An Agent-Based Simulation of How Promotion Biases Impact Corporate Gender Diversity

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This paper describes the application of an agent-based simulation to understanding and managing the impact of diversity and inclusion on organizations.

Abstract

Diversity and inclusion (D&I) is a topic of increasing relevance across virtually all sectors of our society, with the potential for a significant impact on corporations and more broadly on our economy and society. While people are typically the most valuable asset of every organization, human resources (HR) in general, and D&I in particular, are dominated by qualitative approaches. This paper introduces an agent-based simulation that can quantify the impact of certain aspects of D&I on corporate performance. The simulation provides a parsimonious and compelling explanation of the impact of hiring and promotion biases on the resulting corporate gender balance, accurately replicating real-world data about gender imbalances across multiple industry sectors. In addition, the paper shows that the simulation can be used to predict the likely impact of different D&I interventions. Specifically, once a company has become imbalanced, even removing all promotion biases is not sufficient to rectify the situation, and it can take decades to undo the imbalances initially created by these biases. These and other results demonstrate that agent-based simulation is a powerful approach for managing D&I in corporate settings and could become an invaluable tool for the strategic and tactical management of human resources.

1. Introduction

In spite of the growing body of evidence showing that companies with greater gender representation in leadership roles tend to outperform companies with fewer women [1,2,3], many industries continue to exhibit a sharp gender imbalance, with senior and executive levels being dominated by men. These imbalances contribute to some of the severe problems seen across a variety of industries, ranging from gender pay gaps [4,5] and high churn rates [6,7], to discrimination lawsuits [8,9]. In turn, these problems lead to high costs and internal instabilities, and expose companies to significant reputational risk. Beyond the private sector, gender imbalances also impact academia and the public sector [10,11].

Given the extensive studies showing that greater gender inclusion can lead to corporate, economic and societal benefits [12], the tangible negative implications of gender imbalances and the ongoing efforts ranging from individual activism to legislation, why are there still such significant gender imbalances across virtually all industry sectors?

We believe that the relative lack of progress is primarily due to the sheer complexity of the problem, and to the lack of tools that can deal with this degree of complexity. Even the most advanced workforce analytics platforms are unable to quantify the myriad activities, interactions, attitudes and subjective preferences of employees.

The complexity that so challenges workforce analytics, we believe, is also the reason why certain D&I initiatives that seem to work within one organization often fail to produce results—or even backfire—at other organizations; each organization is a unique “ecosystem” whose macroscopic behavior emerges from the complex web of interactions among its staff, leadership, customers, suppliers and partners. Borrowing a D&I initiative from another company and hoping that it will have a positive impact on our company is analogous to copying a specific advertisement that was successful for another company in a different sector and expecting it will sell our products.

When you also consider that the impact of personnel initiatives can take months or years to be observed, and that missteps can be extremely costly, it is no wonder that leaders are reluctant to take decisive action. What is needed is a quantitative approach that can capture and analyze the behavior of complex systems.

We have successfully begun using agent-based simulation [13], one of the primary tools of Complexity Science [14], to replicate the complex behaviors of people within organizations. In this paper, we show the results of applying an agent-based simulation to a particular aspect of corporate gender imbalance: by simulating the hiring and career advancement of employees at typical companies, we can analyze the impact of introducing gender biases.

Under reasonable assumptions, we find that gender biases in promotions can yield the kinds of gender imbalances that are typical of many companies, with a decreased representation of women at higher corporate ranks. We also find that by adjusting gender biases in hiring as well as promotions, it is possible to simulate gender imbalances that match the real-world patterns observed in different industries.

Agent-based simulations capture the causal relationships that link individual behaviors and interactions to the resulting system-level behaviors [15]. As a result, once an agent-based simulation has been shown to capture a real-world phenomenon with some fidelity, it can be used to predict the likely impact of different initiatives, even novel initiatives for which no data are available. Here, we use our simulation to test the likely impact of removing all promotion biases in a company that is already gender-imbalanced. Under a variety of scenarios, we find that simply removing biases is not a very effective strategy, as it can take much longer to eliminate the gender biases than it took to establish them in the first place.

Lastly, we use the simulation to explore how company-level biases lead to different individual day-to-day experiences for employees based on their personal characteristics, and we discuss how these experiences can impact employee satisfaction and productivity.

Our findings, in line with other studies, show that agent-based simulation is a powerful tool for managing a variety of business problems in general [16,17], and workforce analytics in particular [18,19,20], and that this approach holds great promise for theoretical and applied research into D&I—a topic of great current interest with significant economic and societal implications.

After introducing some background material, the remainder of this paper describes the simulation we have developed, and then presents several results that we obtained with the simulation. The paper is brought to a close with some conclusions and suggestions for future opportunities to expand this line of work.

2. Background

2.1. Gender Biases in the Workplace

Although female labor force participation is increasing, women are still severely underrepresented at the top levels of organizations: according to the U.S. Department of Labor, women account for almost half of the total labor force in the U.S., and roughly 45 percent of those have college degrees; however, in 2022, women only held 27% of C-suite, executive and senior-level management positions, roughly one-third of all VP-level positions [21] and just reached 10% of chief executive officer positions in Fortune 500 companies for the first time ever in early 2023 [22].

To understand the causes of these types of gender imbalances, Bielby and Baron [23] categorized the causes of the gender gap in upper management positions into supply-side and demand-side explanations. According to supply-side explanations, the divergence in employment outcomes between women and men is mainly due to differences in gender-specific preferences and productivity [24]; therefore, individual attributes determine the gender inequalities in the workplace [25]. For example, some believe that balancing work and family life lowers women’s promotion rates as women need to take on a larger share of domestic and parental responsibilities [26]; others hypothesize that women, in general, are less competitive than men so they may be reluctant to compete for promotions [27].

