Quantitatively Predictive Logical Models

To theorize minority candidate emergence, I adopt a powerful approach in comparative politics to derive a quantitatively predictive logical model (or logical model) (Li and Shugart Reference Li and Shugart2016; Shugart and Taagepera Reference Shugart and Taagepera2017; Sikk and Taagepera Reference Sikk and Taagepera2014; Taagepera Reference Taagepera2007; Reference Taagepera2008; Taagepera and Nemčok Reference Taagepera and Nemčok2019; Taagepera and Sikk Reference Taagepera and Sikk2010).Footnote ⁷ This approach uses deductive logic (as opposed to statistical estimation or even data) to derive a parsimonious mathematical formula that can both explain and predict the outcome of interest.

A canonical example is the seat product model developed and extended primarily by Taagepera (Reference Taagepera2007) and Shugart and Taagepera (Reference Shugart and Taagepera2017). This model states that the (average) effective number of seat-winning parties in a given country ( $ {N}_s $ ) is a deterministic function of its average district magnitude (M) and assembly size (S): $ {N}_s={(MS)}^{1/6} $ . Research shows that the Seat Product (MS) can also explain and predict the effective number of vote-earning parties, deviation from proportionality, cabinet duration, and even the number of presidential candidates (Shugart and Taagepera Reference Shugart and Taagepera2017, 149).

As this example illustrates, logical models have only few variables, which are (ideally) subject to change in order to assist in (re)designing political institutions. Thus, there is a direct link between theory and application under this approach. Logical models can sharply tell by how much the outcome of interest changes (i.e., quantitative predictions) if one varies specific parts of their formulas without any statistical estimation or data. For this reason, logical models can offer considerably detailed advice for electoral engineering and constitutional design, which have enormous effects on representation (Li and Shugart Reference Li and Shugart2016, 23).

Scope Conditions

I make four assumptions to model district-level minority candidate emergence. Here, I use the term district to refer to a geographically limited area in which a single candidate wins a single seat in the spirit of FPTP. Thus, “districts” considered here include legislative single-member districts, executive offices, and different positions under at-large elections (Abott and Magazinnik Reference Abott and Magazinnik2020).

Assumption 1 (Biracial Elections). Two racial groups (minority and white voters) compete with each other over a single seat.

This assumption clarifies that the proposed model predicts minority candidate emergence in “biracial elections” (Grofman, Handley, and Lublin Reference Grofman, Handley and Lublin2001, 1388).Footnote ⁸ Focusing on minority descriptive representation, I also assume that minority candidates are the only minority-preferred candidates while recognizing that the literature also discusses that white Democrats can be minority-preferred candidates in certain conditions (Handley, Grofman, and Arden Reference Handley, Grofman, Arden and Grofman1998; Schousen, Canon, and Sellers Reference Schousen, Canon, Sellers and Grofman1998).Footnote ⁹

Assumption 2 (Potential Candidates). There is always at least one minority politician who would run for office regardless of the number of minority voters in a district.

This assumption excludes the possibility that we do not observe any minority candidate because there is no minority politician in the “eligibility pool” of potential candidates (Fox and Lawless Reference Fox and Lawless2004; Lawless Reference Lawless2012).

Assumption 3 (Short-Term Instrumentally Rational Candidates). Minority candidates are short-term instrumentally rational such that their primary goal is to get elected in upcoming elections.

This assumption rules out the possibility that minority candidates emerge because of noninstrumental reasons or long-term benefits, such as raising voice for symbolic representation or earning name recognition for future candidacy.

Assumption 4 (The Most Viable Candidate). Whether a district has at least one minority candidate solely depends on the strategic decision of the most viable minority politician in the district.

This assumption states that, in each district, less viable minority politicians would not run for office unless the most viable minority politician runs.Footnote ¹⁰ This assumption allows me to model district-level minority candidate emergence as the decision making of the most viable minority politician whose sole agenda is to win an upcoming biracial election.

