Optimal Gerrymandering in a competitive environment

Abstract

We analyze a model of optimal gerrymandering in which two parties simultaneously redistrict in a competition for influence in a legislature. Parties allocate geographic blocks to districts, in which the median voter determines the winner. The form of the optimal gerrymander involves “slices” of right-wing blocks paired with “slices” of left-wing blocks, as in Friedman and Holden (Am Econ Rev 98(1):113–144, 2008). We also show that, as one party controls the redistricting process in more states, that party designs districts such that the most extreme districts within its control become more extreme. We show that this comparative static holds for a broad class of objective functions.

Appendix

Proof of Proposition 1

This result follows the proof of Proposition 7 in Friedman and Holden (2008). Note that the objective function, for each district a party R must create, can be factored, such that

$$\begin{aligned} EW_{p}=B \left( \mu _{n,s}\right) K_{n,s}+\left( 1-B \left( \mu _{n,s}\right) \right) L_{n,s}, \end{aligned}$$

where $K_{n,s}=E\left[ W_{p}|r_{n,s}=1\right]$ and $L_{n,s}=E\left[ W_{p}|r_{n,s}=0\right]$ denote the expected value if party R were to win or lose district n in state s , respectively. Now, fix the districting plan (for both parties) and consider the change in the objective function resulting from a small deviation from the existing plan in district n with an offsetting change in district m, with both districts in state s . The derivative of the value function, with respect to this change (which, in shorthand, we denote $\chi$), is

$$\begin{aligned} \frac{\partial E\left[ W_{p}\right] }{\partial \chi }=b \left( \mu _{n,s}\right) \left( K_{n,s}-L_{n,s}\right) \frac{\partial \mu _{n,s}}{\partial \chi }-b \left( \mu _{m,s}\right) \left( K_{m,s}-L_{m,s}\right) \frac{ \partial \mu _{m,s}}{\partial \chi }, \end{aligned}$$

which must equal 0 for the plan to be optimal. At this point, we note that, but for the constants $K_{n,s}$, $L_{n,s}$ , $K_{m,s}$, and $L_{m,s}$, this expression is identical to that in equation (7) of Friedman and Holden (2008). Thus, we can directly apply Lemmas 1 through 3 from that paper, which imply Proposition 1 in that paper, which is the result here. Since any optimal strategy must have this form, it must be that all equilibria are such that each party employs a strategy of this form. $\square$

Proof of Proposition 2

The proof follows exactly along the lines of Proposition 2 from Friedman and Holden (2008). Since all optimal districting schemes have this feature, it must be that all equilibria involve strategies with this feature. $\square$

Proof of Proposition 3

Using the definition of $\phi ^{*}$, the maximization problem for party R can then be written as

$$\begin{aligned}&\max _{\left\{ \mu _{nR}\right\} }\int W^{\prime }\left( X\right) \left[ 1-Y\left( \phi ^{*}\left[ \left\{ \mu _{nR}\right\} ,\left\{ \mu _{nD}\right\} ;X\right] \right) \right] dX\text { \ } \\&\text {such that \ }\left\{ \mu _{nR}\right\} \in \Omega _{R}. \end{aligned}$$

In words, the party gets $W^{\prime }\left( x\right)$ if the aggregate shock is higher than $\phi ^{*}\left( X\right)$, and we must add up across all of the values X. At an optimum it cannot be the case that reallocating voters with positive mass between (say) district i to district j increases the value function and is still within the constraint set. However, consider such a reallocation and denote the increase in the median of district i of $\Delta \mu _{i}$ and the decrease in the median of district j of $\Delta \mu _{j}.$ Since the value function is differentiable it must be that for any two districts i and j in the same state

$$\begin{aligned} \frac{\int W^{\prime }\left( X\right) y\left( \phi ^{*}\left( X\right) \right) \frac{\partial \phi ^{*}\left( X\right) }{\partial \mu _{i}}dX}{ \int W^{\prime }\left( X\right) y\left( \phi ^{*}\left( X\right) \right) \frac{\partial \phi ^{*}\left( X\right) }{\partial \mu _{j}}dX} =\lim _{\varepsilon \rightarrow 0}\frac{\Delta \mu _{j}}{\Delta _{\mu i}}, \end{aligned}$$

where the limit is taken such that the profile of switching voters is held constant. But, by our definition of $\phi ^{*}$ above, we know that

$$\begin{aligned} \frac{\partial \phi ^{*}}{\partial \mu _{i}}=\frac{c\left( \mu _{i}^{*}+\phi ^{*}\left( X\right) \right) }{\lambda \sum _{nR}c\left( \mu _{d}^{*}+\phi ^{*} \left( X\right) \right) +\left( 1-\lambda \right) \sum _{nD}c\left( \mu _{d}^{*}+\phi ^{*} \left( X\right) \right) }. \end{aligned}$$

