Skip to main content
Log in

Padding and pruning: gerrymandering under turnout heterogeneity

  • Original Paper
  • Published:
Social Choice and Welfare Aims and scope Submit manuscript

Abstract

Padding is the practice of adding nonvoters (e.g., noncitizens or disenfranchised prisoners) to an electoral district in order to ensure that the district meets the size quota prescribed by the one man, one vote doctrine without affecting the voting outcome in the district. We show how padding— and its mirror image, pruning—, can lead to arbitrarily large deviations from the socially optimal composition of elected legislatures. We solve the partisan districter’s optimal padding problem.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  • Ainsworth R, Munoz EG, Gomez AM (2023) District competitiveness increases voter turnout: evidence from repeated redistricting in North Carolina

  • Bernstein M, Duchin M (2017) A formula goes to court: partisan gerrymandering and the efficiency gap. Not AMS 64(9):1020–1024

    Google Scholar 

  • Black D (1958) The theory of committees and elections. Cambridge University Press, Cambridge

    Google Scholar 

  • Bouton L, Genicot G, Castanheira M, Stashko A (2024) Pack-crack-pack: gerrymandering with differential turnout

  • Bracco E (2013) Optimal districting with endogenous party platforms. J Public Econ 104(13):1–13

    Article  Google Scholar 

  • Chambers CP, Miller AD, Sobel J (2010) A measure of bizarreness. Q J Polit Sci 5(1):27–44

    Article  Google Scholar 

  • Chambers CP, Miller AD, Sobel J (2017) Flaws in the efficiency gap. J Law Polit 33(1):1–33

    Google Scholar 

  • Chen J, Rodden J (2013) Unintentional gerrymandering: political geography and electoral bias in legislatures. Q J Polit Sci 8:239–269

    Article  Google Scholar 

  • Coate S, Knight B (2007) Socially optimal districting: a theoretical and empirical exploration. Q J Econ 122(4):1409–1471

    Article  Google Scholar 

  • Duggan J (2017) A survey of equilibrium analysis in spatial models of elections

  • Friedman JN, Holden RT (2008) Optimal gerrymandering: sometimes pack, but never crack. Am Econ Rev 98(1):113–144

    Article  Google Scholar 

  • Gilligan TW, Matsusaka JG (2006) Public choice principles of redistricting. Public Choice 129:381–398

    Article  Google Scholar 

  • Gomberg A, Pancs R, Sharma T (2023) Electoral maldistricting. Int Econ Rev (forthcoming)

  • Grofman B, King G, Koetzle W, Brunell T (1997) An integrated perspective on the three potential sources of partisan bias: malapportionment, turnout differences, and the geographic distribution of party vote shares. Elect Stud 16(4):457–470

    Article  Google Scholar 

  • Grofman B, King G (2007) The future of partisan symmetry as a judicial test for partisan gerrymandering after LULAC v. Perry. Elect Law J

  • Grofman B, King G, Cervas JR (2020) The terminology of districting: a user guide to understanding gerrymandering

  • Gul F, Pesendorfer W (2010) Strategic redistricting. Am Econ Rev 100(4):1616–1641

    Article  Google Scholar 

  • Hinich MJ (1977) Equilibrium in spacial voting: the median voter result is an artifact. J Econ Theory 16(2):208–219

    Article  Google Scholar 

  • Khan MA, Rath KP, Sun Y (2006) The Dvoretzky–Wald–Wolfowitz theorem and purification in atomless finite-action games. Int J Game Theory 34:91–104

    Article  Google Scholar 

  • Konishi H, Pan C-Y (2020) Partisan and bipartisan gerrymandering. J Public Econ Theory 22(5):1183–212

    Article  Google Scholar 

  • Moskowitz DJ, Schneer B (2019) Reevaluating competition and turnout in U.S. house elections. Q J Polit Sci 14(2):191–223

    Article  Google Scholar 

  • Owen G, Grofman B (1988) Optimal partisan gerrymandering. Polit Geogr Q 7:5–22

    Article  Google Scholar 

  • Persson T, Tabellini G, Economics P (2000) Explaining economic policy. The MIT Press, Cambridge

    Google Scholar 

  • Puppe C, Tasnadi A (2009) Optimal redistricting under geographical constraints: why ‘pack and crack’ does not work. Econ Lett 105(1):93–96

    Article  Google Scholar 

  • Sherstyuk K (1998) How to gerrymander: a formal analysis. Public Choice 95:27–49

    Article  Google Scholar 

  • Stephanopoulos NO, McGhee EM (2015) Partisan gerrymandering and the efficiency gap. Univ Chic Law Rev 82(2):831–900

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Romans Pancs.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix

Appendix

Lemma A.1

For every nonempty block of locations \(\mathcal{B}\subset \mathcal{L}\) and for every integer \(K'\in \left\{ 2,3,\ldots ,K\right\} \), one can construct \(K'\) equipopulous districts that jointly partition the block \(\mathcal{B}\) so that each district has the same mean voter ideology as \(\mathcal{B}\).

The conclusion of Lemma A.1 would be immediate if we could split a location among multiple districts. The proof shows the conclusion holds even when a location cannot be split, which is a maintained assumption in the paper.

