Exploring women's leadership labyrinth: Effects of hiring and developmental opportunities on gender stratification

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Exploring women's leadership labyrinth: Effects of hiring and developmental opportunities on gender stratification Hannah L. Samuelsona, , Benjamin R. Levineb, Sara E. Barthb, Jennifer L. Wesselc, James A. Grandd ⁎

a

Department of Psychology, 3102 Biology-Psychology Building, University of Maryland, College Park, MD 20742, United States of America Department of Psychology, 3129 Biology-Psychology Building, University of Maryland, College Park, MD 20742, United States of America c Department of Psychology, 3147F Biology-Psychology Building, University of Maryland, College Park, MD 20742, United States of America d Department of Psychology, 3147A Biology-Psychology Building, University of Maryland, College Park, MD 20742, United States of America b

ARTICLE INFO

ABSTRACT

Keywords: Female leadership Gender stratification Gender bias Computational modeling

Many factors have been proposed as potential causes for the underrepresentation of women in leadership positions. The present research leverages computational modeling and simulation to examine the impacts of external hiring and developmental opportunities, which may have consequences at different junctures in women's leadership labyrinth. Two agent-based simulations examined 1) the emergence of gender stratification in genderbalanced organizations and 2) the impact of reducing bias in external hiring and developmental opportunities in gender-stratified organizations. Results revealed that gender differences in external hiring heightened women's sense of tokenism and their turnover rates and that bias in developmental opportunities increased women's turnover rates due to a lack of promotions - a “sticky floor” effect. Further, improving women's leadership representation in gender-stratified organizations may be a rocky road – positive impacts were preceded by elevated turnover for women. Implications for organizational interventions are discussed. All code and datasets are available at https://github.com/grandjam/SamuelsonEtAl_GenderStratModel.

Introduction Exemplary women can be found in leadership positions of the largest U.S. companies. However, they remain far out-numbered by their male counterparts. Women only hold approximately 5% of CEO positions in either the Fortune 500 and S&P 500 lineups (Catalyst, 2018; Fortune 500, 2018). Underrepresentation of female leaders has been linked to a host of undesirable consequences for organizations and employees, including reduced satisfaction and perceptions of networking opportunities, diminished perceptions of organizational attractiveness to high-achieving applicants, and decreased diversity of ideas and organizational strategies (Adler, 2001; Bassett-Jones, 2005; Baumgartner & Schneider, 2010; Catalyst, 2004; Cooper Jackson, 2001; Fields & Blum, 1997; Ng & Burker, 2005). Unfortunately, unpacking the cause of female leadership underrepresentation has proven to be difficult. For example, women now account for 57% of the labor force; thus, disparity in base rates of potential male and female leaders does not offer an adequate account (U.S. Bureau of Labor Statistics, 2018). Furthermore, evidence suggests that the rarity of women in top-level

positions cannot be explained by inadequate leadership skills (Dezso & Ross, 2012; Pew Research Center, 2008; see also Eagly & Carli, 2003, and Eagly, Johannesen-Schmidt, & van Engen, 2003, for reviews). Gender stratification effects in leadership have most commonly been interpreted through a “glass ceiling” metaphor that draws attention to barriers impeding women's vertical movement up the organizational hierarchy (e.g., Baumgartner & Schneider, 2010; Hoobler, Wayne, & Lemmon, 2009; Ragins, Townsend, & Mattis, 1998). However, a key limitation of this explanatory lens is that it tends to presuppose the existence of such “top-down” barriers (i.e., glass ceilings exist that harm women's upward mobility) and directs comparatively less attention towards the processes by which stratified gender representation emerges as a function of experiences in the workplace. More contemporary accounts of gender stratification in leadership have begun to shift towards these more “bottom-up” accounts. For example, Eagly and Carli (2007) highlight differences in the career trajectories of men and women, noting that women's paths often follow more complex and circuitous routes than their male counterparts. Whereas men tend to climb the career ladder, women more often must navigate a career

Corresponding author. E-mail addresses: [email protected] (H.L. Samuelson), [email protected] (B.R. Levine), [email protected] (S.E. Barth), [email protected] (J.L. Wessel), [email protected] (J.A. Grand). ⁎

https://doi.org/10.1016/j.leaqua.2019.101314 Received 31 August 2018; Received in revised form 25 August 2019; Accepted 28 August 2019 1048-9843/ © 2019 Elsevier Inc. All rights reserved.

Please cite this article as: Hannah L. Samuelson, et al., The Leadership Quarterly, https://doi.org/10.1016/j.leaqua.2019.101314

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“labyrinth.” These and similar conceptualizations have begun to draw greater attention to the unique obstacles women face throughout the entire organizational entry-performance-promotion process that contribute to gender stratification in leadership. For example, perceptions that women are more likely to have greater work-family conflict or that being a woman is incompatible with being a leader can influence the social context of organizations in ways that create uniquely difficult obstacles for female employees to navigate on the path towards leadership positions (e.g., Agars, 2004; Cech & Blair-Loy, 2010; Hoobler et al., 2009; Pichler, Simpson, & Stroh, 2008; Reskin & McBrier, 2000). For women, the repeated impediments of these barriers can accumulate over time in ways that greatly reduce their likelihood of reaching senior leadership positions. For organizations, these barriers can manifest on a larger scale as the holding back and shutting out of valuable leadership potential. Consistent with this bottom-up lens and to further facilitate understanding of the emergence of female underrepresentation in organizational leadership, the current research focuses on the impact of two noteworthy barriers that women may face at different points in time while navigating their career labyrinths - leadership development and organizational entry. As we detail below, leadership development and organizational entry are frequent targets for intervention, though not always targeted simultaneously, when organizations seek to improve the representation of women in higher-level positions. Further, research on these targets tends to occur in isolation of not only one another, but also of other critical barriers in organizations faced by women. As such, this leaves gaps in our understanding of these factors, their underlying mechanisms, and their dynamics in organizations at large. In order to advance our understanding of the dynamics of gender stratification in leadership, develop more effective interventions, and better understand the potential effects of these intervention, the scope, complexity, and dynamism of the social system in which leadership development and organizational entry play out should be acknowledged. Thus, we employ computational modeling and simulation techniques to explore how these factors interact and give rise to gender stratification in organizations over time. A key advantage of our approach is the ability to represent and examine the impact of these factors across a wide range of conceptually and practically relevant circumstances that would be difficult or impossible to do using other methodologies (Davis, Eisenhardt, & Bingham, 2007; Harrison, Lin, Carroll, & Carley, 2007; Kozlowski, Chao, Grand, Braun, & Kuljanin, 2013; Wang, Zhou, & Zhang, 2016). Furthermore, this approach facilitates efforts to examine the extent to which barriers in leadership development and organizational entry result in unique patterns and explanations for female leader underrepresentation. We are thus also interested in examining whether these different barriers uniquely impact the ways by which potential female leaders become “lost” in the career labyrinth, potentially pointing to useful targets for future research and intervention. We begin by first reviewing the unique career barriers associated with leadership development and organizational entry for women and existing research on the relation these factors share with gender stratification in leadership. Next, we describe our computational model and two simulation studies designed to explore the impact of these factors. Results and insights gleaned from both simulations are summarized, followed by a discussion of implications from our work for understanding the emergent properties of female leadership underrepresentation.