The Logical Model

I argue that the most viable minority candidates decide to run for office when they see a high probability of winning in the upcoming elections. I then theorize that they calculate the likelihood of winning from two sources of information: (1) the electoral performance of coethnic candidates in the most recent elections and (2) the racial composition of a district. Formally, I introduce the following logical model of minority candidate emergence:

(1) $$ {P}_{\mathrm{run}}=\Phi \left({(MC)}^{1/2}-50\right), $$

where $ {P}_{\mathrm{run}} $ is the probability of minority candidate emergence, $ \Phi $ is a cumulative distribution function (CDF) of the standard normal distribution, and $ MC $ is a product of two terms representing the two sources of information (detailed below). Here, I briefly describe how I obtained Equation 1 while leaving more detailed explanations to Appendix A.

Step III (Two Logical Bounds)

Third, I argue that minority candidates can estimate the future racial margin of victory by considering two logical bounds (or limit cases). One limit scenario is that the future racial margin of victory (at time $ t $ ) is equivalent to the racial margin of victory in the most recent election (at time $ t-1 $ ), which I denote by $ {M}_0=\frac{1}{2}\left({V}_{t-1}^M-{V}_{t-1}^W\right)\hskip1.5pt \in \hskip1.5pt \left(-50,50\right) $ .Footnote ¹² The other extreme case is that it is identical to the racial margin of victory based on the district’s racial composition (with the assumption that minority and white voters cohesively vote for their coethnic candidates), which I denote by $ {C}_0\hskip1.5pt \in \hskip1.5pt \left(-50,50\right) $ . The next subsection details what these two logical bounds mean substantively.

I contend that the future racial margin of victory cannot be larger than $ {M}_0 $ and smaller than $ {C}_0 $ (and vice versa) and it will be located somewhere between the two logical bounds:

(2) $$ {C}_0\hskip1.5pt \le \hskip1.5pt \frac{1}{2}\left({\hat{V}}_t^M-{\hat{V}}_t^W\right)\hskip1.5pt \le \hskip1.5pt {M}_0\hskip1em \mathrm{or}\hskip1em {M}_0\hskip1.5pt \le \hskip1.5pt \frac{1}{2}\left({\hat{V}}_t^M-{\hat{V}}_t^W\right)\hskip1.5pt \le \hskip1.5pt {C}_0. $$

Footnote ¹³

Given that, I argue that the most likely results (or our best guess) can be represented by the geometric mean (i.e., average) of these two extreme cases. Importantly, it is our best guess “in the absence of other information” (Shugart and Taagepera Reference Shugart and Taagepera2017, 105) and it could be off for actual (future) results due to various election or candidate-specific factors (see below). I use the geometric mean (over other metrics such as the arithmetic mean) because while $ {M}_0 $ may take extreme values in some elections (i.e., minority candidates may do extremely well or poorly), the geometric mean is robust to such extreme values and commonly used in the logical model approach (Taagepera Reference Taagepera2008, 120–27). Since $ {M}_0 $ and $ {C}_0 $ may contain negative values and the geometric mean cannot be computed for negative values, however, I take the geometric mean of slightly adjusted quantities ( $ M={M}_0+50 $ and $ C={C}_0+50 $ ) without changing any substantive meaning: $ {(MC)}^{1/2} $ . Finally, I model the estimated racial margin of victory as $ \frac{1}{2}\left({\hat{V}}_t^M-{\hat{V}}_t^W\right)={(MC)}^{1/2}-50 $ , where $ -50 $ readjusts for the above transformation (see, again, Appendix A for more detailed explanations).

Connecting all three steps, I obtain

(3) $$ {P}_{\mathrm{run}}={\hat{P}}_{\mathrm{win}}=\Phi \left(\frac{1}{2}\left({\hat{V}}_t^M-{\hat{V}}_t^W\right)\right)=\Phi \left({(MC)}^{1/2}-50\right). $$

The logical model states that minority candidate emergence can be explained by minority candidates’ decision-making process (as the supply-side theory argues). It also claims that minority candidates estimate the probability of winning by looking at past ( $ M $ ) and expected ( $ C $ ) voter demand (as the demand-side theory contends). Consequently, the model unifies the two competing theories under a single equation.