In contrast, demand-side explanations suggest that gender stratification in the workplace is primarily due to gender-specific barriers; demand-side explanations focus on the institutional constraints and managerial biases faced by women in climbing the career ladder. For example, a male-dominated board of directors may prefer to hire male executives [24], women must meet higher performance standards for promotions than their male colleagues [28,29] and stereotypes of leadership style differences favor men in advancing into leadership roles [30].

In this paper, we introduce a quantitative methodology that provides indirect but compelling evidence for a demand-side explanation. Specifically, we simulate the career advancement of employees in a typical organization and test the hypothetical impact of introducing gender biases in the promotion process. We find that, under a range of simple and realistic assumptions, it is possible to replicate the levels of gender disparity that are observed in typical companies across a variety of industries, with increasing male dominance at increasingly senior levels. Hence, while we cannot conclusively prove that gender imbalances are due to gender-specific barriers, we demonstrate that the existence of gender-specific barriers yields the kinds of imbalances that are observed empirically.

Another gender-specific barrier that we examined is a bias in hiring. As suggested in a recent report by McKinsey & Company [31], in some industries women may remain underrepresented at manager levels and above because they are less likely to be hired into entry-level jobs, in addition to being less likely to be promoted into manager-level positions. To test this hypothesis, we extended our simulation to capture a simple form of hiring bias and found that by adjusting the hiring and promotion biases simultaneously, we are able to replicate industry-specific gender balance patterns across various industry sectors, as reported in the McKinsey study.

2.2. Simulating Corporate D&I

A majority of the research on gender inequality in the workplace is conducted by applying a statistical methodology on collected data. These statistical approaches have significant limitations, including the fact that they hide details about individuals and cannot capture dynamic interactions. In general, statistical approaches capture population-level outcomes but cannot capture the underlying causal relationships and time-dependent interactions that lead to those population-level outcomes. Statistical approaches also lack the ability to investigate longitudinal problems because of a lack of data availability [32].

The human behaviors and interactions that drive the performance of an organization—including the impact of personal interactions that are influenced by D&I—are exactly the types of complex relationships that cannot be captured through statistical, population-based analyses.

A more fruitful approach to analyzing the impact of D&I is to apply methodologies that are designed specifically to analyze complex systems. In this paper, we draw from Complexity Science, a discipline that first took shape in the late 1960s with the seminal work of Thomas Schelling on the emergence of segregation [33], and became a full-fledged area of academic inquiry in 1984 with the creation of the Santa Fe Institute. Complexity Science is a broad field which encompasses a variety of technologies for studying complex systems, i.e., systems whose behavior depends, in complex and often unpredictable ways, on the behaviors of many individuals interacting with one another and with their environment [14].

One of the primary tools for the analysis of complex systems is agent-based simulation, a methodology that combines behavioral science and computer modeling [13,34]. Agent-based simulations capture the behaviors of individuals, as well as their interactions with other individuals and with their environment, to simulate the way in which the overall behavior of a system emerges from these complex chains of interactions.

Agent-based simulations are ideally suited to analyze and predict the performance of human systems such as companies [32,35] because they capture the mutual relationship between individuals and the organization in which they belong: in the case of a company, the behaviors of individual employees combine to determine the success of the organization and, conversely, the environment created by the organization influences the performance of the individual employees. This sort of “feedback loop” is part of what makes workforce management so complex, and it is exactly the type of problem that lends itself to analysis using agent-based simulation.

In this light, agent-based simulations promise to be a valuable tool specifically to capture the impact of D&I on corporate environments: to the extent that a company influences people’s experiences differently based on personal traits, the company’s performance will in turn be impacted by how those people are treated. In fact, the connection between D&I and complexity science has been proposed by others [36], and agent-based simulation has already been used to analyze some issues related to corporate D&I. For example, the simulation developed by Bullinaria [37] shows how ability differences and gender-based discrimination can lead to gender inequality at different hierarchical levels within an organization; Takács et al. [38] found that discrimination can emerge due to asymmetric information between employers and job applicants, even without hiring biases; Robison-Cox et al. [39] used agent-based simulation to test the possible explanations of gender inequality at the top level of corporations, and found that giving men favorable performance evaluations significantly contributes to the gender stratification of top-level management.

Our agent-based simulation takes a step further and simulates the ongoing activities and transitions within a typical company to show the dynamics of how corporate gender imbalances arise at each level of the hierarchy as a result of biases in hiring and promotion processes. Our simulation replicates the day-to-day operations of a typical company with men and women distributed across four levels: entry-level employees, managers, vice presidents (VPs) and executives. In a pilot project, we were able to simulate the impact of gender biases in the promotion process, which lead to the kinds of gender imbalances seen in real companies, with an increasing representation of men in higher levels of the company [40]. In this paper, we report a more complete and extensive set of results and provide a direct comparison to industry data. In addition, we provide additional results showing that removing biases in a company that is already imbalanced is not a very effective strategy, as gender inequalities can persist for significant periods of time. Our findings are in line with the empirical findings of Kalev et al. [41], who found that programs targeting lower levels of management, such as diversity training and performance evaluations, do not help to increase diversity at higher corporate levels, and that imbalances persist after organizations adopt these diversity management programs.

3. The Simulation

One of the most powerful aspects of agent-based simulation is that it captures the way real-world systems work in an intuitive, human-centric fashion. This means that any-one who has familiarity with the problem can contribute to the design of the simulation. In a sense, agent-based simulation democratizes analytics, because it does not require knowledge of advanced mathematical or computational techniques.

Agent-based simulation allows domain experts to be closely involved both with the model design and with the analysis of the results [15], a key departure from more traditional approaches to analytics in which a data scientist analyzes large amounts of data using analytical tools to look for patterns, and the domain expert is relegated to making sense of the identified patterns. In fact, working closely with domain experts, our team has developed dozens of agent-based simulations to solve complex problems across many sectors, including consumer marketplaces [42,43], energy consumption in commercial buildings [44], manpower and personnel management for the U.S. Navy [18], healthcare [45] and computer security [46]. In all these examples, the simulations were developed by asking domain experts to describe individual behaviors and interactions, and translating them into software simulations. The knowledge of the domain experts is explicitly represented in the simulation, which makes the “structure” of the simulation intuitive to anyone using it. Running the simulation then replicates the complex interactions, that are impossible to grasp intuitively or mathematically, over time.