male and female leadership representation. However, much of this research is correlational in nature and does not elucidate the processes by which these factors percolate throughout a dynamic organizational system to shape its gender composition. Second, we wished to focus on factors that were likely to occur at different points in the timeline of the employee-organization relationship. For a single employee, entry into an organization typically only happens once for a single employee, whereas this phenomenon occurs frequently across multiple people and time points from the perspective of an organization. In contrast, developmental opportunities likely to propel upward organizational mobility can presumably happen across numerous occasions during an employee's tenure. In this sense, these factors are likely to be important for both the initial conditions (i.e., organizational entry) as well as potential difficulty (i.e., leadership development) of navigating the leadership labyrinth. Lastly, we believe that indicators of both these factors often are (or could readily be) collected and monitored by organizational decision-makers. Although such statistics are seldom made public or accessible to researchers, making these focal inputs to in the present investigation opens the possibility for the model to be used in a more predictive/prescriptive manner to explore and communicate the impact of possible interventions aimed at improving female leader representation by researchers and practitioners. In the sections below, we provide a brief overview and discussion of existing work on organizational entry and leadership development as they relate to gender stratification among organizational leaders. Leadership development Many organizations use challenging job assignments to develop employees' leadership competence (Day, 2007; Dragoni, Tesluk, Russell, & Oh, 2009; Woodall, Edwards, & Welchman, 1997). Research suggests these experiences have numerous positive impacts on employee learning, evaluations, and advancement (Morrison & Brantner, 1992; Schmidt, Hunter, & Outerbridge, 1986; Silva, Carter, & Beninger, 2012). Diminished access to novel or challenging experiences involving high levels of responsibility (i.e., developmental opportunities) can negatively impact organizational decision-makers' perceptions of an employee's preparedness for leadership (Ohlott, 2004). Research further highlights that differences in the quality of developmental opportunities can lead to disparities in job knowledge and skill as well as recognition and status that facilitates promotion into leadership roles (e.g., McCauley, Ruderman, Ohlott, & Morrow, 1994). Although women and men generally report equal desire to receive high status/high accountability developmental opportunities, research suggests that women are often afforded less challenging, or less “mission critical”, opportunities than those received by men (De Pater, Van Vianen, & Bechtoldt, 2010; Silva et al., 2012). For example, King et al. (2012) found that although men and women reported receiving a similar number of developmental opportunities in their workplace, the prestige and visibility of the opportunities received by females were rated significantly lower than those received by males. Some have attributed these patterns to the persistence of stereotypical beliefs held about women that they should be protected, are less competent at agentic tasks, and thus are less able to succeed at challenging assignments (i.e., ambivalent sexism; Glicke & Fiske, 1999). While prior computational models have examined the effects of perceived gender differences in performance on gender stratification (e.g., Martell, Lane, & Emrich, 1996; Robison-Cox, Martell, & Emrich, 2007), they were not directly connected to developmental opportunities within a leadership context. Thus, we incorporate potential gender differences in the quality of developmental opportunities into our examination of the emergence of low female leadership representation.

Barriers to female leadership attainment Although there are myriad contributing factors and potential barriers to women's upward organizational mobility, we primarily focus on leadership development and organizational entry in the present work for three reasons. First, there is a robust body of both empirical and conceptual work discussing the significance of these factors to both

Organizational entry A second barrier faced by women concerns disproportionate rates of 2

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entry into open organizational positions relative to men. There are numerous reasons why gender imbalances in organizational entry could exist. Some industries possess highly skewed applicant pools that tend to perpetuate this imbalance over time. For example, referral applicants (i.e., applicants referred/encouraged to apply for a job opening by someone in an organization) tend to be of similar demographics as the majority of employees in the hiring organization (Taber & Hendricks, 2003). Furthermore, these individuals are more likely to be placed among the top-ranked candidates competing for an opening relative to non-referral applicants. Thus, if an organization is predominantly male, their applicant pool and hires often remain majority male (Fernandez & Mors, 2008; Fernandez & Sosa, 2005). Selection biases may be another reason for disproportionate hiring rates of men and women. For instance, there is evidence that women are hired less often when selection criteria emphasize stereotypically masculine characteristics (which also tend to be associated with desired leadership qualities) and when men are making the hiring decision (Bosak & Sczesny, 2011; Gorman, 2005; Isaac, Lee, & Carnes, 2009). In the present work, we do not provide an account of the specific process mechanisms that may underlie particular access or selection biases. Instead, we examine organizational systems in which organizational entry for women is more or less proportional to that of men to explore its relative impact on female leadership representation. In line with the “labyrinth” approach to this issue, we acknowledge that external hiring may not only impact women as they seek organizational entry, but also as they try to navigate into leadership positions once in the organization. Specifically, bringing more males into an organization may increase the likelihood that women face negative experiences that accompany being a minority in the workplace, a phenomenon referred to as tokenism. Tokens are members of a numerical social minority in which the numerically dominant members control and shape the culture of a group (Kanter, 1977). Token women in work environments often perceive their organizational climate as inequitable to women (King, Hebl, George, & Matusik, 2009), and report feeling less satisfied with their job, excluded from important networks, and higher levels of job-related depression and lower self-esteem (Krimmel & Gormly, 2003; Maranto & Griffin, 2011). One significant consequence of tokenism for females is an increased likelihood of leaving an organization (King et al., 2009). In contrast, when women are more represented in a given organizational level, they report significantly lower intentions of turning over (Elvira & Cohen, 2001). This suggests that the impact of organizational entry is likely to extend beyond simply whom enters the organization and may also have downstream consequences on whom exits the organization. In summary, the present work integrates how differences in developmental opportunities and external hiring—two key barriers in the leadership labyrinth faced by women in their everyday work experiences (Eagly & Carli, 2007)—may unfold and interact over time to impact the representation of women in the leadership positions of an organization's hierarchy. As stated previously, we acknowledge that other factors also operate in these environments that may hinder women's access to higher-level positions. An advantage of developing a formal model of this process is the capacity to account for and represent some of these systemic and idiosyncratic experiences so as to better contextualize the impact of core model mechanisms. We briefly discuss some of the additional mechanisms we elected to include in the present model in the description below and in Appendix A.

Table 1 Pseudocode for computational model and simulation of gender stratification in organizational leadership. Step

Action

1 2 3 4

Initialize time clock T = 0 Create organizational structure and populate with initial employees Increment time clock T = T + 1 Assign developmental opportunities and determine which employees take assigned opportunities based on risk-taking propensity Calculate base performance score, add opportunity values to employees' performance scores, accumulate total performance scores If remainder of T/12 ≠ 0, return to Step 3 Assign career delays, deduct performance rounds from delay takers, and assign turnover to specified percentage of delay takers Update employee tenure at level and age, calculate likelihood of turning over due to level tenure, age, and tokenism (for women only) Invoke voluntary turnover based on total turnover likelihood Fill specified percentage of open positions with external hires Promote employees into remaining open positions Fill open positions in lowest level of organization with external hires If the number of original employees is > 0, return to Step 3 End

5 6 7 8 9 10 11 12 13 14

Note. T = time period.

organization and incumbent employees are evaluated for promotion. This performance-turnover-hiring-promotion loop iterates until the organization consists of no original employees, at which point the final distribution of men and women in the organization is recorded. A more detailed account of these processes follows.

