# Bathtub Dynamics

Failure to account for bathtub dynamics is a basic misperception of system structure, that occurs even in simple systems that lack feedback. Research shows that pattern matching, a common heuristic, leads even highly educated people to draw incorrect conclusions about systems as simple as the entry and exit of people in a store.

This can occur in any stock-flow system, which means that it’s ubiquitous. Here’s the basic setup:

Replace “Flow” and “Stock” with your favorite concepts – income and bank balance, sales rate and installed base, births and rabbits, etc. Obviously the flow causes the stock – by definition, the flow rate is the rate of change of the stock level. There is no feedback here; just pure integration, i.e. the stock accumulates the flow.

The pattern matching heuristic attempts to detect causality, or make predictions about the future, by matching the temporal patterns of cause and effect. So, naively, a pattern matcher expects to see a step in the stock in response to a step in the flow. But that’s not what happens:

Pattern matching fails because we shouldn’t expect the patterns to match through an integration. Above, the integral of the step ( flow = constant ) is a ramp ( stock = constant * time ). Other patterns are possible. For example, a monotonically decreasing cause (flow) can yield an increasing effect (stock), or even nonmonotonic behavior if it crosses zero:

If you add simple outflow feedback, there are even more possibilities. Here, we have our flow and stock, with an additional outflow that is a fraction of the stock per unit time:

In this case, with the same step up Flow pattern the stock can go up or down, depending on whether its initial level is such that the outflow exceeds the inflow, or not:

To ensure that the universe is as confusing as possible, there are some cases where pattern matching seems to work. For example, in steady state growth, where a flow grows exponentially at a constant rate, its associated stock will grow exponentially at the same rate (because the integral of e^x is e^x). Similarly, the integral of a ramp (c*time) is a parabola (c/2*time^2), which is approximately linear (i.e. resembling a ramp) over short time horizons. So, it’s not uncommon to see vehicle sales (flow) and fleet (stock) plotted together appear to have the same trend (especially with nonzero scales on the axes), or to see a consistent relationship between deficits (flow) and debts (stock). Pattern matching also works when a stock has time to come into equilibrium with the flows driving it. If the time horizon over which one observes the behavior is long with respect to the time constant of the stock’s dynamics, one sees the equilibrium relationship, stock = f(flow), relatively unobscured by integration.

My favorite example of bathtub misperceptions comes from Erling Moxnes work with renewable resource problems. In experiments with managing simulated reindeer herds, dependent on stocks of lichen for grazing, many participants adopt an inappropriately static mental model – that is, they ignore the dynamics of accumulation. They start with a reindeer herd that is too large, so that the flow of grazing depletes the stock of lichen. Taking corrective action, they reduce the reindeer herd, but not enough, so that the lichen stock continues to decline. At this point, they have some data: grazing has gone down, and lichen has gone down. Some question the static model and course-correct in the right direction (but not fast enough), but a subset of participants instead take pattern matching to its absurd logical end. They conclude that if grazing down->lichen down works, it follows that the way to increase lichen is to increase grazing. You can imagine how that works out.

Next up: the extension of all this to naive statistical comparisons.

#### 2 Responses to “Bathtub Dynamics”

• Nice post–illuminating, and not overly complicated.

• Thanks – that’s what I was shooting for.