending much of the qualitative character to every single problem, interest the author most and form the core of this absorbing volume. On the other hand, the reader should not expect an existence analysis in the large (which could not account for the many types of singularities), nor a sophisticated discussion of the underlying classes of strategies, or differentiability properties, which are simply assumed when needed.

5. Review by: Steven Vajda.
The Mathematical Gazette 51 (375) (1967), 80-81.

The structure of such games, and of their treatment, is exhibited in great detail and with commendable clarity. We have here problems which can be solved by a systematic approach, which is rare in the theory of infinite games. If one player is a dummy, or can be ignored, then we are faced with control problems of the type commonly associated with the name of Pontryagin, with the calculus of variations, or with dynamic programming (rather neglected here). But the real meat of the book is in the great variety of types of possible solutions, and they are illustrated by a plethora of examples. Isaacs indicates also the frontier of his researches, and further desirable development. One feels here a challenge to mathematicians who have so far, on the whole, ignored his efforts, since they appeared first in RAND-Reports. In this book we have an easy approach to his studies, perhaps somewhat too explicitly spelled out for mathematicians, in a very personal style. Those who know the author also know that he would be the first to regret it if he remained the only harvester in this field

6. Review by: Hans S Witsenhausen.
Operations Research 14 (5) (1966), 957-960.

The study of differential games began in earnest in the early 50's at the Rand Corporation, motivated by military applications. Dr Issacs developed his theory in a series of Rand memoranda in 1954-55. The present volume is based on these groundbreaking investigations and on some later developments. Its appearance is particularly timely because of promising new applications in control theory. ... this book with its lively style, its profound insight, and its fascinating examples, is a most enjoyable and worthwhile addition to the applied mathematics literature. A long time may elapse before all the results are rigorously justified and all the research problems proposed by the author solved; but this only enhances the value of this pioneering work.

7. Review by: Hans Hermes.
Mathematics of Computation 19 (92) (1965), 700-701.

Although the theory and application of differential games (and control theory) has received much attention since 1950, I feel it will be worthwhile to begin with a short description of the types of problems dealt with in these fields, and in particular in this book. ... As the author remarks, the theory of differential games grew from solving problems, and this is the approach taken in the book. Little time or effort is spent on theorem proving, instead many diversified types of problems are formulated, often completely solved, and a theory introduced which stems from the method of solution. ... The wide range of possible applications of differential games is exemplified in the many examples discussed and solved throughout the text. While obtaining solutions to these intriguing problems, the author has done an excellent job in providing insight into the deep mathematical theories which exist and the difficulties which must still be overcome.

8. Review by: John A Bather.
Journal of the Royal Statistical Society. Series A (General) 129 (3) (1966), 474-475.

The subject is a special class of two-player, zero-sum games which can be represented by the motion of a point x in Euclidean n-space. ... The approach here closely resembles Bellman's method of dynamic programming and it is surprising that the author has not recognized this. There is an even closer correspondence between his construction of solutions, by integrating the equations of motion backwards from the terminal surface along the optimal paths, and the use of Pontryagin's "maximum principle". However, the study of the interaction between two opposing players does require special treatment and the book contains a careful analysis of various singular surfaces which can occur, for example, at a discontinuity in one of the control variables. Particularly interesting is the treatment of games of kind by considering the neutral barrier between the regions dominated by one or the other player. The theory is illustrated throughout by applications to deceptively simple pursuit games. Indeed, these are the main purpose and success of the book. It is perhaps as well, in view of the finality of most of the pay-offs, that the applications are rather more entertaining than practical. The presentation is for the most part admirable, but there are occasional verbal excesses: "chancify" seems a particularly objectionable substitute for "randomize". Unfortunately for the reader interested in stochastic control, this is the most memorable feature of the chapter on incomplete information.

JOC/EFR April 2015