1. if you insist on using the r-squared (i.e., a linear regression measure), then properly center and normalize your data, and model what you actually predict: the difference between the baseline (0.5) and the probability to vote for party 0 or party 1. If you model the outcomes as 0/1 without this, then you are using a model made for gaussian variables on what should be a logistic regression
2. if you can live with something that more accurately captures the idea of "explanatory power", you can use a GLM (logistic link function), do a logistic regression, and then use the log odds or another measure.
In both cases, the variance explained by the state that you are in is 1, because of course it is, that's how the thought experiment is constructed - p(vote for party 1)=0.5+ \delta(state).
"Paradoxes" like this are often interesting in the sense that they point to the math being the wrong math or you using it wrong, but instead people tend to assume that they are obviously understanding things correctly so it must be some weird property of the world (which then sometimes is used to construct some faulty conclusions as in some of the cited papers)
1. if you insist on using the r-squared (i.e., a linear regression measure), then properly center and normalize your data, and model what you actually predict: the difference between the baseline (0.5) and the probability to vote for party 0 or party 1. If you model the outcomes as 0/1 without this, then you are using a model made for gaussian variables on what should be a logistic regression 2. if you can live with something that more accurately captures the idea of "explanatory power", you can use a GLM (logistic link function), do a logistic regression, and then use the log odds or another measure.
In both cases, the variance explained by the state that you are in is 1, because of course it is, that's how the thought experiment is constructed - p(vote for party 1)=0.5+ \delta(state).
"Paradoxes" like this are often interesting in the sense that they point to the math being the wrong math or you using it wrong, but instead people tend to assume that they are obviously understanding things correctly so it must be some weird property of the world (which then sometimes is used to construct some faulty conclusions as in some of the cited papers)