This is one of the benefits of agent-based simulations: because they are literally trying to replicate the individual-level behaviors, they are intuitive to domain experts, but because they replicate the complex interactions over time that occur in real systems, they are able to reproduce the outcomes observed in the real world. We now show how we applied this methodology to the study of corporate D&I, and, more specifically, to corporate gender biases.

3.1. Core Elements of Our Simulation

The simulation we developed for the present study is based on simple assumptions about the operation of a typical company. Figure 1 is a screenshot of the simulation, which was developed with the NetLogo simulation platform [47]. Employees fall into one of four increasing ranks: entry-level employees, managers, VPs and executives. Because we are exploring gender biases, in our simulated company there are two types of employees: men (in blue—or the darker shade if viewed in grayscale) and women (in yellow—lighter shade). Our simulation assumes that men and women have identical abilities and that their performance is also identical (Note that although many other personal traits could easily be added, for the present study we wanted to focus exclusively on how gender biases in promotions and hiring impact the overall behavior of a company. However, this is not meant to suggest that gender should be seen as binary, nor that we suggest that the experiences of all those who do not identify as men are identical).

At the start of each simulation “run,” employees are distributed across ranks in a way that mimics a typical company, with smaller numbers of employees at higher ranks. Unless otherwise stated, the simulation starts with a 50–50 gender split at each level.

The company is assumed to grow over time, creating vacancies within each rank. In addition, employees may be promoted or may leave the company, creating additional vacancies. Vacancies at the entry level are filled through hiring from a hypothetical external pool of candidates, while vacancies at all other ranks are filled by promoting individuals from the rank immediately below.

For this version of the simulation, we do not simulate direct hiring into higher ranks, nor do we allow promotions to skip levels. However, these and other details could be added if we were interested in exploring the effects of such modifications.

While the simulation runs, the employees move back-and-forth across the floor of their rank as time elapses (this movement is not relevant to the results, but it helps to visualize the activities unfolding over time). Employees who separate from the company rise just above the other employees, turn gray and gradually fade. Employees who are promoted are seen floating up to the next level. These animations help to visualize the main events that occur to the simulated employees during each run.

In addition, to help visualize the gender balance for each employee rank, the floor of each level of the simulation acts as a simple histogram, showing the percentage of men and women at that rank. The two vertical bars seen in Figure 1 on the left side of the screen show the overall gender balance across the entire organization.

The company’s overall gender balance is an emergent behavior that results from three main HR activities which take place as the company grows over time: hiring, promotion and separation. We now describe each of these processes.

3.1.1. Simulating the Hiring Process

When a vacancy occurs at the entry level, the simulation assumes that a new employee will be hired from a potentially infinite external candidate pool. By default, there is an equal chance of hiring a man or a woman. Gender biases in hiring can be simulated by setting a “hiring-bias” parameter, which changes the probability of men (or women) being hired.

When an employee is hired, the simulation begins to track their seniority, i.e., the amount of time they have been with the company. At the start of the simulation, employees are assigned a random rank and seniority level to avoid a situation in which all starting employees have identical characteristics.

3.1.2. Simulating the Promotion Process

When a vacancy appears at any rank above the entry level, promotions are simulated by identifying a pool of “promotion candidates” from the rank immediately below, and then randomly choosing one employee from that pool.

By default, the promotion pool is set as a percentage of the total number of employees at that rank and is based on the “promotion score” of each employee, a normalized value based on the amount of time the employee has spent at the current rank (“rank seniority”) relative to the amount of time spent by the most senior employee at that rank. In other words, in the absence of a bias, promotions are based on rank seniority, with some randomness to reflect the fact that seniority is not an exact indicator of merit. It is worth noting that seniority has been widely used as a salient criterion for promotions in many industries, and it has been suggested that promoting the most senior employee instead of promoting the candidate with best performance or ability can reduce possible psychological disturbances [48].

When an employee is promoted, their seniority at the new rank is set to zero, while the simulation still tracks the total amount of time that the employee has been with the organization.

Gender biases in the promotion process are simulated by adding a “promotion-bias” parameter, a value that is added to the seniority of each employee to calculate the promotion scores. The promotion-bias parameter can be positive or negative to simulate biases that favor men or women, respectively. The promotion-bias parameter is applied uniformly at all ranks.

It is important to note that we are not explaining how promotion biases come into play; we are simply interested in understanding how gender biases in the promotion process, if present, would impact the overall outcomes for the company, regardless of their source or nature.

3.1.3. Simulating the Separation Process

In this simulation, we do not distinguish between employees who quit and those who are fired, referring simply to “separation” as the act of an employee leaving the company. While it would be possible to capture the nuances of the different types of separations, for the purposes of this study the key value of separations is to create turnover at a rate comparable to what we see in real companies, so that the company’s employee pool is refreshed over time.

We simulate separations by assuming a certain annual company turnover rate and apply that rate uniformly at each rank at each time step. This means that, in a given year, the absolute number of employees who separate is greatest at the entry level and decreases with increasing rank. This ensures that the ratios of the number of employees across ranks remains consistent (subject to small fluctuations due to the timing of terminations and promotions) as the company grows over time. Notice that this also means that the average “tenure” of employees is directly proportional to rank: if there are twice as many entry-level employees than VPs, then, on average, the tenure of a VP will be about twice as long as that of an entry-level employee. This kind of relationship between rank and average tenure aligns with existing data from corporations [49].

