Implementing partisan symmetry: Problems and paradoxes
Daryl DeFord, Natasha Dhamankar, Moon Duchin, Varun Gupta, Mackenzie McPike, Gabe Schoenbach, Ki Wan Sim
IImplementing partisan symmetry: Problems and paradoxes
Daryl DeFord, Natasha Dhamankar, Moon Duchin, Varun Gupta, Mackenzie McPike,Gabe Schoenbach, and Ki Wan SimFall 2020
Abstract
We consider the measures of partisan symmetry proposed for practical use in the political science literature, asclarified and developed in [10]. Elementary mathematical manipulation shows the symmetry metrics derived fromuniform partisan swing to have surprising properties. To accompany the general analysis, we study measures ofpartisan symmetry with respect to recent voting patterns in Utah, Texas, and North Carolina, flagging problems ineach case. Taken together, these observations should raise major concerns about using quantitative scores of partisansymmetry—including the mean-median score, the partisan bias score, and the more general “partisan symmetrystandard”—as the decennial redistricting approaches.
In the political science literature, there is a long legacy of work on gerrymandering, or the act of drawing politicalboundary lines with an ulterior motive. One of the questions attracting the most attention has been to measure thedegree of partisan advantage secured by a particular choice of redistricting lines. But to counteract partisan gamingrequires a baseline notion of partisan fairness, which has proved elusive. The family of fairness metrics with perhapsthe longest pedigree is called partisan symmetry scores [5, 11, 12, 7, 4, 6], which got a conceptual and empiricaloverview and a timely renewed endorsement in [10]. The partisan symmetry standard is premised on the intuitivelyappealing fairness notion that the share of representation awarded to one party with its share of the vote should alsohave been secured by the other party, had the vote shares been exchanged. For instance, if Republicans achieve 40%of the seats with 30% of the vote, then it would be deemed fair for the Democrats to also achieve 40% of the seats with30% of the vote.At the heart of the symmetry ideal is a commitment to the principle that half of the votes should secure half of theseats. There are several metrics in the symmetry family that derive their logic from this core axiom. The mean-medianmetric is vote-denominated: it produces a signed number that is often described as measuring how far short of half ofthe votes a party can fall while still securing half the seats. A similar metric, partisan bias , is seat-denominated. Giventhe same input, it is said to measure how much more than half of the seats will be secured with half of the votes. Theideal value of both of these scores is zero. These are two in a large family of partisan metrics that can be described interms of geometric symmetry of the “seats-votes curve.”The focus in the current work is to show that there are serious obstructions to the practical implementation ofsymmetry standards. This is of pressing current interest because, as we write, states are racing to adopt redistrictingreform measures. In 2018 alone, voter referenda led four states to pass constitutional amendments (CO, MI, MO, OH),and another to write reforms into state law (UT) in anticipation of 2021 redistricting. In Utah, partisan symmetry hasnow been adopted as a criterion to be considered by the new independent redistricting commission before plans can beapproved. We sound a note of caution here, showing that the versions of these scores that are realistically useable areeminently gameable by partisan actors and do not have reliable interpretations. To be precise: in each state we studied,the most extreme partisan outcomes for at least one political party are still achievable with a clean bill of health fromthe full suite of partisan symmetry scores. Furthermore, the signed scores (like mean-median and partisan bias) makesystematic sign errors in terms of partisan advantage. “The Legislature and the Commission shall use judicial standards and the best available data and scientific and statistical methods, includingmeasures of partisan symmetry, to assess whether a proposed redistricting plan abides by and conforms to the redistricting standards” that bar partyfavoritism. Cf. Utah Code, Chapter 7 Title 20A, Chapter 19 Part 1, Para. 103, https://le.utah.gov/xcode/Title20A/Chapter19/C20A-19_2018110620181201.pdf . a r X i v : . [ phy s i c s . s o c - ph ] A ug tah itself gives strong evidence of the interpretation problems: with respect to recent voting patterns, a goodsymmetry score can only be achieved by a plan that secures a Republican congressional sweep; what’s more, the pop-ular symmetry scores described above flag all possible plans with any Democratic representation as major Republican gerrymanders—arguably a sign error.
We consider an election in a state with k districts and two major parties, Party A and Party B. A standard constructionin the political science literature is the “seats–votes curve,” a plot representing the relationship of the vote share forParty A to the seat share for the same party. Observed outcomes generate single points in V - S space—for instance,( . , .
4) represents an election where Party A got 30% of the votes and 40% of the seats—but various methods havebeen used to extend from a scatterplot to a curve, such as by fitting a curve from a given class (linear or cubic, forinstance) to observed data points. We will focus on a second construction of seats-votes curves that is emphasizedin [10]: linear uniform partisan swing. Beginning with a single data point derived from a districting plan and a votepattern, the vote share is varied by a uniform partisan swing, so that the district vote shares ( v , . . . , v k ) will shift to( v + t , . . . , v k + t ). This generates a step function spanning from (0 ,
0) to (1 ,
1) in the V - S plot, with a jump in seatshare each time a district is pushed past 50% vote share for Party A. (See Figures 1-2 below for examples.)Linear uniform partisan swing is the leading method proposed for use in evaluation. Katz–King–Rosenblatt ex-plicitly make it Assumption 3 in their symmetry survey, noting that the curve-fitting alternative is more suited “foracademic study... than for practical use” in evaluation of plans. Grofman noted in 1983 that linear swing is preferredin practice to more sophisticated models [5, n.14], it is touted as the standard technique in [3], and it has been invokedas recently as 2019 in expert reports and testimony [13].
