Finding the set of k-additive dominating measures viewed as a flow problem

n this paper we deal with the problem of obtaining the set of k-additive measures dominating a fuzzy measure. This problem extends the problem of deriving the set of probabilities dominating a fuzzy measure, an important problem appearing in Decision Making and Game Theory. The solution proposed in the paper follows the line developed by Chateauneuf and Jaffray for dominating probabilities and continued by Miranda et al. for dominating k-additive belief functions. Here, we address the general case transforming the problem into a similar one such that the involved set functions have non-negative Möbius transform; this simplifies the problem and allows a result similar to the one developed for belief functions. Although the set obtained is very large, we show that the conditions cannot be sharpened. On the other hand, we also show that it is possible to define a more restrictive subset, providing a more natural extension of the result for probabilities, such that it is possible to derive any k-additive dominating measure from it.

Fixed vs. variable noise in 2AFC contrast discrimination: lessons from psychometric functions.

Recent discussion regarding whether the noise that limits 2AFC discrimination performance is fixed or variable has focused either on describing experimental methods that presumably dissociate the effects of response mean and variance or on reanalyzing a published data set with the aim of determining how to solve the question through goodness-of-fit statistics. This paper illustrates that the question cannot be solved by fitting models to data and assessing goodness-of-fit because data on detection and discrimination performance can be indistinguishably fitted by models that assume either type of noise when each is coupled with a convenient form for the transducer function. Thus, success or failure at fitting a transducer model merely illustrates the capability (or lack thereof) of some particular combination of transducer function and variance function to account for the data, but it cannot disclose the nature of the noise. We also comment on some of the issues that have been raised in recent exchange on the topic, namely, the existence of additional constraints for the models, the presence of asymmetric asymptotes, the likelihood of history-dependent noise, and the potential of certain experimental methods to dissociate the effects of response mean and variance.