When R. E. Moore introduced the idea of interval computation in the later 1950s, his motivation was to automatically validate computational results of floating-point operations performed by digital computers. To accomplish the objective, the basic approach he employed was to enclose a real with its lower and upper bounds, respectively, in machine-presentable floating-points as an interval. A new branch of computing, named interval analysis or interval computing, was then developed. Very different from traditional point-valued computation, interval computing performs all operations on interval-valued data. Consequently, interval computing can bound machine representation and rounding errors caused by finite digits floating-point arithmetic with interval-valued data and operations. Moreover, interval computing enables us to enclose other kinds of uncertainties in computing. For instance, interval-valued payoffs in cooperative games are applied in this book to model uncertainty in economic management.

This book consists of three chapters to expand existing knowledge of cooperative games from point-valued payoff to interval-valued. At the very beginning of the first chapter, the author briefly reviews basic concepts of cooperative games, in terms of traditional point-valued payoffs, such as core, imputation, monotonicity, super-additivity, weakly super-additivity, and so on. These concepts are extended to interval-valued payoffs with the capacity of modeling uncertainty in section 1.3. Solving such an interval-valued cooperative game is then presented as an interval-least squares problem of minimizing interval distance. In section 1.4, interval quadratic algorithmic approaches without and with considerations of the efficiency condition of a game are presented. The last section of this chapter provides constructed illustrative examples with computational results.

The focus of chapter 2 is on satisfactory interval-valued cores of interval-valued cooperative games. The author measures the degree of satisfaction of an interval-valued core with a fuzzy partial order relation between intervals established earlier by other researchers in the field. In this chapter, a bisection approach of finding an interval-valued core is proposed. Constructed examples show that an interval-valued cooperative game may or may not have an interval-valued core if using Moore’s order relations between intervals. Therefore, it is a good idea for using the fuzzy partial order relation.

In cooperative game theory, Shapley value is an important solution that assigns a unique distribution (among the players) of a total surplus generated by the coalition of all players. The last chapter of this book discusses direct and effective methods in finding interval-valued egalitarian Shapley value for interval-valued cooperative games with simplifications such as equal division and equal surplus division. The chapter further generalizes solidarity, and Banzhaf values of cooperative games to interval-valued ones.

Through its three chapters, this book expands cooperative games, theoretically, from point-valued to interval-valued ones. This expansion uses interval-valued data and allows users to take uncertainties into consideration when applying cooperative games theory in decision making. This book can benefit a broad range of readers in computing beyond those using cooperative game theory. One of the most significant challenges in computing is to effectively handle various kinds of uncertainties, both theoretically and practically, ranging from imprecise data measurement to imperfect modeling.

This book presents a good example of expanding existing knowledge with interval methods in addition to many other successful applications of interval computing. The author very correctly points out that interval computing has its own properties. Some of them can be significantly different from traditional point-valued computation and, hence, may raise issues in software implementation. However, recent developments in interval computing have resulted in a formal standard approved by the IEEE Standard Association, the IEEE Standard for Interval Arithmetic (IEEE 1788-2015). Standard-conforming software tools should make implementations of interval computing applications much easier than they were before.

What is computing about fundamentally? It is about nothing but processing data to produce information. Interval-valued data is an addition to traditional point-valued methods. It has made and will continue to make positive impacts on the broad field of computing. To this perspective, this book can be a good reference for those who are interested in bringing interval computing as an additional tool into their computing practice in any field.