The Invisible Primary in an Agent-Based Model: Ideology,Strategy,and Competitive Dynamics

Abstract

Historical accounts of American presidential nominating contests suggest that candidates jockey over ideology and policy in ways that shape the outcomes of these races. Yet this aspect of competition has been difficult to analyze with the formal and statistical methods that dominate this research agenda. To address this gap, this article presents a computational agent-based model (ABM) of candidates’ ideological maneuvering during the invisible primary. We extend the framework developed by Michael Laver to study dynamic party competition in Europe, but recast it for the different context and to enable model fit to be more rigorously determined. Our analysis of data from the 2012 Republican invisible primary suggests the importance of ideological jockeying in this contest. Moreover, its dynamics can be well-explained by a basic version of the ABM in which candidates select between three strategies (aggregator, hunter or sticker) and then maintain that strategy over time. The fit of this model, particularly in the short run, can be improved by introducing a “momentum effect” that allows the candidates’ standing in the race to rise or fall without any accompanying ideological change.

Keywords

agent-based model ideology invisible primary machine-learning presidential nominations

Introduction

American presidential nominations generate a notoriously complex form of competition. In part, this is because of the sequential voting in primary elections and caucuses, which generates dynamic processes—namely, learning, signaling, and momentum—that do not arise with simultaneous voting (Aldrich 1980; Bartels 1988; Battaglini et al. 2007; Knight and Schiff 2010). Crowded, multicandidate fields and the wide variety of primary rules used across the states bring further complexity (Kamarck 2018), while the variables that often bring coherence to the study of presidential election campaigns, such as partisanship, demography and the state of the economy, are often redundant in nominating politics. The result, as Mayer (2003, 154) contends, is that the competition in this context is generally more intractable and unpredictable than the competition in presidential elections.

Despite such obstacles, our understanding of nominating politics has made progress via three methodological approaches. Historical studies and campaign biographies describe specific races (Heilemann and Halperin 2010; Jamieson 2012), and the evolution of formal rules (Kamarck 2018; Shafer 1988). Applied statistics has been widely used to study, among other things, the impact of party elites (Cohen et al. 2008), candidate exit from races (Norrander 2000, 2006), and the influence of the early-stage “invisible primary” (Mayer 1996; Steger 2007). And formal models have been used to examine how momentum affects outcomes such as the stability of races (Aldrich 1980) and candidates’ resource allocation decisions (De Roos and Sarafidis 2018).

In contrast to these traditional approaches, this article develops an agent-based model (ABM) of nominating politics, building on the framework devised by Michael Laver and his collaborators to study interparty competition in Europe (Laver 2005; Laver and Schilperoord 2007; Laver and Sergenti 2012). Though ABMs have a comparative advantage in the analysis of dynamic and complex competition (De Marchi and Page 2014), they have only rarely been used to study American presidential nominations (see Gurian 1993). Seeking to exploit this untapped potential, we develop a model that considers how candidates compete over policy and identity—in other words, their ideological strategies. This focus distinguishes our model from Gurian’s (1993) ABM, which concentrates on resource allocation. It also distinguishes our study in the wider scholarship on presidential nominations because ideology has received little attention in its analytical studies (i.e., those using formal and statistical methods), even though historical accounts of races often describe competition in ideological terms, and despite the increasing prominence of ideology in modern American politics (e.g., Layman et al. 2006; Már 2020; Noel 2013; Rosenfeld 2017).

While seeking to provide a fresh, ideological perspective on the nominating process as a whole, we analyze specifically its early, unofficial stage, often described as the invisible primary, which begins about 1 year out from the primary season proper (e.g., Norrander 2010, 35). Targeting this stage simplifies model construction as it means we need just one (national) model instead of fifty (for all the states), but it also makes substantive sense since the outcomes of nominating races are generally set by coordination dynamics that occur during the invisible primary (Cohen et al. 2008). To calibrate our model, we rely on data from the Republican contest in 2012. We find that a simple version of the ABM, in which candidates select one ideological strategy and then stick with it, adequately captures the long-run dynamics of this invisible primary including the determination of its winner, though a better account of short-run fluctuations emerges when a “momentum effect” is introduced. This analysis suggests that the eventual nominee, Mitt Romney, but also most of his rivals, followed Laver’s hunter strategy—they continually repositioned in the ideological space to broaden their appeal among primary voters.

The article proceeds as follows. The next section reviews the scholarship on presidential nominations and the invisible primary. It also introduces our case study, the Republican invisible primary of 2012. The third section discusses Laver’s ABM framework, which we adapt to analyze this race. The fourth section discusses our model setup, detailing the baseline ABM and several realistic extensions of it. The fifth section explains the evolutionary algorithm (EA) procedure that we apply to select the settings of the best-fitting ABM. The sixth section applies this working model to practical questions about the 2012 Republican race. We conclude by identifying the lines of inquiry that our analysis opens up.

Observing the Invisible Primary

The official presidential nominating process runs from the Iowa caucuses, in January of an election year, until the national conventions of late summer when the parties’ nominees are confirmed. But the competition begins in earnest at least a year before Iowa, during the “invisible primary” (Hadley 1976). During this early, informal stage of the contest, the serious aspirants build campaign organizations, raise funds, seek endorsements from party elites, take stands on the issues of the day, and develop and promote new policy proposals (Norrander 2010). Some early entrants may drop out altogether, after receiving signals that they have no chance of winning. Meanwhile, the media, party elites, campaign financiers and the (attentive) primary voting electorate consume the information emanating from these activities—it will inform their reporting, endorsements, donations and voting as the race unfolds. This activity during the invisible primary cannot be dismissed as merely “noise” or a “false start” because most evidence suggests that it shapes the official contest that follows, making the invisible primary “a fundamental part of the strategic environment” for “all serious candidates” (Buell 1996, 11; see also Cohen et al. 2008; Norrander 2010).