Proof of Lemma A.1

For every location \(\left( \rho ,\tau \right) \) in \(\mathcal{B}\) and for every index k in \(\left\{ 1,2,\ldots ,K'\right\} \), define an auxiliary function \(f_{k}\left( \rho ,\tau \right) \equiv 1/K'\). By the definition of \(f_{k}\), for every k, we have

$$\begin{aligned} \int _{\mathcal {B}}f_{k}\left( \rho ,\tau \right) h\left( \rho ,\tau \right) \textrm{d}\rho \textrm{d}\tau&=\frac{1}{K'}\\ \int _{\mathcal {\mathcal {B}}}f_{k}\left( \rho ,\tau \right) \rho \tau h\left( \rho ,\tau \right) \textrm{d}\rho \textrm{d}\tau&=\frac{1}{K'}\int _{\mathcal {L}}\rho \tau h\left( \rho ,\tau \right) \textrm{d}\rho \textrm{d}\tau \\ \int _{\mathcal {\mathcal {B}}}f_{k}\left( \rho ,\tau \right) \tau h\left( \rho ,\tau \right) \textrm{d}\rho \textrm{d}\tau&=\frac{1}{K'}\int _{\mathcal {L}}\tau h\left( \rho ,\tau \right) \textrm{d}\rho \textrm{d}\tau . \end{aligned}$$

The Dvoretzky–Wald–Wolfowitz purification theorem reported by Khan et al. (2006, Theorem DWW, p.93) implies the existence of functions \(\left( f_{k}^{*}\right) \) with \(f_{k}^{*}:\mathcal{B}\rightarrow \left\{ 0,1\right\} \) such that, for every k, we have

$$\begin{aligned} \int _{\mathcal {L}}f_{k}^{*}\left( \rho ,\tau \right) h\left( \rho ,\tau \right) \textrm{d}\rho \textrm{d}\tau&=\int _{\mathcal {L}}f_{k}\left( \rho ,\tau \right) h\left( \rho ,\tau \right) \textrm{d}\rho \textrm{d}\tau \\ \int _{\mathcal {L}}f_{k}^{*}\left( \rho ,\tau \right) \rho \tau h\left( \rho ,\tau \right) \textrm{d}\rho \textrm{d}\tau&=\int _{\mathcal {L}}f_{k}\left( \rho ,\tau \right) \rho \tau h\left( \rho ,\tau \right) \textrm{d}\rho \textrm{d}\tau \\ \int _{\mathcal {L}}f_{k}^{*}\left( \rho ,\tau \right) \tau h\left( \rho ,\tau \right) \textrm{d}\rho \textrm{d}\tau&=\int _{\mathcal {L}}f_{k}\left( \rho ,\tau \right) \tau h\left( \rho ,\tau \right) \textrm{d}\rho \textrm{d}\tau . \end{aligned}$$

We interpret each function \(f_{k}^{*}\) as an indicator that places a location \(\left( \rho ,\tau \right) \) in its entirely into district k if and only if \(f_{k}^{*}\left( \rho ,\tau \right) =1\). The district map associated with \(\left( f_{k}^{*}\right) \) is \(\textbf{D}\equiv \left( D_{1},D_{2},\dots ,D_{K'}\right) \), where \(D_{k}\equiv \left\{ \left( \rho ,\tau \right) \in \mathcal{B}\mid f_{k}^{*}\left( \rho ,\tau \right) =1\right\} \). Combining the two displays above with the definition of \(\textbf{D}\) we conclude that all districts in \(\textbf{D}\) are equipopulous,

$$ \int _{D_{k}}h\left( \rho ,\tau \right) \textrm{d}\rho \textrm{d}\tau =\frac{1}{K'}\qquad \text {for all}\quad k\in \left\{ 1,2,\ldots ,K'\right\} . $$

We also conclude that all districts share with the block \(\mathcal{B}\) the same mean voter ideology:

$$\begin{aligned} \frac{\int _{D_{k}}\rho \tau h\left( \rho ,\tau \right) \textrm{d}\rho \textrm{d}\tau }{\int _{D_{k}}\tau h\left( \rho ,\tau \right) \textrm{d}\rho \textrm{d}\tau }&=\frac{\int _{\mathcal {L}}f_{k}\left( \rho ,\tau \right) \rho \tau h\left( \rho ,\tau \right) \textrm{d}\rho \textrm{d}\tau }{\int _{\mathcal {L}}f_{k}\left( \rho ,\tau \right) \tau h\left( \rho ,\tau \right) \textrm{d}\rho \textrm{d}\tau }\\&=\frac{\int _{\mathcal {L}}\rho \tau h\left( \rho ,\tau \right) \textrm{d}\rho \textrm{d}\tau }{\int _{\mathcal {L}}\tau h\left( \rho ,\tau \right) \textrm{d}\rho \textrm{d}\tau }. \end{aligned}$$

\(\square \)

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gomberg, A., Pancs, R. & Sharma, T. Padding and pruning: gerrymandering under turnout heterogeneity. Soc Choice Welf 63, 401–415 (2024). https://doi.org/10.1007/s00355-024-01536-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00355-024-01536-z

Navigation