How $ M $ and $ C $ Relate to Other Determinants of Minority Representation

At first glance, the logical model may look “too simple” because it potentially ignores many other important determinants of minority representation. In contrast, this subsection demonstrates that the model is simple because it hierarchically summarizes various factors that affect minority electoral success. Below, I discuss how the two logical bounds $ M $ and $ C $ relate to many other concepts discussed in the literature.

$ M $ Summarizes a Variety of Factors

First, $ M $ is the racial margin of victory in the most recent election. It summarizes a range of information about factors that influence the electoral performance of minority candidates just as the effective number of parties summarizes complex information about “cohesion of parties and their ability to get one’s way in negotiations” (Taagepera Reference Taagepera2007, 285). Recall that $ M $ is a function of how many votes the top minority and white candidates received in the most recent election. These vote shares then offer rich information about how voters in particular districts behave in biracial elections. Figure 1 visualizes this chain of associations. Here, U represents a set of factors that influence $ {V}_{t-1}^M $ and $ {V}_{t-1}^W $ , whereas T denotes a set of elements that affect U (i.e., T $ \to $ U $ \to $ $ {V}_{t-1}^M $ , $ {V}_{t-1}^W $ ).

Figure 1. How the Racial Margin of Victory Relates to Other Factors

Note: The racial margin of victory ( $ M $ ) is a function of the vote shares of the top minority and white candidates in the most recent election ( $ {V}_{t-1}^M,{V}_{t-1}^W $ ). These vote shares are in turn functions of three factors (U): the proportion and turnout of minority voters, degree of minority bloc and white crossover voting, and strategic coordination among minority and white candidates. U depends on various factors (T), including district partisanship, incumbency, geographical concentration of minorities, urbanization, residential segregation, and historical factors.

According to the literature, U has three factors: (1) the proportion of minority voters among those who turn out, (2) the strength of minority bloc and white crossover voting, and (3) the level of strategic coordination within minority and white candidates. It may seem intuitive that the proportion of minority voters affects $ {V}_{t-1}^M $ (and $ {V}_{t-1}^W $ ) (i.e., the more minority voters, the more votes the top minority candidate receives). What seems less obvious is that $ M $ contains information about the turnout gap by race (the difference in turnout rates between minority and white voters) (Fraga Reference Fraga2018) in the most recent election. While accounting for the turnout gap has been a keystone in the literature (e.g., Grofman, Handley, and Lublin Reference Grofman, Handley and Lublin2001, 1404–07), $ M $ incorporates it systematically.

The strength of minority bloc and white crossover voting also matters (Lublin Reference Lublin1997, 47–48). Minority bloc voting means that minority voters cohesively vote for their coethnic candidates over other candidates. In contrast, white crossover voting means that white voters cast ballots on minority candidates instead of white candidates. In the literature on the VRA and redistricting, scholars often use the term racially polarized voting to describe the combination of minority bloc voting and the lack of white crossover (Abosch, Barreto, and Woods Reference Abosch, Barreto, Woods and Henderson2007; Crayton Reference Crayton2012; Engstrom Reference Engstrom2015; Grofman, Handley, and Niemi Reference Grofman, Handley and Niemi1992). The level of racially polarized voting and the percentage of minority voters solely determine $ M $ when there is only one minority and white candidates, respectively (Grofman, Handley, and Lublin Reference Grofman, Handley and Lublin2001, 1407–09).

Strategic coordination within minority and white candidates also affects both $ {V}_{t-1}^M $ and $ {V}_{t-1}^W $ . Recall that the racial margin of victory is not about how many votes minority candidates collectively received, but about how many votes the minority candidate with the highest chance of winning received compared with the most viable white candidate. It suggests that $ M $ reflects how well both minority and white candidates coordinated within each group so that there would be less vote splitting (Cox Reference Cox1997). Thus, minority representation is often a collective action problem (Schousen, Canon, and Sellers Reference Schousen, Canon, Sellers and Grofman1998). Indeed, when multiple minority candidates emerge under a divided minority leadership, it could easily make a less-preferred white candidate the plurality winner even in majority-minority districts (Vanderleeuw, Liu, and Marsh Reference Vanderleeuw, Liu and Marsh2004).