Given a seats-votes curve, many symmetry scores have been proposed; here, we focus on the mean-median score MM , the partisan bias score PB , and the partisan Gini score PG , which have all been considered for at least 35years. (Definitions are found in the next section.) Grofman’s 1983 survey paper [5] lays out eight possible scores ofasymmetry once a seats-votes curve has been set. His Measure 3 is vote-denominated bias, which would equal MM under linear uniform partisan swing; similarly his Measure 4 corresponds to PB , and Measure 7 introduces PG .Because the partisan Gini is defined as the area between the seats-votes curve and its reflection over the center(seen in the shaded regions in Figs 1-2), it is easily seen to “control” all the other possible symmetry scores: when PG =
0, its ideal value, all partisan symmetry metrics also take their ideal values, including MM and PB . This agreeswith Katz–King–Rosenblatt [10, Def 1], where the coincidence of the curve and its reflection, i.e., PG =
0, is calledthe “partisan symmetry standard.” In the current work, our Theorem 3 gives precise necessary and su ffi cient conditionsfor the partisan symmetry standard to obtain.The literature invoking MM and PB as measures of bias is too large to survey comprehensively. We note that theparticular interpretation of median-minus-mean as quantifying (signed) Party A advantage is fairly standard in thejournal literature, such as: “The median is 53 and the mean is 55; thus, the bias runs two points against Party A (i.e.,53 − = − MM “essentially slices the S / V graphhorizontally at the S =
50% level and obtains the deviation of the vote from 50%” [16, p351]. The current work is designed to evaluate the techniques proposed by leading practitioners for practical use. Politicalscientists and their collaborators have advanced these scores in amicus briefs spanning from
LULAC v. Perry (2006)[12] to
Whitford v. Gill (2018) [4] to
Rucho v. Common Cause (2019) [6]. The scores have been claimed to be “reliable Katz et al. also o ff er Assumption 4, a stochastic generalization of uniform swing, as the only alternative to linear UPS mentioned for generaluse. This would add many additional modeling decisions, so would be di ffi cult to carry out in a practical setting. If used with recent election data,its impact would chiefly be to noise the seats-votes curve γ , which changes the precision of our findings but not the basic structure. In particular,this does not impact the prevalence of “paradoxes,” as described below. There is even more work centered on PB (notably [11]), but it is more rarely used in conjunction with linear swing, since that assumptionmakes its values move in large jumps. ffi cult to manipulate” and authors have argued that while “Symmetry tests should deploy actual election out-comes” (as we do here), they will nonetheless “measure opportunity,” i.e., give information about future performance[4, p17,24]. That assertion is drawn from an amicus brief in the Whitford case explicitly proposing mean-median as aconcrete choice of score for this task.As laid out in the influential LULAC brief,Models applying the symmetry standard are by their nature predictive, just as the legislators themselvesare predicting the potential impact of the map on partisan representation. The symmetry standard andthe resulting measures of partisan bias, however, do not require forecasts of a particular voting outcome.Rather, by examining all the relevant data and the potential seat divisions that would occur for particularvote divisions, it compares the potential scenarios and determines the partisan bias of a map, separatingout other potentially confounding factors. Importantly, those drawing the map have access to the samedata used to evaluate it, and no data is required other than what is in the public domain” [12, p11].This paper takes up precisely this modeling task in the manner explicitly proposed by its authors. We begin with definitions and notation needed to state the results in this paper, and particularly Theorem 3, whichcharacterizes when PG = k districts as follows: v = ( v , . . . , v k ) , where 0 ≤ v ≤ . . . ≤ v k ≤
1. Let the mean district vote share for Party A bedenoted v = k (cid:80) v i and the median district vote share, v med , be the median of the { v i } , which equals v ( k + / if k is oddand ( v k / + v ( k / + ) if k is even because of the convention that coordinates are in non-decreasing order. We note that v is not necessarily the same as the statewide share for Party A except in the idealized scenario that the districts haveequal numbers of votes cast (i.e., equal turnout). VS (cid:63) VS Figure 1: Red: The seats-votes curve γ generated by the vote share vector v = ( . , . , . , . , .
87) under uniformpartisan swing. This gives v = .
61 as the average vote share across the five districts. The jump points , which are valueswhere an additional seat changes hands, are marked on the V axis. Blue: the reflection of γ about the center point (cid:63) .Since MM is the horizontal displacement from (cid:63) to a point on γ , this hypothetical election has a perfect MM = PG = . γ and its reflection.Because the step function jumps at V = .