Organizational structure and agent characteristics All simulated organizations contained eight hierarchical levels, with Level 1 representing the lowest/entry-level positions and Level 8 representing the highest/CEO position in the organization. For descriptive purposes, Levels 1 through 4 were collectively considered the lower levels of the organization and Levels 5 through 8 the upper levels (Fig. 1). Each organization consisted of 30,600 positions plus the CEO, a

Computational model description The algorithmic steps and procedural rules describing the logic and flow of activities in the current model are provided in Table 1. As a brief overview, a hierarchically-structured organization is initialized and populated with employees. The accumulation of monthly employee performance is then simulated, and at the end of a 12-month cycle (i.e., one year), employees turn over. New employees are hired into the

Fig. 1. Organizational structure. Note. Numbers within each level signify the number of employees at each level. In all simulations, the number of line versus staff positions was equal at each level. At the beginning of each simulation, the gender breakdown within line positions favored males (70% male) while the gender breakdown within staff positions favored females (70% female). 3

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similar size to that found among major Fortune 500 companies (Fortune 500, 2018). Positions below the CEO were equally designated as line or staff, representing positions that tend to directly impact an organization's core and those that tend to be more supportive in nature, respectively (Hellriegel, Jackson, & Slocum, 2002). Although industries may vary by the percentage of women and men in line and staff positions, the currently simulated organizations were initialized with 70% of the staff positions assigned to women and 70% of the line positions assigned to men to generally reflect prior research findings (Catalyst, 2007). The distribution of men and women in line and staff roles was free to vary as organizations developed and employees were internally promoted and externally hired into positions. In Simulation 1, the gender composition of the organizations was initialized as 50% male and 50% female in order to examine the emergence of gender stratification. In Simulation 2, the gender composition of the organizations was set to equal the gender composition of organizations in their final year from Simulation 1. In addition to gender, agents were assigned an age, a risk-taking propensity, and an ability level. Agent age was randomly sampled from a Poisson distribution with a mean of 35 years for lower-level positions and a mean of 55 years for higher-level positions, corresponding to reported age differences among lower-level employees and executives in U.S. companies (U.S. Department of Labor, Bureau of Labor Statistics, 2018). Agent risk-taking propensity represented the likelihood of accepting a developmental opportunity if offered and was thus operationalized as a probability value between [0, 1]. To reflect the slight gender difference in risk taking reported in prior research (Byrnes, Miller, & Schafer, 1999), 55% of men and 50% of women were randomly assigned propensities above 0.50 (i.e., more likely to take a risk than not), and the remaining below. Lastly, each agent was randomly assigned an ability level from a normal distribution with a mean of 100 (SD = 15), which was used to determine how well they performed each month.

At the end of each year in the simulation, a percentage of agents were designated as having taken a career delay during that year. This percentage and the duration of a career delay were based on archival data on family and medical leave practices in the United States (Klerman, Daley, & Pozniak, 2012). Table 2 provides the percentage of agents taking leave for any reason, broken down by age and gender, used to inform the simulation. The typical leave duration was set to one month, reflecting the average leave time among working American men and women (27.7 days) for any reason. However, women who take leave for childcare reasons tend to take nearly two months leave (57.5 days), while men take closer to one month (21.8 days). Thus, female agents designated as having specifically taken parental leave (3.9% of female leave-takers; Klerman et al., 2012) were flagged as absent from the organization for two months.2 Since agent's monthly performance is randomly sampled, it was not necessary to distinguish which specific month(s) an agent took leave. Consequently, the final month's performance (or final two months for female parental leavetakers) was removed from a leave-taking agent's annual performance score to account for periods during which they were not working. Turnover After each year, a percentage of agents voluntarily left their organization. An agent's decision to turn over from their organization was influenced by a number of factors. First, 10.1% of agents who took any type of career delay in a given year did not return from leave (FML Report, 2012). The likelihood of turning over for all remaining agents was operationalized as a function of an agent's age, tenure at the agent's current level (i.e., the number of years in a given position), and, for female agents only, the extent to which they were a token. Appendix B provides a more detailed description of the formulas, parameterizations, and computations for agents' turnover likelihood. Drawing from existing research, turnover probability was operationalized to increase most rapidly when an agent reached retirement age (65 years old) and when they had been in the same level for 14 years (Lyness & Judiesch, 2001; Quarles, 1994; Weng & McElroy, 2012). With respect to tokenism, Kanter (1977) originally identified a skewed ratio of 85 majority to 15 minority members (i.e., tokens) as the situation in which minority members “become symbols rather than individuals,” facing negative consequences such as increased performance pressure and role entrapment. At a ratio of 65 majority to 35 minority members, Kanter (1977) suggests minorities are able to begin creating alliances with one another and affecting the culture of the group, decreasing the negative effects associated with tokenism. Based on this formulation, the turnover likelihood for female agents was made to increase as the proportion of other women in their workplace decreased. Specifically, each agent was tied to a group of 100 other agents in their level, representing the set of employees an individual employee would theoretically interact or come into contact with in an organization of the size modeled here in a given year. A female agent's likelihood of turning over due to tokenism most rapidly increased when the group was 15% female (Kanter's, 1977, proposed skewed ratio). Based on these inputs, a turnover likelihood ranging from [0, 1] was computed for each agent at the end of each simulated year, and agents departed from the organization with a probability equal to that likelihood.

Performance Agents accrued performance over a 12-month cycle (i.e., one year). At the start of each month, both male and female agents had a 50% chance of being offered a task designated as a developmental opportunity. Agents offered an opportunity chose to complete it at a rate equal to their risk-taking propensity.1 To represent its greater value compared to a typical job activity, performing a developmental opportunity task earned an agent additional performance during that month. The value for any completed opportunity was determined by randomly sampling from a truncated normal distribution bounded between 15 and 150. To reflect the less visible and “mission critical” opportunities given to women compared to men (King et al., 2012), the mean of the normal distribution from which men's opportunity values were drawn was always centered around 82.5, whereas the mean of the normal distributions from which women's opportunity values were drawn differed depending on the simulation condition, as discussed below. Both genders' opportunity value distributions had a standard deviation of 20. The additional points earned by completing a developmental opportunity were added to the agent's monthly performance. The normal performance for all agents was determined by randomly sampling a value from a normal distribution with a mean equal to the agent's ability level (SD = 15).

Hiring and promotion Following agent turnover, open positions were either filled via external hiring or internal promotions. External hiring occurred prior to

1 Given this operationalization and the fact that risk-taking propensities were static over time in the model, the expected percentage of male and female agents performing a developmental opportunity at each time point could be computed a priori. More specifically, (0.55 ∙ 0.5 ∙ (0.51 + 1)) = 41.53% of men and (0.5 ∙ 0.5 ∙ (0.51 + 1)) = 37.75% of women would be expected to take developmental opportunities each month under this parameterization. The impact of the small gender difference in risk-taking on the number of developmental opportunities accepted by male and female agents was thus small and consistent with previous empirical results (King et al., 2012).

2 The technical definition of “day” in the FML Report (2012) is unclear. For the current simulation, “day” was treated as a calendar day, such that 57.5 ÷ 7 = 8.21 weeks, or approximately 2 months of leave for women taking parental leave. We acknowledge that this may be the more conservative calculation compared to treating “day” as a work day, such that 57.5 ÷ 5 = 11.5 weeks, or approximately 3 months.

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affords the opportunity to explore how the focal barriers in organizational entry and developmental opportunities may unfold within the context of and interact with these dynamics and influence the representation of females in organizational leadership positions. To examine these dynamic processes, we conducted two simulations, each. consisting of 10,000 organizations encompassing over 306 million simulated employees. The goal of the first simulation was to isolate the impact of the two focal barriers on the emergence of gender stratification and the intra-organizational dynamics that maintain it. This was accomplished by introducing varying levels of bias in organizational entry and developmental opportunities into initially gender-balanced organizations. The goal of the second simulation was to examine the consequences of removing these biases and the manner by which the gender composition of organizations with varying degrees of existing gender stratification react. Together, these two simulations facilitate identification and description of the bottom-up processes that aid in the explanation of low rates of female leaders, ultimately pointing to future research and the development of interventions to continue to combat gender stratification in organizational leadership.

