The Economics of Partisan Gerrymandering

The most related prior papers on optimal partisan gerrymandering are Owen and Grofman (1988), Friedman and Holden (2008), and Gul and Pesendorfer (2010). Owen and Grofman’s model is equivalent to the special case of our model with two voter types. Gul and Pesendorfer consider competition between two designers who each control districting in some area and aim to win a majority of seats.⁷⁷7Friedman and Holden (2020) study designer competition in the model of their *FH paper. A simplified version of their model with a single designer is equivalent to the special case of our model where vote swings are linear in voter types; we discuss this special case in Section 3.4. Friedman and Holden consider essentially the same model as we do (and in particular allow non-linear swings), but their main results concern the special case where aggregate uncertainty is much larger than idiosyncratic uncertainty. In contrast, we do not restrict the relative amounts of aggregate and idiosyncratic uncertainty, and we show empirically that the practically relevant case is that where idiosyncratic uncertainty dominates (i.e., the opposite of the case emphasized by Friedman and Holden).

The broader literature on gerrymandering and redistricting addresses a wide range of issues, including geographic constraints on gerrymandering (Sherstyuk, 1998; Shotts, 2001; Puppe and Tasnádi, 2009), gerrymandering with heterogeneous voter turnout (Bouton, Genicot, Castanheira, and Stashko, 2023), socially optimal districting (Gilligan and Matsusaka, 2006; Coate and Knight, 2007; Bracco, 2013), measuring district compactness (Chambers and Miller, 2010; Fryer and Holden, 2011; Ely, 2022), the interaction of redistricting and policy choices (Shotts, 2002; Besley and Preston, 2007), measuring gerrymandering (Grofman and King, 2007; McGhee, 2014; Stephanopoulos and McGhee, 2015; Duchin, 2018; Gomberg, Pancs, and Sharma, 2023), and assessing the consequences of redistricting (among many: Gelman and King, 1994b; McCarty, Poole, and Rosenthal, 2009; Hayes and McKee, 2009; Jeong and Shenoy, 2022). As the partisan gerrymandering problem interacts with many of these issues, our analysis may facilitate future research in these areas.

The paper is organized as follows: Section 2 presents the model. Section 3 analyzes some benchmark cases. Section 4 contains our main theoretical and numerical results. Section 5 contains our empirical results. Section 6 discusses policy implications of our results. Section 7 concludes. All proofs are deferred to the appendix.

With perfect information, optimal gerrymandering takes a simple and well-known form.

Case (1) says that a designer with majority support wins all the districts (e.g., with uniform districting). Case (2) says that a designer with minority support $m<1/2$ wins $2m$ districts with 50% of the vote, and gets zero votes in the remaining $1-2m$ districts. We call any optimal plan in case (2) pack-and-crack.

When $m<1/2$ and voter types are continuous, there are many pack-and-crack plans. For example, some types of supporters can be assigned to only a subset of cracked districts, and some types of opponents can be assigned only to packed districts. This seemingly pedantic point will become important once we introduce uncertainty, because optimal plans under a small amount of uncertainty will approximate some but not all pack-and-crack plans.

Figure 1 illustrates four pack-and-crack plans that play important roles in our analysis. Panel (a) is what we call traditional pack-and-crack: the strongest opposing voters are pooled in one type of district, while the remaining voters (a mix of supporters and opponents) are pooled in another type of district. Panel (b) is the same, except now each strong opposing type is segregated in a distinct, homogeneous district. We call this plan pack-opponents-and-pool. This plan was previously studied by Gul and Pesendorfer (2010), who called it “$p$-segregation.” Panel (c) is the same as Panel (b), except now favorable voter types are matched in a negatively assortative manner to form distinct districts. We call this plan pack-opponents-and-pair, or POP. This plan plays a central role in our analysis, as we will see that it is optimal for realistic parameter values; however, we will also see that the simpler traditional pack-and-crack and pack-opponents-and-pool plans are approximately optimal for the same parameters.

Finally, we call the plan in Panel (d)—where extreme voter types are matched in a negatively assortative manner, and intermediate voter types are segregated—pack-moderates-and-pair, or PMP. This plan was previously studied by Friedman and Holden (2008), who called it “matching slices.”¹⁶¹⁶16Friedman and Holden did not emphasize the possibility of segregating a non-trivial interval of intermediate voter types under matching slices, but their results allow this possibility, and we will see that this is actually the typical case. We also refer to the extreme form of PMP where the segregation region is degenerate, so that only a single voter type is segregated, as negative assortative districting.

We next consider the case with idiosyncratic uncertainty but no aggregate uncertainty. As we will see, this case is fairly realistic, as empirically idiosyncratic uncertainty is much larger than aggregate uncertainty.