5, it is not clear how PB is defined in this case.The number of districts in which Party A has more votes than Party B in an election with vote shares v is the seatoutcome, { i : v i > } . This induces a seats-votes function γ = γ v defined as γ ( v + t ) = { i : v i + t > } / k , which we caninterpret as the share of districts won by Party A in the counterfactual that an amount t was added to A’s observed voteshare in every district. Varying t to range over the one-parameter family of vote vectors ( v + t , . . . , v k + t ) is knownas (linear) uniform partisan swing . The curve γ , treated as a function [0 , → [0 , seats-votes curve associated to the vote sharevector v . We begin with several scores based on v and γ . 3 S (cid:63) VS Figure 2: The seats-votes curve γ generated by the vote share vector v = ( . , . , . , . , . v = . γ about the center, so it showsseats at each vote share from the Democratic point of view. This could be regarded as a situation with reasonably goodsymmetry, since the red and blue curves are close. Its scores are PG = . MM = − . PB = − .
1. Thesign of the latter two scores is thought to indicate a Democratic advantage.
Definition 1.
The partisan Gini score PG ( v ) is the area between the seats-votes curve γ v and its reflection over thecenter point (cid:63) = ( , ) . PG ( v ) = (cid:90) (cid:12)(cid:12)(cid:12) γ ( x ) − γ (1 − x ) + (cid:12)(cid:12)(cid:12) dx . The mean-median score is MM ( v ) = v med − v. The partisan bias score is PB ( v ) = γ ( ) − . These scores can all be related to the shape of the seats-votes curve γ (see Figures 1,2). Partisan Gini measuresthe failure of γ to be symmetric about the center point (cid:63) = ( , ), in the sense that it is always non-negative, and itequals zero if and only if γ equals its reflection. Mean-median score is the horizontal displacement from (cid:63) to a pointon γ , which is why it is votes-denominated (vote-share being the variable on the x -axis). Similarly, partisan biasis the vertical displacement from (cid:63) to a point on γ , and is therefore seats-denominated. (We note that ( , γ ( )) is awell-defined point unless there is a jump precisely at 1 /
2, which occurs if some v i = v on the nose—this is shown inFigure 1 but should not happen with real-world data.) Below, we will focus on MM instead of PB , but we note that MM > = ⇒ PB ≥ γ passes to the left of (cid:63) and is nondecreasing, thenit must pass through or above (cid:63) .We can see that the curve γ , and consequently the partisan Gini score, is exactly characterized by the points atwhich Party A has added enough vote share to secure the majority in an additional district. For the following analysis,it will be useful to characterize this curve in terms of the v data. Definition 2.
The gaps in a vote share vector v can be written in a gap vector δ = ( δ , δ , . . . , δ k − ) = ( v − v , v − v , . . . , v k − v k − ) . The jump points for vote share vector v are the values of v + t such that some v i + t = . We havet : = − v k , t : = − v k − , . . . t k : = − v as the times corresponding to these jumps, so we can denote the jumps as j i = + v − v k + − i , and the jump vector as j = ( j , . . . , j k ) . By the linear partisan swing definition of γ , these jump points j i , marked in the figures, are exactly the x -axislocations (i.e., the V values) at which γ jumps from ( i − / k to i / k . To see this, plug in t = / − MM − v = / − v med to deduce that (1 / − MM , /
2) is on γ . Warning to the reader: if you try to draw your own examples to test some of these results, note that not just any step function can be generatedby a vote vector. The jump points must satisfy (cid:80) j i = k /
2, which follows directly from summing j i = + v − v k + − i . γ and its reflection have height 2 PB and width 2 MM , allowing the derivation of inequalitiesrelating these scores. For small k , these reduce to extremely simple expressions: PG = | MM | when k =
3, and PG = | MM | when k =
4. (Proved in the Supplement.)For any number of districts, we obtain a very clean characterization of precisely which distribution of vote sharesto districts satisfy the Partisan Symmetry Standard [10, Def 1].
Theorem 3 (Partisan Symmetry Characterization) . Given k districts with vote shares v , jump vector j , and gap vector δ , the following are equivalent under uniform partisan swing: PG ( v ) = j i + j k + − i − = ∀ i (jumps)12 ( v i + v k + − i ) = v ∀ i (mean vote)12 ( v i + v k + − i ) = v med ∀ i (median vote) δ i = δ k − i ∀ i (gaps)That is, this partisan symmetry standard is nothing but the requirement that the vote shares by district are distributedon the number line in a symmetric way. In particular, this tells you at a glance that an election with vote shares( . , . , . , .
67) in its districts rates as perfectly partisan-symmetric, while one with vote shares ( . , . , . , . PG = = ⇒ MM = Recall that mean-median and partisan bias are signed scores that are supposed to identify which party has an advantageand by what amount. A positive score is said to indicate an advantage for Party A (the point-of-view party whose voteshares are reported in v ). Let us say that a paradox occurs when the score indicates an advantage for one party eventhough it has a very low number of seats—the fewest seats it can possibly earn with its vote share, say. In other words,a paradox means that the score makes an apparent sign error.When there is an extremely skewed outcome (with a vote share for one party exceeding 75%), we will show thatparadoxes always occur, just as a matter of arithmetic. But even for less skewed elections with a vote share between62.5 and 75% for the leading party—which frequently occur in practice!—mundane realities of political geographycan force these sign paradoxes.To illustrate these observations, we will begin with the case of k = k , however: in the empirical section we will find paradoxes of this kind in k =
13 and k =
36 cases as well.
