After the release of the 2020 Census, legislatures across the country redrew their states&#x2019; congressional district maps, just like they do every decade. And, just like every decade, <a href="https://www.washingtonpost.com/politics/interactive/2022/redistricting-mini-golf/?itid=lk_inline_enhanced-template" target="_blank">aggrieved citizens sued them for gerrymandering</a> &#x2014; the process whereby politicians craft district boundaries to ensure their own parties&#x2019; victory.

But this time around, something has changed. A technological revolution, decades in the making, has added a sharp new arrow to those citizens&#x2019; quiver of legal arguments. Known as algorithmic redistricting, the technology has persuaded judges to throw out gerrymandered maps in several states, including <a href="https://www.washingtonpost.com/politics/2022/04/27/ny-congressional-map-is-illegal-must-be-redrawn-court-says/?itid=lk_inline_enhanced-template" target="_blank">New York</a> and <a href="https://www.washingtonpost.com/politics/ohio-redistricting/2022/01/14/338009aa-755d-11ec-8b0a-bcfab800c430_story.html?itid=lk_inline_enhanced-template" target="_blank">Ohio</a>.<b> </b>And it will be part of a case before the Supreme Court <a href="https://www.washingtonpost.com/politics/2022/06/30/supreme-court-federal-elections-state-legislatures/?itid=lk_inline_enhanced-template" target="_blank">in October</a> that could play a role in the 2024 election and the future of voting rights.

The governor of your state<span class="governor-icon"><svg width=19 height=50><g transform="scale(0.45)">
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          <path class="fill-yellow-3 stroke-black" d="M3.6,34.5l2.1-0.8l-1.2-2.8c0,0,2.6-2.6,4.6-3.6c0,0,0,0,0,0c0.3,0.7,0.7,1.4,1.2,2.1c-0.1,0.7-0.2,1.1-0.2,1.1C3.4,41.3,9.2,55.7,9.2,64.1l0,0C7.5,56.7-0.9,45.1,3.6,34.5z"/>
          <circle class="fill-yellow-skin-1 stroke-black" cx="20.7" cy="22" r="12.8"/>
          <path class="fill-black stroke-black" d="M20.6,7.3c4.4,0,8.3,1,10.5,2.4c0.3-0.2,0.6-0.3,0.9-0.3c1.8,0,3.3,3.7,3.3,8.3c0,4.6-1.5,8.3-3.3,8.3c-1.8,0-3.3-3.7-3.3-8.3c0-0.3,0-0.7,0-1c-2.2,0.8-5,1.3-8.1,1.3c-6.9,0-12.5-2.4-12.5-5.4S13.7,7.3,20.6,7.3z"/>
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          <path class="fill-yellow-2 stroke-black" d="M37.6,8.1c-0.4,1.8-6.7,2.1-14.1,0.7S10.3,4.9,10.7,3.2l-0.5,5C9.8,10,15.5,12.5,23,13.8s13.8,1,14.1-0.7L37.6,8.1z"/>
          <path class="fill-yellow-4 stroke-black" d="M37.3,11.2c-0.4,1.8-6.7,2.1-14.1,0.7S10,8.1,10.4,6.3l-0.2,1.9C9.8,10,15.5,12.5,23,13.8s13.8,1,14.1-0.7L37.3,11.2z"/>
          <ellipse transform="matrix(0.1769 -0.9842 0.9842 0.1769 14.4047 28.2439)" class="fill-yellow-2 stroke-black" cx="24.1" cy="5.5" rx="3.2" ry="13.7"/></g></svg></span>, who is from the Yellow Party, has approved the state legislature&#x2019;s plan that packs lots of Purple Party-leaning voters into a single district. The plan fulfills the requirement, followed by every state, that each district has the same number of people as all the others. Still, it results in a map that skews the partisan balance of your state in favor of the governor&#x2019;s party.

In other words, this map is a gerrymander. Arguably more than any other practice &#x2014; more than voter suppression or even outright fraud &#x2014; gerrymandering undermines the democratic principle that each citizen&#x2019;s vote is worth the same as everyone else&#x2019;s.

