## Escalator Solutions

As promised, here’s my solution to the escalator problem … several, actually.

Before getting into the models, a point about simulation vs. analytic solutions. You can solve this problem on pencil and paper with simple algebra. This has some advantages. First, you can be completely data free, by using symbols exclusively. You don’t need to know the height of the stair or a person’s climbing speed, because you can call these Hs and Vc and solve the problem for all possible values. A simulation, by contrast, needs at least notional values for these things. Second, you may be able to draw general conclusions about the solution from its structure. For example, if it takes the form t = H/V, you know there’s some kind of singularity at V=0. With a simulation, if you don’t think to test V=0, you might miss an important special case. It’s easy to miss these special cases in a parameter space with many dimensions.

On the other hand, if there are many dimensions, this may imply that the problem will be difficult or impossible to solve analytically, so simulation may be the only fallback. A simulation also makes it easier to play with the model interactively (e.g., Vensim’s Synthesim mode) and to incorporate features like model-data comparisons and optimization. The ability to play invites experimentation with parameter values you might not otherwise think of. Also, drawing a stock-flow diagram may allow you to access other forms of visual thinking, or analogies with structurally similar systems in different domains.

With that prelude, here’s how I conceived of the problem:

• You’re in a building, at height=0 (feet in my model, but the particular unit doesn’t matter as long as you have and check units).
• Stairs rise to height=100.
• There’s an escalator from 100 to 200 ft.
• Then stairs resume, to infinite height.
• The escalator ascends at 1ft/sec and the climber at 1ft/sec whether on stairs or not.
• At some point, the climber rests for 60sec, at which point their rate of climb is 0, but they continue to ascend if on the escalator.

Of course all the numbers can be changed on the fly, but these concepts at least have to exist.

I think of this as a problem of pure accumulation, with height as a stock. But it turned out that I still needed some feedback to determine where the climber was – on the stairs, or on the escalator:

At first it struck me that this was “fake” feedback – an accounting artifact – and that it might go away with an alternate conception. Here’s my implementation of Pradeesh Kumar’s idea, from the SDS Discussion Group on Facebook, with the height to be climbed on the stairs and escalator as a stock, with an outflow as climbing is accomplished:The logical loop is still there, and the rest of the accounting is more complex, so I think it’s inevitable.

Finally, I built the same model in Ventity, so I could use multiple entities to quickly store and replicate several scenarios:

Looking at the Ventity output, resting on the escalator is preferable:

While resting on the stairs, nothing happens. While resting on the escalator, you continue to make gains.

There’s an unstated assumption present in all the twitter answers I’ve seen: the escalator is the up escalator. I actually prefer to go up the down escalator, though it attracts weird looks. If you do that, resting on the escalator is catastrophic, because you lose ground that you previously gained:

I suspect there are other interesting edge cases to explore.

The models:

Vensim (any version): Escalator 1.mdl

Vensim, alternate conception: Escalator 1 alt.mdl

Vensim Pro/DSS/Model Reader – subscripted for multiple experiments: escalator 2.mdl

Ventity: Escalator 1.zip

JJ Lauble has also created a version, posted at the Vensim forum. I haven’t had a chance to explore it yet, but it looks like he may have used Vensim to explore the algebraic solution, with the time axis as a way to scan the solution space with Synthesim overrides.

## Encouraging Moderation

An interesting paper on Arxiv caught my eye the other day. It uses a simple model of a bipolar debate to explore policies that encourage moderation.

Some of the most pivotal moments in intellectual history occur when a new ideology sweeps through a society, supplanting an established system of beliefs in a rapid revolution of thought. Yet in many cases the new ideology is as extreme as the old. Why is it then that moderate positions so rarely prevail? Here, in the context of a simple model of opinion spreading, we test seven plausible strategies for deradicalizing a society and find that only one of them significantly expands the moderate subpopulation without risking its extinction in the process.

This is a very simple and stylized model, but in the best tradition of model-based theorizing, it yields provocative counter-intuitive results and raises lots of interesting questions. Technology Review’s Arxiv Blog has a nice qualitative take on the work.

The model runs in discrete time, but I’ve added implicit rate constants for dimensional consistency in continuous time.

These should be runnable with any Vensim version.

If you add the asymmetric generalizations in the paper’s Supplemental Material, add your name to the model diagram, forward a copy back to me, and I’ll post the update.

## Bifurcations from Strogatz’ Nonlinear Dynamics and Chaos

The following models are replicated from Steven Strogatz’ excellent text, Nonlinear Dynamics and Chaos.

These are just a few of the many models in the text. They illustrate bifurcations in one-dimensional systems (saddle node, transcritical, pitchfork) and one two-dimensional system (Hopf). The pitchfork bifurcation is closely related to the cusp catastrophe in the climate model recently posted.

Spiral from a point near the unstable fixed point at the origin to a stable limit cycle after a Hopf bifurcation (mu=.075, r0 = .025)

These are in support of an upcoming post on bifurcations and tipping points, so I won’t say more at the moment. I encourage you to read the book. If you replicate more of the models in it, I’d love to have copies here.

These are systems in normal form and therefore dimensionless and lacking in physical interpretation, though they certainly crop up in many real-world systems.