With the impact and significance of the invisible primary widely acknowledged, the research in this agenda has tried to identify what specific trends from this period best predict the outcomes of the primary season proper (e.g., Aldrich 2009; Dowdle et al. 2009; Dowdle et al. 2016; Mayer 1996, 2003; Steger 2007). In a pioneering study of this problem, Mayer (1996) found that the candidate who leads in the national polls at the end of the invisible primary almost always wins the official contest and, more generally, that the candidates’ standing in the polls at the end of the invisible primary roughly predicts their eventual primary vote share. The polls, he argues, capture more than name recognition—they reveal assessments by party activists and the press, whose impressions “are apparently filtering through to the voters” (Mayer 1996, 48; see also Mayer 2003). Fundraising capacity, the other variable in Mayer’s model, is correlated with standing in the polls, but on its own is a weak predictor of vote share. Adkins and Dowdle (2000) report similar results but discover that cash reserves at the end of the invisible primary matter more than total money raised (see also Steger 2000). As they explain, raising a lot of money shows that a candidate has support among financiers, but their need to spend it immediately, before the official race begins, reveals their low standing among the party’s rank-and-file members. Especially in Democratic races, the strongest candidates build a war chest that is deployed strategically when the official race begins. Although party elites feature prominently in Mayer’s explanation of results, his model has no variable to capture their impact. Steger (2007) addresses this gap by including a measure of endorsements by federal and state politicians in his model, and finds that “elite signaling prior to the primaries does matter, especially in the Republican Party, where elites pick a horse early in the invisible primary … [who] solidifies his lead as the primaries approach” (Steger 2007, 98; see also Dowdle et al. 2009).

As a result of this work, Aldrich (2009, 37) summarizes, “we know a fair amount about what sorts of resources [from the invisible primary] are helpful.” But the upstream process—that is, how candidates end up with different resource profiles by the end of the invisible primary—remains unclear. Organizational capacity has often been decisive, argues Aldrich (2009), while Steger et al. (2002, 550) highlight the importance of struggle over “the definition of the party’s policy agenda.” We explore the latter conjecture by analyzing the extent to which an invisible primary was shaped by the competitors’ ideological jockeying.

The Case Study

The Republican invisible primary in 2012 was “won,” according to all the relevant metrics (see Dowdle et al. 2016; Sides and Vavreck 2014), by Mitt Romney, a US senator from Utah, who previously served as governor of Massachusetts. Consistent with the literature’s predictions, this success foreshadowed Romney’s victory in the primary season proper. This race is a good one to showcase the value of our ABM approach. On the one hand, it is fairly representative of modern nominating contests because Republican elites largely “decided” who would emerge as the winner (Cohen et al. 2008). But the convergence on Romney was not straightforward (Cohen et al. 2016; Steger et al. 2012). This reflected, among other things, the deepening of ideological divisions within the Republican party (Blum 2020), and an unusually large number of candidate debates that year, which disrupted the coalescence on Romney (Masket 2020, 148). Romney’s tricky path to victory meant that this invisible primary had turbulent spells that a satisfactory model should be able to capture and explain.

The 2012 race featured seven serious candidates alongside Romney. They were: Michelle Bachman, then a congresswoman from Minnesota; Herman Cain, a business executive and Tea Party favorite; Newt Gingrich, a congressman from Georgia and former Speaker of the House of Representatives; Jon Huntsman, a former governor of Utah who had more recently served as Ambassador to China; Ron Paul, a congressman from Texas; Rick Perry, then Governor of Texas; and Rick Santorum, a former US senator from Pennsylvania. The polls sometimes included other prominent figures who ultimately did not enter the race (Donald Trump) or left it very early (Fred Karger, Thaddeus McCotter, and Timothy Pawlenty).

Figure 1 shows the changing fortunes of these candidates during the invisible primary as revealed by the Real Clear Politics national polls for the period March 14, 2011 until January 16, 2012.¹ The fluctuations in the candidates’ standing are the main “dynamics” that we aim to explain. Romney led at the outset and finished in pole position. But his standing declined midway through the race and this coincided with a surge for Perry, whose fortunes soon collapsed following a string of poor performances in the candidate debates. Both Cain and Gingrich also briefly led the race, before Romney reclaimed the frontrunner position. The figure also shows that there was considerable flux in the lower ranks of the contest.

Figure 1.

The national polls during the Republican invisible primary of 2012.

Given our focus on the competitors’ maneuvering in the ideological space, we also need data from this race to map the primary voters’ preferences (which we assume do not change) and the candidates’ initial positions. These preferences and positions, we assume, are dominated by an economic dimension (i.e., “liberal” versus “conservative”) and a cultural dimension, which we reduce to “pro-life” and “pro-choice” stances on abortion. We obtain information for these dimensions from Bonica’s (2014) database of ideological positions, which is generated from official data on donations to 50,000 candidates between 1979 and 2012. The database includes scores for all eight competitors (see Table 1). While it does not include the directly measured preferences of ordinary Republican voters, it does have scores for most party officeholders, which we use to approximate the profile of the Republican electorate.

Table 1.

The Candidates’ Initial Ideological Positions.

Candidate	Economic dimension (original data)	Cultural dimension (transformed data)
Jon Huntsman	0.327	−0.870
Herman Cain	1.140	0.654
Rick Perry	0.879	0.352
Rick Santorum	0.835	1.000
Michelle Bachmann	1.158	0.790
Ron Paul	1.606	0.428
Newt Gingrich	0.976	0.816
Mitt Romney	0.617	−0.478

The primary voters’ preferences, understood in this sense, are shown in Figure 2. The economic dimension runs from −2 (liberal) to +2 (conservative) and the cultural dimension from −1 (pro-choice) to +1 (pro-life). Bonica’s study includes raw data for only the economic dimension, which we extract. It is therefore necessary to construct a cultural dimension. We assume that this curve has a Gaussian distribution with the same standard deviation as the economic curve (0.65) and a mean of 0.17, based on polling data from 2012 suggesting that 70% of Republicans have pro-life views (i.e., they believe that abortion should be illegal in most circumstances) (CNN/ORC 2012).² Since it would be problematic to treat the two dimensions as independent (Stimson 2012), we assume they correlate at 0.5 based on a test of the relationship between indicative questions in the American National Election Study.³

Figure 2.