Table 2 Rates of male and female leave-taking (FML Report, 2012). Age (years)

Gender

Average % taking leave per year

≤33

Male Female Male Female Male Female

10.2 15.3 11.5 15.2 12.5 14.8

34–49 ≥50

internal promotions in order to avoid the potential for all employees remaining in a given level post-turnover to be promoted into the next highest level as an increasing number of positions are opened due to the promotion process cascading down the organizational hierarchy – an unrealistic representation. Macro-level examinations of labor markets and organizational hiring practices note that a larger proportion of openings in lower-level organizational positions tend to be filled by external hires relative to openings in higher-level positions, though a non-trivial proportion of upper-level openings are still filled with employees external to the organization rather than through internal promotion (e.g., Dohmen, Kriechel, & Pfann, 2004; Fernandez & Abraham, 2010; Treble, van Gameren, Bridges, & Barmby, 2001). To represent these circumstances, 60% of the open positions in Levels 2 through 4 and 40% of open positions in Levels 5 through 8 were filled via external hiring (Fernandez & Abraham, 2010).3 The age and risk-taking propensities of all external hires were assigned using the same procedures as during the initialization of each organization. The ability level of new agents was determined by sampling a random value from a normal distribution centered at the mean ability of the level into which the new agent was being hired (SD = 15) to reflect a new hire possessing performance potential commensurate with existing agents at that level. The probability that an external hire was male or female was a free parameter and manipulated based on the simulation condition. All remaining open positions were filled internally through promotions in a cascading fashion, starting with higher-level openings, continuing through the lower levels. Agents were only eligible for promotion if they had held their position for at least one year. The promotion process occurred by rank ordering agents in the level immediately below the level to be filled based on their accumulated performance from the previous year. Those with the highest scores were promoted until all open positions were filled. Finally, Level 1 openings were filled via external hiring using the same procedure described above. The simulation repeated this basic performance-turnover-hiringpromotion cycle until no original agents remained. This stopping rule allowed sufficient time for the organizational dynamics to unfold under different conditions and meaningful observations about their impact to be evaluated. The computational model and computer simulations were programmed and performed in R (R Core Team, 2017). All code necessary to run the model, replicate the reported simulations/analyses, and final datasets are available at https://github.com/grandjam/ SamuelsonEtAl_GenderStratModel.

Simulation 1 Organizations in Simulation 1 were comprised of 50% male and 50% female employees to evaluate the effects of male-female differences in organizational entry and leadership development under conditions of gender parity. The values for these parameters in each organization were determined by randomly sampling values in a fully crossed Monte Carlo design. External hiring rates were randomly selected from a uniform distribution bounded by 20% (i.e., 20% of external hires were male) at the lower end and 80% at the upper. Gender differences in opportunity values were operationalized as an effect size representing the difference between the center of the distributions from which male and female opportunity values were sampled. King et al. (2012) report that men's developmental opportunities may be valued by as much as one standard deviation more than women's; thus, opportunity value difference effect sizes were sampled from a uniform distribution bounded by 0 (i.e., no gender difference) and 1.0 (one standard deviation, or a 20-point difference). This translated to opportunity value distributions for female agents' whose means varied between 62.5 points (1 SD difference) and 82.5 points (0 SD difference) compared to the constant mean of 82.5 points for male agents. Simulation 2 Organizations in Simulation 2 were initialized to replicate the gender representation and agent characteristics of organizations at the conclusion of Simulation 1 (i.e., after all original agents had been replaced). In other words, the final organizational composition and agents at the end of Simulation 1 organizations were used as the starting conditions for Simulation 2. To reflect the removal of bias in the two primary barriers of interest, the external hiring rate was set to 50% male (i.e., equal hiring) and the opportunity value difference effect size to 0 (i.e., equal opportunity values) for all Simulation 2 organizations.

Simulations

Outcome variables and analysis plan

The computational model described above offers a representation of the normally occurring performance/promotion, turnover, and hiring dynamics that take place in an organizational environment. Thus, it

The primary outcome measure recorded in both simulations was the percentage of simulated male and female employees within each organizational level at the end of each simulated year. Given that the manipulated parameters accounted for essentially all between-organization variance, no inferential statistical tests were conducted. Instead, the analyses for addressing our primary interests proceeded in two stages. First, the combined effect of the developmental opportunity and external hiring barriers in Simulation 1 and their removal in Simulation 2 was explored by examining how the percentage of female employees was affected at different organizational levels across different

3 While the overall percentage of external hires at any given level may differ based on labor market and industry, we intended to capture the relative difference between the external hiring rates of lower and upper levels of organizations. Different percentages that reflect different labor market forces could result in different gender stratification dynamics; however manipulating and examining these particular parameters were beyond the scope of the current simulations.

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combinations of these parameters. A significant advantage of studying complex emergent phenomena using computational modeling and simulation is the potential to drill down into observed effect patterns and tease apart how particular factors operate to engender particular outcome patterns. Consequently, the second stage of our analyses pursued a more detailed examination of how the two factors differentially affected male and female agents across conditions and gave rise to observed patterns of gender representation in leadership. Because the CEO level (Level 8) contained only a single employee and was therefore prone to substantial random variation across simulation runs, only model results from Levels 1 through 7 of the organizations are presented. Color version of all plots can be accessed through the GitHub repository for this paper (https://github.com/grandjam/ SamuelsonEtAl_GenderStratModel).

example, even in conditions when only 20% of the external hires into an organization were male, the largest discrepancies in women's developmental opportunities were associated with up to a 50% decrease in the final percentage of women in Level 7 beyond what would be expected based on the external hiring rate for females (i.e., 80% of external hires are women, but only 40% of top leadership positions held by women). From an intervention perspective, these results demonstrate that although an absolute increase in the representation of female leaders could be attained by increasing the number of female hires, doing so in the presence of developmental opportunity differences is akin to simply “pouring more water into a leaky bucket.” Organizational entry As noted above, organizational entry rates exerted a direct relation to the flow of employees into organizations; what is less transparent is that this mechanism also exhibited a sizable impact on the flow of employees out of organizations. In the present model, this pattern emerged as the number of male agents in an organization reached the point at which female agents increasingly found themselves in the numerical minority (Kanter, 1977). This experience of tokenism intensifies as females become increasingly under-represented, resulting in a higher propensity for female agents to perceive their organization's climate as inequitable to women and ultimately turn over (King et al., 2009). Fig. 3 illustrates this basic effect and shows how female agents' likelihood of turning over due to tokenism changed with respect to the percentage of males hired into the organization. When male agents were disproportionately hired, the impact on female turnover propensity was robust and consistent across all non-entry levels of the organization. Notably, the overall pattern exhibits characteristics of a (bounded) positive feedback loop in which hiring more males into the organization resulted in decreased female representation, which caused even more females to depart. A similar though more subtle effect on female turnover propensity was observed even when equal numbers of male and female agents were hired into the organization, but only at the highest leadership position. The explanation for this effect is more complex, but emerges as a result of less upward mobility by female agents through the organization (described further below) coupled with

Results Simulation 1 To examine the extent to which gender stratification in organizational leadership was influenced by differences in organizational entry and developmental opportunities, Fig. 2 presents the proportion of women in the final year of each organization in Simulation 1 for each organizational level. Across all levels, an increase in the percentage of men hired into an organization was strongly associated with a decrease in the final percentage of women in any given level, thus reflecting the relationship between the gender composition of an organization's new hires and its final gender composition. While this relationship may seem obvious, Fig. 2 also demonstrates that the representation of females at each organizational level could vary significantly from the hiring rates as a function of differences in developmental opportunities for males and females. More specifically, decreases in the value of developmental opportunities received by female agents were associated with a notable decrease in the overall number of females in each organizational level beyond the first two levels. Furthermore, the size of this disparity increases at successively higher levels of the organizational hierarchy and persists even in situations in which women were hired into the organization at significantly higher rates than men. For

Fig. 2. Final percentage of female agents in the final year of Simulation 1 across organizational levels 1 through 7. Note. Red line with a slope of −1 represents inflection point for equal representation at each percentage of male external hires. 6