In case (1), the designer wins all districts under uniform districting. In case (2), the designer assigns all voter types $s>s^{*}$ to cracked districts that he wins with exactly 50% of the vote, and packs the remaining voters arbitrarily. The intuition is that the designer wins a district iff the mean vote share $v(s,r^{0})$ among voters in the district exceeds 50%, so to win as many districts as possible the designer assigns only voter types above $s^{*}$ to cracked districts. This plan approximates the pack-and-crack vote share pattern as closely as possible, given the uncertainty facing the designer.

The optimal plans in Proposition 2 coincide with the subset of optimal perfect-information plans that pack opponents (e.g., the plans in Figure 1(a)–(c)). Hence, pack-and-crack plans that pack opponents can be optimal with idiosyncratic uncertainty but no aggregate uncertainty, but plans that pack moderates (e.g., PMP) cannot. In Sections 4 and 5, we will see that idiosyncratic uncertainty dominates aggregate uncertainty in practice. Hence, any optimal plan in Proposition 2—for example, traditional pack-and-crack—will prove to be approximately optimal for realistic parameters.

We now turn to the case with aggregate uncertainty but no idiosyncratic uncertainty.

That is, for each voter type $s$ above the population median, the designer creates a district consisting of 50% voters with this type and 50% voters with below-median types. Note that, for every realization of aggregate uncertainty $r\in(\underline{s},\overline{s})$, the designer wins some districts with exactly 50% of the vote and wins zero votes in all other districts. This is precisely the pack-and-crack vote share pattern.

The intuition for Proposition 3 is easy to see with a finite number $N$ of districts. With no idiosyncratic uncertainty, the probability that the designer wins a given district is determined by the median voter type in that district. The strongest district the designer can possibly create is formed by combining the $1/(2N)$ highest voter types with any other voters: that is, it is impossible to create a district where the median voter is above the $1-1/(2N)$ quantile of the population distribution. Similarly, it is impossible to create $n$ districts where the median voter is everywhere above the $1-n/(2N)$ quantile of the population distribution. But, by creating districts one at time by always combining the $1/(2N)$ highest remaining voters with $1/(2N)$ below-median voters, the designer ensures that the median voter in the $n^{\text{th}}$ strongest district is exactly the $1-n/(2N)$ quantile. So this plan is optimal.

The optimal plans in Proposition 3 are a subset of optimal perfect-information plans. For example, the PMP plan in Figure 1(d) remains optimal when $v(s,r)=\mbox{\bf 1}\{s\geq r\}$ but $r$ is not degenerate, while the plans in Figures 1(a)–(c) that pack opponents are not optimal in this setting. This result is consistent with Friedman and Holden (2008), who show that matching slices is optimal when idiosyncratic uncertainty is sufficiently small, under some additional assumptions which we discuss in Section 4.1.¹⁷¹⁷17Note that in every optimal plan in Proposition 3, all voters with the highest type $s$ are assigned to the same district: in Friedman and Holden’s words, “one’s most ardent supporters should be grouped together.” This is what Friedman and Holden mean when they write that “cracking is never optimal” and summarize their findings as “sometimes pack, but never crack.”

Our last benchmark case is when vote shares and swings are linear in voter types. There are two equivalent ways to define this case. The simplest definition is that vote shares $v(s,r)$ are linear in $s$:

An alternative, equivalent definition is that vote swings are linear in $s$. To state this definition, first define the swing of a voter type $s$ when the aggregate shock changes from $r^{\prime}$ to $r$ by

We then say that swings $\Delta_{s}^{r,r^{\prime}}$ are linear in $s$ if

It is easy to see that, up to a rescaling of $s$, vote shares are linear iff swings are linear.

The linear case nests the uniform swing case where $\Delta_{s}^{r,r^{\prime}}$ is independent of $s$ (for each $r,r^{\prime}$), so the aggregate shock shifts the vote share equally for all voter types. Political scientists often assume uniform swing to study how a given districting plan would perform under different electoral outcomes.¹⁸¹⁸18See, e.g., Katz, King, and Rosenblatt (2020) for a recent discussion of this methodology. The linear case also nests the case where voter types are binary (i.e., $\operatorname{supp}(F)=\{\underline{s},\overline{s}\}$), as well as the no-aggregate-uncertainty case considered in Section 3.2. However, the no-idiosyncratic-uncertainty case considered in Section 3.3 cannot be linear, unless voter types are binary.