Example 4 (Paradoxes forced by arithmetic) . Suppose we have k = districts and an extremely skewed election infavor of Party A, achieving < v < . . With equal turnout, Party B can get at most one seat. However,every vote vector v achieving this outcome (one B seat) yields MM ≥ v − > . In particular, such districting plansall have positive MM and PB , paradoxically indicating an advantage for Party A in every case where Party B getsrepresentation. The demonstration is simple arithmetic. Since ( v + v ) = v med , we have v =
14 ( v + v + v + v ) = v + v + v + v = ⇒ v med − v = v − v + v . Since v ≤ (for B to win a seat) and v ≤
1, we get MM = v med − v ≥ v − , as needed.A stronger statement can be made if one takes political geography into account. It was shown by Duchin–Gladkova–Henninger-Voss–Newman–Wheelen in a study of Massachusetts [2] that, if the precincts are treated asatoms that are not to be split in redistricting, then several recent elections have the property that no choice of district5ines can create even one district with Republican vote share over 1 /
2. This is because Republican votes are distributedextremely uniformly across the precincts of the state. While other states are not as uniform as Massachusetts, it is stilltrue that there is some upper bound Q on the vote share that is possible for each party in any district. When this boundsatisfies Q < v − , even moderately skewed elections are forced to exhibit paradoxical symmetry scores. As we willsee below, having all v i < v − ensures both that one seat is the best outcome for Party B and that the median voteshare is greater than the mean. Example 5 (Paradoxes forced by geography) . Suppose we have k = districts and a skewed election in favor of PartyA, with . ≤ v < . Suppose the geography of the election has Party A support arranged uniformly enough thatdistricts can not exceed a share Q of A votes, for some Q < v − . Then with equal turnout, Party B can get at mostone seat. However, every vote vector v achieving this outcome (one B seat) has a positive MM and PB , paradoxicallyindicating an advantage for Party A in every case where Party B gets representation.Proof of paradox. First, it is easy to see that Party B can’t achieve two seats: in that case, we would have v , v ≤ .Since we also have v , v ≤ Q < v − , we can average the v i to get the contradiction v < v .To see that MM >
0, we write MM = v med − v = v + v − v + v + v + v = v + v + v + v − v + v = v − v + v . Since v < and v ≤ Q < v − , we have MM > v − Q + > v − v = (cid:3) In this section we illustrate the theoretical issues from above, using naturalistically observed election data togetherwith a Markov chain technique that produces large ensembles of districting plans. All data and code are public andfreely available for inspection and replication [9, 8].In each case, we have run a recombination (
ReCom ) Markov chain for 100,000 steps—long enough to comfortablyachieve heuristic convergence benchmarks in all scores that we measured—while enforcing population balance, con-tiguity, and compactness. Note that some Markov chain methods count every attempted move as a step, even thoughmost proposals are rejected, so that each plan is counted with high multiplicity in the ensemble; in our setup, theproposal itself includes the criteria, and repeats are rare. 100,000 steps produces upwards of 99,600 distinct districtingplans in each ensemble.We have run trials on multiple elections in our dataset and all results are available for comparison [9]. Below,we have highlighted the most recent available Senate race from a Presidential election year in each state, to matchconditions as closely as possible. The data bottleneck is access to a precinct shapefile matching geography to votingpatterns, which is surprisingly di ffi cult. A database of our research group’s vetted shapefiles is available at [15]. Forthe results we present here, we prefer to use the statewide U.S. Senate election to the endogenous Congressional votingpattern because the latter is subject to uncontested races and variable incumbency e ff ects. For instance, Utah’s 2016Congressional race had all four seats contested, but a Republican vote in District 3 went for hard-right Jason Cha ff etz,while on the other side of the invisible line to District 4, the vote went to Mia Love, a Black Republican who isoutspoken on racial inequities. When the district lines are moved, it’s not clear that a Love voter stays Republican.The U.S. Senate race had a comparable number of total votes cast to the Congressional race (1,115,608 vs. 1,114,144)and o ff ers a consistent choice of candidates around the state, making it better suited to methods that vary district lines.The algorithmically generated plans are not o ff ered as a statistical experiment and come with no probabilisticclaims about plan selection, but mainly provide existence proofs to illustrate how readily gameable partisan symmetrystandards will be for those engaged in redistricting. The methods also produce many thousands of examples of plansthat are paradoxical in the senses described above, where partisan symmetry metrics identify the wrong party as thegerrymanderers, relative to the common understanding of that term. The population balance imposed here is 1% deviation from ideal district size. Such plans are easily tuneable to 1-person deviation by refinementat the block level without significant impact to any other scores discussed here. Contiguity is enforced by recording adjacency of precincts.Compactness, at levels comparable to those observed in human-made plans, is an automatic consequence of the spanning-tree-based recombinationstep. For more information about the Markov chain used here, see [1]. This is exactly the use of ensemble methods that is endorsed in [10, p176] as productive and compelling: a demonstration of possibility andimpossibility. .2 Utah and the “Utah Paradox” We begin with Utah, where the elections that were available in our dataset all come from 2016. Utah has only fourcongressional districts and has a heavily skewed partisan preference, with a statewide Republican vote share of 71.55%in the 2016 Senate race. Figure 3 shows outcomes from our 100,000-step