Therefore the above ratio can be rewritten as

$$\begin{aligned} \frac{\int W^{\prime }\left( X\right) y\left( \phi ^{*}\left[ \left\{ \mu _{nR}^{*}\right\} ,\left\{ \mu _{nD}^{*}\right\} ;X\right] \right) c\left( \mu _{i}+\phi ^{*}\left[ \left\{ \mu _{nR}^{*}\right\} ,\left\{ \mu _{nD}^{*}\right\} ;X\right] \right) dX}{\int W^{\prime }\left( X\right) y\left( \phi ^{*}\left[ \left\{ \mu _{nR}^{*}\right\} ,\left\{ \mu _{nD}^{*}\right\} ;X\right] \right) c\left( \mu _{j}+\phi ^{*}\left[ \left\{ \mu _{nR}^{*}\right\} ,\left\{ \mu _{nD}^{*}\right\} ;X\right] \right) dX}=\lim _{\varepsilon \rightarrow 0}\frac{\Delta \mu _{j}}{\Delta \mu _{i}}, \end{aligned}$$

where $\phi ^{*}$ is that value associated with the equilibrium strategies. But these are the just the necessary conditions to the problem in which the gerrymanderer maximizes the alternative objective function

$$\begin{aligned}&\max _{\left\{ \mu _{nR}\right\} }\int W^{\prime }\left( X\right) y\left( \phi ^{*}\left[ \left\{ \mu _{nR}^{*}\right\} ,\left\{ \mu _{nD}^{*}\right\} ;X\right] \right) \sum C\left( \mu _{n,s}+\phi ^{*} \left[ \left\{ \mu _{nR}^{*}\right\} ,\left\{ \mu _{nD}^{*}\right\} ;X\right] \right) dX. \\&\text {such that }\left\{ \mu _{nR}\right\} \in \Omega _{R}. \end{aligned}$$

The proof for party D follows precisely the parallel logic. $\square$

Proof of Corollary 1

Consider the situation in which party R’s value function is

$$\begin{aligned} W_{n}= & {} \frac{x^{n}}{x^{n}+\left( 1-x\right) ^{n}} \\ W_{n}^{\prime }= & {} \frac{\left( \left( x-1\right) x\right) ^{n}\left( \log x-\log \left( 1-x\right) \right) }{\left( x^{n}+\left( 1-x\right) ^{n}\right) ^{2}} \end{aligned}$$

Note that, as $n\rightarrow \infty ,$W limits to the desired function. By Proposition 3, party R solves the alternative maximization

$$\begin{aligned}&\max _{\left\{ \mu _{dR}\right\} }\int \left[ \frac{W_{n}^{\prime }\left( x\right) }{W_{n}^{\prime }\left( \frac{1}{2}\right) }y\left( \phi ^{*}\left( x\right) \right) \sum C\left( \mu _{n,s}+\phi ^{*}\left( \left\{ \mu _{dD}^{*}\right\} ,\left\{ \mu _{dR}^{*}\right\} ,x\right) \right) \right] dx \\&\text {such that }\left\{ \mu _{nR}\right\} \in \Omega _{R}. \end{aligned}$$

which is identical to equation (3) above but for scaling by the constant term $W_{n}^{\prime }\left( \frac{1}{2}\right)$ . But as $n\rightarrow \infty$, the weights

$$\begin{aligned} \lim _{n\rightarrow \infty }\frac{W_{n}^{\prime }\left( x\right) }{ W_{n}^{\prime }\left( \frac{1}{2}\right) }\rightarrow \left\{ \begin{array}{cc} 0 &{} x\ne \frac{1}{2} \\ 1 &{} x=\frac{1}{2} \end{array} \right. . \end{aligned}$$

Thus the necessary conditions are simply that

$$\begin{aligned} \frac{c\left( \mu _{i}+\phi ^{*}\right) }{c\left( \mu _{j}+\phi ^{*}\right) }=\frac{\Delta \mu _{j}}{\Delta \mu _{i}} \end{aligned}$$

