# Why the properties of individual agents may be ignored?

Human behavior is perplexingly complex. Why their collective behavior is so well described by rather general mathematical models using very few parameters? Why do they not need deeper insight into the human psychology or decision making? One of the simple answers - if the statistical signature is not present in the data, which is usually aggregated at least to some degree, we can do nothing about it. Namely, usually we do not observe individuals making decisions and as such we are will not be able to differentiate between different mechanisms of human decision making. There are two major mechanisms of human decision making - homophily (selecting your peers) and peer pressure (adopting your peers behavior). Mathematically there usually will be no difference between them, both mechanisms can be be described using the same Kirman's model.

In this text we will consider Bass diffusion model with heterogeneous agents (each of them having his own independent parameters). We will show that the heterogeneous model produces similar macroscopic dynamics as homogeneous model. To simplify matter even further we will use unidirectional Kirman's model.

Let me remind you that Bass diffusion model describes diffusion of new durable products (or new technologies) in the market. Namely, agents in this model make a choice to adopt the product (technology) or not to. After becoming adopter the agent can no longer go back. The original model is driven by two parameters - idiosyncratic behavior parameter \( \sigma \) (how much agents are influenced by ad campaigns) and herding behavior parameter \( h \) (how much agents are influenced by their peers). Here we will assume that each agent has his own intrinsic \( \sigma_i \) and \( h_i \) values. In the app below these values are sampled from uniform distribution (the lower and upper bound of the value distribution are model parameters).

In this case agent identified by index \( i \) will adopt the product (technology) with probability:

\begin{equation} p_i = \left( \sigma_i + h_i \frac{X}{N} \right) \Delta t, \end{equation}

here \( X \) is a current number of adopters, while \( N \) - total number of agents, \( \Delta t \) - short intrinsic model time step. If agent has already adopted, then his transition probability is zero, \( p_i = 0 \). The one step transition probability for a system as whole:

\begin{equation} p(X \rightarrow X+1) = \frac{1}{N} \sum_i p_i =\frac{1}{N} \sum_i (1-S_i) \left( \sigma_i + h_i \frac{X}{N}\right) \Delta t = (N-X) \left({\bar \sigma} + {\bar h}\frac{X}{N} \right) \Delta t, \end{equation}

here we have introduced variable \( S_i \), which describes the current state of agent identified by index \( i \) (\( S_i=1 \) if agent has adopted, while \( S_i=0 \) if he is not). This expression should hold well if there are not qualitative differences and/or significant correlations between the intrinsic parameter values held by agents. It is rather hard to define what kind of differences would constitute a "qualitative difference", e.g., if significant number of agents have \( \sigma_i=0 \) ir \( h_i =0 \), then they will distort the dynamics in a different way - they will decrease the effective number of agents acting in the system, \( N \).

According to the discussion above the heterogeneous should be well approximated by the Bass diffusion equation with parameter values equal to the averages of agents' intrinsic values.

In the app below we have implemented the heterogeneous model as well as original Bass diffusion equation. As the heterogeneous model is being evaluated (red circles are drawn to show its evolution), the original Bass diffusion equation is also being solved (blue line show its prediction). Note that dynamics of heterogeneous model and the prediction matches rather well. Before trying the app out note that \( \Delta t \) here only sets the plotting time step and not the intrinsic model time step, which is actually set automatically.

We invite our readers to think critically about this topic. Not all models remain unaffected by the introduction of heterogeneity in agents' parameters. In some models, e.g., kinetic models of wealth distribution, parameter heterogeneity plays a crucial role in producing the main result.