1.1. Review by: Darald J Hartfiel.
Mathematical Reviews MR0414573 (54 #2674).

A Boolean matrix is an $m \times n$ matrix composed of zeros and ones. The sum and product of such matrices are to be computed by using the sum and product over the Boolean algebra {0,1} of order two. This lecture note, consisting of four chapters, is concerned with the algebraic theory of such matrices. The major sources of material for the note include the author's dissertation On (0,1)-matrix semigroups, Ph.D. Thesis, George Washington Univ., Washington D.C. 1970, a privately circulated paper of G Markowsky, and a publication of R J Plemmons (1971).

2.1. Review by: Ronald H Atkin.
Mathematical Reviews MR0567330 (81c:00003).

This book is a readable introduction to the kind of mathematics which some social scientists are now using. The first part of the book (nine chapters) discusses the basic ideas of set theory, matrices (Boolean and others), graphs, some topics in combinatorics, difference and differential equations, Markov chains, renewal theory, cluster analysis, and ends with an inadequate critique of modelling theory. The second part consists of seven chapters on "applications'', viz., demography, economics, management, game theory, psychology, sociology and informatics.

3.1. Review by: John Baylis.
The Mathematical Gazette 69 (449) (1985), 241-242.

The general problem of consensus theory is that of amalgamating the wishes or preferences of the individual members of a society in order to decide what is the consensus view by which all the members agree to abide. The theory is interesting outside mathematics, particularly in politics and economics and other social sciences, because of the wide range of interpretations of its basic terms; for example, the members of the "society" may be the people of the British electorate, or sets of such people like parliamentary constituencies or departments in a university or the nations of the E.E.C. etc. The mechanisms by which committees reach their decisions provide an interesting small-scale application of the theory. Whatever the "society" is, its members have to make choices, and these could be ranking a set of candidates in an election, choosing a policy for a club, choosing between economic options etc. Mathematically the theory is interesting because concepts like democracy, dictatorship, liberalism, manipulation of voting systems and their consistency and equilibrium can all be formulated mathematically. The mathematical machinery involved is a rich mixture of the theory of order relations, graph theory and other topology, multi-variable calculus, game theory, Boolean matrices, Markov chains etc. The theory also has its share of counter-intuitive surprises, the most important of which is Arrow's impossibility theorem - the single result which detonated the explosion of interest in the subject since the early 1950s. ... The reader whose interest in this subject stems from political or economic reality will be disappointed since the main interest of the authors is clearly mathematical, and their book is principally about mathematics. The word "introduction" in the title is misleading as the book is very thin on motivation and definitions tend to be unnecessarily general and abstract. It is difficult to judge the relative importance of the closely packed theorems; some theorems which are interesting because of their obvious interpretation in "the real world" are only stated whilst purely technical results are proved in full. If you already know quite a lot about the subject and your interest in it is mainly mathematical then this is a useful book, especially as a guide to the literature ...

3.2. Review by: Mark A Aïzerman.
Mathematical Reviews MR0591681 (82e:90012).

The book is mainly dedicated to the review and analysis of the results of the last few years that have been obtained under the stimulus of the Arrow theorem well known in the theory of collective choice. The authors are concerned less with the completeness of their list of published results than with the attempt to reveal their underlying logic. Remaining as they do in the framework of the Arrow paradigm the authors are deprived of the possibility of clarifying the causes of Arrow-type paradoxes. They concentrate on the analysis of different combinations of conditions imposed on such cases, discussing the connections between Arrow paradoxes and the manipulation problem (in the sense of Gibbard's contributions) and some game-theoretic aspects of the problem. ... The brevity and simplicity of exposition make this a useful book for beginners. At the same time, some results of Chapters IV and V will be of interest even to specialists.

