Foundations of the theory of evidence: Resolving conflict among schemata

Schematic conflict occurs when evidence is interpreted in different ways (for example, by different people, who have learned to approach the given evidence with different schemata). Such conflicts are resolved either by weighting some schemata more heavily than others, or by finding common-ground inferences for several schemata, or by a combination of these two processes. Belief functions, interpreted as representations of evidence strength, provide a natural model for weighting schemata, and can be utilized in several distinct ways to compute common-ground inferences. In two examples, different computations seem to be required for reasonable common-ground inference. In the first, competing scientific theories produce distinct, logically independent inferences based on the same data. In this example, the simple product of the competing belief functions is a plausible evaluation of common ground. In the second example (sensitivity analysis), the conflict is among alternative statistical assumptions. Here, a product of belief functions will not do, but the upper envelope of normalized likelihood functions provides a reasonable definition of common ground. Different inference contexts thus seem to require different methods of conflict resolution. A class of such methods is described, and one characteristic property of this class is proved.     

Words versus actions as a means to influence cooperation in social dilemma situations

We use a sequential voluntary contribution game to compare the relative impact of a first-mover’s non-binding announcement versus binding commitment on cooperation. We find that a non-binding announcement and a binding commitment increase individual contributions to a similar extent. Since announced contributions systematically exceed commitments, in sessions with a non-binding announcement, second-movers tend to contribute more to the group activity than in sessions with a binding commitment. Yet, second-movers appear to be more motivated towards achieving a social optimum when the first-mover uses commitment. We also find that a non-binding announcement has a higher impact on individual propensity to cooperate than the ex post contribution of the first-mover. However, the failure to make announced contributions decreases cooperation even though the first-mover is reassigned in every period.     

Maximally even sets and configurations: common threads in mathematics,physics, and music

Convex (concave) interaction weighting functions are combined with circular configurations of black and white sites to determine configurations that have minimum (maximum) weight. These configurations are called maximally even configurations. It is shown that for a given number of black and white sites, all maximally even configurations are equivalent under rotation and reflection, and a simple algorithm is constructed that generates these configurations. A number of equivalent conditions that determine a maximally even configuration are established. These equivalent conditions permit maximally even configurations to apply to a number of seemingly disparate problems including the dinner table and concentric circles problems, the one-dimensional antiferromagnetic Ising model, and musical scales. This paper is dedicated to the memory of John Clough (1928–2003). Without his seminal works in music theory and patient encouragement of others, this work and much of the work referenced herein would never have been started, much less completed. The field of mathematical music theory owes a great debt to John Clough. The authors are privileged to have known and worked with John Clough.     

The Relation Between Probability and Evidence Judgment: An Extension of Support Theory

We propose a theory that relates perceived evidence to numerical probability judgment. The most successful prior account of this relation is Support Theory, advanced in Tversky and Koehler (1994). Support Theory, however, implies additive probability estimates for binary partitions. In contrast, superadditivity has been documented in Macchi, Osherson, and Krantz (1999), and both sub- and superadditivity appear in the experiments reported here. Nonadditivity suggests asymmetry in the processing of focal and nonfocal hypotheses, even within binary partitions. We extend Support Theory by revising its basic equation to allow such asymmetry, and compare the two equations' ability to predict numerical assessments of probability from scaled estimates of evidence for and against a given proposition. Both between- and within-subject experimental designs are employed for this purpose. We find that the revised equation is more accurate than the original Support Theory equation. The implications of asymmetric processing on qualitative assessments of chance are also briefly discussed.