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.