Mediators of History

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).