3.1.1 Simulating cognitive style

The agents’ exploration of the solution space depends on their total KAI score, as well as the three sub-factor scores of sufficiency of originality (SO), efficiency (E), and rule/group conformity (RG). People with higher KAI scores tend to make larger changes to a design in search of a different solution, while people with lower KAI scores tend to make smaller changes to refine an existing solution. In the simulation, the distance an agent moves in the solution space on one turn (iteration) is positively correlated with the total KAI score. Figure 3 illustrates the differences in step size for agents of adaptive, mid-range, and innovative styles in a 2-dimensional problem. When searching for new solutions, a more adaptive agent moves in smaller, incremental steps, tweaking the solution with marginal adjustments. In contrast, a more innovative agent moves in larger leaps, often generating ideas that are distant and very distinct from their current solution.

Figure 3. Agents with more adaptive styles (lower KAI scores) move in smaller steps on each iteration, while agents with more innovative styles (higher KAI scores) take larger steps in the solution space.

Each agent has an individual speed parameter with an initial value dependent on their total KAI score. Speed determines how closely or distantly related an agent’s subsequent solutions are. Speed is roughly analogous to novelty of solutions in real-world problems, in that high speed will lead to a series of very distinct solutions, while low speed will lead to a series of very similar solutions. The speed decays geometrically throughout the course of the simulation. The distance, $D$ , from the current solution to the next solution an agent generates is drawn from a chi ( $\unicode[STIX]{x1D712}$ ) distribution and scaled by the agent’s current speed,  $s$ :

(1) $$\begin{eqnarray}D=s\cdot \unicode[STIX]{x1D712},\end{eqnarray}$$

where $\unicode[STIX]{x1D712}$ is a random variable characterized by the chi probability density function with $k=1.9$ degrees of freedom. The minimum travel distance in one step is set to one ten-thousandth of the entire space, and there is no upper bound. The agent’s start speed, $s_{0}$ , is calculated as

(2) $$\begin{eqnarray}s_{0}=\unicode[STIX]{x1D707}_{s}+KAI^{\ast }\cdot \unicode[STIX]{x1D70E}_{s},\end{eqnarray}$$

where $KAI^{\ast }$ is the standardized KAI score (re-scaled for a mean of 0 and a standard deviation of 1), $\unicode[STIX]{x1D707}_{s}$ is the average starting speed for all agents, and $\unicode[STIX]{x1D70E}_{s}$ is the standard deviation of starting speed across all agents.

3.1.2 Model parameters

Parameters in the model are generally tuned to reflect observed human behavior. Except where noted, the parameters were held constant at the values listed in Table 1. An appendix contains a table listing all parameter values for each figure in the paper.

Table 1. List of model parameters and their default values

3.1.3 Efficiency

Efficiency describes an individual’s preference for structure in their working methods, which range from applying highly detailed and incremental improvements to a solution (more adaptive) to dramatically or tangentially altering a solution, with less regard for detail and quality (more innovative). The temperature parameter in simulated annealing determines the probability of an agent accepting a candidate solution that does not improve the objective function relative to their own current solution. In a real-world design setting, temperature corresponds to exploration. Early on in the design process, designers explore new solutions with little regard for quality in order to expand the known solution space. As the design process continues, solution quality becomes more important and exploration transitions to exploitation.

In KABOOM, an agent’s starting temperature is correlated with the E sub-score. Temperature always decreases geometrically over the course of the simulation, but both the start temperature $T_{0}$ and the cooling rate are a function of E. An agent with a higher (more innovative) E sub-score starts with a higher temperature, and thus a higher probability of accepting solutions that do not improve the objective function. This can cause them to leave good solutions behind and miss high-quality nearby solutions, but it can also allow them to escape local minima to find better solutions in a “rough” optimization topology. Agents with higher efficiency scores also cool down slower, meaning they might never reach a very low temperature or refine a single local solution. In contrast, an agent with a lower (more adaptive) E sub-score starts with a lower temperature, and thus a lower probability of accepting solutions that do not improve the objective function. This can cause them to quickly achieve locally optimal results, but it can also lead them to become stuck in local minima. The resulting behavior of the agents overall is that the more adaptive agents choose a solution early on and polish it to perfection, while the more innovative agents spend most of their time exploring very diverse solutions without refining them. The agent’s start temperature $T_{0}$ is calculated as