You decide to fight back. While ensuring that each district has the same number of people, you find you can draw an alternative map that puts you and your neighbors in a majority-Purple district and helps the Purple Party statewide.

With your new map in hand, you are ready to sue. You tell the judge that the map put together by the governor&#x2019;s party<b> </b>misrepresents the voters of your state and that it is simple to draw a fairer map.

The judge hears you out, and then she dismisses your case. Where did you get the idea it was your job to draw the congressional map? Who elected you, buddy?

You need a better plan. Fortunately, you have some computer skills. You decide to create a redistricting <i>algorithm</i>. That&#x2019;s a fancy word for a simple thing. An algorithm is a recipe, where each of the steps is computational.

Now, you don&#x2019;t have an alternative map to show the judge. You have something better: a distribution. You can show the judge that the governor&#x2019;s map doesn&#x2019;t just underrepresent Purple voters. It is an outlier in the universe of possible maps, drawn by a computer that knows nothing of politics.

You also tell the judge that this trick, where you tell a computer to repeatedly perform randomized simulations, was <a href="https://permalink.lanl.gov/object/tr?what=info%3Alanl-repo%2Flareport%2FLA-UR-88-9068&amp;itid=lk_inline_enhanced-template" target="_blank">invented by the legendary scientists Stanislaw Ulam and John von Neumann</a>, who needed a better way of understanding how neutrons behaved during nuclear chain reactions. Their first simulations were run in 1948 on a computer program written by Kl&#xE1;ra D&#xE1;n von Neumann, who translated her husband&#x2019;s flow diagrams into some of history&#x2019;s earliest computer code.

The technique, which came to be known as &#x201C;Monte Carlo,&#x201D; after the casino in Monaco, would go on to influence nearly all of science. &#x201C;Many basic scientific problems are now routinely solved by simulation,&#x201D;<b> </b>the mathematician Persi Diaconis <a href="https://math.uchicago.edu/~shmuel/Network-course-readings/MCMCRev.pdf?itid=lk_inline_enhanced-template">wrote in 2009</a>, saying it has &#x201C;revolutionized applied mathematics.&#x201D;

The distribution produced by your ensemble of <span class="inline-simulation-count"></span> computer-generated maps convinces the judge that the governor&#x2019;s map, with just one Purple district, was not drawn by chance alone. Rather, the governor must have intended to draw the map in a way that discriminated against Purple voters. The judge orders the state legislature to draw a new map that better reflects the partisanship of your state. Congratulations, you won.

Back in the real world, plaintiffs in gerrymandering cases across the country have hired experts to build ensembles of computer-generated maps that show how state-approved maps were drawn unfairly.

<a href="https://www.washingtonpost.com/politics/interactive/2021/redistricting-faq/?itid=lk_inline_enhanced-template" target="_blank">In 37 states</a>, redistricting is a deeply political process: The legislature passes the map, and the governor either vetoes it or signs it into law. Still, many states ban partisan gerrymandering, and their courts are empowered to reject gerrymandered maps. (The Supreme Court, for its part, <a href="https://www.washingtonpost.com/politics/courts_law/supreme-court-says-federal-courts-dont-have-a-role-in-deciding-partisan-gerrymandering-claims/2019/06/27/2fe82340-93ab-11e9-b58a-a6a9afaa0e3e_story.html?itid=lk_inline_enhanced-template" target="_blank">has decided not to concern itself with partisan gerrymandering</a>.)

During this redistricting cycle, map ensembles have been used as evidence in litigation in 11 states.

In a case in Ohio, Harvard&#x2019;s Kosuke Imai, the rare combination of mathematician and political scientist, served as an expert witness for a group of plaintiffs led by the League of Women Voters of Ohio. Imai used <a href="https://arxiv.org/abs/2008.06131?itid=lk_inline_enhanced-template" target="_blank">an algorithm he created</a> with fellow Harvard mathematician Cory McCartan to produce an ensemble of 5,000 maps. &#x201C;Of those simulated plans, <i>none</i> was as favorable to Republicans as the adopted plan,&#x201D; <a href="https://redistricting.lls.edu/wp-content/uploads/oh_lwv-210112-order.pdf?itid=lk_inline_enhanced-template" target="_blank">the court noted as it ordered</a> the Republican-controlled redistricting commission to draw a fairer map.

