Authors: Matthew P. Coleman, Benjamin Fine, Barry Mittag, Leslie Schaer Leslie Schaer
A common activity in popular culture is the formation of top lists: the top 100 songs of all time, the 50 favorite movies, the top 100 movie stars and so on. These top lists are then publicized on radio and on the inter- net. In this paper, which is part research, part expository and part computer simula- tion, we examine the mathematics of form- ing such lists in the situation where there are no measurable head-to-head comparisons and the compilation of the ranked list is formed by respondents' choices. There is an implied assumption that the compiled lists fomr a representative list of a true ranked list. statistical and therefore form a rep- resentative picture of a "true" ranked list. We construct statistical models for these list formations and de ne a true ranking in two di erent ways. We then prove several para- doxical results that show non-signi cant cor- relation between the true ranked list and the compiled list. For example, we show that there exist populations which contain an item that has a true ranking of number one but for which this item will not appear on any compiled list. These results are related to Arrow's impossibility theorem (see ) in the theory of elections. A computer simula- tion was conducted in conjunction with the study to determine whether these paradoxes are aberrations. The simulations show that these aberrations only occur with low prob- ability and on average there is a positive but weak correlation between the compiled list and the true ranked list.