The key simplification afforded by linearity is that the threshold shock $r^{*}(P)$ for winning a district $P$ depends only on the district mean voter type $x=\mathbb{E}_{P}[s]$. Under linearity, the designer thus effectively chooses a distribution $H(x)$ of mean types $x$, rather than a distribution $\mathcal{H}(P)$ of distributions of types $P$. With this formulation, the constraint $\int Pd\mathcal{H}(P)=F$ simplifies to the requirement that $H$ is a mean-preserving contraction of $F$, which we denote by $F\succsim H$.¹⁹¹⁹19One way to see this is by analogy to statistics, where if a state $s$ is distributed according to $F$ then there exists an experiment such that the distribution of posterior expectations of $s$ is given by $H$ iff $H$ is a mean-preserving contraction of $F$ (e.g., Blackwell, 1953; Kolotilin, 2018).

Slightly abusing notation, the designer wins districts with mean voter type at least $x$ iff $r\leq r^{*}(x)$. The probability of this event is

We can interpret $U$ as the distribution of a re-scaled aggregate shock $z$ where the designer wins a district with mean type $x$ iff $z\leq x$. The designer’s problem thus becomes

Clearly, uniform districting is optimal if $U$ is concave, and segregation is optimal if $U$ is convex. However, a more realistic assumption is that $U$ is strictly S-shaped, so the marginal impact of replacing a less favorable voter with a more favorable one on the probability of winning a district is first increasing and then decreasing. Formally, this means that there is an inflection point $x^{i}\in[0,1]$ such that $U$ is strictly convex on $[0,x^{i}]$ and strictly concave on $[x^{i},1]$; equivalently, the re-scaled aggregate shock $z$ is unimodal.

We will see that $U$ being S-shaped is closely related to the optimality of pack-opponents-and-pool districting (i.e., $p$-segregation, see Figure 1(b)), where voter types below some cutoff $s^{*}$ are segregated, and voter types above $s^{*}$ are pooled in districts with mean voter type $x^{*}=\mathbb{E}_{F}[s|s\geq s^{*}]$. Under pack-opponents-and-pool districting with cutoff $s^{*}$ and pool mean $x^{*}=\mathbb{E}_{F}[s|s\geq s^{*}]$, the designer’s expected seat share is

The best pack-opponents-and-pool plan is the one where $s^{*}$ is chosen to maximize this expectation. When the optimal value of $s^{*}$ is interior, it is characterized by the first-order condition

The intuition for this equation is that a marginal increase in $s^{*}$ increases the pool mean, which increases the designer’s expected seat share by $u(x^{*})(1-F(s^{*}))dx^{*}/ds^{*}=u(x^{*})(x^{*}-s^{*})f(s^{*})$; but also decreases the mass of pooled voters, which decreases the designer’s expected seat share by $(U(x^{*})-U(s^{*}))f(s^{*})$. The first-order condition equates the marginal benefit and marginal cost. See Figure 2.

A simple result is that pack-opponents-and-pool is optimal when $U$ is strictly S-shaped.

Intuitively, when $U$ is S-shaped the designer is risk-loving in the pool mean $x$ for $x\in[0,s^{*}]$ and is risk-averse in $x$ “on average” for $x\in[s^{*},1]$, so voters below $s^{*}$ are segregated and voters above $s^{*}$ are pooled. Similar results were established by Gul and Pesendorfer (2010) and, in the persuasion literature, Kolotilin (2018) and Kolotilin, Mylovanov, and Zapechelnyuk (2022).

As aggregate uncertainty vanishes, the best pack-opponents-and-pool plan converges to the plan characterized in Proposition 2 with segregated packed districts.²¹²¹21Note that as $G$ converges to the step function $\mathbf{1}\{r\geq r^{0}\}$, $U$ converges to the step function $\mathbf{1}\{x\geq{x}^{i}\}$, where ${x}^{i}$ is the solution to $v({x}^{i},r^{0})=1/2$. The first-order condition then reduces to the condition that $x^{*}={x}^{i}$, which yields the same condition for $s^{*}$ as in Proposition 2. Thus, traditional pack-and-crack (where packed districts are pooled) and pack-opponents-and-pool and POP (where packed districts are segregated) are all optimal without aggregate uncertainty, but only the latter two plans remain optimal with a small amount of aggregate uncertainty.²²²²22Intuitively, the designer optimally segregates packed districts to have a respectable chance of winning the strongest of these districts. Note that pack-opponents-and-pool and POP induce the same distribution of district mean types, and hence may both be optimal even when the optimal distribution of means is unique. However, the designer’s indifference among different ways of creating cracked districts with the same mean type is not robust to introducing slightly non-linear swings, as we show in the next section.

