A dutiful decision maker may not be persuaded to adopt a satisficing solution just because the gains exceed the losses. The satisficing options should also conform to a sense of fairness or equanimity. There are three additional criteria that should govern the ultimate selection of a satisficing option. First, if time and resources permit, a decision maker should neither sacrifice quality needlessly nor pay more than is necessary. Second, a decision maker should be as certain as possible that the decision really is good enough, or adequate. Third, a decision maker should not foreclose against optimality; that is, the optimal decision, should it exist, ought to be satisficing.

Equilibria

Although the satisficing set Σq contains all possible options that satisfy the PLRT and, in that sense, are legitimate candidates for adoption, they generally will not be equal in overall quality. Consider the following example.

It is a platitude that decisions should be optimal; that is, that decision makers should make the best choice possible, given the available knowledge. But we cannot rationally choose an option, even if we do not know of anything better, unless we know that it is good enough. Satisficing, or being “good enough,” is the fundamental desideratum of rational decision makers – being optimal is a bonus.

Can a notion of being “good enough” be defined that is distinct from being best? If so, is it possible to formulate the concepts of being good enough for the group and good enough for the individuals that do not lead to the problems that exist with the notions of group optimality and individual optimality? This book explores these questions.

Decision makers are usually not hermits and do not function in isolation from others. They are usually influenced by the opinions (i.e., preferences) of other decision makers, and their problem is one of how to select a course of action, not only for themselves, but for the community. If each individual were to possess a notion of rationality for the group as well as a notion of rationality for itself, it might be in a position to improve its behavior.

Group rationality, however, is not a logical consequence of rationality based on individual self-interest. Under substantive rationality, where maximization of individual satisfaction is the operative notion, group behavior is not usually optimized by optimizing each individual's behavior, as is done in conventional game theory. Unfortunately, those who put their final confidence in exclusive self-interest may ultimately function disjunctively, and perhaps illogically, when participating in collective decisions.

The point of departure for conventional game theory is games of pure conflict, with the prototype being constant-sum games, where one player's loss is another player's gain. For a constant-sum game, any notion of group-interest is vacuous; individual self-interest is the only appropriate motive.

The basic principle of decision making based on substantive rationality is very simple: one seeks to maximize expected utility. This principle has led to a body of mathematics that accommodates ways to rank-order expectations and to search or to solve for the option (or options) that meet the optimality criteria. The major mathematical components of this approach are utility theory, probability theory, and calculus.

The basic principle of decision making based on intrinsic rationality is also very simple: one seeks acceptable tradeoffs. To be useful, this principle must be supported by a body of mathematics that accommodates ways to formulate tradeoffs and to identify the options that meet the satisficing criteria. This chapter introduces such a mathematical structure. It also is composed of utility theory, probability theory, and calculus, but with some important differences in the structure and the application of these components (Stirling and Morrell, 1991; Stirling, 1993; Stirling, 1994; Stirling et al., 1996a; Stirling et al., 1996c; Stirling et al., 1996b; Goodrich et al., 1999; Stirling and Goodrich, 1999a; Goodrich et al., 2000).

That decision makers can rarely be certain is obvious. The question is, what are they uncertain about? The most well-studied notion of uncertainty is epistemic uncertainty, or uncertainty that arises due to insufficient knowledge. Such uncertainty is usually attributed to randomness or imprecision. The effect of epistemic uncertainty is to increase the likelihood of making erroneous decisions. The worst thing one can do in the presence of epistemic uncertainty is to ignore it. It is far better to devise models that account for as much of the uncertainty as can be described. Consequently, a large body of theories of decision making in the presence of less than complete information has been developed over many decades. I do not propose to give an exhaustive treatment of the way epistemic uncertainty is accounted for under classical decision-making paradigms, but instead summarize two ways to deal with uncertainty. The first is the classical Bayesian approach of calculating expected utility, and the second is the use of convex sets of utility functions.

Probability theory developed as a means of analyzing the notion of chance and, as a mathematical discipline, it has developed in a rigorous manner based on a system of precise definitions and axioms. However, the syntax of probability theory exists independently of its use as a means of expressing and manipulating information and of quantifying the semantic notions of belief and likelihood regarding natural entities. In this book we employ the syntax of probability theory to quantify semantic notions that relate to the synthesis of artificial entities. In this appendix we present the basic notions of probability theory. However, since much of the terminology is motivated by the historical semantics, it will be necessary to supplement the standard treatment with terminology to render the definitions more relevant to our usage.

We begin by establishing the notation that is used in this book.

Definition C.1

A set is a collection of simple entities, called elements. If A is a set and the points ω1, ω2, … are its elements, we denote this relationship by the notation

A = {ω1, ω2, …}.

If ω is an element of the set A, we denote this relationship by the notation ω ∈ A, where ∈ is the element inclusion symbol.

Often, we will specify a set by the properties of the elements. Suppose A comprises the set of all points ω such that ω ∈ S possesses property P.

The first topologies defined on a lattice directly from the lattice ordering (that is, Birkhoff's order topology and Frink's interval topology) involved “symmetrical” definitions – the topologies assigned to L and to Lop were identical. A guiding example was always the unit interval of real numbers in its natural order, which is of course a highly symmetrical lattice. The initial interest was in such questions as which lattices became compact and/or Hausdorff in these topologies. The Scott topology stands in strong contrast to such an approach. Indeed it is a “unidirectional” topology, since, for example, all the open sets are always upper sets; thus, for nontrivial lattices, the T0 separation axiom is the strongest it satisfies. Nevertheless, we saw in Chapter II that the Scott topology provides many links between domains and general topology in such classical areas as the theory of semicontinuous functions and in the study of lattices of closed (compact, convex) sets (ideals) in many familiar structures.