4.1. Review by: Sergiu Rudeanu.
Mathematical Reviews MR0655414 (84a:15001).

This book deals with matrices over the Boolean algebra $B_{0}$ = {0,1}. The theory of such matrices arises from at least two facts: (1) addition and multiplication can be defined as for matrices over an arbitrary ring, except that the role of scalar addition is played by the Boolean join
   0 + 0 = 0,
   0 + 1 = 1 + 0 = 1 + 1 = 1,
and (2) the multiplicative group $B_{n}$ of square Boolean matrices of order $n$ is isomorphic to the semigroup of binary relations over an $n$-element set. Numerous applications of 0-1 matrices within and outside mathematics are described in this book or mentioned with appropriate references.

5.1. From the publisher.

This book contains all the necessary subject matter of abstract algebra, featuring the most recent topics and illustrative applications it approaches the subject from a broad, non numerical viewpoint. The material has been worked as a simple and concrete introduction, leading to a good cross-section of applications which include voting theory, automata theory, mathematical linguistics, sociology, symmetries in physics and geometry, kinship systems, geometrical constructions, and design of experiments for statistical analysis The authors include newer topics connected with relational structures, to complement the traditional emphasis on operational structures. Delving into their 30 years of combined teaching and research experience, the authors organise the material into major areas of abstract algebra, such as groups or rings, which are divided into sections which can be covered in one, two or three lectures. They add exercises to test the student's comprehension, graded in three levels to accommodate varying different backgrounds. Advanced exercises provide important results beyond those given in the text. Among the major topics covered here, which are neglected in other books at this level, are binary relations, lattices, semigroups. Boolean matrices, directed graphs, network theory, group representation (both finite and continuous), all Important tools for the applied mathematician.

5.2. Review by: John Baylis.
The Mathematical Gazette 68 (445) (1984), 231-232.

The title sounds hopeful and indeed the contents make impressive reading. The book is basically about abstract algebra but there is an unusually wide range of applications and an unusual relative emphasis amongst the purely mathematical topics. These non-standard features make it a useful text for any lecturer contemplating giving his algebra course a new slant, but for the reasons outlined below it cannot be recommended to students as a lecture back-up, and especially not for unassisted private study. ... I turned in hope rather than confidence to Arrow's theorem only to find its explanation and proof virtually incomprehensible. Polya's enumeration which demands clarity of exposition failed completely, and at this point I ran out of patience and energy. Have the author's really proof-read their book, or even read it?

5.3. Review by: Boris M Schein.
Mathematical Reviews MR0708279 (84j:00001).

This book is intended for a two-semester course in abstract algebra. It emphasizes relational structures as well as operational structures; basic ideas valid for all types of structures are given parallel treatment. Applications in many areas illustrate the usefulness of abstract algebra; exercises are graded in three levels to accommodate students with varying mathematical backgrounds. The book contains some important open problems in algebra; students would understand the problems after the course, thus having some idea of what research is like in mathematics. There are six chapters: Sets and binary relations, Semigroups and groups, Vector spaces, Rings, Group representations, and Field theory. As a whole this is one of the best books on applications of abstract algebra the reviewer has ever seen. However, its rather terse style of a mathematical monograph without long verbal explanations may make its reading difficult for students with no previous exposure to abstract algebra ...

6.1. Review by: John Asafu-Adjaye.
The Journal of the Operational Research Society 35 (8) (1984), 777-778.

Economists today are a sharply divided group of professionals, and I am sure that an increasing number of people is becoming sceptical about economic theory in general. It was once thought that the ideas of the great Lord Keynes were the ideal cure for our economic ills. Let governments increase spending (incurring huge deficits in the process), and the economy will be stimulated and jobs created. We do know now that this kind of policy tends to cause other problems, such as inflation. Then came the monetarists, who preached that these problems could be better solved by cutting government spending and controlling money supply. Recent experience in the U.K. and elsewhere has shown that this policy does a marvellous job on inflation but tends to increase unemployment. The fact of the matter is that there is a need for new concepts to deal with today's economic problems. Classical economic concepts have failed to address adequately the problems of stagflation, income distribution, the environment, the struggle between governments and powerful unions and the allocation of public goods. This book presents a comprehensive survey of results of recent mathematical economics research and a discussion of their relevance and application to some of the above economic problems. The two main approaches adopted are game theory (situations in which n agents make an input into a process, with each trying to maximize a different function of the resulting output) and social welfare theory (what is best for a group?). The book is targeted at graduate students, professional economists, management scientists, mathematicians, planners and administrators, as well as the lay public.

