Jonathan K. Hodge

Separable discrete preferences

Abstract An ordering of multidimensional alternatives is separable on a set of dimensions if fixing values on the complementary dimensions always produces the same induced ordering. Most often, studies of separability assume continuous alternative spaces; as we show, separability has different properties when alternative spaces are discrete. For instance, two well-known theorems of Gorman—that common set operations preserve separability and that separable preferences are additive—fail for binary alternative spaces. Intersection is the only set operation that preserves separability. For binary alternative spaces, separability is equivalent to additivity if and only if there are four or fewer dimensions.

Classifying interdependence in multidimensional binary preferences

When individual preferences over multiple dimensions are interdependent, the resulting collective decisions can be unsatisfactory and even paradoxical. The notion of separability formalizes this idea of interdependence, and preferences that are completely free from interdependence are said to be separable. In this paper, we develop a mechanism for classifying preferences according to the extent to which they achieve or fail to achieve the desirable property of separability. We show that binary preferences over multiple dimensions are surprisingly complex, in that their interdependence structures defy the most natural attempts at characterization. We also extend previous results pertaining to the rarity of separable preferences by showing that the probability of complete nonseparability approaches 1 as the number of dimensions increases without bound.

Permutations of separable preference orders

The notion of separability is important in economics, operations research, and political science, where it has recently been studied within the context of referendum elections. In a referendum election on n questions, a voters preferences may be represented by a linear order on the 2n possible election outcomes. The symmetric group of degree 2n, S2n, acts in a natural way on the set of all such linear orders. A permutation σ ∈ S2n is said to preserve separability if for each separable order ≻, σ(≻) is also separable. Here, we show that the set of separability-preserving permutations is a subgroup of S2n and, for 4 or more questions, is isomorphic to the Klein 4-group. Our results indicate that separable preferences are rare and highly sensitive to small changes. The techniques we use have applications to the problem of enumerating separable preference orders and to other broader combinatorial questions.

The Top Ten Things I Have Learned about Discovery-Based Teaching.

ABSTRACT In recent years, there has been much lively debate about the merits (or lack thereof) of discovery-based approaches to the teaching and learning of mathematics. In this article, I discuss what I have learned from my own experiments with discovery-based teaching, and offer suggestions to others who may be interested in implementing a more discovery-based pedagogy.

Preseparable Extensions of Multidimensional Preferences

Throughout much of the literature in economics and political science, the notion of separability provides a mechanism for characterizing interdependence within individual preferences over multiple dimensions. In this paper, we show how preseparable extensions can be used to construct certain classes of separable and non-separable preferences. We prove several associated combinatorial results, and we note a correspondence between separable preference orders, Boolean term orders, and comparative probability relations. We also mention several open questions pertaining to preseparable extensions and separable preferences.

The Mathematics of Referendum Elections and Separable Preferences

Summary Voters in referendum elections are often required to cast simultaneous ballots on several possibly related questions or proposals. The separability problem occurs when a voters preferences on one question or set of questions depend on the known or predicted outcomes of other questions. Nonseparable preferences can lead to seemingly paradoxical election outcomes, such as a winning outcome that is the last choice of every voter. In this article, we survey recent mathematical results related to the separability problem in referendum elections. We explore the structure of interdependent preferences, consider related combinatorial and algebraic results, and examine the practical impact of separability on the outcomes of referendum elections.

Single-peaked preferences over multidimensional binary alternatives

Single-peaked preferences are important throughout social choice theory. In this article, we consider single-peaked preferences over multidimensional binary alternative spaces-that is, alternative spaces of the form {0,1}^n for some integer n>=2. We show that preferences that are single-peaked with respect to a normalized separable base order are nonseparable except in the most trivial cases. We establish that two distinct base orders can induce the same single-peaked preference order if any only if they differ by a transposition of their two central elements. We then use this result to enumerate single-peaked binary preference orders over a separable base order.