An axiomatic theory of political representation

We discuss the theory of voting rules which are immune to gerrymandering. Our approach is axiomatic. We show that any rule that is unanimous, anonymous, and representative consistent must decide a social alternative as a function of the proportions of agents voting for each alternative, and must either be independent of this proportion, or be in one-to-one correspondence with the proportions. In an extended model in which voters can vote over elements of the unit interval, we introduce and characterize the quasi-proportional rules based on unanimity, anonymity, representative consistency, strict monotonicity, and continuity. We show that we can always (pointwise) approximate a single-member district quota rule with a quasi-proportional rule. We also establish that upon weakening strict monotonicity, the generalized target rules emerge.