Rain falls on a topographic map. Drops landing on one slope flow into River A; those on the other slope, into River B. From the twists of the rivers you can infer the terrain. But an engineer sees only the outflow points and knows which slope the water came from, not where each drop originally fell. That’s the economist’s task: given a binary instrument (like program participation), recover the hidden structural model.

Once it was thought that you’d need a rich, continuous instrument. It turns out one property, borrowed from nature, is enough: water always flows downhill and never loops back. This is called cyclical monotonicity (no whirlpools). It’s what allows you to extract a unique structure even from a simple yes/no.

The key lies in optimal transport: a method for matching distributions at minimal cost. For each slope, a rank map is built: it pairs each point on one slope with a point on the other that has the same “amount of water.” The watershed is the set where the map doesn’t shift the point ( $R_z$(x) = x ).

Take any point and repeatedly switch the slope, applying the rank map; it inevitably slides down to the watershed. Meanwhile, the spread of the invisible factors remains intact. Thus, from a binary instrument, a single causal model is born. Two researchers looking at the same data will arrive at the same conclusion. Now, gazing at the river map, we see not just meanders but secret relief lines etched by the runoff of invisible drops.