ensemble.We note in passing that the amount of linear partisan swing needed to reverse the partisan advantage could beviewed as unreasonably large under these conditions. With respect to SEN16, fully 199 out of 2123 precincts in thestate have Republican vote share that reaches zero under this amount of swing. This is one of the reasons (though notthe only reason) that this style of quantifying partisan advantage is poorly suited to Utah.The vast majority (94 . PG scores above 0.06. Below, we explore and explain these bounds on seats and scores.When looking at the full PG histogram, we see a large bulk of plans with nearly-ideal PG scores, all giving aRepublican sweep (four out of four R seats). This is surprising enough to deserve a name. The Utah Paradox • Partisan symmetry scores near zero are supposed to indicate fairness, and signed symmetry scores are supposedto indicate which party is advantaged. • There are many trillions of valid Congressional plans in Utah, and under reasonable geographical assumptions,every single one of them with PG close to zero is mathematically guaranteed to yield a Republican sweep of theseats. In particular, even constraining symmetry scores to better than the ensemble average (for any reasonablydiverse algorithmic ensemble) would deterministically impose a partisan outcome: the one in which Democratsare locked out of representation. • Furthermore, the signed scores make an apparent sign error: they report all plans with Democratic representationto be significant pro-Republican gerrymanders.
Geographic assumptions:
The UT-SEN16 election has a statewide R share of . v (or exactly equal in the equal-turnout case). If we can upper-bound the possible Republican share ofa district by any Q < . MM >
0, by the derivation in the last section.We consider the assumption that no district can exceed 93% Republican share to be very reasonable. Indeed, even agreedy assemblage of the 608 precincts with the highest Republican share in that race (which is the number neededto reach the ideal population of a Congressional district) only produces a district with R share . . Example 6 (The Utah paradox, empirical) . The UT-SEN16 vote pattern can be divided into 4R-0D seats or 3R-1Dseats. However, even though MM , PB , and PG scores can all get arbitrarily close to zero, there are no reasonablysymmetric plans that secure a Democratic seat. In our algorithmic search, every plan with nonzero Democraticrepresentation has PG > . , MM > . , and PB ≥ . , which is in the worst half of scores observed for each ofthose scores. Thus even a mild constraint on partisan symmetry stands to lock Democrats out of representation, andall plans with D representation are reported as significant R-favoring gerrymanders. As described in the introduction, Utah recently became the first state to encode partisan symmetry as a districtingcriterion in statute. This makes the Utah Paradox quite a striking example of the worries raised by using partisansymmetry scores in practice. Out of GOV16, SEN16, and PRES16, none gives an especially pure partisanship signal, because the Democratic candidates for Governor andSenate were quite weak, while the partisanship in the Presidential race was complicated by the presence of a very strong third-party candidate inEvan McMullen, giving that race an extremely di ff erent pattern. The Governor’s race shows similar results to the Senate, as the reader can verify in[9]. Recall that v is the average of the district vote shares, which will not generally equal the statewide share except under an equal-turnoutassumption. MM and PG histograms restricted to the plans with a D seat. Recall that the mean-median scorereports a Republican advantage when MM >
0. The PG score is unsigned, but larger magnitude indicates greaterasymmetry. 8igure 4: Ensemble outputs for Texas Congressional plans with respect to SEN12 votes. Republicans received 58.15%of the two-way vote in this election, which is marked in the plots to show the corresponding seat share. There are 1646plans in the ensemble that have an outlying number of seats for one party or the other; these are shown in red andblue in the top row and their relative frequency can be observed in the next two rows, which focus on plans with thebest symmetry scores. The last row of the figure shows the MM and PG histograms restricted to the 1646 outlier plansflagged above. Recall that the mean-median score reports a Republican advantage when MM >
0. The PG score isunsigned, but larger magnitude indicates greater asymmetry.9 .3 Texas Next, we turn to Texas, creating a chain of 100,000 steps to explore the ways to divide up the 2012 Senate votedistribution. With 36 Congressional districts, Texas has one of the highest k values of any state (only California hasmore seats). The 2012 Senate race was won by a Republican with ∼
58% of the vote.Figure 4 shows the partisan properties in the ensemble of plans, allowing us to compare extreme symmetry scores(the ostensible indicator of partisan gerrymandering) to extreme seat shares (the explicit goal of partisan gerrymander-ing). We find no evidence of correlation or any kind of correspondence.Over 98% of all plans give Republicans 22 to 27 seats out of 36, seen in gray in the histogram. The red bars markthe outlying plans with the most Republican seats (28 or more R seats), while the blue bars mark the outlying planswith the most Democratic seats (21 or fewer R seats). We can then study the histograms formed by the winnowedsubsets of the ensemble with the best PG and MM scores, which in each case fall in the top 6%. Note that theseseverely winnowed subsets not only have a shape similar to the full ensemble (indicating a lack of correlation), butthat the partisan outlier plans are not proscribed by strict symmetry thresholds. Plans with the extreme outcome of28 or more Republican seats occur with higher frequency among the plans with MM ≈ ≤
21 Republican seats and red for plans with ≥
