Socially Optimal Districting: An Empirical Investigation

This paper provides an empirical exploration of the potential gains from socially optimal districting. As emphasized in the political science literature, districting matters because it determines the seat-vote curve, which relates the fraction of seats parties obtain to their share of the aggregate vote. Building on the theoretical work of Coate and Knight (2006), which develops and analyzes the optimal seat-vote curve, this paper develops a methodology for computing actual and optimal seat-vote curves and for measuring the potential welfare gains that would emerge from implementing optimal seat-vote curves. This method is then applied to analyze districting plans in place during the 1990s to elect U.S. State legislators. The analysis shows that the plans used by the states in our data set generate seat-vote curves that are overly responsive to changes in voters' preferences. While there is significant variation across states, the potential welfare gains from implementing optimal seat-vote curves are on average small relative to the overall surplus generated by legislatures. This appears to be because seat-vote curves are reasonably close to optimal rather than because aggregate welfare is insensitive to varying districting plans. Interestingly, implementing proportional representation would produce welfare levels quite close to those achieved by implementing optimal seat-vote curves.

Socially Optimal Districting

This paper provides a welfare economic analysis of the problem of districting. In the context of a simple micro-founded model intended to capture the salient features of U.S. politics, it studies how a social planner should allocate citizens of different ideologies across districts to maximize aggregate utility. In the model, districting determines the equilibrium seat-vote curve which is the relationship between the aggregate vote share of the political parties and their share of seats in the legislature. To understand optimal districting, the paper first characterizes the optimal seat-vote curve which describes the ideal relationship between votes and seats. It then shows that under rather weak conditions the optimal seat-vote curve is implementable in the sense that there exist districtings which make the equilibrium seat-vote curve equal to the optimal seat-vote curve. The nature of these optimal districtings is described. Finally, the paper provides a full characterization of the constrained optimal seat-vote curve and the districtings that underlie it when the optimal seat-vote curve is not achievable.

The political districting problem: A survey

Computer scientists and social scientists consider the political districting problem from different viewpoints. This paper gives an overview of both strands of the literature on districting in which the connections and the differences between the two approaches are highlighted

Optimal partisan districting on planar geographies

We show that optimal partisan districting in the plane with geographical constraints is an NP-complete problem

Axiomatic Districting

In a framework with two parties, deterministic voter preferences and a type of geographical constraints, we propose a set of simple axioms and show that they jointly characterize the districting rule that maximizes the number of districts one party can win, given the distribution of individual votes (the \optimal gerrymandering rule"). As a corollary, we obtain that no districting rule can satisfy our axioms and treat parties symmetrically

Optimal redistricting under geographical constraints: Why "pack and crack" does not work

We show that optimal partisan redistricting with geographical constraints is a computationally intractable (NP-complete) problem. In particular, even when voter's preferences are deterministic, a solution is generally not obtained by concentrating opponent's supporters in \unwinnable" districts ("packing") and spreading one's own supporters evenly among the other districts in order to produce many slight marginal wins ("cracking")

Socially Optimal Districting: A Theoretical and Empirical Exploration

This paper investigates the problem of optimal districting in the context of a simple model of legislative elections. In the model, districting matters because it determines the seat-vote curve, which describes the relationship between seats and votes. The paper first characterizes the optimal seat-vote curve, and shows that, under a weak condition, there exist districtings that generate this ideal relationship. The paper then develops an empirical methodology for computing seat-vote curves and measuring the welfare gains from implementing optimal districting. This is applied to analyze the districting plans used to elect U.S. state legislators during the 1990s.

Measuring the Compactness of Political Districting Plans

The United States Supreme Court has long recognized compactness as an important principle in assessing the constitutionality of political districting plans. We propose a measure of compactness based on the distance between voters within the same district relative to the minimum distance achievable -- which we coin the relative proximity index. We prove that any compactness measure which satisfies three desirable properties (anonymity of voters, efficient clustering, and invariance to scale, population density, and number of districts) ranks districting plans identically to our index. We then calculate the relative proximity index for the 106th Congress, requiring us to solve for each state's maximal compactness; an NP-hard problem. Using two properties of maximally compact districts, we prove they are power diagrams and develop an algorithm based on these insights. The correlation between our index and the commonly-used measures of dispersion and perimeter is -.22 and -.06, respectively. We conclude by estimating seat-vote curves under maximally compact districts for several large states. The fraction of additional seats a party obtains when their average vote increases is significantly greater under maximally compact districting plans, relative to the existing plans.