To determine which employees separate at each rank, at each time step we determine the number of employees who are likely to separate, and then we randomly select the required number of employees from a “separation pool,” a portion of the employees with the lowest promotion scores at that rank; in other words, the decision to terminate is influenced both by seniority (or lack thereof) and—if biases are present—by gender.

Applying the same gender bias for separations as well as for promotions reflects two assumptions: first, that the sources of bias that drive promotion decisions are also behind termination decisions. This seems reasonable, given that typically the same manager is in charge of both decisions. Second, having access to career advancement opportunities is a crucial part in shaping job satisfaction, and in turn, will influence an employee’s intention to leave the organization [50,51]. Therefore, if biases cause women to be held back, they are more likely to become frustrated, which may increase the probability that they will separate voluntarily (i.e., quit). In preliminary studies, we have tried other separation processes, such as purely random selection. Including gender bias in separation heightens and accelerates some of the results we report, but does not alter our overall findings.

4. Simulation Setup

4.1. Simulation Time Step and Duration

The simulation is time-based, meaning that we simulate the passage of time, with each time step equal to one week of operations. The software is designed to allow any level of temporal resolution, but we found that a weekly time step gives the best balance between capturing typical personnel fluctuations and being able to sufficiently simulate long operating periods to see meaningful results. For all the results reported below, we ran the simulations for the equivalent of either 10 or 40 years (520 or 2080 time steps, respectively). Using a standard laptop, we can simulate four decades of company operations in a few seconds. We have tested the impact of reducing the time step to one day, or even hours, and found no significant differences—other than taking proportionately longer to run. All time-dependent variables described herein are specified as annual rates, and then scaled automatically to the time-step size to ensure consistent behavior.

4.2. Randomness

At each time step, the simulation includes several operations that invoke a random number generator, which results in variance across simulations (e.g., in selecting promotion and separation candidates). To ensure the reproducibility of individual simulation runs, we have the ability to select the seed for the random number generator. Unless otherwise specified, each result reported here was obtained by averaging the results from ten runs with different random seeds at each parameter setting, and it includes error bars corresponding to one standard deviation. We found that the results do not vary meaningfully as we increase the number of simulations beyond ten.

4.3. Company Size and Employee Distributions

For all the results reported here, the company begins with a total of 300 employees, 150 women and 150 men. The employees are distributed across the four levels in a way that roughly simulates a typical company: 40% at the entry level, 30% at the manager level, 20% at the VP level and 10% at the executive level. We have tested the simulation with other settings and found that the overall company size and exact distribution across the levels have no significant impact on the results. However, the smaller the company, the more fluctuations that will be seen across simulation runs with different random seeds.

4.4. Company Growth and Separation Rates

As mentioned earlier, the simulation time step is set to correspond to a 7-day period. At each time step, random numbers are drawn to determine whether separations, hires or promotions need to take place. The frequencies of each of these occurrences are set so that, over the course of a simulated year, the overall growth and churn rates are within a range that would be consistent with a “generic” company: the company grows by 10% each year (linearly, meaning that it will double in size in ten years, and triple in size in 20 years), while we target an annual churn rate of 20%, which is in line with national average separation rates [52].

4.5. Promotion and Separation Pools

For promotions, we set a candidate pool size of 15%: in other words, when someone needs to be promoted to a higher rank, we select the 15% of employees with the highest promotion scores, and then randomly choose one of them to be promoted. For separations, we set the candidate pool size to 50%, meaning that someone is selected randomly from half of the employees at each rank with the lowest promotion scores. We have tested different sizes of candidate pools and found that the results do not change significantly as these parameters are changed.

4.6. Employee Characteristics and Variables

Our simulation treats each employee as an “agent” with certain characteristics and variables. For this study, at the start of each run, every agent is assigned a gender value that remains unchanged during the simulation; each employee is also randomly assigned to an initial rank based on the distributions given above, and is given a start date that is set randomly (proportional to the starting rank). The start date is used to calculate the seniority of each employee.

All other agent properties, such as seniority and promotion score, are variables used to track the state of each agent over time and to make decisions about promotions and separations.

Table 1. List of employee agent variables tracked in the simulation.

Variable Name	Variable Description
rank	The level of the employee within the organization
start-date	The date when the employee joined the company
start-date-in-rank	The date when the employee first reached a certain rank
tenure-in-rank	How much time the employee has been at a certain rank
seniority	A normalized variable [0–1.0] that reflects how long the employee has been at this rank, relative to all employees at the same rank
promotion-score	A score that combines the employee’s seniority and promotion biases
times-passed-over	The number of times someone from the same level with lower seniority was promoted
hr-events	A list of HR-related events such as being hired or promoted

describes all the agent characteristics and variables that we used.

4.7. Scenario-Testing Parameters

To test the impact of gender biases, we varied certain parameters systematically to explore scenarios of interest. Specifically, all the scenarios reported here manipulated one or more of the following three parameters:

• Hiring Bias: This parameter determines the proportion of men and women that are hired when vacancies occur at the entry level. A positive bias means men are favored and a negative bias means women are favored.
• Promotion Bias: This parameter is added to the seniority of each simulated employee to influence the probability that the employee will be included in the candidate pool when a promotion takes place. A positive bias means men are favored and a negative bias means women are favored. The bias is expressed as a number between 0.0 and 1.0 which is added directly to each employee’s normalized seniority in their rank.
• Promotion Bias Duration: For some of the scenarios we tested, the promotion bias was set to zero after a certain amount of time (typically 10 years) to simulate the removal of all gender biases in promotions and separations. This parameter specifies the time when the promotion bias is turned to zero.

A complete list of parameter settings used for all simulations (except as noted) is provided in

Table 2. Parameter settings for our promotion bias in Experiments 1 and 3.