These are the same necessary conditions as if party R simply maximized the number of seats won at critical value $\phi ^{*}\left( \left\{ \mu _{nD}^{*}\right\} ,\left\{ \mu _{nR}^{*}\right\} ,\frac{1}{2}\right)$, which could be written

$$\begin{aligned} \max _{\left\{ \mu _{nR}\right\} }\sum _{nR}C\left( \mu _{n,s}+\phi ^{*}\left( \left\{ \mu _{nD}^{*}\right\} ,\left\{ \mu _{nR}^{*}\right\} ,\frac{1}{2}\right) \right) \\ \text {such that }\left\{ \mu _{nR}\right\} \in \Omega _{R}. \end{aligned}$$

$\square$

Proof of Proposition 4

First consider a state redistricted by party R. Suppose $N_s=2.$ Following Corollary 1, there are two FOCs that combine to imply

$$\begin{aligned} \frac{c\left( \mu _{1}+\phi ^{*}\right) }{c\left( \mu _{2}+\phi ^{*}\right) }=\frac{\Delta \mu _{2}}{\Delta \mu _{1}}. \end{aligned}$$

Writing $\mu _{2}\left( \mu _{1}\right)$ one can substitute into the objective function above, so that the FOC becomes

$$\begin{aligned} \mu _{1}^{*}= & {} \arg \max _{\left\{ \mu _{1}\right\} }\left\{ C\left( \mu _{_{1}}+\phi ^{*} \right) +C\left( \mu _{2}\left( \mu _{_{1}}\right) +\phi ^{*} \right) \right\} \\\Rightarrow & {} \frac{c\left( \mu _{1}+\phi ^{*}\right) }{c\left( \mu _{2}+\phi ^{*}\right) }=-\frac{d\mu _{2}\left( \mu _{1}\right) }{\mathrm{d}\mu _{1}}. \end{aligned}$$

Of course, $\frac{d\mu _{2}\left( \mu _{1}\right) }{d\mu _{1}}<0$. Then, by the implicit function theorem, we know that $\frac{ \partial \mu _{1}^{*}}{\partial \phi ^{*}}<0$ if and only if the LHS is decreasing in $\phi ^{*},$ which is true if and only if

$$\begin{aligned} \frac{c^{\prime }\left( \mu _{1}+\phi ^{*}\right) }{c^{\prime }\left( \mu _{2}+\phi ^{*}\right) }<\frac{c\left( \mu _{1}+\phi ^{*}\right) }{c\left( \mu _{2}+\phi ^{*}\right) } \end{aligned}$$

Note that by the equal mass constraint it must be that $\frac{ \partial \mu _{2}}{\partial \phi ^{*}}$ is of the opposite sign as $\frac{\partial \mu _{1}^{*}}{\partial \phi ^{*}}$. Moreover, $\frac{\partial \mu _{1}^{*}}{\partial \phi ^{*}}$ depends entirely on whether the ratio $\frac{c^{\prime }\left( \psi \right) }{c\left( \psi \right) }$ is increasing or decreasing in $\psi .$ This ratio being decreasing is precisely the definition of log concavity and hence $\frac{\partial \mu _{1}^{*}}{\partial \phi ^{*}}<0 .$

To prove the result for $N_{s} \ge 2,$ note that we maximize the objective function

$$\begin{aligned} \max _{\left\{ \mu _{nR}\right\} }\sum _{nR}C\left( \mu _{n}+\phi ^{*}\right) . \end{aligned}$$

Consider a deviation in which one shifts $\mu _{1}$ upwards by amount $\Delta \mu _{1}$ and then shifts all other medians down by $\Delta \mu _{-1}$. The no-benefit condition from such a deviation is

$$\begin{aligned} \frac{c\left( \mu _{1}+\phi ^{*}\right) }{\sum _{n\ne 1}c\left( \mu _{n}+\phi ^{*}\right) }=\frac{\Delta \mu _{-1}}{\Delta \mu _{1}}. \end{aligned}$$

(4)

Note, at this point, that the medians $\left\{ \mu _{2},\ldots ,\mu _{N}\right\}$ are chosen optimally. Therefore, we can implicitly differentiate this expression to obtain the impact of $\phi ^{*}$ on $\mu _{1}$, since all deviations within the medians $\left\{ \mu _{2},\ldots ,\mu _{N}\right\}$ have a second order impact on the value function, by the Envelope Theorem. From the $N_{s}=2$ case we know that the ratio $\frac{c\left( \mu _{1}+\phi ^{*}\right) }{c\left( \mu _{n}+\phi ^{*}\right) }$ is falling with $\phi ^{*} ,$ and therefore we know that the LHS of equation (4) is also decreasing in $\phi ^{*}$ . Therefore, we know that $\frac{\partial \mu _{1}^{*}}{\partial \phi ^{*}}<0 .$ A parallel argument establishes that $\frac{\partial \mu _{N}^{*}}{\partial \phi ^{*}}>0 .$ The argument for states controlled by party D follows exactly the same logic. $\square$