In <a href="https://law.justia.com/cases/new-york/court-of-appeals/2022/60.html?itid=lk_inline_enhanced-template" target="_blank">a lawsuit opposing New York&#x2019;s alleged gerrymander</a>, meanwhile, Sean Trende, an elections analyst who works for the news website RealClearPolitics, used Imai and McCartan&#x2019;s algorithm to produce 10,000 maps. Trende, who was hired by plaintiffs opposing the state&#x2019;s map, testified that his ensemble showed the map drawn by New York state&#x2019;s democratic legislature was an &#x201C;extreme outlier.&#x201D; The court ordered the legislature to go back to the drawing board.

Imai and McCartan, along with a team of six other scientists, have created <a href="https://alarm-redist.github.io/fifty-states/?itid=lk_inline_enhanced-template" target="_blank">the 50-State Redistricting Simulations project</a>, which aims to produce 5,000-map ensembles of every state. The ensembles produced for Ohio and New York as part of that project show how the gerrymandered maps fell outside the middle range of the distributions.

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			<p>Note: Expected seats were calculated based on multiple elections from 2016, 2018 and 2020, and are reported in terms of fractional seats.</p>
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	</div>

	
	<div id="g-histograms-xsmall" class="g-artboard" style="width:384px; height:662.027887342279px;" data-aspect-ratio="0.58" data-min-width="384" data-max-width="639">
		<img id="g-histograms-xsmall-img" class="g-aiImg g-aiAbs" alt="" data-src="https://www.washingtonpost.com/wp-stat/graphics/ai2html/democracytech/F5OBL5ON6ZDT3CFHRCYOZMZ6O4/histograms-xsmall.jpg?v=11" src="data:image/gif;base64,R0lGODlhCgAKAIAAAB8fHwAAACH5BAEAAAAALAAAAAAKAAoAAAIIhI+py+0PYysAOw=="/>
		<div id="g-ai1-1" class="g-content g-aiAbs g-aiPointText" style="top:1.62%;margin-top:-12.7px;left:0.6511%;width:107px;">
			<p class="g-pstyle0">New York</p>
		</div>
		<div id="g-ai1-2" class="g-content g-aiAbs g-aiPointText" style="top:4.9907%;margin-top:-10px;left:0.6511%;width:189px;">
			<p class="g-pstyle1">Democratic gerrymander</p>
		</div>
		<div id="g-ai1-3" class="g-content g-aiAbs g-aiPointText" style="top:11.7125%;margin-top:-18.5px;left:27.224%;margin-left:-51px;width:102px;">
			<p class="g-pstyle2">State&rsquo;s map</p>
			<p class="g-pstyle2">was here</p>
		</div>
		<div id="g-ai1-4" class="g-content g-aiAbs g-aiPointText" style="top:39.1283%;margin-top:-10px;left:49.884%;margin-left:-98px;width:196px;">
			<p class="g-pstyle3">Expected number of seats</p>
		</div>
		<div id="g-ai1-5" class="g-content g-aiAbs g-aiPointText" style="top:42.1391%;margin-top:-9px;left:3.5548%;width:120px;">
			<p class="g-pstyle4">More Democrats</p>
		</div>
		<div id="g-ai1-6" class="g-content g-aiAbs g-aiPointText" style="top:42.1391%;margin-top:-9px;right:3.8226%;width:128px;">
			<p class="g-pstyle5">More Republicans</p>
		</div>
		<div id="g-ai1-7" class="g-content g-aiAbs g-aiPointText" style="top:48.7479%;margin-top:-12.7px;left:0.6516%;width:64px;">
			<p class="g-pstyle0">Ohio</p>
		</div>
		<div id="g-ai1-8" class="g-content g-aiAbs g-aiPointText" style="top:52.1187%;margin-top:-10px;left:0.6516%;width:187px;">
			<p class="g-pstyle1">Republican gerrymander</p>
		</div>
		<div id="g-ai1-9" class="g-content g-aiAbs g-aiPointText" style="top:57.1789%;margin-top:-18.5px;left:82.7881%;margin-left:-51px;width:102px;">
			<p class="g-pstyle2">State&rsquo;s map</p>
			<p class="g-pstyle2">was here</p>
		</div>
		<div id="g-ai1-10" class="g-content g-aiAbs g-aiPointText" style="top:84.5946%;margin-top:-10px;left:49.884%;margin-left:-98px;width:196px;">
			<p class="g-pstyle3">Expected number of seats</p>
		</div>
		<div id="g-ai1-11" class="g-content g-aiAbs g-aiPointText" style="top:87.7565%;margin-top:-9px;left:3.5548%;width:120px;">
			<p class="g-pstyle4">More Democrats</p>
		</div>
		<div id="g-ai1-12" class="g-content g-aiAbs g-aiPointText" style="top:87.7565%;margin-top:-9px;right:3.8226%;width:128px;">
			<p class="g-pstyle5">More Republicans</p>
		</div>
		<div id="g-ai1-13" class="g-content g-aiAbs" style="top:91.5369%;left:0.3627%;width:98.9583%;">
			<p>Note: Expected seats were calculated based on multiple elections from 2016, 2018 and 2020, and are reported in terms of fractional seats.</p>
		</div>
	</div>