The linear swing case considered in Section 3.4 is a natural benchmark, but it makes the counterfactual prediction that the “swingiest” voters—those with the largest $\Delta_{s}^{r,r^{\prime}}$—are “extremists” with $s\in\{\underline{s},\overline{s}\}$. In contrast, election forecasters (and presumably sophisticated gerrymanderers) take into account that moderate voters are usually swingier than extremists. As Nathaniel Rakich and Nate Silver put it when describing the “elasticity scores” in the FiveThirtyEight.com forecasting model, “Voters at the extreme end of the spectrum—those who have a close to a 0 percent or a 100 percent chance of voting for one of the parties—don’t swing as much as those in the middle,” (Rakich and Silver, 2018). We provide some evidence for this claim in Section 5.

The following assumption formalizes the idea that moderates are swingier than extremists.

To see why Assumption 1 corresponds to moderates being swingy, note that integrating (1) gives, for all $s<s^{\prime}<s^{\prime\prime}$ and $r<r^{\prime}$,

or equivalently

Recall that the linear case is defined by having equality in (2). Thus, Assumption 1 says that, for any pair of aggregate shocks $r<r^{\prime}$ and any triple of voter types $s<s^{\prime}<s^{\prime\prime}$, when the aggregate shock improves from $r^{\prime}$ to $r$, type $s^{\prime}$ voters swing toward the designer more than type $s$ and $s^{\prime\prime}$ voters, relative to the linear case.

We mention an equivalent condition and an implication of Assumption 1.

Many common distributions have strictly log-concave densities, including the normal, logistic, and extreme value distributions (see, e.g., Table 1 in Bagnoli and Bergstrom 2005), so part 1 of the proposition shows that Assumption 1 is a standard property. The property in part 2 of the proposition gives another sense in which moderates are swingier than extremists. For example, for any $s<s^{\prime}<s^{\prime\prime}$, this property implies that (letting $v_{r}=\partial v/\partial r$) if $v_{r}(s,r)=v_{r}(s^{\prime\prime},r)$, then $v_{r}(s^{\prime},r)<v_{r}(s,r)=v_{r}(s^{\prime\prime},r)<0$ (recalling that $v_{r}<0$), so type $s^{\prime}$ is swingier than types $s$ and $s^{\prime\prime}$.

We now show that Assumption 1 implies that every optimal districting plan is “strictly single-dipped,” in that more extreme voters are assigned to stronger districts. Formally, a districting plan $\mathcal{H}$ is strictly single-dipped if any district $P\in\operatorname{supp}(\mathcal{H})$ containing any two voter types $s<s^{\prime\prime}$ is stronger than any district $P^{\prime}\in\operatorname{supp}(\mathcal{H})$ containing any intervening voter type $s^{\prime}\in(s,s^{\prime\prime})$, in that $r^{*}(P^{\prime})<r^{*}(P)$.²³²³23Formally, we say that a district $P$ “contains” a voter type $s$ if $s\in\operatorname{supp}(P)$. Note that if districting is strictly single-dipped then each district consists of at most two distinct voter types.

Similar results were established by Friedman and Holden (2008) and, in the persuasion context, Kolotilin, Corrao, and Wolitzky (2023).²⁴²⁴24Assumption 1 is equivalent to Friedman and Holden’s “informative signal property.” Friedman and Holden assume a finite number of districts, and also assume that the median and mode of $Q$ coincide. Kolotilin, Corrao, and Wolitzky (2023) give sufficient conditions for single-dippedness in a more general model that allows state-dependent designer preferences. To see the intuition, suppose a districting plan creates two districts, 1 and 2, with the same threshold aggregate shock $r^{*}$, but where District 1 consists entirely of moderates and District 2 consists of a mix of left-wing and right-wing extremists. With linear swings, the distribution of vote shares in the two districts are identical. However, under Assumption 1, the vote share is swingier in District 1 than in District 2. Thus, conditional on the aggregate shock being close to $r^{*}$, a marginal voter is more likely to be pivotal in District 2 than in District 1. The designer can then profitably exploit this asymmetry by re-allocating some unfavorable voters to District 1 and re-allocating some favorable voters to District 2, thus weakening the moderate District 1 and strengthening the extreme District 2. Breaking all ties in favor of extreme disticts in this manner leads to strictly single-dipped districting.

Proposition 6 implies that, under Assumption 1, the designer should never pool more than two voter types in the same district. Thus, among the plans in Figure 1, only POP and PMP can be optimal under Assumption 1 (and, moreover, more extreme paired districts under these plans must be stronger than more moderate districts). In particular, while pack-opponents-and-pool is optimal with linear swings and unimodal aggregate shocks, if moderates are even slightly swingier than extremists then the designer is better-off splitting the pool into distinct districts each consisting of at most two types such that more extreme districts are strictly stronger.