28 Republican seats. We find that a significant number of extremeD-favoring plans paradoxically register as major Republican gerrymanders under the MM score. In terms of the overallsymmetry measured by PG , extreme plans for both parties can be found with scores that are as good as nearly anythingobserved in the ensemble. So from this perspective as well, neither MM nor PG signals anything with respect to seatcounts. Even if the proponents of symmetry standards never intended to constrain extreme seat imbalances, this runscounter to the common expectations of anti-gerrymandering reforms in popular discourse, in legal settings, and evenin much of the political science literature. Finally, we move to a state with a much closer to even partisan split: North Carolina ( k =
13 seats), with respect to the2016 Senate vote ( ∼
53% Republican share).In this case, mean-median does much better than in Texas in terms of distinguishing the seat extremes: Figure 5shows consistently higher scores for the maps with the most Republican seat share than the ones with the most Demo-cratic outcomes. However, the extreme Republican maps still straddle the “ideal” score of MM =
0, and both sidescan still find very extreme plans whose PG scores report that their symmetry is essentially as good as anything in theensemble.Overall it is fair to say that partisan symmetry imposes no constraint on partisan gerrymandering in North Carolina,at least for one side: this method easily produces hundreds of maps with 10-3 outcomes (which was clearly reportedin the Rucho case to be the most extreme that the legislature thought was possible) while securing nearly perfectsymmetry scores. Indeed, the ensemble even finds highly partisan-symmetric maps that return an 11-2 outcome forthis particular vote pattern. Four of these are shown in Figure 6.10igure 5: Ensemble outputs for North Carolina Congressional plans with respect to SEN16 votes. Republicans re-ceived 53.02% of the two-way vote in this election, which is marked in the plots to show the corresponding seat share.There are 1202 plans in the ensemble that have an outlying number of seats for one party or the other; these are shownin red and blue in the top row and their relative frequency can be observed in the next two rows, which focus on planswith the best symmetry scores. The last row of the figure shows the MM and PG histograms restricted to the 1202outlier plans flagged above. Recall that the mean-median score reports a Republican advantage when MM >
0. The PG score is unsigned, but larger magnitude indicates greater asymmetry.11igure 6: Each of these 13-district plans has 11 Republican-majority seats with respect to the SEN16 voting data,while having nearly perfect partisan symmetry: the PG score that describes the di ff erence between the seats-votescurve and its reflection is near zero and in the best 2% of scores in the ensemble. These maps have PG scores of0.0096, 0.0099, 0.0107, and 0.0115, respectively. This figure also illustrates the diversity of districting plans achievedby this Markov chain method. In this note, we have characterized the partisan symmetry standard from [10] mathematically: under uniform partisanswing, it turns out to amount simply to a prescription for the arrangement of vote totals across districts (Theorem 3,Partisan Symmetry Characterization). We follow this with examples of realistic conditions under which the adoptionof strict symmetry standards not only (a) fails to prevent extreme partisan outcomes but even (b) can lock in unforeseenconsequences on these partisan outcomes. Finally, again under realistic conditions, signed partisan symmetry metrics(c) can plainly mis-identify which party is advantaged by a plan. None of these findings gives a theoretical reason for rejecting partisan symmetry as a definition of fairness. Abeliever in symmetry-as-fairness can certainly coherently hold that symmetry standards do not aim to constrain partisanoutcomes, but merely to reinforce the legitimacy of district-based democracy by reassuring the voting public that thetables can yet turn in the future. This reasoning would tell us not to worry that Democrats in Utah may for now belocked out of Congressional representation by the symmetry standard itself; this is still fair because Democrats wouldenjoy a similar advantage of their own if election patterns were to linearly swing by 40 percentage points in their favor.For those who do want to constrain the most extreme partisan outcomes that line-drawing can secure, these inves-tigations should serve as a strong caution regarding the use of partisan symmetry metrics, whether in the plan adoptionstage or in plan evaluation after subsequent elections have been conducted.If symmetry metrics measured something that was obviously of inherent value in the proper functioning of demo-cratic representation, then we might reasonably choose to live with the consequences of the definition, no matter thepartisan outcomes. However, the Characterization Theorem shows that a putatively perfect symmetry score is noth-ing more and nothing less than a requirement that the vote shares v i in the districts be arranged symmetrically onthe number line. Stated this way, it is more di ffi cult to argue that symmetry captures an essential ingredient of civicfairness. One possible response is to try to preserve the partisan symmetry standard but abandon linear uniform partisan swing in favor of a di ff erent wayof drawing seats-votes curves. However, we know of no other detailed method that has been proposed for this task in practical or legal applications.See footnote 2. While this is beyond the scope of the current paper, there is also every reason to believe that partisan symmetry metrics can (d) give answers thatdepend unpredictably on which vote pattern is used to assess them: endogenous or exogenous? Senate race or attorney general? Most single-scoreindicators have this problem. cknowledgments MD is deeply grateful to Gary King for illuminating conversations about the partisan symmetry standard. All authorsthank the other participants of the 2019 Voting Rights Data Institute at Tufts University, especially Cleveland Waddell,Brian Morris, Michelle Jones, and Thomas Weighill, for stimulating discussions. We are grateful to the Prof. Amar J.Bose Research Foundation at MIT and the Jonathan M. Tisch College of Civic Life at Tufts for their generous supportof the Voting Rights Data Institute.