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Fig. 3. Female agents' average likelihood of turning over due to tokenism in the final year of Simulation 1 at organizational levels 1 through 7. Note. In Level 7, the number of agents used to calculate women's likelihood of turning over due to tokenism is equal to the size of the level. Thus, there is no variance in this outcome within organization. Further, the rate of women's likelihood of turning over due to tokenism increases at the fastest rate with each 1% decrease in the number of other women between the likelihoods of approximately 1% and 9%. This results in the banding seen in Level 7 in the Fig. 3.

an increasingly smaller number of employees at higher levels that magnifies the cumulative effects of gender composition differences. Thus, even when the number of males and females entering into an organization were equal, if the movement of males and females through the organization (i.e., up the hierarchy) was not similar, gender imbalanced workgroups and experiences of tokenism emerged that tended to drive women out of leadership positions.

number of promotions received by male and female employees in the lower levels of the organization across differing degrees of developmental opportunities (note that Level 1 is excluded since all agents in this level were external hires and therefore had not yet had an opportunity to earn promotions). In general, these results reveal that men tended to accumulate more and women fewer promotions on average as the size of the gender discrepancy in developmental opportunities increased. For example, the results shown for Level 5 reveal that the typical male agent at this level in organizations with the largest developmental opportunity differences had already been promoted between 2 and 3 times at that stage in their career. In contrast, the typical female agent at this level in these same organizations was more likely to have been promoted only 1–2 times. Consequently, the overall pattern of findings revealed that men

Developmental opportunities Like external hiring, differences in developmental opportunities also impacted the departure of female agents from organizations. This departure, however, was primarily caused by a lack of promotions received by women rather than the changing demographics of the organization. This effect is illustrated in Fig. 4, which presents the average

Fig. 4. Male and female agents' average promotions earned by the final year of Simulation 1 across organizational levels 2 through 5. 7

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Fig. 5. Male and female agents' average likelihood of turning over due to level tenure in the final year of Simulation 1 in organizational levels 2 through 5.

tended to ascend the organizational hierarchy via internal promotions more quickly than women and that this effect was amplified by larger gender disparities in developmental opportunity values. Notably, the lack of promotions received by female agents increased their likelihood of being held back in lower levels of the organization for longer periods of time relative to men. This phenomenon, described by some gender researchers as the “sticky floor” effect (e.g., Booth, Francesconi, & Frank, 2003; Yap & Konrad, 2009), further increased female agents' likelihood of leaving the organization before reaching the upper levels of the organization (Fig. 5).4 Though the process described above reduced the number of females reaching the upper levels of the simulated organizations, a percentage of the available positions in Levels 5 through 7 were still occupied by women each year even in organizations with heavily male-favored developmental opportunities (see Fig. 2). This perhaps suggests that a few talented (or lucky) females managed to overcome differences in developmental opportunities to successfully navigate the leadership labyrinth. However, an examination of the women that reached the higher organizational levels reveals that they attained those positions primarily through externally hiring rather than promotion. Fig. 6 shows the percentage of positions in Levels 5 through 7 that were held by agents who had been promoted from lower levels versus those who were externally hired into those positions. These results show that the percentage of men who reached upper-level leadership positions through promotion tended to be greater than women when external hiring rates were equal or favored men and that this difference increased up the organizational hierarchy. For example, only 32.63% of the women that ended up reaching Level 7 did so as a result of rising through the ranks via internal promotions, compared with 49.75% of men.

Fig. 6. Average percentage of male and female agents promoted versus hired into upper organizational levels in the final year of Simulation 1. Note. Data includes organizations with external hiring rates of 50% or greater male.

ripple throughout an organization via both structural and social mechanisms in ways that contribute to female underrepresentation in upper level organizational positions. Notably, although increases in the proportion of females hired into an organization—even to the point of hiring disproportionately more females than males—could attenuate experiences of tokenism and reduce female turnover, they were not sufficient to fully buffer against potential differences in the quality of developmental opportunities that reduced upward mobility for females. The presence of these disparities increased—in some cases dramatically—the likelihood that women would be pushed out of their organizations either due to negative consequences of the demographic makeup of their workgroups or decreased access to internal promotions before they could reach leadership positions. Furthermore, females seldom reached the upper levels of an organization by climbing the corporate ladder; instead, these positions were more likely to be attained by a woman being externally hired into these positions. These

Summary of simulation 1 Overall, the results from Simulation 1 revealed how gender differences in organizational entry and developmental opportunities can 4

By the time agents reached Level 5, age often became a predominant factor in their likelihood of turning over before they had been in the same level for enough years for the likelihood of turning over due to level tenure could become a significant impact. Thus, opportunity value differences played less of a role in men's and women's turnover likelihoods due to level tenure in Level 5. 8

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Fig. 7. Percentage of female agents across time in Simulation 2 across organizational levels 1 through 7. Note. Data includes organizations with an initial gender breakdown of 49% or greater male. Red line represents equal representation of male and female agents (50–50%). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

overall pattern of findings thus provide a descriptive account of the unique complexities and challenges involved in women's efforts to navigate the career labyrinth towards organizational leadership positions.

Table 3 Average female representation in simulation 2 after stabilization.

Simulation 2 Whereas the previous simulation results provide insight into how gender stratification can emerge as a result of organizational entry and developmental opportunity differences in organizations that begin with perfect gender parity, Simulation 2 examines how removing these barriers (i.e., hiring equal proportions of male and female employees and no gender differences in development opportunity values) impacts gender stratification in organizations where imbalances in leader representation already exist. As depicted in Fig. 2, a proportion of organizations in Simulation 1 were majority female by the final year of the simulation and thus were initially female-dominated in Simulation 2. However, because Simulation 2 was intended to examine the removal of barriers in gender imbalanced organizations and the reality of leadership representation today is male dominated (Catalyst, 2018; Fortune 500, 2018), we focus only on organizations that begin as majority male in Simulation 2. Fig. 7 provides a high-level summary of these results and plots how the proportion of women in each organizational level changed over time once organizational barriers were removed. By the end of a single generation of employees (that is, after all original employees had turned over), women represented an average of 48.05% (SD = 2.54) of the workforce across all organizations and organizational levels. The slightly lower proportion of female relative to male agents was attributable to the additional factors that remained in organizations which still disproportionately affected female agents (i.e., gender differences in career delays, risk-taking propensity, and tokenism). Despite this near gender parity, a number of noteworthy patterns in the changes of organizational demographics following the removal of organizational entry and developmental opportunities differences were observed. For example, although the average percentage of women was relatively equal across all organizational levels once the change patterns had stabilized, the standard deviation in female representation tended to increase at higher organizational levels (Table 3). The result of this effect was that the male-female distribution in the upper organizational levels still tended to (slightly) favor male representation.

Level

M (%)

SD

1 2 3 4 5 6 7

50.43 48.90 48.34 48.05 46.94 46.58 46.30

0.29 0.43 0.66 1.09 1.58 2.27 5.00

Note. A cut-off of 25 years was used to determine when an organizational level stabilized. This cut-off was the approximate number of years required to reach gender parity in Level 7 – the maximum required of any given level.

Furthermore, the return to near gender parity was a slow one in most organizations. For example, an average of 23.39 years (SD = 8.22) was needed before near gender parity was first achieved in Level 7 across all simulated organizations.5 As one might expect, organizations that initially contained more males required more time to reach this balance; for example, the correlation between an organization's initial percentage of men and the time required to reach gender parity in upper organizational levels was r = 0.43. In summary, a large percentage of organizations were still male-dominated by the end of Simulation 2, it took a considerable length of time to reach gender parity for those that did, and the extent to which an organization was initially male-dominated impacted their speed of “recovery.” The dynamics of women's navigation into leadership positions within organizations from which barriers in external hiring and opportunity differences were removed was similarly complex. The Simulation 1 results revealed that developmental opportunities were tied to female agents' promotability and external hiring tied to female agents' turnover via their experiences as tokens (Kanter, 1977; King et al., 2009). These dynamics also provide insights into the reversal, rather than emergence, of gender stratification in Simulation 2.