By now, readers may wonder how other factors that may relate to minority representation fit into the logical model. These factors are considered as elements of T in Figure 1. For example, research shows that the relative size of coethnic voters (and the sense of “electoral influence” from it) would affect the turnout gap by race (Fraga Reference Fraga2018). Scholars have also argued that strategic coordination among minority candidates may become more difficult in majority-minority districts (Schousen, Canon, and Sellers Reference Schousen, Canon, Sellers and Grofman1998). Also, research finds that heavily democratic districts often yield a high level of white crossover voting, which in turn increases $ {V}_{t-1}^M $ (and decreases $ {V}_{t-1}^W $ ), resulting in a larger value of $ M $ (Juenke and Shah Reference Juenke and Shah2016; Lublin et al. Reference Lublin, Handley, Brunell and Grofman2019). Moreover, incumbency may affect minority and white vote choices (Liu Reference Liu2001a; Voss and Lublin Reference Voss and Lublin2001), while term limits may attenuate such incumbency advantage (Casellas Reference Casellas2011).

Meanwhile, the “racial threat” (or “social contact”) hypothesis contends that the geographical concentration of racial minorities decreases (or increases) white crossover (Avery and Fine Reference Avery and Fine2012; Key Reference Key1949; Liu Reference Liu2001a; Reference Liu2001b; Oliver and Wong Reference Oliver and Wong2003), while other research shows that such an influence may be contingent upon urbanization and residential segregation (Rocha and Espino Reference Rocha and Espino2009; Voss and Miller Reference Voss and Miller2001;Weaver and Bagchi-Sen Reference Weaver and Bagchi-Sen2015). Alternatively, Liu and Vanderleeuw (Reference Liu and Vanderleeuw2007) theorize that the highest level of racially polarized voting occurs when minority and white voters have an equal share of the electorate. In addition to (or instead of) contemporary factors, the history of slavery-based local economy may also make white crossover less likely (Acharya, Blackwell, and Sen Reference Acharya, Blackwell and Sen2020). Finally, many election and candidate-specific factors could influence turnout rates and voting behaviors of minority and white voters, such as candidates’ names (Casellas Reference Casellas2011, 91–93), scandals (Parent and Perry Reference Parent, Perry, Bullock and Rozell2017), deracialized campaigns (Vanderleeuw, Liu, and Marsh Reference Vanderleeuw, Liu and Marsh2004), and natural disasters (Liu and Vanderleeuw Reference Liu and Vanderleeuw2007). All of these factors are inside T and thus incorporated in the logical model.

The logical model contends that by only looking at $ M $ , minority candidates can learn about their likely electoral results because it summarizes a set of factors that directly or indirectly determine the relative performance of minority candidates. I define $ M $ as a logical benchmark for the upcoming election because there is likely a temporal dependency between voting behavior in two consecutive elections. That is, I assume that a set of factors that determined the racial margin of victory at time $ t-1 $ (“last time”) is likely to produce similar results at time $ t $ (“this time”) since these factors are not likely to change in a short period and may even persist for a long time (Acharya, Blackwell, and Sen Reference Acharya, Blackwell and Sen2020). This idea is also consistent with previous studies, including Marschall, Ruhil, and Shah (Reference Marschall, Ruhil and Shah2010, 114–15) and Shah (Reference Shah2014, 269, 271), that argue that prior minority candidacy and electoral performance may affect future minority representation.

Illustration of Model Predictions

To illustrate the quantitative predictions of the logical model, I first provide a simple example. Suppose that the top minority and white candidates obtained 30% and 50% of vote shares in the most recent election, respectively.Footnote ¹⁷ Suppose also that 60% of the district are minority voters. In this example, $ M=\frac{1}{2}\left(30-50\right)+50=40 $ and $ C=60 $ . The model suggests that the probability of minority candidate emergence in this district becomes: $ \Phi \left({\left(40\times 60\right)}^{1/2}-50\right)=\Phi \left(48.99-50\right)=\Phi \left(-1.01\right)=0.156 $ .