References [1] Daryl DeFord, Moon Duchin, and Justin Solomon. Recombination: A family of Markov chains for redistricting. https://arxiv.org/abs/1911.05725 , 2019.[2] Moon Duchin, Taissa Gladkova, Eugene Henninger-Voss, Ben Klingensmith, Heather Newman, and Han-nah Wheelen. Locating the representational baseline: Republicans in Massachusetts.
Election Law Journal ,18(4):388–401, December 2019.[3] James C. Garand and T. Wayne Parent. Representation, swing, and bias in U.S. presidential elections, 1872-1988.
American Journal of Political Science , 35(4):1011–1031, November 1991.[4] Heather K. Gerken, Jonathan N. Katz, Gary King, Larry J. Sabato, and Samuel S.-H. Wang. Amicus brief inWhitford v. Gill, 2018. No. 16-1161.[5] Bernard Grofman. Measures of bias and proportionality in seats-votes relationships.
Political Methodology ,9(3):295–327, 1983.[6] Bernard Grofman and Ronald Keith Gaddie. Amicus brief in Rucho v. Common Cause, 2019. Nos. 18-422,18-726.[7] Bernard Grofman and Gary King. The future of partisan symmetry as a judicial test for partisan gerrymanderingafter LULAC v. Perry. , 6:2–35, 2007.[8] Voting Rights Data Institute. GerryChain.
GitHub repository , 2018. https://github.com/mggg/gerrychain .[9] Voting Rights Data Institute. Partisan Symmetry.
GitHub repository , 2020. https://github.com/mggg/Partisan-Symmetry .[10] Jonathan N. Katz, Gary King, and Elizabeth Rosenblatt. Theoretical foundations and empirical evaluations ofpartisan fairness in district-based democracies.
American Political Science Review , 114(1):164–178, 2020.[11] Gary King and Robert X. Browning. Democratic representation and partisan bias in congressional elections.
American Political Science Review , 81(4):1251–1273, December 1987.[12] Gary King, Bernard Grofman, Andrew Gelman, and Jonathan N. Katz. Amicus brief in LULAC v. Perry, 2005.Nos. 05-204, 05-254, 05-276, 05-439.[13] Jonathan Mattingly. Expert report in Common Cause v. Lewis, 2019. https://sites.duke.edu/quantifyinggerrymandering/files/2019/09/Report.pdf see also https://sites.duke.edu/quantifyinggerrymandering/2019/06/26/the-animated-firewall/ .[14] Michael B. McDonald and Robin E. Best. Unfair partisan gerrymanders in politics and law: A diagnostic appliedto six cases.
Election Law Journal , 14(4):312–330, 2015.[15] MGGG. MGGG States.
GitHub repository , 2020. https://github.com/mggg-states .[16] John F. Nagle. Measures of partisan bias for legislating fair elections.
