Constraints on the turnout gap between high and low knowledge (or income) voters: Combining the Duncan-Davis method of bounds with the Taagepera method of bounds

Countries differ quite substantially in mean turnout levels, and it is equally well known that there may be substantial within-country variation as well, for example, between high income and low-income groupings or between high political knowledge and low political knowledge groupings. It has been hypothesized that the size of such between-group gaps will fall as turnout rises, and conversely (Franklin, 2004. Blais, 2000). However, as Franklin (2004) also noted, there are mathematical constraints on the size of the turnout gap that are related to the level of turnout. For example, in the limit, if turnout is 100%, then all groups must have identical turnout. Here we build on this insight by adapting the classic work on boundary conditions done by two sociologists (Duncan and Davis, 1953) to show precisely what the boundary constraints look like over the entire range of turnout values. Then we show how these constraints can help make sense of the strong relationship found between overall turnout and the size of the gap between voters above and below the median in political knowledge in the Fisher et al. (2008) cross-national study. To do so we draw on ideas in Rein Taagepera, 2007 and Taagepera, 2008 about how to use boundary condition information to develop better theoretical models.