Next, I visualize the quantitative predictions of the model under varying conditions. Panel A of Figure 2 plots the probability of minority candidate emergence against $ {(MC)}^{1/2}-50 $ . It shows that the probability is lower than 0.5 when the entire term is negative, whereas it becomes higher than 0.5 when the value is positive. Again, this formalizes the theoretical story that minority candidates run for office when they see a high probability of winning.

Figure 2. Quantitative Predictions of the Logical Model

Note: This figure visualizes the probability of minority candidate emergence against $ {(MC)}^{1/2}-50 $ (the estimated future racial margin of victory) (Panel A) and $ M $ with varying values of $ C $ (% minority voters) (Panel B).

Panel B plots the probability of minority candidate emergence against $ M $ with varying values of $ C $ . While researchers have long debated whether minority candidates can run and win in majority-white districts (Hicks et al. Reference Hicks, Klarner, McKee and Smith2018; Lublin et al. Reference Lublin, Handley, Brunell and Grofman2019), the logical model suggests that it depends on the value of $ M $ . For example, when $ C=30 $ , the probability of minority candidate emergence is mostly 0, but it starts to increase after $ M $ is beyond 75. It means that, in 30% minority districts, minority candidates do not usually emerge, but they could run if their coethnic candidates won with large margins or “did extremely well” relative to their white counterparts in the most recent election. Similarly, the logical model suggests that having majority-minority districts does not automatically guarantee the emergence and electoral success of minority candidates. When $ C=60 $ , for instance, the probability of minority candidate emergence is generally high, but it gets increasingly low after $ M $ is less than 40. It suggests that, in 60% minority districts, minority candidates usually appear, but they may not run if their coethnic candidates lost with great margins or “did extremely poorly” relative to their white counterparts in the most recent election.Footnote ¹⁸ Indeed, this result reconciles influential and competing theories on the necessity of majority-minority districts in electing minority candidates (Cameron, Epstein, and O’Halloran Reference Cameron, Epstein and O’Halloran1996; Grofman, Handley, and Lublin Reference Grofman, Handley and Lublin2001; Lublin Reference Lublin1997; Reference Lublin1999; Lublin et al. Reference Lublin, Handley, Brunell and Grofman2019).

Data

To test how well the model can predict minority candidate emergence in the real world, I leverage data on mayoral elections in 313 Louisiana municipalities from 1986 to 2016 (N = 2,037).Footnote ¹⁹ Louisiana mayoral elections offer a desirable context to evaluate the model predictions for multiple reasons. First, more than 96% of registered voters in Louisiana are either African American or white, yielding ideal conditions (biracial elections) to assess the model predictions (Appendix B.2). Second, the cultural, religious, and political differences between northern and southern Louisiana whites yield a rich variation in the degree of white crossover voting (Parent Reference Parent2006, 20). Third, Louisiana adopts a unique electoral system where all candidates participate in a single election regardless of party affiliation (Parent and Perry Reference Parent, Perry, Bullock and Rozell2017), which allows me to assess the model predictions without having to account for multiple stages of partisan primaries and general elections (Grofman, Handley, and Lublin Reference Grofman, Handley and Lublin2001, 1409–11).Footnote ²⁰ Finally, Louisiana is one of the states with extensive studies on and court rulings about redistricting and voting rights (Adegible Reference Adegible2007; Engstrom and Kirksey Reference Engstrom, Kirksey and Grofman1998; Engstrom et al. Reference Engstrom, Halpin, Hill, Caridas-Butterworth, Davidson and Grofman1994), which is a kind of environment where the model can be most relevant.

Model Prediction and its Novelty

To assess the model predictions, I first compare them with the actual minority candidate emergence in the data. Figure 3 visualizes the model predictions along with the observed data in a contour plot, where the $ x $ -axis is $ M $ and the $ y $ -axis is $ C $ measured by the percentage of African Americans in each municipality. The gradation of color represents the predicted probability of Black candidate emergence based on the logical model. The figure visually confirms that the logical model predicts observed Black candidate emergence quite accurately in Louisiana municipalities, including New Orleans (Liu Reference Liu2001a; Reference Liu2001b; Liu and Vanderleeuw Reference Liu and Vanderleeuw2007). Namely, most elections with Black candidates ( $ \circ $ ) are located in the upper-right region (where $ {P}_{\mathrm{run}}>0.9 $ ) and most elections without Black candidate ( $ \times $ ) appear in the lower-left area (where $ {P}_{\mathrm{run}}<0.1 $ ).Footnote ²¹