	
	<div id="g-histograms-medium" class="g-artboard" style="width:640px; height:334.3787529895px;" data-aspect-ratio="1.914" data-min-width="640">
		<img id="g-histograms-medium-img" class="g-aiImg g-aiAbs" alt="" data-src="https://www.washingtonpost.com/wp-stat/graphics/ai2html/democracytech/F5OBL5ON6ZDT3CFHRCYOZMZ6O4/histograms-medium.jpg?v=11" src="data:image/gif;base64,R0lGODlhCgAKAIAAAB8fHwAAACH5BAEAAAAALAAAAAAKAAoAAAIIhI+py+0PYysAOw=="/>
		<div id="g-ai2-1" class="g-content g-aiAbs g-aiPointText" style="top:2.9083%;margin-top:-12.7px;left:54.0243%;width:64px;">
			<p class="g-pstyle0">Ohio</p>
		</div>
		<div id="g-ai2-2" class="g-content g-aiAbs g-aiPointText" style="top:3.2074%;margin-top:-12.7px;left:0.3126%;width:107px;">
			<p class="g-pstyle0">New York</p>
		</div>
		<div id="g-ai2-3" class="g-content g-aiAbs g-aiPointText" style="top:9.881%;margin-top:-10px;left:54.0243%;width:187px;">
			<p class="g-pstyle1">Republican gerrymander</p>
		</div>
		<div id="g-ai2-4" class="g-content g-aiAbs g-aiPointText" style="top:10.1801%;margin-top:-10px;left:0.3126%;width:189px;">
			<p class="g-pstyle1">Democratic gerrymander</p>
		</div>
		<div id="g-ai2-5" class="g-content g-aiAbs g-aiPointText" style="top:21.0958%;margin-top:-18.5px;left:92.0971%;margin-left:-51px;width:102px;">
			<p class="g-pstyle2">State&rsquo;s map</p>
			<p class="g-pstyle2">was here</p>
		</div>
		<div id="g-ai2-6" class="g-content g-aiAbs g-aiPointText" style="top:21.0958%;margin-top:-18.5px;left:12.5398%;margin-left:-51px;width:102px;">
			<p class="g-pstyle2">State&rsquo;s map</p>
			<p class="g-pstyle2">was here</p>
		</div>
		<div id="g-ai2-7" class="g-content g-aiAbs g-aiPointText" style="top:75.9737%;margin-top:-10px;left:23.0484%;margin-left:-98px;width:196px;">
			<p class="g-pstyle3">Expected number of seats</p>
		</div>
		<div id="g-ai2-8" class="g-content g-aiAbs g-aiPointText" style="top:75.9737%;margin-top:-10px;left:76.8385%;margin-left:-98px;width:196px;">
			<p class="g-pstyle3">Expected number of seats</p>
		</div>
		<div id="g-ai2-9" class="g-content g-aiAbs g-aiPointText" style="top:82.2338%;margin-top:-9px;left:55.3698%;width:120px;">
			<p class="g-pstyle4">More Democrats</p>
		</div>
		<div id="g-ai2-10" class="g-content g-aiAbs g-aiPointText" style="top:82.2338%;margin-top:-9px;right:1.6936%;width:128px;">
			<p class="g-pstyle5">More Republicans</p>
		</div>
		<div id="g-ai2-11" class="g-content g-aiAbs g-aiPointText" style="top:82.5329%;margin-top:-9px;left:1.6581%;width:120px;">
			<p class="g-pstyle4">More Democrats</p>
		</div>
		<div id="g-ai2-12" class="g-content g-aiAbs g-aiPointText" style="top:82.5329%;margin-top:-9px;right:55.4053%;width:128px;">
			<p class="g-pstyle5">More Republicans</p>
		</div>
		<div id="g-ai2-13" class="g-content g-aiAbs g-aiPointText" style="top:94.067%;margin-top:-20.5px;left:0.2665%;width:651px;">
			<p class="g-pstyle6">Note: Expected seats were calculated based on multiple elections from 2016, 2018 and 2020,</p>
			<p class="g-pstyle6">and are reported in terms of fractional seats.</p>
		</div>
	</div>