In this chapter we introduce a new topology, called the Lawson topology, which is crucial in linking continuous lattices and domains to topological algebra. Its definition is more in the spirit of the interval and order topologies, and indeed it may be viewed as a mixture of the two. However, it remains asymmetrical – the Lawson topologies on L and Lop need not agree. But, even if one is seeking an appropriate Hausdorff topology for continuous lattices, this asymmetry is not at all surprising in view of the examples we have developed.

In Chapter I we encountered the rich order theoretic structure of complete lattices and of continuous lattices. Wherever it was feasible to express statements on the level of generality of dcpos and domains we did so. Perhaps even more typical for these partially ordered sets is their wealth of topological structure. The aim of the present chapter is to introduce topology into the study – a program to be continued in Chapter III.

Section II-1 begins with a discussion of the Scott topology and its connection with the convergence given in order theoretic terms by lower limits, or liminfs. This leads to a characterization theorem for domains in terms of properties of their lattices of Scott open sets (II-1.14) – a type of theorem that will become a recurrent theme (see Chapter VII). One motivation for such considerations arises from the appearances of domain theory in theoretical computer science: one typically needs the generality of domains to model the structures and constructions under consideration, while continuous lattices enter the scene as their lattices of open sets.

In Section II-2 we determine that the functions continuous for the Scott topology are those preserving directed sups. We can thus express one and the same property of a function between dcpos either in topological or in order theoretical terms. The space [S → T] of all Scott-continuous functions between continuous lattices is itself a continuous lattice, and the category of continuous lattices proves to be cartesian closed.

Our final chapter is devoted to exploring further links between topological algebra and continuous lattice and domain theory. This theme has already played an important role: the Fundamental Theorem of Compact Semilattices (VI-3.4) is just one example. In this chapter, however, the methods of topological algebra occupy a more central role, while the methods of continuous lattices are somewhat less prominent.

Section VII-1 is devoted to somewhat technical results about certain non-Housdorff topological semilattices; they are included primarily to facilitate the proof of later results concerning separate continuity of semilattice and lattice operations implying joint continuity. Section VII-2 makes various observations about topological lattices and their topologies, with a particular focus on completely distributive lattices.

Section VII-3 introduces the class of continuous lattices for which the Lawson topology is equal to the interval topology: the hypercontinuous lattices. The distributive ones are paired with the quasicontinuous domains via the spectral theory of Chapter V. Thus several earlier themes are nicely rounded out.

Section VII-4 characterizes those meet continuous complete lattices which admit a compact semilattice topology as being exactly those lattices whose lattice of Scott open sets forms a continuous lattice; this augments II-1.14, which shows that the continuous lattices are exactly those complete lattices whose Scott open sets form a completely distributive lattice. The final part of Section VII-4 is devoted to a proof that a compact semitopological semilattice is in fact topological.

A mathematics book with six authors is perhaps a rare enough occurrence to make a reader ask how such a collaboration came about. We begin, therefore, with a few words on how we were brought to the subject over a ten-year period, during part of which time we did not all know each other. We do not intend to write here the history of continuous lattices but rather to explain our own personal involvement. History in a more proper sense is provided by the bibliography and the notes following the sections of the book, as well as by many remarks in the text. A coherent discussion of the content and motivation of the whole study is reserved for the introduction.

In October of 1969 Dana Scott was led by problems of semantics for computer languages to consider more closely partially ordered structures of function spaces. The idea of using partial orderings to correspond to spaces of partially defined functions and functionals had appeared several times earlier in recursive function theory; however, there had not been very sustained interest in structures of continuous functionals. These were the ones Scott saw that he needed. His first insight was to see that – in more modern terminology – the category of algebraic lattices and the (so-called) Scott-continuous functions is cartesian closed. Later during 1969 he incorporated lattices like the reals into the theory and made the first steps toward defining continuous lattices as “quotients” of algebraic lattices.

BACKGROUND. In 1980 we published A Compendium of Continuous Lattices. A continuous lattice is a partially ordered set characterized by two conditions: firstly, completeness, which says that every subset has a least upper bound; secondly, continuity, which says that every element can be approximated from below by other elements which in a suitable sense are much smaller, as for example finite subsets are small in a set theoretical universe. A certain degree of technicality cannot be avoided if one wants to make more precise what this “suitable sense” is: we shall do this soon enough. When that book appeared, research on continuous lattices had reached a plateau.

The set of axioms proved itself to be very reasonable from many viewpoints; at all of these aspects we looked carefully. The theory of continuous lattices and its consequences were extremely satisfying for order theory, algebra, topology, topological algebra, and analysis. In all of these fields, applications of continuous lattices were highly successful. Continuous lattices provided truly interdisciplinary tools.

Major areas of application were the theory of computing and computability, as well as the semantics of programming languages. Indeed, the order theoretical foundations of computer science had been, some ten years earlier, the main motivation for the creation of the unifying theory of continuous lattices. Already the Compendium of Continuous Lattices itself contained signals pointing future research toward more general structures than continuous lattices.