Distribution of Republican voters on economic and cultural issues.

The Invisible Primary in Laver’s Framework

In a series of studies, Michael Laver and his collaborators have developed a powerful ABM framework for analyzing dynamic party competition (Laver 2005; Laver and Schilperoord 2007; Laver and Sergenti 2012). That framework follows traditional spatial theory in its treatment of voters, who are assumed to prefer the party closest to them in a well-defined ideological space. But the parties’ position-taking is more complicated than spatial models allow. Laver follows Kollman et al. (1992) by assuming adaptive parties, and introduces a variety of simple but realistic strategies. At the most basic level, parties can be stickers that never change their initial policy stance, aggregators who move to the central position of their existing supporter base, or hunters who make incremental moves in a direction that increases their support. The interactions of parties thus motivated are explored with computational techniques to shed light on how ideological strategies relate to the evolution of party competition, especially the fragmentation and representativeness of the competitive field.

We apply this framework to study intraparty competition and, specifically, the invisible primary. Since Laver used his model to investigate interparty dynamics in Europe, our analysis effectively integrates this research agenda with American Politics scholarship on the invisible primary. This is an unusual synthesis that crosses the often sharply drawn boundaries between American and comparative research on parties (see Katz 2019), and between interparty and intraparty competition (see Budge et al. 2010). Nonetheless, we see significant potential in drawing together these lines of research because of the dynamic nature of competition in both settings. Laver recognized that party competition “never stops,” which causes problems for static formal models (Laver and Sergenti 2012, 3). And, likewise, scholars of nominating politics see this context as a prime example of “the powerful dynamics forces at work in many real processes of collective choice” (Bartels 1988, 310).

While exploiting Laver’s ABM toolkit, our modeling aims to be sensitive to the ways in which the invisible primary differs from European multiparty competition. Three contrasts stand out. The first is that invisible primary competition involves (individual) candidates rather than (collective) parties. This difference is easy to account for because Laver reduces parties’ strategies to those of their leaders—or, putting it another way, parties are totally controlled by their leaders (Laver and Sergenti 2012, 42). The reality in many parties clashes against this assumption (Budge et al. 2010), but it is more accurate as a description of nominating politics, where candidates are typically able to exercise strategic control over their campaign organizations (Gurian 1993). A second difference is that party system dynamics play out over decades while the invisible primary occurs within 1 year. This difference can be handled by simply adjusting the time frame and intervals in our model. Thirdly, while ideological conflict has been central to studies of European interparty competition, this factor has received little attention in analyses of the invisible primary. So, it might be tempting to conclude that the role of ideology is an important contrast between these two settings. We suspect, however, that the conventional views of the invisible primary might underestimate the significance of ideology in this context. And, for this reason, Laver’s framework provides a basis to explore a potentially critical but neglected aspect of invisible primary competition.

The Invisible Primary ABM

We aim to develop an ABM that captures the dynamics of the 2012 Republican race from an empirically calibrated start. With Laver’s (2005) model serving as our point of departure, we first create a basic (i.e., baseline) ABM. This setup can be expanded in various directions that would make it more superficially realistic, but also more complicated. We present several of these potential extensions in this section, and their leverage is assessed using an evolutionary algorithm (EA) in the next section to identify how much and what forms of complexity generate the best-fitting model, which we then use to reexamine the 2012 Republican race.

The Baseline Model

The agents in our model are candidates who compete for the support of voters who are attracted to the most ideologically congruent candidate. The interaction between candidates and voters is dynamic, with moves set by an algorithm describing how candidates respond to information about their standing among voters. These steps are summarized in Figure 3 and fleshed out below. However, in broad terms, each candidate takes up an initial position set using real-world data (see Table 1) and they can move away from that location over 45 iterations, a duration that matches the series of weekly polls we use to assess model outputs.

Figure 3.

Setup and iterations in the invisible primary ABM.

Initial Positions and Candidate Vote Share

The candidates’ initial position is set using data from the 2012 Republican invisible primary, which we also use to map the distribution of voters. As it would be intractable to consider a population of agents approximating the size of the Republican electorate, much like a professional poll we use a smaller, representative sample of 1000 voters.⁴ At the start of the simulation and for every iteration, the (Euclidian) policy distance between each voter and each candidate is computed. Each voter prefers the candidate who minimizes this distance. Thus, once the candidate preference of each voter has been determined, the candidates’ expected vote share can be calculated.

Candidate Utility Functions and Strategies

To understand the candidates’ strategies, it is useful to assume that they follow a decision rule, prescribing an action in the current round, t, in response to outcomes from the previous period, t-1. Underlying the rule is a candidate’s preferences, which can be specified as a utility function giving the “quantity” that candidate j tries to maximize. Following Laver and Sergenti, it can be expressed as follows

U_{j} = - φ_{1, j} d_{p, j}^{2} - φ_{2, j} d_{c, j}^{2} + φ_{3, j} v_{j}

(1)

where

d_{p, j}

gives the distance between the candidate’s current position and original position, which we assume is also her personal ideal;

d_{c, j}

gives the distance between the candidate’s current position and the centroid of their current supporters; and

v_{j}

, the candidate’s vote share. Parameters

φ_{1, j}

φ_{2, j}

and

φ_{3, j}

therefore capture a candidate’s propensity to be policy-oriented (sticker), base-oriented (aggregator) or vote-oriented (hunter). So, if

φ_{2} = φ_{3} = 0

, the candidate minimizes the distance between current and original positions; with

φ_{1} = φ_{3} = 0

, she moves toward the mean position of her base (i.e., current supporters); and if

φ_{1} = φ_{2} = 0

, she maximizes vote share. Equation (1) suggests that candidates can have mixed preferences, a possibility we explore later when discussing complex versions of the ABM.