</div>



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But the courtroom successes have some of algorithmic redistricting&#x2019;s enthusiasts worried. They fear that computer-generated map ensembles can seem convincing to judges even when they were produced with insufficient mathematical rigor.

&#x201C;Judges are not mathematical people,&#x201D; said Zachary Schutzman, a postdoctoral researcher at MIT&#x2019;s Institute for Data, Systems, and Society who has worked on redistricting algorithms and consulted in litigation. Sometimes, he said, &#x201C;there is this effort to win by wowing the judge while also not really going into any of the detail that might bore the judge.&#x201D;

Recall that your computer drew <span class="inline-simulation-count"></span> maps. In a state like yours, with 25 towns arranged in a square, it turns out that <a href="http://oeis.org/A172477?itid=lk_inline_enhanced-template" target="_blank">there are exactly 4,006 ways</a> to draw five equally-sized districts. That means you drew just <span class="inline-simulation-percent"></span>% of all the possible maps. If you had drawn more maps, isn&#x2019;t it possible that your distribution would have looked different?

In the real world, the problem is more serious. Real states are composed not of five-by-five grids of towns, but of massive jigsaw puzzles of administrative units called census blocks.

Even after imposing constraints on mapmakers &#x2014; for instance, that all districts should have about the same population and should avoid splitting counties and towns &#x2014; the world&#x2019;s fastest computer could not count all the ways to combine New York&#x2019;s 350,169 census blocks into its 26 congressional districts. There are likely more valid maps than the number of atoms in the observable universe.

The trick, then, is not to draw every possible map, but to draw enough maps to collect a representative sample.

Moon Duchin, the mathematician from Tufts University who has done more than perhaps anyone else to advance the mathematical theory underlying algorithmic redistricting, has produced as many as 1 billion possible maps for a single state. Most of those maps, she said, were unnecessary. After you draw about 100,000, the distribution stops changing.

&#x201C;That&#x2019;s what tells you with very high confidence, but not a proof, that 100,000 steps was enough to get a good sample,&#x201D; Duchin said.

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Duchin is skeptical about some of the smaller ensembles that have persuaded judges, characterizing 5,000-map samples as &#x201C;wildly inadequate.&#x201D;

Imai and McCartan designed their algorithm, which they dubbed Sequential Monte Carlo, to more efficiently explore the universe of possible maps. That allows them, at least in theory, to get away with generating smaller samples. But Sequential Monte Carlos has not yet withstood the rigor of academic peer review, which sometimes uncovers errors.

&#x201C;Unfortunately, academic publishing can take a long time especially in technical fields like statistics,&#x201D; Imai said in an email, noting that he and his colleagues had published <a href="https://imai.fas.harvard.edu/research/files/DAS.pdf?itid=lk_inline_enhanced-template" target="_blank">an applied paper</a> using the method and made the code and data publicly available.

