### Conquering a Market by Killing It

One of the simplest models of the spread of innovations is the Bass Diffusion Model, proposed by Frank Bass in the 1960s.

Suppose we have a population of people using a product, perhaps a drug to treat a chronic illness that patients will have for the rest of their lives.

A new drug is developed and we would like to estimate how quickly the population will adopt the new treatment.

Bass’s model says that there are two ways by which users of the old product will switch to the new one: Some will naturally ‘innovate‘ – that is, they will switch because they have a special need or an extreme case of the disease and are not being properly served by the current solution. Maybe they switch of their own volition, or a doctor or advertisement convinces them to try the new drug. It is suggested that the rate at which this occurs is simply some portion (p) of the fraction of the population that use the old drug.

Others will switch because they ‘imitate‘ a friend or acquaintance using the new drug. By word-of-mouth, perhaps, when a user of the old solution bumps into a user of the new one, there is a chance (q), that the old user will switch to the new product.

Visually, these two switching processes (innovation and imitation) can be represented by the following diagram:

There is a population of users of the old solution (Potential Adopters) who switch to the new solution (Adopters) by either innovating or imitating. For this exercise, I have assumed that the initial number of Potential Adopters is 99% and initial number of Adopters is 1%, and p = 0.05 and q = 0.6.

I have also modified Bass’s model to consider the case where Adopters use the new product only for a while, and then exit the market entirely. For example, perhaps the new drug cures people of their chronic disease, meaning that they no longer need any treatment at all. For this case, I say that 1% of the Adopters ‘discard’ the treatment each time period.

The graph below shows how the fraction of users using the new drug (that is, Adopters / (Adopters + Potential Adopters)) changes over time.

Interestingly, the time it takes for say, 98% of the market to be Adopters of the new drug is shorter when the Adopters ‘discard’ the treatment (due to being cured) than when no Adopters discard the treatment (Bass’s original model). In the long run, having Adopters discard the new treatment beats having them not do so (if our goal is only to have the largest percentage of active users using the new drug).

What this graph does not show is that, when discarding occurs, the total market shrinks by the end of 1000 days to just a couple of percentage points of its original size. In the original case, the market size stays constant. That is, we are conquering the market simply by killing it off – which in the case of a chronic disease is a good thing.

Personally, I don’t know how realistic this situation is, but it shows how even fairly simple systems can display nonintuitive behavior.