Strict single-dippedness is an important property of a districting plan, but many plans can be strictly single-dipped. This subsection argues that, among strictly single-dipped plans, it is natural to focus on “pack-and-pair” districting, where weaker districts are segregated and stronger districts consist of exactly two voter types. Formally, a strictly single-dipped districting plan $\mathcal{H}$ is pack-and-pair if $\delta_{s}\in\operatorname{supp}(\mathcal{H})$ implies that any $P\in\operatorname{supp}(\mathcal{H})$ such that $r^{*}(P)<r^{*}(\delta_{s})$ takes the form $P=\delta_{s^{\prime}}$ for some $s^{\prime}<s$.

For simplicity, for the remainder of the current section, we restrict attention to the additive taste shock case, and assume that the taste shock density is strictly log-concave and symmetric about $0$. The symmetry assumption has the convenient implication that the threshold shock to win a packed district $P=\delta_{s}$ is just $r^{*}(P)=s$.

We first show that any pack-and-pair plan $\mathcal{H}$ can be described in a simple way. First, there exists a bifurcation point $r^{b}\in[\underline{s},\overline{s}]$ such that a district $P\in\operatorname{supp}(\mathcal{H})$ is packed if $r^{*}(P)\leq r^{b}$ and is paired if $r^{*}(P)>r^{b}$. The bifurcation point thus divides the packed and paired districts. Second, the assignment of voters to paired districts is described by a decreasing function $s_{1}$ and an increasing function $s_{2}$ where, for each paired district $P$, the two voter types in district $P$ are $s_{1}(r^{*}(P))$ and $s_{2}(r^{*}(P))>s_{1}(r^{*}(P))$. Stronger paired districts thus contain more extreme voters, as single-dippedness requires.

Examples of pack-and-pair districting include segregation, POP, PMP, and negative assortative districting. Note that segregation and negative assortative districting represent the extreme pack-and-pair plans where all voter types are segregated and where only a single type is segregated. We first give conditions under which these extreme districting plans are optimal.

The intuition for the first part of the result is as follows. First, any strictly single-dipped districting plan that never segregates any two voter types is negative assortative. So, it suffices to show that if $G$ is concave (and the taste shock density is strictly log-concave and symmetric), it is sub-optimal for the designer to segregate any two voter types $s<s^{\prime}$. To see this, suppose the designer pools a few type-$s$ voters in with the type-$s^{\prime}$ voters. The marginal effect of this change on the designer’s expected seat share among type-$s$ voters is

which is the increased probability of winning a type-$s$ voter’s district when she moves from the weak district $\delta_{s}$ to the strong district $\delta_{s^{\prime}}$. On the other hand, the marginal effect of this change on the designer’s expected seat share among type-$s^{\prime}$ voters is

This follows because the first term is the marginal effect on the threshold shock to win the strong district, where this comes from using the implicit function theorem (and $Q(0)=1/2$) to calculate $dr/d\rho$ at $\rho=0$ from the equation

and the second term is the density of the aggregate shock at $r^{*}(\delta_{s^{\prime}})=s^{\prime}$. Finally, the sum of the two effects is positive, because

where the first inequality is by concavity of $G$, and the second inequality is by symmetry and strict convexity of $Q$ on $(-\infty,0]$ (which follows from strict log-concavity of $q$).

The intuition for the second part of the result is that if $G$ is sufficiently convex then, for any two voter types $s$ and $s^{\prime}$, we have

which by a similar logic as above implies that it is optimal for the designer to separate any two voter types rather than pooling them.

Proposition 8 expresses the intuition that concavity of $G$ favors pooling (which, under strict single-dippedness, takes the form of pairing types, rather than pooling intervals of types), while convexity of $G$ favors segregation. In the realistic case where $G$ is strictly S-shaped (i.e., the aggregate shock is unimodal), segregation and negative assortative districting are both sub-optimal, unless the two parties are substantially asymmetric.²⁵²⁵25Proposition 9 can be compared to Proposition 1 of Friedman and Holden (2008). Friedman and Holden show that PMP (“matching slices”) is optimal when idiosyncratic uncertainty is sufficiently small, but their discussion focuses on the extreme case of negative assortative districting, where only a single voter type is segregated. Proposition 9 shows that this extreme case never arises when the distribution of the aggregate shock is unimodal and the two parties are symmetric.

The intuition is simple. By Proposition 8, the designer prefers pooling any two voter types above the inflection point $r^{*}(F)$, so segregation is suboptimal. Moreover, for any negative assortative districting, there exist nearby voter types that are paired in a district $P$ with $r^{*}(P)<r^{*}(F)$, but the designer prefers segregating such types.