6.2. Review by: Zvi Artstein.
Mathematical Reviews MR0709425 (86g:90002).

Game theory was created, and is still being developed, as a tool for the modelling and analysis of situations of conflict including economic systems. The mathematical questions and models addressed in this theory are, to a great extent, isolated from the real world. This is so partly since only in isolation can the participating forces be fully understood, and partly because the real social world is too involved for a transparent model. Yet the gap between the rigorous mathematical analysis of the models and the verbal description of real life is bothersome. This gap is wide in particular in economic applications, and it is this gap that the book attempts to bridge. The choice of subjects is very attractive: History of some economic conflicts; Equilibria; Noncooperative strategies; Cooperative actions and bargaining; Disequilibrium and dynamics; Arbitration and social welfare functions. In each of these subjects the core mathematical development is presented, and along with it, associated real life situations are exhibited. Integration of the two subjects is a different story.

7.1. Review by: Siegfried Gottwald.
Mathematical Reviews MR0769102 (86e:06001).

An incline is an algebraic structure with two associative operations, called addition and multiplication, such that addition is commutative and idempotent, multiplication distributes over addition, and the special incline property $x + xy = x$ holds. Thus, inclines are special cases of semirings and they generalize Boolean algebras, fuzzy algebras and even distributive lattices. ... The book offers a new kind of algebraic structure whose value on its own (and not only as a special case of semirings) is hard to evaluate. Overall, the presentation is easy to grasp, though sometimes a bit sketchy, but the reviewer felt a lack of examples which would convince the reader that the introduction of inclines is a necessity.

7.2. Review by: Sergi V Ovchinnikov.
Mathematical Reviews MR0921235 (89b:08001).

The book studies properties and applications of incline algebras. An incline is defined as a set $S$ with associative, commutative and idempotent addition, and associative, commutative and distributive multiplication satisfying the following incline condition: $x + xy = x$, for all $x$ and $y$ in $S$. Chapter 1 is concerned with basic properties of inclines. ... Vectors and matrices over inclines are studied in Chapter 2. ... In Chapter 3 idempotent matrices over inclines are studied. ... Chapter 4 is concerned with topological inclines. ... Powers of matrices over inclines are studied in Chapter 5. The last chapter of the book is an overview of applications of inclines in such areas as linear systems, probabilistic reasoning, social choice, automata and clustering.

8.1. Review by: Edward W Packel.
SIAM Review 30 (4) (1988), 676-677.

The last several decades have seen increasingly sophisticated use of a growing variety of mathematics in the social sciences. Statistical methods and a variety of optimization techniques in classical analysis have long been fundamental tools in economic theory. However, recent work in economics, political science, and other social sciences is making increasingly powerful use of probability theory, linear algebra, topology, graph theory, functional analysis, complexity theory, and other branches of mathematics. In addition to more complex and abstract models, both deterministic and probabilistic, and theorems developing their properties, axiomatic characterizations are playing an important role in laying bare the assumptions behind a particular approach. The book under review illustrates the assertions made above in its treatment of the theory of teams, a branch of economics initiated in the early 1970s by Marschak and Radner. A "team" can be thought of as a set of members with common goals but having individual information and courses of action. ... Unfortunately, the book is beset with a variety of technical problems that will prevent all but the most tolerant, persistent, and experienced readers from reaping the benefits of the later chapters. The writing is poor in terms of both grammar and style. Equally seriously, there are numerous errors, both typographical and mathematical. Examples, though they are offered, are often perfunctory and unenlightening. One is willing to tolerate a degree of expository license in a research monograph in order to get to the new ideas, but Team Theory exceeds the bounds, eroding the reader's tolerance of and respect for the material presented. Sections too often read like an outline, doing little to illuminate except for those who need no illumination.

