We review our current understanding of the interior structure and thermal evolution of Saturn, with a focus on recent results in the Cassini era. There has been important progress in understanding physical inputs, including equations of state of planetary materials and their mixtures, physical parameters like the gravity field and rotation rate, and constraints on Saturnian free oscillations. At the same time, new methods of calculation, including work on the gravity field of rotating fluid bodies, and the role of interior composition gradients, should help to better constrain the state of Saturn’s interior, now and earlier in its history. However, a better appreciation of modeling uncertainties and degeneracies, along with a greater exploration of modeling phase space, still leave great uncertainties in our understanding of Saturn’s interior. Further analysis of Cassini data sets, as well as precise gravity field measurements from the Cassini Grand Finale orbits, will further revolutionize our understanding of Saturn’s interior over the next few years.

General remarks on the study of preference and choice

There is a huge theoretical literature on preference and choice, with a corresponding large literature on testing theories. In this chapter we focus almost entirely on theoretical work. In this section we review some empirical applications that motivate some of our theoretical primitives, such as the “sample spaces” for which probabilistic models are designed. In later sections, we provide references to relevant empirical work.

Recent important books, review articles, and handbooks, with a theoretical emphasis, include Barberá et al. (1999, 2004, 2013); Gilboa (2009); Luce (2000); and Wakker (2010). We focus on the past 15 years in this chapter and, for a significant part, restrict the content to general multiattribute options. While we provide some in-depth details for choice under uncertainty, we do not go into specialized models of intertemporal or sequential choice. Also, to keep the bibliography manageable, we often cite only one or two most relevant articles on a topic.

Historically, research on preference and choice has placed an emphasis on deterministic representations, most notably, algebraic models of preference, utility, and choice. A notorious challenge, for over 50 years, has been the question of adequate representation for data showing different choices in repeated presentations of the same task, both within a person and across decision makers.

This formal modeling challenge is sometimes associated with Duncan Luce, whose 1959 choice axiom was a major milestone in probabilistic characterizations of utility.Many other leading scholars have also studied and discussed ways to generalize algebraic models into probability models (see Blavatskyy and Pogrebna, 2010; Stott, 2006; Wilcox, 2008, and their citations). We will emphasize probabilistic representations of preference, utility and choice, and we will consider deterministic representations as the special case where all probability mass is concentrated on a single preference, a single utility function, or a single way of making a choice, and choice based on such a deterministic representation is errorfree. The probabilistic generalizations we consider are based either on the premise that preferences or utilities are deterministic but responses are probabilistic (e.g., due to probabilistic error), or on the premise that preferences are probabilistic and responses are deterministic.

The focus is on theory, with data and model fitting introduced only when such is intimately connected to the theory.

4.1 Introduction

This chapter describes and discusses the final BWS case (Case 3), which is the multi-profile case. This case is closely related to the literature on discrete choice experiments, with two key differences: (1) the response task associated with DCEs for this case is a choice of the best and the worst profile (and/or choices of the best and the worst profile plus second-best, second-worst, etc.); and (2) an emphasis on individual-level analysis. The plan of the chapter is as follows: we briefly review traditional DCE literature to provide some familiarity with this stream of research; we then show how Case 3 models logically extend prior work, and discuss departures noted above; we next describe, discuss and illustrate Case 3 design, implementation and analysis; and conclude with unresolved issues and future research opportunities.

DCEs are not new; the method of paired comparisons was formalized by Thurstone (1927). DCE developments since 1980 (1) concern multiple and paired choices, (2) rely on random-utility-theory-based probabilistic discrete choice models developed after 1970 (for example, McFadden, 1974), and (3) typically are constructed using principles from the design of statistical experiments. The latter stream of work was initiated by Louviere and Woodworth (1983), who pioneered a way to design multiple choice experiments consistent with McFadden's (1974) conditional logit model and more general models such as “mother logit” (McFadden, Train and Tye, 1978). Prior to Louviere and Woodworth (1983) there was little work on designed experiments for multiple choices, and researchers often used ad hoc combinations of pairs, triples, quadruples, etc. as choice experiments. Louviere and Woodworth provided a systematic way to design DCEs based on ideas from the design of statistical experiments. The focus in this chapter continues this tradition, and we describe and discuss relevant new developments when appropriate.