Since convexity of $G$ favors segregation, concavity of $G$ favors pairing, and it is natural to assume that $G$ is S-shaped (first convex, then concave), a natural conjecture is that pack-and-pair districting (first segregation, then pairing) is optimal. We can verify this conjecture numerically (for an extremely wide range of parameters) in the special case where $G$ and $Q$ are both normal. The following proposition states this result, as well as giving a general sufficient condition for pack-and-pair districting to be uniquely optimal.

Condition (3) can be explained as follows. For any (strictly single-dipped) non-pack-and-pair plan, there exist $s<r<s^{\prime}\leq s^{\prime\prime}$ such that voter types $s<s^{\prime}$ are paired in a district $P$ with $r^{*}(P)=r\in(s,s^{\prime})$ and voter type $s^{\prime\prime}$ is segregated. By a similar logic to Proposition 8, if the first inequality in (3) fails, the designer prefers to segregate a few type-$s$ voters from district $P$; and if the second inequality in (3) fails, the designer prefers to move a few type-$s$ voters from district $P$ to district $\delta_{s^{\prime\prime}}$. Thus, if there do not exist $s<r<s^{\prime}\leq s^{\prime\prime}$ that satisfy (3), then any optimal plan must be pack-and-pair.

Having provided some arguments for pack-and-pair districting, the last part of our analysis compares two key forms of pack-and-pair—POP and PMP—as well as mixed versions of these districting plans. The mixed versions of POP and PMP that we will encounter fall into a class of districting plans that we call “Y-districting.” Formally, a pack-and-pair plan $\mathcal{H}$ is Y-districting if there exists a positive number $\varepsilon>0$ such that

1. (1)

   For all $r\in[r^{b}-\varepsilon,r^{b}+\varepsilon]$ (where $r^{b}$ is the bifurcation point), there exists $P\in\operatorname{supp}(\mathcal{H})$ such that $r^{*}(P)=r$.

2. (2)

   Districts $P\in\operatorname{supp}(\mathcal{H})$ with $r^{*}(P)\in[r^{b}-\varepsilon,r^{b}]$ are segregated (i.e., $\operatorname{supp}(P)=\{r^{*}(P)\}$).

3. (3)

   Districts $P\in\operatorname{supp}(\mathcal{H})$ with $r^{*}(P)\in(r^{b},r^{b}+\varepsilon]$ are paired (i.e., $\operatorname{supp}(P)=\{s_{1}(r^{*}(P)),s_{2}(r^{*}(P))\}$ for some $s_{1}(r^{*}(P))<s_{2}(r^{*}(P))$).

4. (4)

   The functions $s_{1}$ and $s_{2}$ describing the voter types in paired districts are twice differentiable and satisfy $\lim_{r\downarrow r^{b}}s_{1}(r)=\lim_{r\downarrow r^{b}}s_{2}(r)$.²⁶²⁶26The differentiability condition is used in the proof of Proposition 11. It may be possible to drop it.

We will see that Y-districting encompasses a mixed version of POP, where there exists $\hat{s}\in(\underline{s},r^{b})$ such that voter types in $[\underline{s},\hat{s})$ are always segregated and types in $(\hat{s},r^{b})$ are sometimes segregated and sometimes paired, as well as a mixed version of PMP, where there exists $\hat{s}\in(\underline{s},r^{b})$ such that types in $[\underline{s},\hat{s})$ are always paired and types in $(\hat{s},r^{b})$ are sometimes segregated and sometimes paired. (In contrast, recall that under POP there exists $\hat{s}\in(\underline{s},r^{b})$ such that types in $[\underline{s},\hat{s})$ are always segregated and types in $(\hat{s},r^{b})$ are always paired, while under PMP there exists $\hat{s}\in(\underline{s},r^{b})$ such that types in $[\underline{s},\hat{s})$ are always paired and types in $(\hat{s},r^{b})$ are always segregated.) We will give theoretical and numerical results that indicate that POP is optimal when idiosyncratic uncertainty is much larger than aggregate uncertainty, PMP is optimal when aggregate uncertainty is larger than idiosyncratic uncertainty, and Y-districting (and, in particular, mixed POP or mixed PMP) is optimal in the intermediate range.

We first discuss how POP, PMP, and Y-districting relate to the set of all pack-and-pair plans. POP and PMP are both pure districting plans, in that each voter type $s$ is assigned to a single district $P$: formally, for each $s\in[\underline{s},\overline{s}]$, there exists a unique $P\in\operatorname{supp}(\mathcal{H})$ such that $s\in\operatorname{supp}(P)$. They are not the only pure districting plans: for example, a pack-and-pair plan could segregate voter types below a cutoff $s_{0}$ and match slices (including with an intermediate segregation region) among voter types above $s_{0}$. However, POP and PMP are the simplest such plans, as they involve only a single non-degenerate interval of segregated voter types. We are not aware of any parameters for which a more complex pure pack-and-pair plan is optimal.