Parameter Name	Parameter Value
Simulation duration (in years)	40
Number of repetitions of each simulation	10
Initial company size	300
Initial gender balance (male-female)	50–50
Promotion bias level	0, 0.1, 0.3, 0.5
Promotion bias duration	N/A (exp. 1), 10 (exp. 3)
Hiring bias	0
Promotion pool size	15%
Termination pool size	50%

.

5. Results

5.1. Experiment 1: The Impact of Gender Biases in Promotion

In the first set of experiments, we wanted to establish a baseline and then test the impact of systematically increasing the degree of gender bias in the promotion process. In all these simulations, the hiring bias is set to zero.

Figure 2 shows the gender balance at each rank during a 40-year simulation when there is no promotion bias or hiring bias. The figure shows that, in the absence of biases, the gender balance stays at 50–50 throughout the simulation, with only small oscillations due to the inherent randomness of the simulations.

Next, we tested the impact of increasing the promotion bias to 0.1, 0.3 and 0.5. As mentioned earlier, in the absence of biases, each employee’s promotion score is simply its seniority relative to the most senior employee at that level (rank seniority). Hence, all the promotion scores prior to the application of a gender bias are between 0.0 and 1.0.

Adding a promotion bias of 0.1 thus means that while women’s scores will still be in the range [0.0, 1.0], men’s promotion scores will be in the range [0.1, 1.1]. Similarly, at the highest level of bias reported here (0.5), men’s promotion scores will be in the range [0.5, 1.5], while women’s promotion scores will stay in the range [0.0, 1.0].

As can be seen in Figure 3, a bias of 0.1 in promotions begins to show an interesting pattern: while the entry and manager levels continue to stay roughly at 50–50, the VP level (dashed line with no symbols) is starting to show an imbalance in favor of men, and men now make up roughly 60% of the executive level (solid line with diamond symbols).

The pattern becomes much more evident in Figure 4 and Figure 5, which show the gender balance at each company level when the promotion bias is set to 0.3 and 0.5, respectively. Several interesting phenomena are worth pointing out.

First, we see that even though the promotion bias is a single parameter that works uniformly at each level, the successive promotions compound the effect, so that the gender imbalance is greatest at the executive level: after several simulated years, the executive level shows an imbalance of approximately 80% men when the bias is at 0.3 (Figure 4), and exceeds 90% men when the bias is at 0.5 (Figure 5).

Second, increasing the bias has the effect of increasing the degree of imbalance, but also the speed with which the imbalance spreads through the organization: notice that with a bias of 0.3, the imbalance at the executive level builds gradually over a span of nearly 20 years, but when the bias is 0.5, the proportion of men at the executive level crosses the 80% mark in less than five years, and has essentially leveled off by year 10.

Third, there is a surprising effect at the entry level: the percentage of female employees goes up as the promotion bias increases, even though there is no hiring bias, and even though women are being terminated more often than men because the promotion score influences separations. We can see in Figure 4 and Figure 5 that women make up an increasing percentage of the entry-level population, reaching 60% when the bias is 0.5. The reason for this “reverse imbalance” is that men are being promoted at a much higher rate than women, so that women are being left behind. In reality, this is not uncommon in the real world: many industries have large numbers of women in entry-level positions, and women often describe the negative experience of being “stuck” while their male colleagues get promoted [53,54].

This last observation illustrates another great aspect of agent-based simulations: unlike typical “black-box” models, with an agent-based simulation it is possible to dig into the detailed activities to understand the origin of observed macroscopic phenomena, i.e., emergent behaviors at the system level.

Overall, the results of the first experiment show that, starting with very simple assumptions, we can capture some qualitative phenomena that match our observations of real-world companies: the presence of gender biases in promotions leads to an increasing gender imbalance at higher levels, and women being stuck in lower levels.

In the next section, we show how manipulating both hiring and promotion biases can accurately replicate real-world data on gender imbalances observed for specific industries.

5.2. Experiment 2: Combining Promotion and Hiring Biases to Match Industry-Specific Imbalances

While the patterns shown in Experiment 1 qualitatively look remarkably like those we observe in real companies, we wanted to see whether, using a minimal set of assumptions, we could match real-world data on gender imbalances for more specific cases. To this end, we used our simulation to match data from McKinsey’s and LeanIn’s “Women in the Workplace” report [31].

We ran simulations for ten years and measured the gender (im)balance at each rank. In all simulations, we modified two parameters from the baseline case: the promotion bias and the hiring bias. In general, as mentioned earlier, higher promotion biases create larger imbalances at higher ranks, and can lead to a reverse-imbalance at the entry level. In other words, if we think of the company’s gender balance as having the shape of a funnel going from the entry level up towards the executive level, that funnel is straight in the absence of biases, it becomes a bit narrower at higher ranks when the promotion bias is low and becomes dramatically narrower when the promotion bias is high. In contrast, the hiring bias has a direct impact on the number of women at the entry level, which will have a uniform impact on all subsequent ranks; hence, we expect that increasing the hiring bias but not the promotion bias would lead to a narrower but straight funnel.

In Figure 6, we show the comparison of ten-year gender balance data from our simulation and from the McKinsey report. The McKinsey report uses six levels (entry, manager, director, VP, SVP and C-suite), so we selected the four levels that match the ranks used in our simulation: entry, manager, VP and Executive (C-suite).

Starting with Figure 6A, we see that setting the promotion bias to 0.3 and the hiring bias to 0.4 results in just over 30% women at the entry level, and less than 20% women in the top ranks. This shape closely matches the gender imbalances observed in the Engineering and Industrial Manufacturing sector in the McKinsey study.

In Figure 6B, the promotion bias remains at 0.3, but the hiring bias is lowered to −0.1 so that the recruitment at entry level favors women. As expected, the bias favoring the hiring of women widens the base of the “funnel,” leading to approximately 60% women at the entry level; however, the high promotion bias narrows the funnel, leading to only 20% women in the top ranks. These results match closely the gender data reported by McKinsey for the Insurance sector.