5 Due to the continuous nature of the outcome of gender breakdown, we considered parity to be between 49 and 51% female.

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MF = 0.10 (SD = 0.10) in Year 40) and variance in the number of promotions (SDYear 1 = 0.25, SDYear 40 = 0.07) earned by men and women substantially decreased over time. In sum, completely removing gender differences in developmental opportunities greatly closed the gap in upward organizational mobility for females. Turnover In organizations' final years in Simulation 1, female agents had an increased likelihood of turning over due to tokenism as the male external hiring rate (i.e., bias in external hiring) increased. Tokenism, however, is not solely tied to the gender make-up of external hires, but more directly to the gender make-up of all employees in the organization. Thus, while women's tokenism experience should be expected to decrease over time as more women enter the organizations via a gender-balanced hiring rate, women may still be impacted by tokenism in male-dominated organizations before that shift takes place. Fig. 9a depicts the percentage of women who turned over from each level of the organizational hierarchy over the entire course of Simulation 2. Whereas female agents' turnover in organizations that were more gender balanced remained relatively constant, a large proportion of female agents were still departing the organizations the male-dominated organizations even after female barriers to advancement were removed. This was especially visible in the upper-most levels of organizations. For example, 18.90% (SD = 11.74) of women in Level 7 turned over from male-dominated organizations compared to 16.11% (SD = 6.22) in gender-balanced organizations in Year 1. By the end of the simulation, the female turnover rate in Level 7 of initially maledominated organizations reached 14.87% (SD = 5.22), which was largely comparable to the relatively stable rate in gender-balanced organizations (M = 13.70, SD = 5.48). These shifting turnover dynamics meant that the percentage of women who turned over in Level 7 in male-dominated organizations eventually reached levels equal to the turnover percentage of their male counterparts by the end of Simulation 2, but it took many years for this equilibration to be reached. In sum, although things ended up improving substantially for female agents once barriers were removed, times still appeared comparatively rough for females during the early years of this transition. (See Fig. 9b.)

Fig. 8a. Male and female agents' promotions earned across time in Simulation 2 across organizational levels 2 through 5. Note. Data includes organizations with an initial gender breakdown of 49% or greater male.

Fig. 8b. Male and female agents' average promotions earned in years 1, 20, and 40 in Simulation 2 in level 5. Note. Data includes organizations with an initial gender breakdown of 49% or greater male.

Promotions Research suggests that developmental opportunities have positive impacts on employees' promotability (Morrison & Brantner, 1992; Schmidt et al., 1986; Silva et al., 2012). In Simulation 1, women's upward movement in organizations decreased as gender differences in developmental opportunities increased, pushing women out of the organizations due to lack of growth. Consequently, the removal of opportunity differences suggests women should see more advancement up the organizational ladder over time. To examine this effect in Simulation 2, Fig. 8a summarizes the average number of promotions men and women earned in organizational Levels 2 through 5 over time, with Fig. 8b providing a snapshot of this pattern in Level 5 at the beginning, the middle, and end of the simulation. Consistent with the conclusions from Simulation 1, these findings demonstrate that the average number of promotions earned by men remained stable while it increased for women over time. Notably, the discrepancy in both the average number of promotions (e.g., MM – MF = 0.52 (SD = 0.29) in Year 1 versus MM –

Fig. 9a. Percentage of female agents who turned over across time in Simulation 2 across organizational levels 1 through 7 in organizations initialized as 50% or more male. Note. Data includes organizations with an initial gender breakdown of 49% or greater male.

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Even when men and women were externally hired into the organization at an equal rate, differences in opportunity values resulted in women holding as few as 15% of top leadership positions in Level 7. As such, if an organization fails to support the development of leadership skills for both men and women, they still risk losing high-potential women. Whereas Simulation 1 revealed the mechanisms by which bias in the factors of external hiring and developmental opportunities impact the emergence of gender stratification, Simulation 2 examined how combatting these biases in already stratified organizations may impact the representation of women leaders over time. In general, Simulation 2 revealed that improving women's developmental opportunities and hiring rates has a positive impact on their representation, bringing organizational leadership to near parity by increasing women's promotability. However, the simulation also revealed that the journey to parity may be a complex and lengthy one, particularly in severely maledominated organizations. In the present simulations, it took approximately 25 years for female representation to reach an equal level to male representation in the highest leadership level (Level 7). Further, the immediate years following the removal of bias in the focal factors still saw high rates of women leaving male-dominated organizations due to tokenism. Finally, gender differences in career delays and risktaking propensity resulted in high variability among women's leadership rates by the end of the simulation and resulted in a majority of organizations failing to reach parity within one full generation of employees. Thus, the reduction of gender stratification may be possible, but requires time and is not without hurdles along the way.

Fig. 9b. Percentage of female agents and male who turned over in years 1, 20, and 40 in Simulation 2 in level 7. Note. Data includes organizations with an initial gender breakdown of 49% or greater male.

Discussion The present modeling and simulation work sheds light on the ways by which external hiring and developmental opportunities – two significant contributors to the complexity of women's career labyrinths – impact leadership development and diversity at the organizational level. By considering these two factors both in the context of one another and of other barriers to women's leadership potential in organizations, the current research advances our understanding of the interactive dynamics often missed in research on these factors in isolation. The simulation results revealed a number of unique insights into the dynamic processes and driving forces through which gender stratification in organizational leadership may unfold and be combatted. A growing number of researchers acknowledge that computational modeling techniques provide an important lens for summarizing and guiding future research and practices as well as improving the precision and transparency of organizational research (Harrison et al., 2007; Kozlowski et al., 2013; Vancouver & Weinhardt, 2012; Wang et al., 2016). In this vein, the present model marks a useful theoretical and practical contribution by contributing a flexible and generalizable platform for integrating future findings/processes relevant to this domain and exploring potential strategies for improving gender diversity in leadership positions. Results from the first simulation revealed that gender stratification emerged above and beyond what would be expected solely based on an organization's external hiring rate. A male-skewed external hiring rate not only brought more men into the organizations, but also pushed women out due to a heightened sense of tokenism (Elvira & Cohen, 2001; King et al., 2009). Thus, biases in external hiring may not only keep women with leadership potential out of an organization, but may also push top performing women out of their organization due to a negative, inequitable work environment. Previous research has found that although the number of developmental opportunities tends to differ little across male and female employees, females often report receiving fewer high visibility and high value opportunities compared to males (e.g., Catalyst, 2004; King et al., 2012; Ohlott, Ruderman, & McCauley, 1994; Silva et al., 2012). The first simulation demonstrated that this inequality tended to create a “sticky floor” for female employees (Booth et al., 2003; Yap & Konrad, 2009) in which female agents received fewer promotions out of lower organizational levels that subsequently drove up their turnover rates and resulted in large disparities in female representation in leadership.