We can assume that candidates will try to maximize their utility functions. But how to do this may not be obvious as there may be various plausible utility-maximizing strategies. For instance, a vote-maximizer could move toward the mean position of voters or toward the candidate with the highest vote share. Fowler and Laver (2008) consider 25 different decision rules, which they test in an ABM tournament. Laver and Sergenti refine this menu down to just three rules. We take a further step toward integration by proposing one, all-purpose rule:

If the last move increased the candidate’s utility function, move a distance γ in the same direction; else, choose a random direction from the arc ±90° from the direction currently being faced, and move a distance γ in that new reverse direction.

This rule adapts the hunter rule proposed by Laver, but it has the advantage that it always leads to a candidate optimizing their utility function, no matter the set of $φ$ values. A candidate with $φ_{2} = φ_{3} = 0$ (i.e., a sticker), for instance, will always move back toward their ideal policy point if they ever venture away from it. Finally, when a candidate makes moves in the ideological space (a possibility that applies only to aggregators and hunters), we assume that they will move an identical and constant distance, γ, at each iteration.⁵

Model Outputs and Fit

The steps above are repeated until the last iteration. Every run will yield a different outcome due to the random component in the candidates’ movements. But as more and more runs are simulated, the ensemble average results, computed over all runs, will converge to a unique solution. Our preliminary testing showed that 2500 runs (similar to the 1000 runs obtained by Laver and Sergenti) is adequate to yield consistent results. These ensemble results can be used to assess the model’s fit against reality. There are four outcomes of particular interest. First, we would ideally like to compare the simulated time series of the candidates’ policy positions with the real-life evolution of their positions. Unfortunately, this data is not readily available (and would be difficult to estimate).

Secondly, we are interested in the time series of the candidates’ standing in the race, where the model output can be assessed against real-world polling data from the invisible primary. For this purpose, we compute an individual fit value $f_{j}$ for each candidate giving the average difference between candidate j’s polling and their simulated vote share at every time point; it will be low when this gap is small.⁶ By aggregating the individual $f_{j}$ values we can compute a global model fit statistic, F, that captures the overall correspondence between simulated and actual vote shares. Thus, F gives the average percentage difference, across all candidates and time periods, between the model projection and the actual race (as captured by polling data). We use this statistic to evaluate the performance of the model and, particularly, the extent to which it captures the evolution of competition. This outcome cannot be captured with the approaches to model fit used by Laver (2005) and Laver and Sergenti (2012).⁷

Thirdly, we consider the spread of primary voter support across competitors understood as the effective number of candidates (ENC), paralleling Laakso and Taagepera’s (1979) effective number of parties (ENP) measure of party system fragmentation. In our case, ENC will be high when there are many candidates that have similar levels of support, and low (and equal to 1) when support is concentrated on a few (or just 1) candidates. We calculate ENC for both the ABM’s simulated dynamics and the real-world dynamics revealed by the polls, so the difference between these series provides another check on model fit.

Fourth, since there has been debate about whether nominating races capture the range of intraparty views (Bartels 1988; Kamarck 2018), we analyze the representativeness of the competitive field (RP). This statistic captures the alignment of the candidates’ positions with the wider party’s preferences. Representation is better the smaller the distance between a voter and her preferred candidate. So RP, which may be shaped by the number of candidates, their ideological strategies and the presence/absence of momentum, is optimized when equal to 0. We can only simulate RP because we would need granular (and typically unavailable) information about the evolution of candidates’ positions to compute the real-world series.

Model Extensions

We consider four extensions, again amending procedures developed by Laver (and especially Laver and Sergenti), which make the ABM more realistic but also less parsimonious.⁸

Satisficing Hunters

Some hunters, rather than always trying to gain more voters, may be satisfied with a return above a certain comfort threshold, κ. For instance, a candidate may not pursue a vote-maximizing strategy in order to maintain “good” relations with other competitors. We can assume that such a candidate moves only if and when her vote share falls below the comfort threshold (i.e., $v_{p} < κ$ ). We test three $κ$ values (0.1, 0.25, and 1.0).⁹

Variable Mobility

The baseline model assumes that all candidates move the same distance at each iteration. We now consider the possibility that some candidates move further than others at each turn, by letting γ vary across candidates. We test three γ values (0.05, 0.1, 0.15), specified as a function of the voters’ standard deviation (VSD).

Mixed Strategies

Our definition of the utility function (Equation (1)) allows candidates to hold multiple motivations, instead of being purely vote-, base- or policy-oriented. We can therefore consider “ambitious” (i.e., vote-seeking) candidates who are willing to venture away from their ideal position to gain votes but only up to a point; they are not prepared to totally compromise their “authenticity” (i.e., policy goals).

Valence

The basic model excludes (or assumes that candidates are identical in terms of) charisma, competence and fundraising capacity. Taking account of a non-ideological, or valence (Stokes 1963), dimension means that a voter may prefer a candidate who is not ideologically proximate but is unusually trustworthy or charismatic. We assume that each voter prefers the candidate who minimizes $d_{i j} / λ_{j}$ , where $λ_{j}$ represents candidate j’s valence. In the case where $λ_{j} = 1$ for all candidates, the baseline ABM is retrieved. But when $λ_{j} > 1$ , a voter might prefer candidate j ahead of a more proximate candidate. We set Romney’s valence at “1” and other competitors at a level above ( $1 + Δ λ_{j}$ ) or below ( $1 - Δ λ_{j}$ ) his score, where $Δ λ_{j}$ ranges from 0.1 to 0.3. We assume that a candidate’s valence does not change during the race (though we later relax this assumption to analyze the impact of momentum).