In Figure 6C, the promotion bias is lowered to 0.2. With the absence of a hiring bias, women make up more than half of the entry level, while the somewhat lower promotion bias leaves nearly 25% women in the top ranks. These results closely match the gender data reported by McKinsey for the Banking and Consumer Finance sector.

Finally, Figure 6D shows a pattern that resembles the gender imbalances observed in the Retail sector, which tends to be dominated by women at the entry level, but with only a modest female representation at the top ranks. We obtained this graph by keeping the promotion bias to 0.2 but setting the hiring bias to −0.2 so that women receive favorable treatment when hiring into the entry level.

Accuracy of the Industry Simulations

To test the accuracy of our simulations of industry gender imbalances, we calculated the root-mean-squared deviation (RMSD) between each simulation and the data provided from the McKinsey study, given by the formula:

$$RMS{D}_{i }=\sqrt{\frac{{\left(L{O}_D-L{O}_S\right)}^2+{\left(L{1}_D-L{1}_S\right)}^2+{\left(L{2}_D-L{2}_S\right)}^2+{\left(L{3}_D-L{3}_S\right)}^2}{4}}$$

where RMSD_i is the RMSD for a given industry i; each squared term captures the difference between the gender balance data (subscript D) and the simulation (subscript S) at a given level Ln (with n representing rank, from 0 for the entry level to 3 for the executive level); and the division by 4 represents the fact that we are averaging the result across the four levels.

The analysis of the accuracy of the simulations shown in Figure 6 is summarized in

Table 3. Bias parameters and simulation accuracy for the four industries simulated in Experiment 2.

Promotion Bias	Hiring Bias	Industry Simulated	Variability Score
0.3	0.4	Engineering and Industrial Manufacturing	0.007
0.3	−0.1	Insurance	0.024
0.2	0	Banking and Consumer Finance	0.017
0.2	−0.2	Retail	0.015

. In all cases, the RMSD was below 3%.

Taken together, the results shown in Figure 6 and Table 3 show that, by changing just two parameters—promotion bias and hiring bias—our simulation is able to replicate real industry data with high accuracy.

5.3. Experiment 3: The Impact of Removing All Biases

Many companies nowadays have implemented diversity management policies in an attempt to create an inclusive environment for women and members of underrepresented minorities. One of the most important policies targeted at lowering managerial bias in promotions is diversity training. However, most of the research that examines the efficacy of diversity training such as seminars and workshops reveals that implementing diversity training as a single initiative will not lead to a more diverse organization, at least not in the short run [41,55,56]. Some studies show that bias awareness training could even cause biases to be strengthened after the training [57].

We wanted to test what would happen in our simulation if we completely removed all biases (hiring and promotion) for a company that was already showing a significant gender imbalance. In other words, we wanted to see whether creating equal hiring and promotion opportunities by eliminating all biases is sufficient to undo the gender imbalances originally caused by the biases. To test this, we repeated two of the same scenarios we tested in Experiment 1, but we set the promotion bias to zero after 10 years of operations. Once the promotion bias drops to zero, the promotion score is solely based on an employee’s seniority, meaning that advancement depends only on how long an employee has been at a given rank, and not on gender. The results are shown in Figure 7 and Figure 8.

The scenarios we chose for this experiment used the same parameters used to generate Figure 4 (promotion bias of 0.3) and Figure 5 (promotion bias of 0.5). Figure 7 and Figure 8 show that even if all biases could be removed, it can take a very long time to “undo” the damage done, especially at the upper levels.

Specifically, we can see in Figure 7 that, after ten years of operations with a gender bias of 0.3, men occupy 80% of the positions at the executive level, 75% at the VP level, 60% at the manager level and 45% at the entry level (as we mentioned earlier, the reason why there are more women than men at the entry level is because more women are left behind as men have a higher chance of being promoted to the manager level).

When all biases are removed after ten years, we see that the entry level and the manager level return to 50–50 within three years, as women are quickly promoted from entry level to manager level. However, it takes nearly ten years for the VP level to return to approximately a 50–50 level. As to the executive level, we see a gradual decline over the two decades following bias removal, but even after a full 30 years, men still maintain a 60–40 majority over women.

The results are similar when the promotion bias is initially set to 0.5: as shown in Figure 8, after ten years, the representation of men across the four levels is about 40% (entry level), 70% (manager), 85% (VP) and 90% (executive). As in the previous example, the two lower levels return to a 50–50 gender balance within about three years, the VP level reaches parity in just under ten years, while the executive level takes about 12 years to drop from 90% to 70%, but then declines only very gradually, and is still around 60% after three full decades without bias.

These results are relatively easy to understand when we think about the compounding effects of two different factors. First, as explained in an earlier section—and as is generally the case in real companies—the average tenure of employees is related to their rank, with employees at higher ranks staying longer than employees at lower ranks. Hence, it simply takes longer to “flush out” the imbalances at the higher ranks. Second, simply removing promotion biases does not cancel the existing disparities. Consider for example what happens at the manager level in Figure 8: just when the biases are turned off in year ten, 70% of the managers are male. If employees are promoted “fairly,” then for every ten managers promoted to VP level, seven of them will be men, which will continue to support gender disparity at the VP level.

This experiment leads us to a general conclusion about corporate gender biases; namely, that simply removing biases is not an effective way to create equality. Put another way, when systemic biases have created disparities, equal opportunity is not an adequate remedy. Our results suggest strongly that some sort of affirmative action that favors the placement of women into higher ranks—whether through more aggressive promotion or through external hiring—is needed, if balance is to be restored.

5.4. Experiment 4: The Impact of Promotion Biases on Individual Employees

One of the great benefits of agent-based simulations is that they make it possible to analyze at the same time the experiences of individuals and the overall company-level outcomes. This is perhaps the most powerful argument for using agent-based simulation as a core tool in managing talent in general: especially in the context of D&I, we tend to hear anecdotes about individual experiences, but most of the data reported is based on statistics measured at the company level, which hide the details of individual experiences. With agent-based simulation, we can do both: we can calculate population-level data just as we would in the real world, without losing any of the information about individual experiences. This makes it possible to explore the causal relationships that link individual experiences with company-level, emergent outcomes.