Model prescriptions and practical insights As with any new theoretical framework or research, it is prudent to caution overgeneralizing findings from the present model to practice without systematic empirical testing. Nevertheless, the simulations highlight promising targets for intervention that may be explored in future research. Importantly, the simulations represent the potential effects over time of interventions in complex organizations with other potentially gender-biased practices/systems. One notable insight revealed by both simulations is that potential strategies for improving female leadership representation via external hiring or leadership development opportunities are likely to operate and see their impacts unfold in qualitatively different ways, and that the potency of any particular strategy will be dependent on the effects of other factors in the organizational system. To make these discussions more tangible, Fig. 10 shows data from the simulation on how the representation of female employees in top leadership (Level 7) changes when both external hiring and developmental opportunity conditions were altered in ways that make circumstances more equitable across males and females. These data highlight two key points with respect to the patterns of impact that external hiring and developmental opportunities may have as targets of intervention. First, there is a clear positive slope associated with reducing male-favored organizational entry. Irrespective of differences in male and female opportunity values, an increase in the female hiring rate of 30% (i.e., moving from 80% male hires to 50%) increased the percentage of women in leadership positions by 174.19%, on average. This impact was linear in the simulations, such that any reduction in bias had the potential to improve women's representation. Extrapolating from the simulation results examining women's tokenism experience, changes in external hiring also have the potential to positively influence the organization's climate over time and thus decrease women's likelihood of leaving due to a perceived negative climate for women. Improvements in external hiring thus have the potential to bolster an organization by bringing in women with leadership potential and supporting those who are already in leadership positions. Second, rectifying developmental opportunity value discrepancies may be more or less impactful depending on the context driven by external hiring. Reducing differences in the value of developmental 11

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Fig. 10. Average final percentage of female agents in Level 7 for organizations in Simulation 1 with less versus more equitable hiring and developmental opportunity practices. Note. Data includes organizations with external hiring rates between 50% and 80% male and opportunity value difference effect sizes between 0 and 0.5.

opportunities for future research to refine, improve upon, and extend the logic and representations of any computational model, and the present is no exception. The current simulations shed light on the potential connections between gender differences in opportunity values and external hiring rates, women's turnover, and gender differences in career trajectories. When female employees faced barriers to promotion due to a lack of valuable developmental experiences in Simulation 1, the primary route through which females reached the upper levels of that organization was through being externally hired into those positions. This pattern of results is supported by existing data. For example, Favaro, Kalrsson, and Neilson (2013) provide evidence that compared to men, female CEOs are more likely to be “outsiders” hired from a different company. Similarly, Valcour and Tolbert (2003) report that women tend to experience more inter-organizational mobility (i.e., moving between organizations) and men more intra-organizational mobility (moving within their organization). However, because the present model assumed that individuals externally hired into the upper levels of an organization were older (on average) than those hired into the lower levels of an organization, female employees in the upper organizational levels in Simulation 1—who were more often externally hired into those positions—tended to be older than their male counterparts who could reach those levels through promotion at a comparatively younger age. This resulted in externally hired females in leadership positions in the simulation often possessing a higher likelihood of turnover due to age/retirement. This pattern does not coincide with some existing research which reports that female managers and CEOs tend to be younger on average than their male counterparts (Bertrand & Hallock, 2001; Martin, Nishikawa, & Williams, 2009; Withisuphakorn & Jiraporn, 2017). Though the older average age of senior level women in our simulation results does not affect the substantive interpretations of the gender stratification processes represented, it does highlight an area where future refinements could be made to improve model fidelity. We believe the model and simulation results also highlight the need to consider the larger, inter-organizational system when examining issues of gender stratification and leadership development. Within a single organization, male-stereotypic constructions of work and career success (such as progression up the corporate ladder) dominate and

opportunities from a medium effect size to no difference between men and women had a greater impact as the external hiring rate approached 50% from majority male. When a greater proportion of men are hired into an organization, improvements in the visibility and challenge of opportunities given to women may be less powerful due to the negative climate theoretically engendered by the imbalance in the demographic make-up. However, if an organization employs a multifaceted strategy to fostering women's leadership potential, the benefit is dramatic. When an equal rate of men and women are hired, women's representation in the highest level of leadership more than doubles to a level equal to men's when given equally valuable opportunities. Thus, these results suggest that while both approaches are valuable in boosting the leadership diversity of an organization, the greatest benefit is gained when the issue is tackled from both directions simultaneously. Results from the second simulation further revealed that genderstratified organizations may face additional barriers as they implement interventions to improve the hiring rate and leadership development of women. Specifically, women in male-dominated organizations may still appear to be turning over in relatively higher numbers until a critical mass of female representation is reached that reduces the adverse effects of tokenism (King et al., 2012). In other words, a history of gender stratification in an organization may continue to have consequences even after steps have been taken to reverse it. Further, Simulation 2 suggests that even after the experience of tokenism is reduced and women begin earning promotions at a relatively equal rate to men as a result of equal developmental opportunities, perfect gender parity is not likely to be reached if other factors exist that disproportionately impact females exist in organizations. Model refinements and extensions Significant efforts were made to instantiate the core concepts and process mechanisms of the model in ways consistent with existing empirical data and relations. Nevertheless, developing dynamic representations of process by integrating narrative theory and cross-sectional empirical data necessarily involves making a variety of assumptions and operationalizations. These areas represent clear 12

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mark the accumulation of traditionally-defined developmental experiences as critical to success (Dragoni et al., 2009; McDonald, Brown, & Bradley, 2005; Ohlott et al., 1994; O'Neil, Hopkins, & Bilimoria, 2008). The findings presented here emphasize the potential detriment this conceptualization of success could have on female leadership rates, but also suggest alternative pathways women take to reach high-status positions. The majority of women in upper leadership positions in the organizations in Simulation 1 were externally hired rather than promoted into those positions; and indeed, women tend to experience more inter-organizational mobility (Favaro et al., 2013; Valcour & Tolbert, 2003;). In this sense, the labyrinth of obstacles women face in their path to leadership likely expands beyond the walls of a single organization. One organization, independent of the organizational network at-large, may provide only a limited picture of the mechanisms underlying the emergence of gender stratification and its consequences. Thus, the expansion of the model from single-organization to multi-organization may better illuminate additional obstacles faced by women seeking leadership positions. Further, it may provide insight into how alternative definitions of success that value flexibility and skill diversity in addition to the accumulation of traditional developmental experiences could improve female representation if the related markers of competence were taken into consideration in the promotion process. Finally, while our findings demonstrate that the mechanisms included in the model and manipulated in Simulation 1 sufficiently result in large gender stratification effects, other factors may be of interest that could be incorporated into the model or manipulated in future simulations. The factors included in the current model were operationalized at a relatively structural level. However, gender differences in organizational entry and movement within organizations themselves likely emerge from more “invisible” factors, such as stereotypes that

individuals may not even realize they hold, but impact how women are viewed as leaders. Computational models are an apt methodology to not only examine the processes related to within-individual factors, but to connect within-individual mechanisms to structural factors. Thus, while these invisible factors may be assumed antecedents of the factors of interest presented here, future researchers could focus on these antecedents in further developments of the model. With respect to the current findings, the focal mechanisms were sufficient to produce levels of gender stratification observed across current Fortune 500 companies that release their diversity statistics, such as Amazon and Intel (Fortune 500, 2018). Conclusion Despite approximately equal representation in the workforce, the percentage of women holding leadership positions in organizations continues to remain low in the American workforce (Catalyst, 2018; U.S. Bureau of Labor Statistics, 2018). The present research examines how external hiring and developmental opportunities act as barriers in women's navigation of the leadership labyrinth into higher-level positions. As the simulations indicate, barriers to an organization's leadership diversity are complex and multiply determined—as will be any strategies to counteract those barriers. However, with a comprehensive approach, organizations have the potential to support both men and women's leadership development, as well as improve organizational performance and organizational members' satisfaction and wellbeing. Continued efforts to understand and expand upon the dynamic processes responsible for gender stratification represent a high potential focus for advancing both research and evidence-based practices in the leadership domain.