Selecting a Working Model

This section compares several versions of the ABM to identify a best-fitting working model. To do so, we apply an evolutionary algorithm (EA) that tries out parameter settings for each model, progressively “mutating” the promising ones to identify those that generate the closest fit to real-world trends and therefore the lowest F score. The procedure is not an out-of-sample test of the models, but a trial-and-error process to exclude calibrations that are incompatible with real-world dynamics (see Bliss et al. 2014).¹⁰ It is similar to Laver and Sergenti’s approach—they manually select 1000 “tolerable” calibrations and identify the 50 best performers—but it allows for a more comprehensive and efficient testing of the range of parameter settings. Once the best-fitting model has been identified, it can then be analyzed in depth for substantive insights.

Table 2 summarizes the different versions of the ABM and results from tests of their performance. The only parameters in the baseline setup, Model 1a, are Laver’s pure strategies (sticker, hunter and aggregator), so the EA examines all the profiles that are possible within this set of strategies. In one of these profiles all the competitors are hunters, while others include candidates with different strategies. A particularly simple profile casts all candidates as stickers, which means there is no ideological maneuvering. This model performs uniquely badly, confirming that ideological movement is a driver of invisible primary dynamics. Among the more complex specifications, Model 1b includes the pure strategies but also a satisficing threshold, κ, so the EA searches for the best combination of strategies and thresholds. A third specification (Model 1c) includes a mobility parameter, γ; Model 1d includes a valence parameter, λ; and Model 1e allows candidates to adopt mixed strategies.

Table 2.

Model Fit Tests of the Baseline ABM and its Complex Variants.

	Model 1a (Baseline)	Model 1b	Model 1c	Model 1d	Model 1e
Parameters	Pure strategies	Pure strategies + satisficing threshold κ	Pure strategies + mobility γ	Pure strategies + valence λ	Mixed strategies
Model fit (F)	3.90	3.81	3.66	3.75	3.71

The F scores give the disparity between the simulated dynamics with the best version of a model and the real-world trends revealed in the polls. A high score (i.e., a large disparity) therefore indicates a weak model. As Table 2 shows, the baseline model performs worst among the setups, which is not surprising because it is also the simplest—we would expect model fit to only ever improve as parameters are added. More interesting, then, is the fact that the steps towards greater “realism” (i.e., the introduction of satisficing, mobility, valence and mixed strategies) have no more than a slight impact on explanatory leverage. Indeed, if we compare the baseline model to Model 1c, the best performing setup, the difference in their F scores is less than a quarter of one point. Thus, the models all generate decent approximations of the dynamics of the race, missing the mark by slightly less than 4 percentage points across all candidates and time periods. Moreover, the small difference between the models is not substantively significant because they produce similar practical results—for instance, Gingrich and Romney are always identified as hunters. These fit results do not change if the number of rounds (i.e., model ticks) is varied within a moderate range above/below weekly intervals, which, substantively speaking, would be tantamount to more/less frequent polls.¹¹

Figure 4 plots the simulated ENC and RP functions for the various models, giving a visual comparison of their performance. The similarity of their outputs is evident, which in turn suggests the particular merits of the parsimonious baseline model (Model 1a). The figure also shows that the ENC fit of all the simulations declines sharply two-thirds of the way through the race, during a turbulent spell from late August until November.¹² Before then, the contest had settled into a pattern in which Romney’s early lead was eroded by the other competitors, who gained in roughly even proportions from this trend, resulting in a steadily rising ENC function. Beginning in September, however, ENC begins to fluctuate a lot, reflecting the short-lived momentum experienced by several candidates, especially Perry, Cain and Gingrich. The baseline model underestimates these booms and therefore overestimates ENC. These surges all collapsed, so the turbulent period was over by November. It was followed by strong convergence again on Romney (and thus steadily declining ENC). The ABMs are able to capture the dynamics before and after the turbulent spell, but not what happens during it.

Figure 4.

Dynamics of ENC and representativeness in the ABM specifications.

As noted earlier, due to the absence of granular data on representativeness we are unable to compare the simulated RP functions against the real-world trend. However, we can make observations about their projection of RP dynamics. They indicate that RP increases over time as a result of the candidates’ ideological movement, but there are diminishing returns—the RP improvement is greatest early in the race and tails off in the middle of the contest. RP then declines toward the end of the invisible primary as several candidates drop out of the race. That these trends are replicated, nearly identically, across the five simulations indicates that the various model extensions have no impact on the depiction of representativeness.

Making Sense of the Turning Point: Strategy Change and Momentum

To account for the turbulent spell in the evolution of fragmentation, we explore the possibility that its dynamics reflected strategy change (e.g., from sticker to hunter) by one or more competitors. To do so, we revise Laver and Sergenti’s theorization of satisficing behavior. In their account, satisficing applies only to hunters so a candidate’s response to their vote share falling below the comfort threshold, κ, is always for them to relocate according to the hunter movement rule.¹³ Adapting this notion, we allow candidates to transition from one strategy to another, across all three strategies, once the comfort threshold has been reached.

So, how does strategy change affect the ABM’s performance? In short, it produces a favorable but insufficient response. The dynamics in the revised model resemble the direction of change in the polls, but the simulated transitions happen too slowly in comparison to real-world trends. For instance, the mid-September turning point plays out over 4–5 months in the simulations. Thus, we conclude that the turbulent spell is not well-explained by strategy change.

As an alternative, we consider the possibility that the race was transformed by a surge-and-collapse dynamic, similar to the idea of “momentum” that features prominently in race narratives (e.g., Heilemann and Halperin 2010; Sides and Vavreck 2014). A “surge” might be due to a candidate’s stronger-than-expected performance in a televised debate, while “collapse” might follow an embarrassing gaffe or poorly received interview or speech. To analyze this mechanism, we divide the race into 4-week periods, as shown in Figure 5. In each period, as a proxy for momentum, we increase the valence of the surging candidate from a baseline value of λ to λ + Δλ.¹⁴ Loss of momentum is modeled by a linear decrease and a return to the baseline valence value. As depicted in the figure, no candidate surges until July, when Bachman does and her valence increases by Δλ (= 0.5) over a 4-week period before collapsing. This is followed by periods of surge-and-collapse for Perry, Cain and Gingrich.