To illustrate this point, we conducted one final experiment to highlight how a company-wide promotion bias manifests itself in terms of the lived experiences of individual employees. In particular, while running simulations we noticed—not surprisingly—that women were often stuck at each rank, while men with less seniority were getting promoted ahead of them. Anecdotally, this is an experience that is often reported by women.

To quantify this phenomenon, we tracked how many times each simulated employee is passed over for promotion, and how much time it spends at a given rank. Specifically, each time an employee is promoted from a certain level, all other employees who are more senior increment a counter that tracks how many times they have been passed over for promotion. When an employee is promoted, we record the total amount of time it had spent at that rank (its “tenure-in-rank”) and the number of times it was passed over, and then we reset the counter to zero before heading to the higher rank. Note that because there is no promotion beyond the executive level, this analysis can only be performed for the first three ranks (entry level, manager and VP).

Because the number of employees differs at different ranks, and because the simulation tracks every promotion, entry-level employees were being passed over many more times than managers, who in turn were being passed over more times than VPs. To correct this confounding factor, we normalized the data by dividing the number of times passed over by the number of people at that rank, and multiplying by ten. This roughly corresponds to assuming that, at each rank, employees are only comparing themselves to a group of ten “peers.” We also normalized the passed-over data by the average number of years that employees spend at each rank, to obtain an average annual figure, which made the comparison across ranks more meaningful.

Figure 9 shows the results of this analysis. The top row shows data for VPs, the middle row for managers and the bottom row for entry level. Within each row, the left chart shows the average number of times passed-over in one year, while the right chart shows the average tenure-in-rank in years. Within each chart, there are four pairs of bars representing, from left to right, the level of promotion bias (0.0, 0.1, 0.3, 0.5); each pair of bars represents data for men (dark grey) and women (light gray). As with all other experiments, each data point was obtained by running the simulation ten times with different random seeds, and error bars show one standard deviation of the mean.

Several things are worth noting. First, when there is no promotion bias (the leftmost pair of bars in each plot), we see that the results for men and women are very similar, as expected. Note also that because promoted individuals are selected randomly from a promotion pool (as described in the section Simulating the Promotion Process), even in the absence of biases there will always be some employees that are passed over, which is why the leftmost bars at all three ranks are non-zero.

Focusing now on each rank (each row in the figure), we see consistent patterns both in the passed-over data and in the time-in-rank data. First, as the bias increases, all simulations show that women are passed over more times per year than their male peers, as expected (charts on the left). Second, we see that the average tenure-in-rank increases for women, while staying flat or even decreasing for men (charts on the right).

If we now compare the passed-over results across ranks (the charts in the left column of Figure 9), we notice another important trend. While at the entry level (bottom row) the average number of times passed-over for women seems to grow in a roughly linear proportion with the amount of promotion bias, for higher ranks this relationship seems to grow more geometrically: at the entry level (bottom row), raising the bias from 0.1 to 0.5 increases the number of times passed-over for women from 0.63 to 1.91—a ratio of 3.0; at the manager level (middle row), this ratio increases to 6.2 (from 0.5 to 3.1); at the VP level (top row), the ratio further increases to 15.1 (from 0.32 to 4.83). This, in retrospect, makes sense, because being passed over is compounded by the increasing proportion of men at higher ranks: the promotion bias gives each individual man an increasing chance of being promoted over a female peer, and there are a lot more men in each woman’s peer group. Hence, even in the absence of biases, more men than women would be promoted.

If we now focus on the tenure-in-rank data (the charts in the right column of Figure 9), we notice that the average time spent at each rank increases for women with increasing promotion bias, but this increase is fairly consistent across ranks. What is more interesting about the tenure-in-rank data is that, especially at the lower ranks, the average tenure-in-rank of men decreases while that of women increases. This result, while not entirely surprising, further quantifies the differences in the individual experiences of men and women that result when gender biases exist in the promotion process.

There is one aspect of our simulations that could be made even more realistic. Namely, in our current simulation, employees only separate from the company at random times. In contrast, being passed over repeatedly in real life would likely cause a decrease in satisfaction, which would in turn lead to higher turnover rates for women than for men [58]. Hence, the idea that women will remain stuck at a level for a much longer time than men is somewhat unrealistic, because in real life, women in those situations would be more likely to quit. This observation does not change our results but suggests that gender biases may have an even more acute negative impact on women in the workplace than shown in the simulations.

We are currently developing a version of this simulation in which each agent has an internal “satisfaction” variable that is impacted by being passed over for promotion, and which, in turn, increases the probability that an employee will voluntarily leave the company. In this case, we expect to see lower retention rates for women and immediate changes in the organization’s turnover rate [59]—phenomena that are common across male-dominated industries. We leave this additional analysis to future work, and simply point out another advantage of agent-based simulation, viz, the ability to add details to the simulation without compromising or invalidating the results obtained with a simpler simulation.

6. Discussion

We have introduced an agent-based simulation that captures, albeit in a simplified form, some of the gender imbalances that are observed across a variety of industries, supporting a demand-side explanation of observed corporate gender biases [23]. What is perhaps most striking about our findings is that we are able to capture several phenomena through very simple assumptions, and by varying only a small number of parameters.

Of course, the fact that our model is able to reproduce some of the observed phenomena does not mean that we are accurately capturing the true causes of these phenomena: it is possible that the mechanisms we hypothesized are not representative of real-world corporate functions, and that the similarity between our results and real-world observations are purely coincidental.

However, what we have been able to show is that if a company has gender biases in the way it promotes its employees, then, over time, the company will exhibit growing levels of gender imbalance, and that this imbalance will be increasingly pronounced at higher ranks within the organization.