Recent reviews of DCE literature include those by Louviere, Hensher and Swait (2000), Louviere (2001), Louviere and Street (2000), Louviere, Street and Burgess (2003), Street, Burgess and Louviere (2005), Burgess et al. (2006), Street et al. (2008), Street and Burgess (2007), Louviere and Lancsar (2009) and Rose and Bliemer (2009). Thus, there is no need to review what is already well known and well worn, so we give only brief details when they are relevant.

3.1 Introduction

This chapter describes, discusses and illustrates the BWS profile case (Case 2). Researchers familiar with discrete choice experiments will recognize that this case has many features in common with DCEs. For example, like traditional DCEs, a good or service is described by a common set of attributes and levels; the descriptions are called profiles (hence, the name for this case). As in traditional ratings-based conjoint analysis (Green and Rao, 1971; Louviere, 1988a; 1994), the profiles are presented to experimental participants one at a time. In this sense, this case is somewhat like binary response DCEs that present profiles one at a time and ask for “Yes”/“No” (accept/reject, etc.) responses.

However, this case differs from traditional conjoint analysis and DCEs because experimental participants choose, respectively, the best and worst attribute levels in each profile. In other words, participants do not make holistic profile choices. Our experience is that this is an easier task for participants, as they evaluate only one profile at a time instead of choosing among two or more profiles, as in most DCEs. The one-profile-at-a-time aspect may be attractive in applications in which DCEs rarely have been used, such as the United Kingdom's National Health Service. Indeed, the authors have a fair amount of experience applying Case 2 to health-related research problems.

In the past researchers wanting to analyze data from Case 2 had to write special-purpose estimation software or develop a “work-around” in statistical software products such as SPSS, STATA and SAS. We suspect that the latter problem and the fact that formal proofs of the measurement properties appeared only recently (Marley, Flynn and Louviere, 2008) are responsible for less exposure for Case 2 than Case 1 or Case 3. Accordingly, we discuss these and other issues in this chapter to give this case more exposure; indeed, as we later discuss, Case 2 has unique advantages not shared by other conjoint, DCE or utility theoretic approaches for preference elicitation.

The plan of the rest of the chapter is as follows. We first describe and illustrate BWS profile case choice tasks (hereafter, “Case 2”). We then discuss issues associated with designing and implementing profile case projects.

Jordan J. Louviere first proposed best-worst scaling (BWS) in the late 1980s as a way to capitalize on humans' tendency to be more reliable and accurate at identifying extreme options. Louviere first called the method maximum difference scaling, to describe what he hypothesized as the underlying process, namely choosing the pair of stimuli in a set of stimuli that exhibited the largest subjective difference on the underlying continuum of interest. Since that time BWS has been adopted by academics and practitioners in many fields globally. However, marketing researchers continue to refer to it as maximum difference scaling (or “maxdiff”), while academics have overwhelmingly now begun to call it best-worst scaling. Louviere and colleagues changed the name to reflect the fact that years of academic research had made it clear that no one actually used a maximum difference choice process, so a much better general term for the method was BWS.

So, BWS now is almost 25 years old. The current authors began receiving numerous requests for assistance and explanations about how to do BWS around 2005; such requests have continued unabated since then. It became clear from the requests, comments and interactions in BWS and more conventional choice modelling short courses that there was a need for a book that brought BWS theory and methods together in such a way that as many people as possible could learn the basic theory and ways to design, implement and analyze BWS experiments in as simple a pedagogical manner as possible. Therefore, this book began with many discussions between Louviere, Flynn and Marley about the need for such a book, leading to them spending time together in the Seattle, Washington, area in 2009 to begin the writing process. That led to discussions about the need for application chapters, which in turn led to invitations to various researchers, principally academics, who were early adopters of BWS, to contribute such chapters.

So, our key reason for writing the book was to introduce as many people as possible to choice-based measurement methods (of which BWS is one type) with the hope of eventually eliminating the many atheoretical and ad hoc measurement methods that are applied in the social and business disciplines. BWS provides a theoretical framework to measure latent, subjective quantities that can produce measurement values with known properties.