Figure 5.

Valence change as a proxy for momentum.

Valence change, our proxy for momentum, introduces a second dimension of competition into the ABM, alongside ideological repositioning. Indeed, in a race where all the competitors are stickers, valence change must account for all of the competitive dynamics. To investigate the effects of ideology and valence change, separately and jointly, we perform several additional EA simulations, which are summarized in Table 3. The baseline ABM (Model 1a), shown again for reference, allows ideological jockeying but not valence change. Model 2 is a simplified version of this baseline model with only stickers, so there is no ideological repositioning or valence change. Model 3 is also a static setup (i.e., all candidates are stickers) but it incorporates valence change. Model 4 is fully dynamic and incorporates both ideological jockeying and valence change.

Table 3.

Model Fit of Best Competitive Field under Ideological and Valence Models.

	Model 1a (Baseline)	Model 2 (Static)	Model 3 (Optimized static)	Model 4 (Optimized baseline)
Parameters	Pure strategies; no valence change	All stickers; no valence change	All stickers; valence change	Pure strategies; valence change
Bachmann	3.77	4.82	4.40	2.54
Gingrich	4.60	8.38	3.40	2.07
Huntsman	0.98	1.41	1.41	0.99
Paul	2.33	4.54	4.36	2.30
Romney	3.10	3.75	6.95	2.55
Santorum	3.59	2.90	3.02	2.49
Cain	4.83	4.96	3.77	3.34
Perry	5.86	29.98	6.50	2.58
Model fit (F)	3.90	11.30	4.55	2.43

Comparing these models using the EA, the static setup (Model 2) performs particularly poorly; though the large overall disparity is not evenly distributed among the candidates as the model gives a better account of the Huntsman and Santorum campaigns than it does of Perry’s campaign. The leverage gained by incorporating valence change is confirmed by comparing this model to the optimized static model (Model 3). Its model fit (F) score is approximately 7 points better than the performance of the simple static model, and it predicts each campaign more accurately with the additional leverage greatest for the candidates whose standing fluctuated a lot during the race (i.e., Gingrich, Cain and Perry). Yet it still performs worse than the baseline ABM. Given what these two models represent, this suggests that the race was shaped more by ideological maneuvering than by valence change. The optimized baseline model (Model 4) is the best performer, both overall and in its predictions of each campaign, which is what we might expect because it is the only setup that incorporates both ideological maneuvering and valence change. The overall prediction generated from this setup is approximately 1.5 points better than the Model 1a F score.

The red line in Figure 6 below shows the ensemble average vote share for each candidate as predicted by the optimized baseline ABM, in comparison to their vote share as revealed by the polls (black line). The shaded area around each red line gives the standard deviation of the simulated vote share, showing the variation obtained across 2500 runs of the model used to compute the predicted vote share. The figure confirms that the ABM generates decent predictions on the whole, but some campaigns are better projected than others. For instance, it provides a particularly good account of Huntsman’s consistently low-performing campaign while the dynamics of the Cain, Gingrich and Perry campaigns are captured with more noise.

Figure 6.

Candidates’ vote share dynamics in the optimized baseline ABM.

Figure 7 charts the simulated ENC dynamics obtained from the four models alongside the real-world trends captured by the polls. As can be seen, the performance of the static model (Model 2) declines as the race progresses. Though a better performer, the optimized static model (Model 3) consistently underestimates the fragmentation of the competitive field. The baseline model (Model 1a) performs well during the first half of the race but not in its second half. The fluctuations it misses are, however, well captured by the optimized baseline model (Model 4), as illustrated in Figure 6.

Figure 7.

Dynamics of ENC and representativeness in models with contrasting ideology and valence parameters.

Figure 7 also displays the evolution of representativeness. The static model (Model 2) predicts that representativeness will improve over time, until near the end of the race when it drops off due to the mass exit of losing candidates. This happens even though the competitors make no ideological moves, so this projection can be attributed to sorting by voters (Már 2020). Interestingly, the competitive field in this simplified static model is more representative than that of the optimized static model (Model 3). As the difference between these setups is the presence/absence of valence change, this comparison suggests that momentum can undermine representation, as Bartels (1988: 310) speculated. We can gain further insights into the effects of momentum by comparing the baseline and optimized baseline ABMs (i.e., Models 1a and 4)—their only difference is that the latter incorporates valence change while the former does not. Since the two models generate nearly identical RP values, this suggests that the candidates’ ideological maneuvering can offset momentum’s adverse effects on representation.

Figure 8 provides a candidate-level perspective on these observations. Its left panel shows the candidates’ final position in Model 1a, the baseline model (which does not have a valence parameter), and the right panel shows their position at the end of the Model 4 simulated race, which does include a valence parameter but is otherwise identical to the baseline model. As can be seen, the candidates take up very similar positions in both panels, which suggests that momentum does not affect the ideological configuration of the contest. This finding contrasts with the expectations that emerge from Groseclose’s (2001) model, which considers candidate location when one candidate has a valence advantage. In Groseclose’s model, the valence difference triggers a game of ideological pursuit and the candidates end up in positions that they wouldn’t otherwise have moved to. That such positional effects do not occur in our model may be due to the transient and short-lived nature of a candidate’s valence advantage in our depiction of the invisible primary.

Figure 8.

Valence change and the candidates’ ideological positioning.

Applications of the ABM

In this section, we use the optimized baseline ABM (Model 4) to identify candidate strategies in the 2012 race and predict its final outcomes with early-stage information. These have been challenging tasks for analysts of this contest and, more generally, in the study of nominating politics. Exploiting the flexibility of the ABM, we also explore the counterfactual question of how the race might have unfolded had there been a reduction in the number of candidates.