(3) $$\begin{eqnarray}T_{0}=\unicode[STIX]{x1D707}_{T}+E^{\ast }\cdot \unicode[STIX]{x1D70E}_{T},\end{eqnarray}$$

where $E^{\ast }$ is the standardized efficiency sub-score (re-scaled for a mean of 0 and a standard deviation of 1), $\unicode[STIX]{x1D707}_{T}$ is the average temperature for all agents, and $\unicode[STIX]{x1D70E}_{T}$ is the standard deviation of temperature across all agents.

The agent’s ratio of start temperature $r_{0}$ to final temperature $r_{f}$ is

(4) $$\begin{eqnarray}\frac{r_{0}}{r_{f}}=\frac{1}{\exp (2-E^{\ast })}\end{eqnarray}$$

which is bounded by $10^{-10}$ and 1. The geometric decay ratio for each time step is calculated based on start temperature, start-to-end ratio, and the number of steps in the simulation. Figure 4(A) shows the temperature over the course of a simulation for agents with more innovative (high E), mid-range (mid-range E), and more adaptive (low E) efficiency sub-scores. The agent’s speed decays with the same ratio as the temperature.

Figure 4. (A) Cooling schedules for agents with different efficiency sub-scores. (B) In calculating the perceived memory location, an agent’s past memories are weighted based on the serial position effect reflected in this curve: the most recent memories and earliest memories are recalled more easily than intermediate memories.

In simulated annealing, the probability that an agent will accept or reject a solution that is inferior to its current solution depends on its temperature and the difference in solution quality according to Equation (5) (Kirkpatrick et al. Reference Kirkpatrick, Gelatt and Vecchi1983). At high temperatures, an agent will accept solutions regardless of quality, but at low temperatures, agents only accept solutions that improve the objective function, as shown below:

(5) $$\begin{eqnarray}P_{\text{accept}}=\exp \left(\frac{f(\vec{x}_{n})-f(\vec{x})}{k_{B}T}\right),\quad f(\vec{x}_{n})<f(\vec{x}),\end{eqnarray}$$

where $P_{\text{accept}}$ is the probability that an agent with current solution $\vec{x}$ accepts a new solution $\vec{x}_{n}$ that does not improve the objective function $f$ as a function of current temperature $T$ and a constant $k_{B}$ . We can absorb $k_{B}$ into the calculation of temperature.

Figure 5 illustrates the effect of the efficiency style sub-factor on three agents’ exploration of the solution space. Compared to the mid-range agent (black), the more adaptive agent (blue) refines an early solution, while the more innovative agent (red) continuously explores without refining a solution.

Figure 5. Effect of E sub-factor on solution space exploration.

3.1.4 Sufficiency of originality

An agent’s desire to retain and modify known solutions or, conversely, to explore more unfamiliar ideas, is related to the SO sub-score of KAI. When agents evaluate a candidate solution in KABOOM, more adaptive agents are more likely to accept a candidate solution that directs them toward their previous solutions and are less likely to accept a solution that leads them away from previous solutions. On the other hand, more innovative agents prefer to choose ideas that move them further from ideas they have explored in the past and to reject ideas that lead them toward their previous solutions. An agent’s perception of its previous solutions is represented by a single point in the solution space, which is a weighted mean (centroid) of all of their previous positions or “memories”. The weights on an agent’s memories are strongest for the most recent memories, weakest for middle memories, and intermediate strength for the earliest memories (Figure 4(B)). This reflects the serial position effect (Colman Reference Colman2015), which describes how people tend to remember early memories (primacy effect) and recent memories (recency effect) more readily than intermediate memories.

