Epistemic note: all of the examples in this post are very simplified for ease of consumption. The core idea applies just as well to the real systems in all their complicated glory, however.

When oil prices change, oil producers adjust in response - they drill more wells in response to higher prices, or fewer wells in response to lower prices. On the other side of the equation, oil prices adjust to production: when OPEC restricts output, prices rise, and when American shale wells expand, prices fall. We have a feedback loop, which makes it annoying to sort out cause and effect - do prices cause production, or does production cause prices?

The narrative: from roughly 2010-2014, OPEC successfully restricted their production enough to keep oil prices around $100/barrel. But at that price, American shale wells are extremely profitable, and they grew rapidly - the dots in the map below are each an oil/gas well in the Eagle Ford basin in Southern Texas, and the graph above shows American oil production in red. This situation was not sustainable; prices eventually dropped to around $50/barrel, which is roughly the marginal cost of American shale. Since then, prices rose above $50/barrel again around 2018, and American shale once again grew rapidly in response.

In this case, there is a useful sense in which production capacity causes prices, not the other way around - at least if we omit OPEC agreements.

Oil is a fairly liquid commodity. When there’s a shock in supply (e.g. OPEC agreeing to restrict output) or demand (e.g. lockdowns), the markets respond and prices rapidly adjust. Production capacity, on the other hand, adjusts slowly: drilling new wells and building new pipelines takes time, and once a well is built it rarely makes sense to shut it down before it runs dry. So (ignoring OPEC) prices right now are caused by production capacity right now, but production capacity right now is not caused by prices right now - it’s the result of prices over the past several years, when the wells were drilled.

Or, to put it differently: production capacity mediates the effects of historical prices on current prices. It’s a “mediator of history” - a variable which changes slowly enough that it carries information about the past. Other variables equilibrate more quickly, so they depend on far-past values only via the mediators of history.

(Incorporating OPEC into this view is an exercise for the reader.)

Another example: each of our cells’ DNA is damaged hundreds or thousands of times per day - things like strand breaks or random molecules stuck on the side. Usually this is rapidly repaired, but occasionally it’s misrepaired and a mutation results - a change in the DNA sequence. On the other side, some mutations can increase DNA damage, either by increasing the rate at which it occurs, or reducing the rate at which it’s repaired. So damage causes mutations, and mutations can cause damage.

Visualizations of some kinds of DNA damage, and keywords to google if you want to know more about them.

Here again, there is a useful sense in which mutations cause damage, not the other way around: the damage right now is caused by the mutations right now, but the mutations right now were caused by damage long ago. The mutations are a mediator of history.

This has important implications for treating disease: we can use antioxidants to suppress (some types of) DNA damage, but that won’t remove the underlying mutations. As soon as we stop administering antioxidants, the damage will bounce right back up. Worse, we probably won’t prevent all damage, so mutations will still accumulate (albeit at a slower rate), and eventually the antioxidants won’t be enough. On the other hand, if we can fix the problematic mutations (e.g. by detecting and removing cells with such mutations), then that “resets” the cells - it’s like the earlier damage never happened at all.

Change the mediators of history, and it’s like history never happened.

A third example: a robot takes actions and updates its world model in response to incoming data. It uses the world model explicitly to decide which actions to take, but the actions chosen will also indirectly influence the world model - e.g. the robot will see different things and update the model differently depending on where it goes. However, the action being taken right now does not influence the world model right now; the world model depends on actions taken previously. So, the world model mediates history.

Here, it’s even more obvious that changing the mediator of history makes it like history never happened: if we reset the robot’s world model to its original state (and return it to wherever it started in the world), then all the influence of previous actions is erased.

In general, looking for mediators of history is a useful tool for making sense of systems containing feedback loops. In chemistry, it’s the fast equilibrium approximation, in which the overall kinetics of a reaction are dominated by a rate-limiting step. In physics more generally, it’s timescale separation, useful for separating e.g. wave propagation from material flows in fluid systems.

In plasmas, charged particles follow a wheel-like motion - they orbit around magnetic field lines with drift superposed. When the orbital motion is on a fast timescale relative to the drift, we can average it out - see gyrokinetics.

The most common application of the idea in chemistry and physics is to simplify equations when we’re mainly interested in long-term behavior. We can just assume that the fast-equilibrating variables are always in equilibrium, and calculate the rate-of-change of the mediators of history under that assumption. In many systems, only a small fraction of the variables are mediators of history, so this approximation lets us simulate differential equations in far fewer dimensions.

From a modelling perspective, the mediators of history are the “state variables” of the system on long timescales. This is especially important in economic models, since the state variables are what agents in the models need to forecast - e.g. stock traders mainly need to know how mediators of history will behave in the future. If they know that, then the rest is just noise plus an equilibrium calculation.

Finally, in terms of engineering, mediators of history are key targets for control. For instance, if we want to cure aging, then identifying and intervening on the mediators of history is the key problem - they are both a necessary and a sufficient set of intervention targets. That actually simplifies the problem a lot, since the vast majority of biological entities - from molecules to cells - turn over on a very fast timescale, compared to the timescale of aging. So there are probably relatively few mediators of history, relative to the complexity of the whole human body - we just need to look for things which turn over on a timescale of decades or slower (including things which don’t equilibrate at all).


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Note to self: "rate-limiting step" is the analogy you want here.