Proof of Example 1

We can rewrite the expected number of seats won by party R as

$$\begin{aligned} X\left( \phi \right)= & {} k\int \left( \mu -\phi \right) \mathrm{d}M\left( \mu \right) \\= & {} k\left( \lambda {\overline{\mu }}_{R}+\left( 1-\lambda \right) {\overline{\mu }}_{D}+\phi \right) . \end{aligned}$$

and the expected value function for party R as

$$\begin{aligned} EW=\int W\left( k\left( \lambda {\overline{\mu }}_{R}+\left( 1-\lambda \right) \overline{ \mu }_{D}+\phi \right) \right) \mathrm{d}Y\left( \phi \right) . \end{aligned}$$

This equation shows that ${\overline{\mu }}_{R}$ is a sufficient statistic for the impact of party R districts on the aggregate outcome. Therefore each party does best simply to maximize the average of the median voters in the districts in their control. There is no strategic interaction at all between the parties in this special case. As a result, the share of districts $\lambda$ under the control of party R can have no impact on the optimal gerrymander. $\square$

Proof of Proposition 5

Karlin (1968, p.30) shows that the convolution $h=f\cdot g$ is $PF_{n}$ if f and g are $PF_{n}.$ By a theorem of Schoenberg (1947, (1951), $PF_{2}$ of a density is equivalent to log-concavity. The assumption of uniformity of y means that we are left with the term $\int W^{\prime }\left( x\right) c\left( \mu _{i}+\phi ^{*}\left( x\right) \right) \mathrm{d}x,$ which is $PF_{2}$ since $W^{\prime }\left( x\right)$ is $PF_{2}$ and c is log-concave. Since $W^{\prime }\left( x\right)$ is $PF_{2}$ it is integrable and hence continuous. Now the proof of Proposition 4 applies. $\square$

Example 2

Suppose that the objective function takes on the form of the double-discontinuity function

$$\begin{aligned} W\left( X\right) =\left\{ \begin{array}{cc} 0 &{} X<\frac{1}{3} \\ \frac{1}{4} &{} X=\frac{1}{3} \\ \frac{1}{2} &{} \frac{1}{3}<X<\frac{2}{3} \\ \frac{3}{4} &{} X=\frac{2}{3} \\ 1 &{} X>\frac{2}{3} \end{array} \right. . \end{aligned}$$

Let $N_{s}=2$, and $\phi ^{*}$ be distributed uniformly for simplicity. Suppose that c is log-concave. By Proposition 3, we know that the ratio

$$\begin{aligned} DDR=\frac{c\left( \mu _{1}+\phi ^{*}\left( \frac{1}{3}\right) \right) +c\left( \mu _{1}+\phi ^{*}\left( \frac{2}{3}\right) \right) }{c\left( \mu _{2}+\phi ^{*}\left( \frac{1}{3}\right) \right) +c\left( \mu _{2}+\phi ^{*}\left( \frac{2}{3}\right) \right) } \end{aligned}$$

(5)

must be monotonic in $\lambda$ for the comparative static to hold. By log-concavity, we know that the ratio $\frac{c\left( \mu _{1}+\phi ^{*}\left( x\right) \right) }{c\left( \mu _{2}+\phi ^{*}\left( x\right) \right) }$ is increasing in $\lambda$ for all x. But it will not generally be the case that the combined ratio in expression (5) is increasing. For instance, suppose that the following values hold for $\lambda _{H}>\lambda _{L}.$

	$\frac{c\left( \mu _{1}+\phi ^{*}\left( \frac{1}{3}\right) \right) }{ c\left( \mu _{2}+\phi ^{*}\left( \frac{1}{3}\right) \right) }$	$\frac{ c\left( \mu _{1}+\phi ^{*}\left( \frac{2}{3}\right) \right) }{c\left( \mu _{2}+\phi ^{*}\left( \frac{2}{3}\right) \right) }$	DDR
$\lambda _{H}$	$\frac{8}{2}$	$\frac{100}{100}$	$\frac{108}{102}\approx 1$
$\lambda _{L}$	$\frac{3}{1}$	$\frac{1}{2}$	$\frac{4}{3}>1.$

Intuitively, it is important to note that the double-discontinuity objective function is an extreme example of a function that is not $PF_{2},$ since it is the limit of an extremely bimodal function. When “convoluted” with $W^{\prime },$c loses its log-concavity, and so a fall in $\phi ^{*}$ is no longer enough to guarantee an increase in the value of the higher median.