The direction of the weighted memory position from the current position is a weighted average of the directions to all previous solutions the agent has visited. We call $\vec{v}_{\text{mem}}$ the perceived memory direction, where the weights are given by the primacy–recency bias function $Q$ :

(6) $$\begin{eqnarray}\vec{v}_{\text{mem}}=\mathop{\sum }_{n=1}^{N}(\vec{x}-M_{n})\ast Q(n)\end{eqnarray}$$

and where the memory $\boldsymbol{M}$ is a list of $N$ previous solutions (each memory being a vector of the same shape as the current solution $\vec{x}$ ). The primacy–recency bias function for memory $n$ of $N$ total memories is:

(7) $$\begin{eqnarray}Q(n)=n^{3}+0.4(N-n)^{3}\end{eqnarray}$$

which gives a U-shaped curve with stronger biases for the earliest and most recent memories, respectively, than for other memories (Figure 4(B)). The weights are calculated with this equation then normalized so that they sum to 1.

The direction of the candidate solution from the current solution is the vector difference between the candidate solution and the current solution, $\vec{v}_{n}=\vec{x}_{n}-\vec{x}$ . The dot product of these two directions indicates whether the new solution is in the direction of the perceived memory or away from the perceived memory. We call this dot product the originality  $\unicode[STIX]{x1D6FA}$ :

(8) $$\begin{eqnarray}\unicode[STIX]{x1D6FA}=\vec{v}_{\text{mem}}\cdot \vec{v}_{n}.\end{eqnarray}$$

Then, the SO preference $P_{SO}$ is:

(9) $$\begin{eqnarray}P_{SO}=\unicode[STIX]{x1D6FA}\cdot SO^{\ast }\cdot W_{SO},\end{eqnarray}$$

where $SO^{\ast }$ is the standardized SO score (re-scaled for a mean of 0 and a standard deviation of 1), and $W_{SO}$ is a global scaling constant that determines the strength of the SO preference. For this paper, we set $W_{SO}$ to 2, which creates a range of behaviors for agents across the KAI cognitive style spectrum. When an agent evaluates the quality of a candidate solution, the SO preference $P_{SO}$ and RG preference $P_{RG}$ (explained below) influence the perceived solution quality: the preferences are added to the true value of the objective function for the candidate solution. In other words, the perceived solution quality $f_{P}$ of a candidate solution $\vec{x}_{n}$ is related to the true solution quality $f(\vec{x}_{n})$ by

(10) $$\begin{eqnarray}f_{P}(\vec{x})=f(\vec{x})+P_{SO}+P_{RG}.\end{eqnarray}$$

Figure 6 illustrates the effect of the SO cognitive style sub-factor on three agents’ exploration in a 2-dimensional solution space. The black line shows the solution path of an agent with a mid-range cognitive style. The red line shows the path of a more innovative agent in the same space, with only the SO preference active (and otherwise identical to the mid-range agent). The innovative SO style gives the agent a preference for solutions that move it away from its previous solutions, leading the agent to explore further into the corner of the solution space. On the other hand, the blue line shows the more adaptive agent’s preference for solutions that are close to its previous solutions, leading to a dense packing of solutions near its original starting point.

Figure 6. Effect of SO sub-factor on solution space exploration.

3.1.5 Rule/group conformity

The third cognitive style sub-score, rule/group conformity (RG), describes an individual’s preference for managing structure in both impersonal and personal contexts. More adaptive individuals prefer to leverage existing impersonal structures, such as rules, guidelines, and precedents, while more innovative individuals are more likely to bend or violate these structures. A similar pattern of behavior emerges for personal structures, such as groups or teams. A person with an adaptive RG style tends to cohere with the group and seek group consensus, while a more innovative person tends to diverge from and may even disrupt the group.