In contrast, Y-districting plans are mixed, because voter types $s$ just below the bifurcation point are sometimes segregated and sometimes paired with higher types. Somewhat surprisingly, we will see that such plans are uniquely optimal for a range of parameters, even though voter types are continuous. While not every mixed pack-and-pair plan is Y-districting, we will see that, at least numerically, optimal plans always take one of the three forms we consider.

We would like to have general necessary and sufficient conditions for the optimality of POP, PMP, and Y-districting. Unfortunately, this seems very challenging, because the form of optimal districting is driven by global constraints that are difficult to analyze. We instead present a seemingly modest result, which is that if Y-districting is optimal, then the ratio of idiosyncratic uncertainty to aggregate uncertainty must fall in an intermediate range. However, numerically it appears that this result actually characterizes when all three forms of districting are optimal: at least in the case where aggregate and idiosyncratic shocks are both normally distributed, our necessary conditions for optimality of Y-districting are also approximately sufficient, and when the ratio of idiosyncratic uncertainty to aggregate uncertainty is below (resp., above) the range where Y-districting is optimal, then PMP (resp., POP) is optimal.

To facilitate a comparison of the amount of aggregate and idiosyncratic uncertainty, the distributions $G$ and $Q$ should have the same shape. We therefore assume that there exists a parameter $\gamma>0$ such that $G(r)=Q(\gamma r)$ for all $r$. The parameter $\gamma$ thus meaures the ratio of the standard deviation of the idiosyncratic shocks (which is normalized to $1$) to that of the aggregate shock (which equals $\gamma^{-1}$). The following is our key result.

The proof of Proposition 11 proceeds by deriving three necessary conditions for optimal districting to involve a bifurcation point at $r$ (which are based on linear programming duality), and then showing that these conditions imply that the bifurcation point must coincide with the inflection point, and the ratio of idiosyncratic to aggregate uncertainty must lie in an intermediate range. The first condition (equation (12) in Appendix B) says that it is optimal to pair voter types just below and just above $r$. The second condition (equation (13)) says that it is optimal to segregate types just below $r$. The third condition (equation (14)) says that the proportions of favorable and unfavorable voters in each district $P$ with $r^{*}(P)=r^{\prime}$ just above $r$ actually generate the desired cutoff $r^{\prime}$. Intuitively, for it to be optimal to pair nearby voter types around $r$, $G$ must be weakly concave at $r$; and for it to be optimal to segregate voter types just below $r$, $G$ must be weakly convex at $r$. Hence, bifurcation can occur only at the inflection point of $G$, which by symmetry equals $0$. Moreover, if we take parameters where Y-districting is optimal and increase aggregate uncertainty, it eventually becomes optimal to always segregate voter types just below $0$ rather than pairing them with higher voter types, at which point optimal districting becomes PMP (with a bifurcation point below $0$). On the other hand, if we take parameters where Y-districting is optimal and decrease aggregate uncertainty, it eventually becomes optimal to always pair voter types just below $0$ with higher voter types rather than segregating them, at which point optimal districting becomes POP (with a bifurcation point above $0$). We discuss the mechanics of the transition from PMP to POP as $\gamma$ increases below.

If we take for granted that the condition $\gamma\in(1,1.65)$ is sufficient as well as necessary for Y-districting to be optimal, the above intuition suggests that:

1. (1)

   PMP is optimal when $\gamma\leq 1$.

2. (2)

   Y-districting is optimal when $\gamma\in(1,1.65)$.

3. (3)

   POP is optimal when $\gamma\geq 1.65$.

Figure 3 presents numerical solutions that verify this heuristic. In the figure, $Q$ is the standard normal distribution, $G$ is the centered normal distribution with standard deviation $\gamma^{-1}$, and $F$ is the uniform distribution on $[-1,1]$.²⁷²⁷27More precisely, we approximate the designer’s problem by a finite-dimensional linear program and then solve it using Gurobi Optimizer. Our approximation specifies that $s$ is uniformly distributed on $\{-1,-.99,\ldots,.99,1\}$ and that the designer is constrained to create districts $P$ satisfying $r^{*}(P)\in\{-1,-.99,\ldots,.99,1\}$. Voter types are on the $x$-axis, and the threshold shocks to win the districts to which each voter type is assigned are on the $y$-axis. (Thus, packed districts lie on the $45^{\circ}$ line, while paired districts straddle the $45^{\circ}$ line.) For mixed districting plans (i.e., Y-districting, the middle row of the figure), the shading intensity indicates the probability that a voter type is assigned to each district. We see that optimal districting takes exactly the conjectured form: PMP is optimal for $\gamma\in\{0.2,0.5,1\}$, Y-districting is optimal for $\gamma\in\{1.2,1.4,1.6\}$, and POP is optimal for $\gamma\in\{1.7,3,6\}$. The highest value of $\gamma$ in the figure, $\gamma=6$, is the value closest to our empirical estimates. When $\gamma=6$, POP remains optimal but now closely resembles $p$-segregation. Thus, for what we will see is the empirically relevant parameter range, $p$-segregation is approximately optimal.