Because our model is capturing the causal links between the behaviors of individuals and the emergent behaviors of a company, and because our model is very parsimonious in its assumptions, we are therefore confident that our model, while certainly simplistic, is capturing fundamental aspects of corporate functions that reflect real-world contexts.

There is another way in which our simulation offers indirect but strong support for a demand-side explanation of corporate gender biases: as we discussed in our description of Experiment 2, our simulation shows that reducing the number of women that are hired at the entry level (which could result from a smaller supply of women or from hiring biases) would result in an overall reduction in the representation of women that is consistent at all ranks. In other words, if we start the simulation assuming that only 40% of applicants are women, over time there will be 40% of women at every rank. A supply-side hypothesis would have to explain the progressive reduction in representation at higher ranks, and why this progressive reduction is observed so consistently across every industry. Any single proposed factor, such as child-bearing or competitiveness, would be unable to explain the observed results.

6.1. The Nature of Gender Biases

Our simulation treats gender bias as a single parameter. We are not making any claims about what is actually causing this bias, or whether the bias results from a single factor or from multiple factors. Rather, we believe that each company has a combination of individual biases and structural biases which jointly impact the probability that women will be selected for promotion.

At the individual level, studies have shown that managers often fall victim to a range of unconscious biases, such as implicit prejudice based on stereotypes, group favoritism or overclaiming credit for their own achievements [60]. At the structural level, a company may consider “face time” or other performance metrics that implicitly favor men as part of their promotion criteria. The idea behind our use of a single parameter is to show that, taken together, these kinds of biases combine to shape the experiences of individual employees in a way that differs for men and women. Although it is likely that on one hand, certain forms of bias are widespread, and on the other hand, each company may have unique forms of biases, the fact that different industries exhibit different levels of imbalances suggests that certain types of biases may be endemic to specific industries.

In the future, we hope that companies will be able to use agent-based simulations such as the one we have used here to estimate the level of gender biases that are likely to exist in their organizations, and then use that information as guidance to identify—and remove—specific sources of bias that impact their employees. For instance, by comparing the actual representation levels and advancement rates of men and women, the organization’s leadership could use the simulation to determine the level of bias that is likely to exist within specific levels of the organization. They could then enhance the simulation to include realistic elements of their organization, such as the size and demographic makeup of their staff as well as labor costs and the cost of hiring talent. Armed with this information, companies could use the simulation to obtain accurate estimates of the monetary losses that result from increased churn rates due to biases. In turn, they could estimate the financial benefit of identifying and removing those biases. In other words, they could use the simulation as a decision-support platform to guide their D&I strategy and to estimate the ROI of investing in targeted D&I initiatives.

6.2. Benefits of Agent-Based Simulation as a Management Tool

It is worth noting that most research on gender inequality in the workplace is conducted by applying statistical methods to observed data. This common approach has a number of limitations, including the fact that it is only capturing present conditions, and it removes any information about dynamics. Additionally, traditional analytical methods cannot model micro-interactions among individuals [61] and they tend to identify correlations that may or may not be due to causal relationships. As we highlighted in Experiment 4, our agent-based simulation preserves the individual experiences, and makes it possible to explore the causal relationships that link individual behaviors to the emergent behaviors observed at the company level. This is one of the greatest benefits of using agent-based simulation to study corporate workforce management in general, and D&I in particular.

Another significant benefit of agent-based simulation as a management tool is that, because it captures causal relationships rather than correlations, it can also be used to explore the likely impact of different initiatives. In other words, agent-based simulation can serve both as an explanatory tool and as a predictive modeling tool. For example, in this paper we used the simulation to estimate the likely impact of removing all promotion biases. In fact, we believe that the inability of corporations to predict the likely impact of different D&I initiatives, and the risks that D&I missteps can create, are some of the main reasons why, almost half a century after the passage of equal opportunity, corporate D&I across most industries has made little, if any, noticeable progress [62].

6.3. Future Research Directions

We see this project as the beginning of a systematic study of the impact of D&I on corporate performance. For instance, while the simulation we used in this paper focused only on advancement, we can use a similar approach to simulate other aspects of workforce management, including retention or recruitment. In fact, a former graduate student of one of the authors, for his Master’s thesis, developed an agent-based simulation that shows how job candidates are influenced by the perceived level of inclusion and diversity of a company, and how this will impact the talent pool available to any company and influence the cost of hiring [63]. We have also begun to simulate the impact of many other facets of D&I that impact the day-to-day experience of individual employees, how these experiences impact the satisfaction and productivity of individuals, and how this in turn impacts the overall performance of the company across a variety of performance indicators.

Even within the promotion-bias model itself, there are many ways in which we could increase the fidelity of the model to explore the impact of different assumptions and of different initiatives. We already mentioned our plans to extend the simulation to track employee satisfaction and its impact on retention and productivity. Similarly, we could also simulate how the presence of managers of a different gender may impact satisfaction and career advancement, or we could test alternative hiring practices, such as hiring people directly into higher ranks. In other words, this model can serve as the basis to explore a large number of hypotheses about the sources of gender disparities, and to test the likely impact of different interventions.

Finally, although in this paper we have focused on gender, and we treated gender only as a binary (male/female) variable, it is possible to capture a more nuanced and realistic representation of gender identity and to represent other personal characteristics that impact an employee’s experience, such as ethnicity, race, religious beliefs, sexual orientation, physical abilities, cognitive abilities and any other characteristic that may impact an individual’s experience within an organization. We have actually begun to develop some agent-based simulations that include other facets of diversity, and have already encountered some complex, fascinating issues that suggest entirely different ways of thinking about D&I and its impact on employee satisfaction and corporate performance.

In all, we are optimistic that our work can lead to a dramatic shift not only in how corporate leaders think about D&I, but, more importantly, how they manage it.