Appendix A This appendix presents broader conceptual rationales for including and operationalizing the mechanisms incorporated into the simulations described above. A.1. Career delays Men and women differ in their leave-taking behavior. A Family and Medical Leave Act (FMLA) technical report in 2012 suggests that a slightly higher percentage of women (14.9%) take leave to provide family care compared to men (11.4%). However, when women take leave for childcare reasons, they tend to take a longer leave (2 months) compared to men (1 month or less). Such differences are often attributed to societal beliefs and stereotypes that women should be responsible for caregiving while men should provide financial stability, and the consequences that violations of these expectations carry (Stolzenberg, 2001). For example, men who request leave may be viewed as poor organizational citizens and less masculine, which can decrease their eligibility for rewards and promotions in the workplace (Rudman & Mescher, 2013). Although it is more socially normative for women to take leave, this activity carries career consequences for them as well. Women frequently identify family responsibilities as a barrier to their advancement (Wellington, Kropf, & Gerkovich, 2003), and perceptions of female subordinates' work-family balance has been shown to impact supervisor evaluations of employees' perceived job fit, commitment and promotability (Hoobler et al., 2009). Thus, we incorporate leave-taking into the current model. In the current model and consistent with existing data (FMLA, 2012), female employees are more likely to take leave than men; in addition, we model differences in the likelihood and duration of leave for childcare for males versus females. The net effect of career delays is that employees are temporarily absent from their job, which reduces their ability to remain competitive for promotions. In the present model, 3.9% of female employees who were assigned a career delay were specifically assigned parental leave (FML Report, 2012). Finally, 10.1% of employees who were assigned a career delay did not “return”, effectively turning over, and their positions were left open to be filled in the subsequent hiring and promotion period (FML Report, 2012). A.2. Line-Staff designation Although counts of organizational demographics often aggregate across both line and staff positions, line positions are the “feeding pool” for senior levels in organizations (Catalyst, 2007). The responsibilities of line employees tend to directly impact their organization's core work, whereas those of staff employees tend to be more supportive in nature (Hellriegel et al., 2002). Potentially due to the male gender-typing of line responsibilities, women have been shown to be underrepresented in line positions and lack access to line experience relative to their male counterparts (Catalyst, 2007; Lyness & Heilman, 2006; Ragins & Sundstrom, 1989; Wellington et al., 2003). Consistent with these findings, we model both line and staff positions, and assign a greater proportion of staff roles to women and a greater proportion of line roles to men. Further, although employees in both line and staff positions can climb the corporate ladder (Wellington et al., 2003), line employees are much more likely to be promoted into the upper organizational echelons (Catalyst, 2007). We operationalized this difference as 70% of the staff positions being assigned to women (30% men), and 30% of the line positions being assigned to women (70% men). However, the distribution of men and women in line and staff roles was free to vary after the simulation began, as organizations developed, and employees were internally promoted and externally hired into positions. 13

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A.3. Tokenism Tokens are members of a numerical social minority in which the numerically dominant members control and shape the culture of the group (Kanter, 1977). Token women in work environments often perceive their organizational climate as inequitable to women (King et al., 2009), report feeling less satisfied with their job, excluded from important networks, and increased levels of job-related depression and lower self-esteem (Krimmel & Gormly, 2003; Maranto & Griffin, 2011). One significant consequence of tokenism for females is an increased likelihood of leaving an organization (King et al., 2009). In contrast, when women are more represented in a given organizational level, they report significantly lower intentions of turning over (Elvira & Cohen, 2001). Experiences of tokenism by females thus represent a potentially important factor in the underrepresentation of females in higher levels of organizations. The present model accounts for this possibility by including the percentage of women in her associated group of 100 agents as a determinant of her turnover decision. A.4. Risk taking Risk taking in the workplace involves taking an unsure action over a sure action (MacCrimmon & Wehrung, 1990) or taking an action that makes oneself vulnerable (Colquitt, Scott, & LePine, 2007; Mayer, Davis, & Schoorman, 1995). In the present model, risk-taking propensity is included as an individual difference variable that impacts the manner by which difficult developmental opportunities are taken up by employees. Carrying out developmental opportunities can have uncertain consequences. Such experiences may involve new, unfamiliar responsibilities, managing large budgets, supervising and coordinating a large number of people, and being accountable to senior-level management or the executive suite (Dragoni et al., 2009; Silva et al., 2012). Although successfully completing such opportunities can contribute to one's perceived competence, the potential for failing a high visibility/high impact opportunity can place the employee in a more vulnerable position compared to carrying out typical performance tasks. Women may be in an especially vulnerable position when carrying out a developmental opportunity due to potential backlash for engaging in behavior that could be perceived as role incongruent (Maxfield, Shapiro, Gupta, & Hass, 2010; Roberts, Roberts, O'Neill, & Blake-Beard, 2008; Tannen, 1994, 1995) or due to the lack of workplace support systems that aid in navigating challenging work experiences (Neves & Eisenberger, 2014; Ohlott et al., 1994). Thus, an employee's risk-taking propensity in the current model influences their likelihood of carrying out developmental opportunities offered to them and differs across gender. Appendix B B.1. Specification of turnover in computational model and simulation Turnover decisions are impacted by a variety of factors. In the present model, turnover was computed as a function of agent age, their tenure in a given organizational level, and for females only, experiences of tokenism. The impact of each of these factors on turnover likelihood was computed separately and then combined into an overall turnover probability. The turnover likelihood function for each factor was represented using a 4parameter logistic function:

y=a+

b a 1 + e c (x

(1)

d)

where x represented an employee's standing on a given factor, a represented the minimum possible likelihood of turning over for a given factor, b represented the maximum possible likelihood of turning over for a given factor, c represented the rate at which the likelihood of turning over for a given factor increases as a function of x, and d represented the point at which the likelihood of turning over for a given factor changed most rapidly. A 4-parameter logistic function is useful for the representation of an agent's turnover decisions because 1) it allows for the specification of a reasonable minimum (i.e., 0% likelihood of turning over) and, for some factors, a reasonable maximum (i.e., 100% likelihood in the case of age), and 2) allows users of the model to weight agents' reasons for turnover relative to one another. Further, it represents a gradual change in an agent's decision based on relevant, dynamic information, such as age, tenure, and the make-up of one's group. Table B1 provides the values used for each for the parameters for each turnover factor, and Figs. B1 through B3 provide a graphical depiction of the turnover functions for comparison. Turnover due to age was intended to represent retirement, such that an employee's likelihood of leaving increased rapidly from 0% to 100% as they age from 55 years old to 75 years old and where the likelihood of turning over due to age increased most rapidly at age 65 (Fig. B1). Turnover due to tenure level was intended to capture the effect of low potential for promotion opportunities on an employee's likelihood of leaving their organization (Lyness & Judiesch, 2001; Quarles, 1994; Weng & McElroy, 2012). After three years at the same level, an employee's likelihood of leaving gradually reached a maximum of 25% after they had spent 12 years in the same level (Fig. B2). Lastly, female agents' likelihood of leaving due to tokenism was parameterized such that turnover probability increased rapidly as the percentage of women in their workgroup dropped below 25% and increased most rapidly when her workgroup was only 15% female (consistent with Kanter, 1977; Fig. B3). An agent's overall total likelihood of turning over was subsequently calculated by summing together their likelihood of turning over due to age, due to tenure at their current level, and, for women, due to tokenism plus a baseline turnover likelihood (0.08) intended to capture reasons for turning over besides the ones specified here. An agent voluntarily turned over from an organization during any given year at a probability equal to their total turnover likelihood. Table B1

Parameter values of employee turnover likelihood functions. Reason for turning over

Employee variable (x)

Minimum likelihood (a)

Maximum likelihood (b)

Slope (c)

Inflection point (d)

Retirement Level tenure Tokenism (female employees only)

Age (years) Time in current level (years) Percentage of women in workgroup

0 0 0

1 0.25 0.10

2 2 −50

65 14 0.15

Note. j = employee's age in years; k = employee's time in current level in years.

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Fig. B1. Employee likelihood of turning over due to retirement as a function of age.

Fig. B2. Employee likelihood of turning over due to level tenure.

Fig. B3. Female employee likelihood of turning over due to tokenism.

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