Election Law Journal , 14(4):346–360,2015. 13
Supplement: Proof of characterization theorem
We briefly recall the needed notation from above: vote share vector v with i th coordinate v i ; gap vector δ with δ i = v i + − v i ; and jump vector j with j i = + v − v k + − i , where v is the mean of the v i . These expressions define j , δ in termsof v ; neither j nor δ completely determines v because they are invariant under translation of the entries of v , but oneadditional datum (such as v or v ) su ffi ces, with j or δ , to fix the associated v . In this section, we begin by expressing PG in terms of the jumps j , then giving equivalent conditions for PG = j , δ , or v .As outlined above, PG measures the area between the seats-votes curve γ and its reflection. The shape of theregion between those curves depends directly on the points j = ( j , j , . . . , j k ), since each j i is the x value of a verticaljump in the curve and the 1 − j i are the values of the jumps in the reflection. But looking at Figure 1 makes it clear thatit is complicated to decompose the integral into vertical rectangles in the style of a Riemann sum, because the { j i } andthe { − j k − i } do not always alternate. Fortunately, it is always easy to decompose the picture into horizontal rectangles(analogous to a Lebesgue integral), where it is now clear which red and blue corners to pair as the seat share changesfrom i / k to ( i + / k . The curve contains the points ( j i , i − k ), ( j i , ik ) as well as ( j k + − i , k − ik ), ( j k + − i , k − i + k ). The rotatedcurve therefore contains the points (1 − j k + − i , i − k ) and (1 − j k + − i , ik ), which means that the i th rectangle has height1 / k and width (cid:12)(cid:12)(cid:12) j i + j k + − i − (cid:12)(cid:12)(cid:12) . Summing over these rectangles gives us the expression PG = k k (cid:88) i = (cid:12)(cid:12)(cid:12) j i + j k + − i − (cid:12)(cid:12)(cid:12) . Recall that the set of vote share vectors V is the cone in the vector space R k given by the condition that the v i arein non-decreasing order in [0 , j vector is simply the v vector reversed and re-centered at 1 / v . Theonly condition on the gap vector δ is that its entries sum to less than one. Putting these observations together we maydefine the set of achievable v , δ , j respectively as V = { ( v , . . . , v k ) : 0 ≤ v ≤ · · · ≤ v k ≤ } , D = (cid:110) ( δ , . . . , δ k − ) : δ i ≥ ∀ i , (cid:88) δ i ≤ (cid:111) , J = (cid:40) ( j , . . . , j k ) : 0 ≤ j ≤ · · · ≤ j k ≤ , (cid:88) j i = k (cid:41) . The condition on j is of interest because it exactly identifies the possible seats–votes curves Γ = { γ v : v ∈ V} . (That is,not just any step function is realizable as a valid γ .)Now we can prove the Characterization theorem. Theorem 3.
Given k districts with vote shares v , jump vector j , and gap vector δ , the following are equivalent: PG ( v ) = j i + j k + − i − = ∀ i (2)12 ( v i + v k + − i ) = v ∀ i (3)12 ( v i + v k + − i ) = v med ∀ i (4) δ i = δ k − i ∀ i (5) Proof.
The condition that PG ( v ) = j above, and converting back to the v i we get1 k k (cid:88) i = | j i + j k + − i − | = k k (cid:88) i = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) v i + v k + − i − v (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = , which immediately gives (1) ⇐⇒ (2) ⇐⇒ (3) since a sum of nonnegative terms is zero if and only if each term iszero. To see (3) ⇐⇒ (4), just consider i = (cid:100) k (cid:101) in (3) to obtain v med = v ; in the other direction, average both sidesover i in (4) to obtain v = v med . Finally, the symmetric gaps condition (5) is clearly equivalent to the symmetry of thevalues of v about the center v med , which is (4). (cid:3) Supplement: Bounding partisan Gini in terms of mean-median
Recall that the mean-median score MM is a signed score that is supposed to identify which party has a structuraladvantage, and by what amount. On the other hand, the partisan Gini PG is a non-negative score that simply quantifiesthe failure of symmetry, interpreted as a magnitude of unfairness. We easily see that PG = = ⇒ MM = i ) = v i + v k + − i − v , measuring the di ff erence between the average of a pair of vote shares fromthe average of all the vote shares. This gives PG = k k (cid:88) i = | v − v i − v k + − i | = k k (cid:88) i = | discrep( i ) | . Note that discrep( (cid:100) k (cid:101) ) = MM , as observed above, and that (cid:80) ki = discrep( i ) = v . Theorem 7.
The partisan Gini score satisfies PG ≥ k | MM | , k odd PG ≥ k | MM | , k even , with equality when k = , .Proof. First suppose k is odd, say k = m +
1. Then discrep( m ) = v m − v = MM , so (cid:80) i (cid:44) m discrep( i ) = − MM . We have PG = k k (cid:88) i = discrep( i ) = k | discrep( m ) | + (cid:88) i (cid:44) m | discrep( i ) | ≥ k | discrep( m ) | + (cid:12)(cid:12)(cid:12)(cid:88) i (cid:44) m discrep( i ) (cid:12)(cid:12)(cid:12) = k ( | MM | + | − MM | ) = k | MM | . The argument for even k = m is very similar, except that discrep( m ) = discrep( m + = v m + v m + − v = MM . Sonow those terms contribute 2 | MM | to the sum and the remaining terms contribute at least 2 | − MM | , for a bound of PG ≥ k | MM | . That completes the proof of the inequalities.For k = k =
4, the term (cid:80) i (cid:44) m | discrep( i ) | is just 2 | discrep(1) | , making the inequality into an equality. (cid:3) By a dimension count, it is easy to see that PG is not simply a function of MM for k ≥
5. A direct calculationconfirms this, and shows that MM is not simply a function of PG either. Let v = ( . , . , . , . , . , v (cid:48) = ( . , . , . , . , . , v (cid:48)(cid:48) = ( . , . , . , . , . , giving PG ( v ) = PG ( v (cid:48) ) while MM ( v ) (cid:44) MM ( v (cid:48) ). On the other hand, MM ( v ) = MM ( v (cid:48)(cid:48) ) while PG ( v ) (cid:44) PG ( v (cid:48)(cid:48)(cid:48)(cid:48)