The AGM postulates for belief revision in sports analytics

Origin: a problem from statute books

In 1985 Carlos Alchourrón, Peter Gärdenfors and David Makinson published "On the Logic of Theory Change" in the Journal of Symbolic Logic. Alchourrón was a legal theorist, and the problem he brought to the collaboration was concrete rather than philosophical: when a legislature repeals a statute, what happens to the rest of the code? You cannot simply delete the repealed clause and leave everything else standing, because other clauses may have been derived from it, cross-referenced it, or been justified in its light. Equally, you cannot rebuild the entire code from nothing every time one law changes. There had to be a principled middle path.

The paper formalised three operations on a belief set closed under logical consequence: expansion, where new information is simply added; contraction, where a sentence is given up along with whatever depended essentially on it; and revision, where an incoming sentence contradicts something already held, forcing both a removal and an addition at once. Eight postulates constrain revision. The result must be consistent whenever consistency is achievable. It must actually contain the new sentence. It must discard no more of the old belief set than the contradiction requires. And it must not depend on the syntactic form in which the new sentence happened to arrive — two logically equivalent inputs must produce the same revised state. The name later given to the whole programme is minimal mutilation: change your mind, but only exactly as much as the evidence compels.

Adam Grove supplied sphere semantics for the postulates in 1988, giving a geometric picture of nested spheres of plausibility around the current belief set, and Gärdenfors and Makinson formalised epistemic entrenchment the same year — an ordering over beliefs that decides which ones fall first when something has to give. The apparatus is fifty years old in spirit and forty in formal detail. It was built for legal codes. It turns out to describe, with uncomfortable precision, what happens on a Monday morning in a football club's analytics department.

The scouting report that outlived its evidence

A performance analyst's working object is not unlike a legal code: a closed set of accepted claims about an opponent, built up over weeks, cross-referenced, used to justify a game plan. The claims come from tracking data — twenty-five frames a second of player and ball position — injury bulletins, transfer windows, and logged tendencies: this full-back overlaps on the outside in the final third seventy per cent of the time when his team is a goal up.

The characteristic failure is not that the data was wrong when collected. It is that the world moved and the belief set did not. The opponent's coaching staff noticed the overlap being exploited, retrained the full-back into an inverted role three matches ago, and the tendency simply stopped occurring. The analyst's dossier, built on last month's tape, still asserts it. The press-training pattern used to spring the winger no longer describes anything. The game plan is executed against a version of the opponent that has already been retracted by reality, and nobody told the belief set.

This is a contraction problem stated exactly in Alchourrón's terms. A single sentence in the accepted set — full-back overlaps outside — has lost its support. The correct response is not to discard the whole report and rebuild from zero, because most of the report (the goalkeeper's distribution bias, the centre-back's recovery pace, the set-piece marking scheme) is still true. The correct response is targeted: retract the one sentence, retain everything not entailed by it, and do so as soon as the evidence — three matches of tracking data showing the overlap gone — arrives. Minimal mutilation, not wholesale re-scouting.

Where the frozen dossier and the matchday feed sit on this axis

A written scouting report is functionally a Large Language Model: a closed set of claims fixed at the point of compilation, with no operator for revising itself against anything. If the opponent changes shape, the report does not know. It can only be replaced — a new report commissioned, the old one binned — and replacement is not revision in any AGM sense. It is not minimal, because there is no mechanism guaranteeing the new report agrees with the old one everywhere the old one was still correct. It simply starts over.

Live tracking data during the match itself behaves more like a Large World Model. It supplies genuine new sentences — this opponent is now pressing with a back three, not four — and a good in-game system can revise the working model of the opposition while the ball is live. But the moment the final whistle sounds, that revised state is not carried forward automatically. The tracking feed for this match ends; next week's opponent generates a fresh scene from zero. Unless someone manually folds this week's revision into next week's dossier, the update is scene-bound and dies with the scene.

What the domain actually needs — and what most professional analytics departments now build toward, awkwardly, with a mixture of spreadsheets, Opta feeds, injury APIs and human judgement — is a belief store that never closes. Tracking data streams continuously, in season and out. Injury reports arrive daily. Transfer activity changes squad composition mid-cycle. Opponent tendencies are logged match by match and expected to update the working model automatically, not on the next scouting trip. This is the third position on the intake axis: a Large Universe Model, in the argued sense, is the first point at which the AGM signature is fully present — a persistent belief set, an unending sequence of incoming sentences, and an entrenchment ordering that decides, every time, what yields.

Two objections worth taking seriously

AGM is a theory of one revision. It says almost nothing about revising a revision, which is precisely what a season-long stream demands.

This is correct and well documented. The original postulates constrain how a belief set changes given a single new sentence; they are silent on what should happen to the entrenchment ordering itself once that first revision is made, which is why Adam Darwiche and Judea Pearl had to add further postulates in 1997 to control iterated revision. A theory that struggles past step two looks like thin support for a claim about a stream that never stops.

But the objection tells against the frozen dossier more than against the standing model. The Darwiche–Pearl repair works by making the whole epistemic state — beliefs plus entrenchment ordering — the object that gets revised, not just the flat set of accepted sentences. That is precisely what a provenance-carrying opponent model is: each tendency is stored with the ordering information (how many matches support it, how recently, against what quality of opposition) needed to decide what yields next time. A once-per-window scouting dossier has no such object to carry forward at all; it is rebuilt, not iterated. The iteration problem is an argument for building the persistent store properly, not an argument against needing one.

Real match data is graded and noisy — expected-threat models, possession-adjusted metrics, probabilistic pressing indices. Bayesian updating already handles this. AGM's talk of consistent sets is a poor fit for a discipline that runs on continuous statistics.

Granted for the numerical layer. Updating a full-back's overlap frequency from 70% to 55% given three new matches is exactly Bayesian conditioning, and no analyst needs axioms of theory change to do that arithmetic. AGM's residue is elsewhere: the cases where conditioning has no prior to update, because the category itself has changed. A player who was scouted as a wing-back gets redeployed by his club as an inverted full-back — the schema slot he occupied no longer describes his role, and no amount of updating the old number fixes that. A long-term injury retires an entire cluster of tendencies built on a player who is no longer selected. These are contractions of the model's structure, not revisions of a probability, and they recur every transfer window and every injury bulletin. That is where minimal mutilation and an explicit retraction record earn their place alongside the statistics, not instead of them.

What is left to build

Nothing in this argument says the entrenchment orderings clubs currently run are good. Most are informal, held in an analyst's judgement rather than an auditable record, and plenty of tendencies survive past their expiry because nobody flagged the contraction. That is a problem of execution: better provenance, faster propagation from tracking feed to game plan, clearer rules for what outranks what when two incoming sentences conflict. None of it is a new kind of input. The stream already includes tracking data, injury status, transfer activity and opponent behaviour — everything a rival's analytics department could conceivably supply. Improving how it is entrenched is real work. It is not a fourth argument to the operator.