The Arm That Was Never Adversarial

I had a formula. It said that friction in a delegated system rises with the stakes, rises with the delegate's uncertainty, and falls with how well the delegate's preferences align with the principal's. Three comparative statics, one expression, and clean enough to put in a preprint.

It is wrong. Not wrong in the way that invites a patch — wrong in a way that took three independent derivations to characterise, and the characterisation is more useful than the formula ever was.

What we actually did

The formula had been fitted, argued for, and used. It had never been derived. So the question was narrow and answerable: is this expression the first-order expansion of the expected delegation gap about the origin, under any of the data-generating processes we actually run?

Three passes, deliberately unshared: an analyst working symbolically, a numericist working from simulation, and an adversarial referee whose only job was to try to break both. The headline is that all three converged. The instructive part is what happened on the way.

The arm that was never adversarial

Start with alignment, because it is the term the whole framework is named for.

The result is not that the alignment effect is small. It is that this design cannot identify it at all. Expected gap as a function of alignment satisfies an exact distributional identity under both of the processes we used to generate it: the value at α is equal to the value at −α. Exactly. Not approximately, not within noise.

Sit with what that means. The condition we had labelled opposition was, distributionally, a mirror image of the condition we had labelled cooperation. The experimental arm built to test whether adversarial preferences make coordination worse was incapable of showing that they do, because the design had a symmetry in it that nobody had looked for. The arm that was supposed to be adversarial was never adversarial.

That is a different kind of failure from a weak result. A weak result tells you the effect is small. An unidentified one tells you the experiment was never a test.

The term with no first-order part

Uncertainty was next, and it fails differently: there is no linear term to expand.

The gap approaches zero uncertainty as a square-root cusp — infinite right-derivative at the origin — and there is a genuine discontinuity at exactly zero, worth between eighteen and thirty-seven per cent depending on the arm, produced by nothing more principled than a tie-breaking convention. A formula linear in uncertainty is claiming a straight line at precisely the point where the curve has no straight part.

There was one concession available and I want to record it, because it is the sort of thing that keeps a dead result alive if you let it. Under one belief model, near the expansion point, the shape is locally close to what the formula says — fitted exponent 1.12 against a claimed 1. It dies anyway: it fails on higher order, it flips sign under the other belief model, and the exponent itself changes sign with alignment, which no product of separate factors can do.

The term that was exactly right and completely empty

Stakes is the one I find hardest to look at, because it was the term I trusted most.

It is exact. The delegated policy is provably independent of the stakes parameter, so the whole expression scales with it in precisely the way the formula says. And that is the problem: a scaling that holds by construction is a statement about units, not a finding about delegation. It cannot fail, so it cannot inform. Effective stakes on its own reaches an R² of 0.60 against the gap while containing no delegation content whatsoever — which is roughly the most efficient way I know to manufacture a result that means nothing.

As a single regressor across all three arms, the full expression achieves negative R². The information criteria prefer the alternatives by margins in the hundreds.

What the referee found that neither builder did

Here is the part I would keep if I could keep only one thing.

The analyst and the numericist agreed on the headline. They also, without either of them noticing, disagreed about the mechanism — one had run the uncertainty ladder under Bayesian beliefs, the other under certainty-equivalent beliefs, and neither had stated which. The two accounts read as mutual confirmation and were nothing of the kind. It took an adversary whose job was to disbelieve both of them to notice that the agreement was a coincidence of headline, not of mechanism.

Agreement is cheap. Agreement for the same reason is the thing worth buying, and you do not get it by adding people. You get it by making someone's job to find the seam.

What survived

Two of six claims, and one of those needed a new proof rather than a defence.

Zero stakes implies zero friction survives, but not for the reason originally given. It was presented as a corollary of the functional form. It is actually a fact about discreteness: agents choosing from a small set of actions against a continuous tolerance cannot land exactly, and the residual is bounded away from zero whenever the stakes are. Same statement, entirely different proof, and the new one is true in three independent codebases.

The bridge from friction to selection pressure survives untouched, because it only ever needed friction to be non-negative, which is now a theorem rather than an assumption.

The signed alignment denominator, the monotone uncertainty effect, and the baseline floor are withdrawn. So are two upstream numbers that depended on them — including one whose value exceeded its own analytic ceiling, which is the kind of thing that is obvious the moment somebody computes the ceiling and invisible for a year if nobody does.

The scope of the correction

The retraction is safe: it is derived in the environment the original experiments actually ran in, so it applies to exactly the claims that were made.

The replacement is not. What sits where the formula used to be is a function with named dependencies — belief model, delegation concept, equilibrium selection rule — and no closed form at all. Anyone quoting it has to name all three, because the sign of the effect changes with them. That is worse to read and considerably better to trust, and the trade is not a coincidence: the formula was quotable because it had suppressed the dependencies that turned out to carry the result.

What I would want taken from it

Nothing new was measured. There was no fresh dataset, no better instrument, no replication in another lab. Every fact above came from doing arithmetic on a design I already had and had already published from. The formula had been fitted and it fit; the question that killed it was whether it could be derived, and nobody had asked.

So: a formula that fits is not a formula that holds. If a term in yours cannot fail, it is telling you about your units. If an experimental arm is symmetric under the thing it was built to vary, it was never a test. And if two of your people agree, find out whether they agree for the same reason — because mine didn't, and it took an adversary to notice.