Figure 3. Model Predictions with Observed Minority Candidate Emergence

Note: This figure shows the observed minority candidate emergence in Louisiana mayoral elections over the model predictions. It illustrates that most elections with Black candidates ( $ \circ $ ) are located in the upper-right region (where $ {P}_{\mathrm{run}}>0.9 $ ) and most elections without Black candidates ( $ \times $ ) appear in the lower-left area (where $ {P}_{\mathrm{run}}<0.1 $ ).

To highlight the novelty of the model, I also mark two observations (with squares) as examples of cases that previous research cannot fully explain, but the logical model can. The left case indicates the absence of minority candidate in a heavily majority-minority district ( $ C=76 $ ), and the right case shows the presence of minority candidate in a majority-white district ( $ C=38 $ ). Both elections are generally puzzling to prior studies that have argued that minority candidate emergence is the norm in majority-minority districts and highly unlikely in less than 40% minority districts. In contrast, the logical model indicates that both of these cases “make sense” (with predicted probabilities being 0.00 and 0.89, respectively). Indeed, the model predicts the absence of minority candidates even in majority-minority districts when their coethnic candidates performed poorly in the most recent elections (relative to their white counterparts). Conversely, it predicts the emergence of minority candidates even in majority-white districts when their coethnic candidates won the last elections with a large racial margin of victory. Consequently, the logical model offers a novel insight that the district’s racial composition is only half the story of minority candidate emergence.

Predictive Performance

Next, I compute the accuracy of the model predictions based on the expected percentage correctly predicted (ePCP). The ePCP provides the percentage of observations for which the model can correctly predict their values (i.e., 0 or 1) while accounting for how close such predictions are (e.g., 0.51 vs. 0.99) (Herron Reference Herron1999, 91–92). The ePCP is an ideal measure of prediction accuracy in this context because “closeness to the predicted value is what matters” for logical models (Taagepera Reference Taagepera2007, 118).

Column 1 of Table 1 shows that the logical model can correctly predict, on average, about 89% of cases. It also reports that the model accurately predicts minority candidate emergence in various contexts defined by $ C $ , types of cities, features of elections, electoral systems for city-councils, and years (before vs. after the Republican Revolution in 1994).

Table 1. Predictive Performance of the Logical Model Relative to Regressions

Note: This table reports the predictive performance of the logical model and linear and logistic regressions in ePCP = $ {N}^{-1}\left({\Sigma}_{y_i=1}{\hat{p}}_i+{\Sigma}_{y_i=0}\left(1-{\hat{p}}_i\right)\right), $ in % where $ N $ is the number of units and $ {y}_i $ and $ {\hat{p}}_i $ are true and predicted values for unit $ i $ , respectively.

Readers may wonder how much the logical model improves upon conventional methods in predicting minority candidate emergence. To investigate, I estimate a set of linear probability models (LPM) and logistic regressions with the observed values of $ M $ , $ C $ , and minority candidate emergence to generate in-sample ePCPs.Footnote ²² Columns 2–3 show that the logical model has a higher predictive performance than these ad hoc regressions in all subsets of data. This is a remarkable result because these regressions are estimated or trained using actual data while the logical model is not. Moreover, I run both regressions with 34 additional variables that previous research tends to include, finding that the logical model performs as good as and even better than such heavily parameterized regressions (Appendix B.3). Finally, I also conduct extensive analyses to ensure the internal validity of the logical model (Appendix B.4).