Mapping Ideological Strategies

Candidates are routinely described in ideological terms based on their policy proposals, employment history, and personal and professional associations. In 2012, Romney was widely regarded as a “pragmatist” or “moderate,” who was trying to boost his conservative credentials by revising his stances on abortion, same-sex marriage and immigration (e.g., Reinhard 2011). These were the “Faustian bargains that Governor Romney had to make in order to become the nominee,” explained David Axelrod (quoted in Jamieson 2012, 22). The purer conservatives in the contest, Bachmann, Gingrich and Santorum, attacked Romney’s ideological movement. For instance, Gingrich dismissed it as “pretense” and countered that Romney was “a Massachusetts moderate who has tax-paid abortions in ‘Romneycare’” (CBS News 2012). Just one candidate (Huntsman) was viewed as more moderate than Romney.

While ideology and ideological movement feature prominently in accounts of this race (e.g., Jamieson 2012; Sides and Vavreck 2014), they are not rigorously analyzed in these discussions. Our ABM contributes insights on this point. It suggests that most candidates—and not just Romney—were ideological movers. In fact, as Table 4 shows, only two candidates (Huntsman and Paul) are not hunters in the best competitive field. To give some indication of the robustness of this finding, Figure 9 shows the distribution of strategy profiles in the set of “plausible” competitive fields, defined as those that generate an F score within 0.5 points of the optimum field. An observation can be considered reasonably robust if it holds across the set of plausible fields and less certain if it does not replicate in this way. Based on this testing, Romney, but also Bachman and Gingrich, are, unequivocally, hunters. Likewise, there can be little doubt that Huntsman is a sticker. Paul is most probably an aggregator, but it would not be unreasonable to describe him as a hunter since he is identified as that type in a minority of the models. Cain is mostly identified as a hunter but sometimes as an aggregator. Perry’s movements are especially unclear as he is identified as each of the types in roughly equal proportions across the set of plausible fields.¹⁵

Table 4.

Candidate Strategies in the Best Competitive Field.

Candidate	Bachman	Gingrich	Huntsman	Paul	Romney	Santorum	Cain	Perry
Strategy	Hunter	Hunter	Sticker	Aggregator	Hunter	Hunter	Hunter	Hunter

Figure 9.

Candidate strategies in the plausible competitive fields (S: Sticker, A: Aggregator, H: Hunter).

The ABM also supplies insights into the candidates’ direction of travel in the ideological space. As Figure 10 shows, they generally moved from a peripheral position towards the party centroid, where the bulk of potential primary support is concentrated. This migration accounts for the improvement in representativeness over time (see Figure 7). These dynamics also advantaged Romney due to the weak competition on his liberal flank, where Huntsman is the only presence (and his position is both extreme and unchanging). The conservative side of the field is more crowded and, evidencing the heightened salience of cultural divisions (Blum 2020; Layman et al. 2006), the candidates here—Bachman, Gingrich, and Cain—moved primarily in the cultural dimension. Romney did too, traveling 0.5 points in a conservative direction in this dimension while remaining in the same economic position. The exception in this regard was Paul, who moderated his economic position as the race progressed.

Figure 10.

Evolution of candidates in the policy space (colorbar represents density of voters).

Our analysis also casts Perry’s candidacy in a new light. As noted earlier, his strategy is unclear because his movements are broadly consistent with all three types. Figure 10 shows that Perry was better positioned (in the sense of having greater vote-winning potential) than any of his rivals at the start of the race. This might explain why the media and Republican elites were initially impressed by his candidacy. It is possible, though, that this seeming advantage was a double-edged sword for Perry and his advisers as they would have faced a strategic dilemma. While hunter is clearly optimal for all the other candidates (assuming they want to win), in Perry’s situation a decent case could be made for another strategy, including staying put as a sticker. Furthermore, as Laver and Sergenti (2012, 94) show, a candidate occupying the central position in the policy space is always vulnerable when surrounded by hunters; the best locations in this scenario are typically nearby but not at the center.

Beyond the Invisible Primary

Forecasting models of nominating races use information from the end of the invisible primary and early in the official contest (e.g., New Hampshire primary) to predict the race winner and candidates’ vote share (e.g., Aldrich 2009; Dowdle et al. 2016; Mayer 1996; Steger 2007). As previously discussed, this work underscores the importance of money, media attention and elite endorsements, all of which can be secured before the official contest begins. Our ABM unpacks dynamics that can shape how these resources are distributed. And it does so with information that is available at the start of the invisible primary, several months before a prediction can be made with the conventional forecasting models. This difference suggests the potential of our model to make earlier predictions about primary season outcomes.

But how accurate are these ABM predictions? In making this assessment it should be noted that our model relies on fewer inputs than the conventional (statistical) forecasting models. The key indicator that it generates is the candidates’ vote share at the end of the invisible primary (VSIP), which is the endpoint of the simulated vote share dynamics. The traditional models also use VSIP but alongside other indicators such as cash on hand, money raised and elite endorsements, making our model look skeletal by comparison. Yet, despite this handicap, there are intuitive reasons to suspect that it should supply a reasonable forecast because VSIP has been shown to be a uniquely powerful predictor of primary season outcomes (Mayer 1996, 2003).

To explore the predictive power of our ABM, Table 5 presents the VSIP projections generated by the baseline and optimized versions of the ABM (i.e., Models 1a and 4), alongside the candidates’ actual vote share. The simulations anticipate the correct winner (Romney), but there is a large disparity between their projections and the actual vote share, which is a result of candidate exit from the race. This “attrition game,” Norrander (2000, 2006) explains, has its own distinctive logic and determinants. But these come into play as the primary season proper unfolds, at a time beyond the scope of our ABM analysis.

Table 5.

Comparing Simulated and Actual Vote Share.