We can give an intuition for how and why optimal districting transitions from PMP to POP as $\gamma$ increases, as illustrated in Figure 3. Along the way, we also mention some additional features of optimal PMP and POP plans, as well as describing the transition from mixed PMP to mixed POP within the Y-districting regime.

First, recall the extreme cases where $\gamma$ is close to $0$ (almost no idiosyncratic uncertainty) and where $\gamma$ is very large (almost no aggregate uncertainty). When $\gamma$ is close to $0$, PMP is optimal; moreover, when $F$ is symmetric about $0$ as in Figure 3, almost all voters are paired, so optimal districting is approximately negative assortative, which implies that the bifurcation point is below $0$ and the range of values of $r^{*}(P)$ across paired districts $P$ is large.²⁸²⁸28Another property of optimal PMP plans is that the left arm of the “Y” is infinitely steep at the bifurcation point, i.e., $\lim_{r\downarrow r^{b}}s_{1}^{\prime}(r)=0$. When $\gamma$ is very large, POP is optimal; moreover, $p$-segregation is approximately optimal, which implies that the bifurcation point is above $0$ and the range of values of $r^{*}(P)$ across paired districts is very small.²⁹²⁹29Another property of optimal POP plans is that pairing at the bifurcation point is smooth, i.e., $\lim_{r\downarrow r^{b}}s_{1}^{\prime}(r)=-\infty$ and $\lim_{r\downarrow r^{b}}s_{2}^{\prime}(r)=\infty$. Now, when $\gamma$ increases from $0$ toward $1$, the range of $r^{*}(P)$ across paired districts decreases (as the range of probable aggregate shocks decreases), and the proportion of packed districts increases. When $\gamma$ reaches $1$, it becomes optimal to pack voters with $s=0$, the inflection point of $G$. Since it cannot be optimal to pack voters above the inflection point, once $\gamma$ crosses $1$ it becomes optimal to pair voters with $s$ just above $0$ with a few slightly less favorable voters. At this point, districting takes the form of mixed PMP.

As $\gamma$ increases farther above $1$, the range of $r^{*}(P)$ across paired districts continues to decrease. This implies a flattening out of the right arm of the “Y”—i.e., an increase in $s_{2}^{\prime}$—which increases the mass of favorable voters assigned to districts where $r^{*}(P)$ is positive but small. To keep $r^{*}(P)$ small in these districts, this effect must be offset by also assigning more unfavorable voters to these districts, which is achieved by assigning more of the “mixed” unfavorable voters type to paired districts rather than packed districts, while the range of unfavorable voter types assigned to each interval of mixed districts actually decreases—i.e., the left arm of the Y gets steeper.³⁰³⁰30The proof of Proposition 11 shows that, for all sufficiently small positive $r$, $|s_{1}^{\prime}(r)|$ is decreasing in $\gamma$ (i.e., the left arm gets steeper) and $s_{2}^{\prime}(r)$ is increasing in $\gamma$ (i.e., the right arm gets flatter). At some point, the right arm of the Y becomes flatter than the left arm so that the most extreme left-wing voters have no right-wing voters to match with, at which point these voters are segregated: this point marks the transition from mixed PMP to mixed POP, which occurs at $\gamma=\sqrt{2}\approx 1.41$ in the uniform case illustrated in Figure 3.³¹³¹31The transition point $\gamma=\sqrt{2}$ is defined as the unique value of $\gamma$ at which $\lim_{r\downarrow 0}|s_{1}^{\prime}(r)|=\lim_{r\downarrow 0}s_{2}^{\prime}(r)$. The $\gamma=1.4$ panel in the figure illustrates a point just before this transition occurs. As $\gamma$ increases further, more and more mixed unfavorable voters are assigned to paired districts, until all such voters are assigned to paired districts, at which point optimal districting becomes POP, and the bifurcation point becomes positive. This occurs when $\gamma\approx 1.65$. Finally, as $\gamma$ increases further beyond $1.65$, the range of $r^{*}(P)$ across paired districts continues to decrease, and the optimal POP plan approximates $p$-segregation more and more closely.