In order to capture one aspect of the RG sub-factor, the present model focuses on conformity to personal structures (i.e. an individual’s desire to converge or diverge from group consensus). Future work could develop other aspects of RG, for instance, by modifying the agents’ adherence to optimization constraints. In KABOOM, the more adaptive agents are more likely to move toward solutions that bring them closer to the team’s average position, thus encouraging group cohesion. More innovative agents, on the other hand, have a preference for solutions that move them away from the mean position of the group.

In KABOOM, this aspect of RG is implemented in a similar way to SO by replacing an agent’s memories with the current solutions. In other words, while SO makes agents favor solutions toward (more adaptive) or away from (more innovative) their own previous solutions, RG gives agents a preference toward (more adaptive) or away from (more innovative) the current solutions of their teammates (Figure 7).

A “team position” $\vec{x}_{\text{team}}$ is represented by the centroid of the current solution of each member of the team:

(11) $$\begin{eqnarray}\vec{x}_{\text{team}}=\frac{1}{N}\mathop{\sum }_{n=1}^{N}T_{n},\end{eqnarray}$$

where $T$ is the team of $N$ agents, and $T_{n}$ selects the current solution of each agent on the team. As before, the direction of the candidate solution from the current solution is the vector difference, $\vec{v}_{n}=\vec{x}_{n}-\vec{x}$ . Likewise, the direction of the team position from the current position is $\vec{v}_{\text{team}}=\vec{x}_{\text{team}}-\vec{x}$ .

The dot product of these two vectors indicates whether the new solution is in the direction of the team (positive dot product) or away from the team (negative dot product). We call this dot product the conformity $C$ :

(12) $$\begin{eqnarray}C=\vec{v}_{\text{team}}\cdot \vec{v}_{n}.\end{eqnarray}$$

Then, the group conformity preference $P_{RG}$ is:

(13) $$\begin{eqnarray}P_{RG}=C\cdot RG\ast \cdot W_{RG},\end{eqnarray}$$

where $RG\ast$ is the standardized RG sub-score (re-scaled for a mean of 0 and a standard deviation of 1), and $W_{RG}$ is a global constant used to change the strength of the RG preference, which is set to 2 to create a range of behaviors.

Figure 7 illustrates the effect of the RG cognitive style sub-factor for adaptive and innovative teams, respectively, of three agents each. Agents with a more adaptive RG style prefer solutions that bring them toward their team, resulting in team convergence (left), while more innovative agents seek solutions away from their team, leading to team divergence (right).

Figure 7. Effect of RG sub-factor on homogeneous teams of adaptive style and innovative style, respectively (blue paths show adaptive agents and red paths show innovative agents).

3.1.6 Interactions and communication

In models of team problem solving, communication can be modeled using organically timed pairwise interactions between team members (McComb et al. Reference McComb, Cagan and Kotovsky2015). In KABOOM, agents collaborate and share solutions by sharing their current positions in the solution space with each other. While communication can be influenced by many social and individual factors (as in Patrashkova-Volzdoska et al. Reference Patrashkova-Volzdoska, McComb, Green and Dale Compton2003; Singh et al. Reference Singh, Dong and Gero2013), this work focuses specifically on the influence of cognitive style on communication. On a given turn, an agent can choose to either explore a new solution individually or communicate with another agent to share solutions. The probability of an agent choosing to communicate in a pairwise interaction on a given turn is set by a model parameter $c$ , which can be constant in time or change over time. On each turn, agents who decide to collaborate are paired randomly, regardless of their sub-team. While future work may give agents preferences for communicating with specific individuals, groups, or networks, the current model chooses the nominal case where any agent interacts with any other without preference. If there is an odd number of agents who wish to collaborate, the unpaired agent explores individually for that turn.

After sharing their current solutions, each agent evaluates the solution shared with them and chooses whether they want to move to that solution. As with evaluating a potential solution in individual exploration, the probability of accepting a solution that does not improve the objective function in favor of the current one is a stochastic function of the agent’s temperature (Equation (5)). When the temperature is zero, there is zero probability of choosing a solution that does not improve the objective function.