	Baseline ABM	Optimized ABM	Baseline attrition ABM	Actual vote share	Early states vote share
Bachmann	11.3	11.7	—	—	—
Gingrich	12.9	12.9	25.5	14.20	21.8
Huntsman	2.5	2.5	—	—	—
Paul	11.6	11.6	18.4	10.89	11.2
Romney	29.0	28.5	36.3	52.14	40.7
Santorum	10.5	11.1	19.8	20.44	24.1
Cain	12.7	13.1	—	—	—
Perry	9.6	8.7	—	—	—

A more accurate interpretation of what our model can predict, then, is the vote share that would be observed in the absence of (or controlling for) attrition. To assess the ABM against this standard, we consider its predictive power if the set of late-stage candidates is correctly anticipated. The results from this test are shown in the third column of Table 5. It confirms that the baseline ABM now generates vote shares that are close to the actual results in the fourth column, with discrepancies of a similar size to those reported by high-performing conventional models (e.g., Mayer 1996). The model accurately predicts the highest (Romney) and lowest (Paul) ranked candidates, though it overestimates Gingrich’s vote share by 4 points and therefore misplaces him above Santorum.

Because a lot of activity takes place in the half-year between the end of the invisible primary and the end of the primary season proper, it is unsurprising that predictions from the ABM (which runs until the end of the invisible primary) are imperfect. Indeed, as we might expect, the model’s predictive power falls as the race progresses. This point can be shown by comparing the predictions of the baseline attrition model with the candidate vote shares at the end of the “early stage” of the race (i.e., on the eve of “Super Tuesday” and following thirteen primaries and caucuses). The discrepancies between these scores are even smaller than those seen in the comparison of the ABM with the observed vote share at the end of the invisible primary (column 4), confirming that the ABM works less well as a tool to predict the late twists and turns of the primary season proper.

Reducing the Number of Competitors

A feature of the 2012 Republican invisible primary was the large number of aspirants, which, coupled with weak initial enthusiasm for Romney, slowed the process of convergence on a candidate.16 One potential way to accelerate that process, and perhaps provide the eventual nominee with a head start in the interparty contest, would be to reduce the number of candidates. While the media have traditionally been a crucial winnowing mechanism (Bartels 1988), supplementary schemes have been proposed that would give party elites or donors control over debate slots (Koger 2019), and thus influence over who is considered a serious contender. Using our ABM we can explore how the invisible primary could have unfolded had the number of candidates been reduced prior to its start. We consider six scenarios with no more than four competitors: Romney and some/all of the three candidates, namely, Gingrich, Santorum, and Paul, who remained in the invisible primary until its end. The results from these simulations are shown in Table 6, alongside the results for these candidates in the optimized baseline (model 4) simulation with the full field (“all candidates”).

Table 6.

Candidate Vote Shares with a Reduced Number of Candidates.

Competitive field at race start	Romney	Gingrich	Santorum	Paul
All candidates	38.5	23.9	18.8	18.8
RGSP	41.8	24.1	17.9	16.2
RGS	50.1	30.1	19.8	—
RGP	47.4	33.9	—	18.7
RSP	51.7	—	27.8	20.6
GSP	—	45.3	29.1	25.6
RG	48.6	51.4	—	—

This analysis suggests that in a four- or three-horse race, Romney benefits from the exclusion of other moderate candidates and the division of support among the conservative candidates. He consistently secures a vote share exceeding his total in the simulated all candidates race and he wins a majority in two scenarios (Romney-Gingrich-Santorum and Romney-Santorum-Paul). These candidate reduction simulations also suggest that the conservative side of the party was handicapped by a failure to coordinate. A conservative candidate could have beaten Romney, but only in a two-horse race (Romney-Gingrich).

Conclusion

This article has developed an agent-based model of the invisible primary. Our depiction of this process is unusual in a literature dominated by statistical, formal and historical methods. For all the insights generated by this work, it has struggled to incorporate ideology into the analysis of candidate strategies and their competitive consequences. The flexibility of an ABM is ideally suited to this task. The starting point for our model-building was Laver’s ABM framework, which supplies a theorization of ideological strategies and procedures to model multiparty competition of the type often seen in the European democracies. We adapted this framework for the purpose of analyzing the American invisible primary and, ultimately, the competitive dynamics of our case study, the Republican race in 2012.

Our ABM analysis supplies novel insights into Romney’s victory in this race. We learn, for instance, that contrary to popular wisdom, it was not only Romney who made ideological moves in the contest—most of his rivals also moderated their initial position to boost their appeal. The fragmentation of the competitive field as this race progressed can be accounted for in a basic version of the ABM in which competitors select one of the three Laver strategies and then stick with that choice over time. Incorporating a momentum dynamic improves model fit in the short run (i.e., its ability to capture fluctuations during the race), though the state of play at the end of the invisible primary is predicted just as well in the basic model as it is in this momentum-enhanced version. Our model suggests that the competitive field became more representative of the party base as the race progressed.

Although most of the “realistic” complications of the baseline model (with the partial exception of valence change, our proxy for momentum) did not materially improve its performance, it would be interesting to test what leverage might be gained by taking account of campaign fundraising, media attention, elite endorsements and other factors that feature prominently in explanatory accounts of nominating politics. Future research might also compare multiple contests to understand how ideological strategies and their effects have evolved over time and in response to institutional and ideological change. It might be useful to explicitly integrate the two stages of the electoral process, to explore how ideological choices in nominating politics affect interparty strategies and outcomes. Finally, this study suggests the potential for machine-learning techniques, such as the EA, to complement analyses of political competition.

Supplemental Material

Supplemental Material—The Invisible Primary in an Agent-Based Model: Ideology, Strategy and Competitive Dynamics

Supplemental Material for The Invisible Primary in an Agent-Based Model: Ideology, Strategy and Competitive Dynamics by Zim Nwokora and Davy Brouzet in Political Research Quarterly

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

Author’s Note

Replication material available at

ORCID iDs

Zim Nwokora

Davy Brouzet

Supplemental Material

Supplemental material for this article is available online.

Notes

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