8.2. Review by: Geert Jan Olsder.
Mathematical Reviews MR0890516 (88k:90001).

This book deals with teams. A team is defined as a group of persons with identical interests but whose information and actions are individual. The book not only deals with the players within one team but also with competition of teams with each other, thus entering the realm of game theory. The preface says that in the past fifteen years, after the appearance of the book by J Marschak and R Radner, Economic theory of teams (1972) (still the standard work), a great deal of work has been done in game theory, incentives and the theory of algorithms. The aim of the authors has been "to extend the theory of games using this work to questions of assignment, efficiency of cooperation, search, coordination and team games''. Elements of team theory are found in many disciplines such as applied mathematics, social science, economics and operations research. A glance through the list of contents will show that many subjects of interest to those disciplines are dealt with.

8.3. Review by: Jonathan H Klein.
The Journal of the Operational Research Society 39 (7) (1988), 695-696.

Twentieth century modern art, with its emphasis upon abstraction of visual and other content, can tell the sophisticated viewer a great deal about the nature of the subject of a painting. On the other hand, if you actually want to recognize who or what your are looking at, you might well be better off with an old master. A lot of mathematical modelling is a bit like modern art. It can provide deep and valuable insight, enhance understanding, and function as a spring-board for further thought. But for more mundane tasks, like solving real, messy problems, many mathematical techniques are inappropriate. Team theory is a mathematical theory intended to describe the way in which a group of individuals, each with their own information sets and possibilities for action, can coordinate their efforts to achieve common goals. It can be regarded as an extension of decision theory and game theory - mathematical models which have recognized worth as abstract descriptions of many classes of decision situation, as well as a reputation as useful, practical decision aids in some circumstances. These qualities are most in evidence in the more simple, transparent renderings of the theories. Thus, team theory is a development of ideas the practical value of which has tended to be to illuminate, structure and conceptualize problems, rather than to provide comprehensive simulations that can prescribe solutions. Does team theory continue in this vein, or depart from it? Kim and Roush's book summarizes the mathematics of the field of team theory, and develops it in various directions. ... Essentially this is a mathematics book, written by mathematicians for mathematicians ... Kim and Roush's book will appeal to those with an interest in relatively pure mathematics, and would not be at all out of place in a library of mathematical operational research. Theorists specifically interested in the topics covered in the later chapters may also find the material useful. Team theory as presented here could have potential for development in more immediately practical contexts, and modellers whose interests lie in such contexts (such as organizational theory or simulation) could find this book worth looking at on a speculative basis. For operational research practitioners, however, the content of the book will be of little interest.

8.4. Review by: Clive W Kilmister
The Mathematical Gazette 73 (465) (1989), 277.

Mathematics owes a debt to economics for the systematic development of game theory, especially zero-sum, two-person games. It is true that more-person games and cooperative games soon get too complicated for the dilettante applied mathematician; none the less, it is worth keeping an eye open for any new developments in the same general field. What then is team theory? A team is a group of individuals with a common goal but individual actions (and information). The basic problem is therefore to maximise a utility function whose variables are, firstly, the state of the environment, and then the actions of each of the members of the team. As early as 1972 Marshak and Radner were able to show that, if the utility function is concave and differentiable in the actions, it is necessary and sufficient to maximise for each member separately. So much for the simple idea; but life is not simple. How is information to be shared amongst members? (Imagine here the team of managers in a large corporation rather than the members of a football team-and we are warned that in the US there is competition even inside the corporation, so that production units in a socialist economy would be better.) Meetings cost money (a fact largely ignored in educational organisations). How much is worth spending to increase team members' knowledge and how much should they rely on guesswork (that is, on knowledge of yesterday's events)? Is there a reasonable theory for two teams taking part in a game? The present book is by two people who have themselves contributed many of the basic ideas of this new theory. ... A new field for applied mathematicians to get into - neatly explained.