Research shows that when the difference in two people’s KAI total scores is greater than 20 points, communicating ideas with each other tends to become increasingly difficult (Kirton Reference Kirton2003). Agent collaboration in KABOOM reflects this increasing difficulty in communication due to the cognitive gap in style. In the model, communication between two agents always has some probability of failing; this likelihood is positively correlated with the difference between the agents’ KAI total scores (i.e., their cognitive style gap). This is implemented by requiring a uniformly distributed random variable to be less than the difference in two agents’ KAI total scores for successful communication. If communication is not successful, no information is shared between the agents, essentially resulting in a wasted iteration.

The probability of successful collaboration is:

(14) $$\begin{eqnarray}P=\left\{\begin{array}{@{}ll@{}}1,\quad & \text{if }\unicode[STIX]{x1D6E5}KAI\leqslant 10\\ 1-(\unicode[STIX]{x1D6E5}KAI-10)/170,\quad & \text{if }\unicode[STIX]{x1D6E5}KAI>10,\end{array}\right.\end{eqnarray}$$

where $\unicode[STIX]{x1D6E5}KAI$ is the difference between the two agents’ total KAI scores. Agents with KAI total score differences of 10 points or less have a 100% success rate, which drops linearly beyond the 10-point just-noticeable difference (JND) established in prior research (Kirton Reference Kirton2003). Communication across extremely large gaps of 100 points is modeled as only being successful 50% of the time (the observed KAI range is 109 points (Kirton Reference Kirton1976)).

3.1.7 Team meetings

In addition to pairwise communication, agent teams have regularly scheduled meetings in which all team members converge to a single solution. First, the team creates an aggregate solution from the specialized sub-teams: each sub-team finds the best solution from any of its agents, then contributes only its dimensions to the aggregate team solution. For example, if there is a 2-dimensional solution space where sub-team 1 controls $x_{1}$ and sub-team 2 controls $x_{2}$ , the aggregate solution is $\langle x_{1}\text{ from sub-team 1},x_{2}\text{ from sub-team 2}\rangle$ . Finally, all agents accept the new aggregate position, regardless of the quality of the new aggregate solution.

3.2.1 Selecting agents for a team

When forming a team of agents, KABOOM can select individuals from the virtual population randomly or according to a team composition rule. This paper uses three team composition strategies:

1. (1) organic composition: team members are drawn randomly from a virtual population that is statistically representative of the true distribution of KAI scores;

2. (2) homogeneous composition: all team members have the same KAI total score (but likely have different sub-scores);

3. (3) linearly distributed heterogeneous composition: the team is composed with a given mean and range of KAI total scores. The team will be composed of agents linearly distributed across the range and centered on the mean. For example, a linearly distributed 5-person team with mean KAI of 100 and a range of 40 will be composed of five agents with KAI scores 80, 90, 100, 110, and 120.

   When teams are divided into specialized sub-teams, each sub-team (rather than the full team) is selected to have a linearly distributed composition with the given mean and range. For example, given a team of 4 with 2 sub-teams of 2, and requiring a KAI mean of 100 and range of 40, each sub-team will be composed of two agents with KAI scores of 80 and 120.

It is worth noting that, in real management scenarios, managers often must compose new teams from a limited group of current employees. This scenario is explored in other work (Lapp, Jablokow & McComb Reference Lapp, Jablokow and McComb2019). In the current work, all teams are composed by drawing from the virtual population. When plots compare agents or homogeneous teams of three styles, the KAI scores are 55 (adaptive), 95 (mid-range), and 135 (innovative). These represent the 1.5th, 50th, and 98.5th percentiles of the population; $\text{KAI}=55$ and $\text{KAI}=135$ represent extremes of cognitive style behavior that are unlikely (but not impossible) to be observed in real life. These are intended to demonstrate the range of possible behaviors across the KAI spectrum. Organic composition gives more realistic team compositions.