In 1961, <a href="https://www.jstor.org/stable/2145973?itid=lk_inline_enhanced-template" target="_blank">the economist William Vickrey first proposed</a> creating &#x201C;mechanical rules which will eliminate the element of human discretion almost entirely from the process of redistricting, and which if followed would make gerrymandering virtually impossible.&#x201D; For Vickrey, the temptation to gerrymander was too great to be left in the hands of humans &#x2014; particularly those who had chosen politics as a profession.

It took six decades, but improvements in mathematical theory and computing power have caught up with Vickrey&#x2019;s vision. Computers can draw millions of maps and scientifically distinguish between those that are typical and those that are not. Yet some now fear computers may be even less trustworthy than humans, that silicon is too ignorant of the flesh-and-blood complexities of the human-populated world.

This is perhaps particularly true in the United States, where Black people long faced intimidation, violence, and legal segregation that made it difficult, and often impossible, to fully participate in the democratic process. In 1965, Congress passed the Voting Rights Act, dictating that racial minorities must be able to elect representatives of their choice.

But if states only care about ensuring that their congressional districts have the same number of people in them, they are unlikely to draw maps that fairly represent minorities. Rather, states have to go out of their way to craft maps that enhance minority representation.

Though well-intentioned, this bending of district boundaries shares some similarities with old fashioned gerrymandering. Like gerrymandering, it involves humans hand-crafting districts to produce a desired political outcome. And, like gerrymandering, it is likely to produce maps that lie at the extremes of distributions created by computer simulations.

In this toy simulation, maps that represent Black people in proportion to their population are incredibly rare. Last year, Duchin and Douglas Spencer, an associate law professor at the University of Colorado who runs <a href="https://redistricting.lls.edu/?itid=lk_inline_enhanced-template">the website All About Redistricting</a>, set out to find out how rare such maps would be in the real world.

<a href="https://www.yalelawjournal.org/pdf/DuchinSpencerEssay_q1g16rki.pdf?itid=lk_inline_enhanced-template" target="_blank">Duchin and Spencer created</a> 2 million map ensembles of congressional, state House and state Senate districts in 19 states &#x2014; 114 million maps in all. Not once did the computer draw a map in which the number of majority-Black districts was proportional to the state&#x2019;s Black population share.

In four states whose Black population is greater than 25 percent &#x2014; Alabama, Louisiana, Mississippi, and South Carolina &#x2014; the computer typically drew zero majority-Black districts. The conclusion was clear: random simulations tended to systematically eliminate opportunities for Black voters to elect candidates of their choice.

Yet some legal scholars and judges have indicated that they favor a &#x201C;race-blind&#x201D; standard for redistricting. Although the Supreme Court has set a precedent that the Voting Rights Act requires electoral opportunities for minorities, the court&#x2019;s present conservative majority has <a href="https://www.washingtonpost.com/politics/2022/06/24/supreme-court-ruling-abortion-dobbs/?itid=lk_inline_enhanced-template" target="_blank">demonstrated a willingness to overturn long-standing legal precedent</a>.

If the court were to impose a ban on any map that fell outside the middle of a computer-generated distribution, many fear it would lead to maps that &#x201C;we might think of as unfair to groups that are historically disenfranchised,&#x201D; as Schutzman put it.

In October, the Supreme Court will hear oral arguments in a redistricting case combining maps from Alabama and Louisiana, with a decision likely to come next year.

Supreme Court stops lower court order requiring Alabama to draw a new district voting map favorable to Black residents

Algorithmic redistricting can create lots of maps to identify extreme gerrymanders, but it can be put to other uses, too. Those drawing maps from scratch can use computers to search the vast universe of redistricting plans to understand what sorts of plans are typical, and also what sorts are possible.

&#x201C;There&#x2019;s a lot of other things to do with&#x201D; redistricting algorithms, Duchin said. &#x201C;So I don&#x2019;t think this is going anywhere.&#x201D;

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