Now, one major payoff is that the logical model of minority candidate emergence is also a model of minority electoral success (i.e., $ {P}_{\mathrm{run}}={\hat{P}}_{\mathrm{win}} $ ). If the theory behind the model is plausible, the model should also predict the presence of minority winners. Column 4 shows that this is exactly the case. The logical model predicts, on average, about 94% of minority electoral success with consistently high predictive performance in all subsets of data. Columns 5–6 suggest that the logical model, again, performs much better than the two regressions in predicting minority winners. Also, the logical model predicts minority electoral success better than it predicts minority candidate emergence. One explanation for this result is that minority candidates may sometimes emerge in districts with a low chance of winning potentially for noninstrumental reasons or long-term benefits (Hardy-Fanta, Pinderhughes, and Sierra Reference Hardy-Fanta, Pinderhughes and Sierra2016, 161–207). Consistent with this idea, I also find that minority candidates are more likely to emerge in districts with low odds of winning than to retreat from elections with high probabilities of winning (Appendix B.5).

External Validity

Readers may wonder how generalizable the above findings are to the broader context of American politics. Indeed, Louisiana is a unique state in many aspects: it has a large African American population (about one third of the state population) (Parent Reference Parent2006), is a Deep South state (Hicks et al. Reference Hicks, Klarner, McKee and Smith2018), is formally covered by Section 5 of the VRA (Engstrom et al. Reference Engstrom, Halpin, Hill, Caridas-Butterworth, Davidson and Grofman1994; Shah, Marschall, and Ruhil Reference Shah, Marschall and Anirudh2013), and has “rarely acutely partisan” elections (Engstrom and Kirksey Reference Engstrom, Kirksey and Grofman1998, 244) that gradually became “deep red, yet unpredictable” (Parent and Perry Reference Parent, Perry, Bullock and Rozell2017).

To address this concern, I construct a novel dataset of state legislative general elections in 2012 and 2014. Specifically, I collect data on the vote shares of the top minority and white candidates and the presence of minority winners for 1,281 general elections from 36 states, including Black-white (N = 642), Latino-white (N = 517), and Asian-white (N = 122) biracial elections in single-member districts. I supplement this data with information on minority candidate emergence and CVAP by race compiled by Fraga, Juenke, and Shah (Reference Fraga, Juenke and Shah2020).Footnote ²³

Figure 4 provides visual confirmation that the logical model accurately predicts minority electoral success. Here, capitalized state names (e.g., TX) denote districts with minority winners, whereas lower-case state names (e.g., tx) represent elections without minority winner. The graph shows that most elections with minority winners concentrate in the upper-right area (where $ {\hat{P}}_{\mathrm{win}}>0.9 $ ) while most elections without minority representative appear in the lower-left region (where $ {\hat{P}}_{\mathrm{win}}<0.1 $ ).

Figure 4. Model Predictions with Observed Minority Electoral Success

Note: This figure shows the observed minority electoral success in state legislative general elections in 36 states over the model predictions. It illustrates that most elections with minority winners are located in the upper-right region (where $ {\hat{P}}_{\mathrm{win}}>0.9 $ ) and most elections without minority winner appear in the lower-left area (where $ {\hat{P}}_{\mathrm{win}}<0.1 $ ).

Table 2 reports the predictive performance of the logical model. Columns 1–3 show that the logical model correctly predicts more than 90% of minority (Black, Latino, and Asian) candidate emergence in various electoral contexts. I find that this result holds even after accounting for the degree of biracial elections, levels of $ C $ , types of office, types of elections, regions, and districts’ histories regarding the VRA and redistricting. Columns 4–6 demonstrate that the logical model also predicts more than 95% of minority electoral success regardless of electoral contexts. The results also suggest that the model can predict minority candidates and winners in general elections without accounting for primaries. Finally, Appendix B.6 demonstrates that the model can also predict the number of minority officeholders at the jurisdiction level (e.g., city).

Table 2. Predictive Performance of the Logical Model in State Legislative General Elections

Note: This table reports the predictive performance of the logical model based on ePCP. A district is coded as a former Section 5 district if it is located in one or more counties at the time of the Shelby County v. Holder (2013) decision and coded as challenged if it is mentioned in several major litigation events related to race and voting rights in the 2010 round of redistricting, including Davis v. Perry (2012), Perez v. Perry (2012), Jeffers v. Beebe (2012), Alabama Legislative Black Caucus v. Alabama (2012), Bethune-Hill v. Virginia State Board of Elections (2015), Covington v. North Carolina (2016), and Thomas v. Bryant (2019).