The Efron dice voting system

About 50?years ago, Efron noted some counterintuitive properties of the long-term behavior of contests involving dice. For instance, consider the 6-sided dice whose sides are labeled (4,4,4,4,0,0), (3,3,3,3,3,3), (6,6,2,2,2,2), and (5,5,5,1,1,1). Each die has a 2/3 probability of rolling a higher number than the next one in the list and the last has the same 2/3 probability of rolling a higher number than the first. The non-transitivity of games involving non-identical dice was popularized by Gardner (Sci Am, 223:110–114, 1970). Although Gardner and other authors have observed that non-transitive dice serve to illustrate the complexities of the theory of voting, it does not seem that much attention has been paid to the corresponding voting system. Our purpose in this article is to present this voting system and compare its properties with those of other voting systems. One of the most interesting properties is the fact that cancellation with respect to the Efron dice voting system can replace cancellation with respect to pairwise preferences in Young’s characterization of the social choice function associated with the Borda Count.     

Variations of Q–Q Plots: The Power of Our Eyes!

In statistical modeling, we strive to specify models that resemble data collected in studies or observed from processes. Consequently, distributional specification and parameter estimation are central to parametric models. Graphical procedures, such as the quantile–quantile (Q–Q) plot, are arguably the most widely used method of distributional assessment, though critics find their interpretation to be overly subjective. Formal goodness of fit tests are available and are quite powerful, but only indicate whether there is a lack of fit, not why there is lack of fit. In this article, we explore the use of the lineup protocol to inject rigor into graphical distributional assessment and compare its power to that of formal distributional tests. We find that lineup tests are considerably more powerful than traditional tests of normality. A further investigation into the design of Q–Q plots shows that de-trended Q–Q plots are more powerful than the standard approach as long as the plot preserves distances in x and y to be the same. While we focus on diagnosing nonnormality, our approach is general and can be directly extended to the assessment of other distributions.