How a Possession Simulator Thinks

Most projection tools do the same thing: they regress to a summary number — a points average, maybe a variance — and then, if they want two players' outcomes to move together, they staple a hand-tuned correlation matrix on top. That last step is where they quietly go wrong, because a correlation you type in by hand is a correlation you can get backwards.

The possession engine here works the opposite way. It never stores a teammate-correlation matrix at all. Instead it simulates the mechanism: a game is a chain of possessions, and each possession is used by exactly one of the five offensive players on the floor. The on-court five are sampled from real stint minutes with a 6.0 minutes-per-game rotation floor; the ball routes to a user in proportion to his usage raised to the 1.25 power, so primaries get fed more, exactly as they do in life. That player draws a turnover, a foul, or a shot; he scores or misses into an offensive-rebound loop; and then the possession is spent. The other four cannot use it. It is gone.

That single constraint — one finite pile of possessions, one user per trip — is the whole trick. Because a shot my teammate takes is a shot I cannot, our scoring is mechanically anti-correlated. Nobody tuned that. It is a consequence of the sampling.

Here is the receipt that makes it worth caring about. Read the teammate points-to-points correlation off the simulated games and compare it to real box scores:

• New possession sim: rho about -0.10, emergent, nothing imposed.
• Realized box scores: rho about -0.10, what actually happens.
• A prior in-house simulator that did impose a matrix: rho +0.65 — wrong sign and wrong magnitude.

The module src/sim/sgp_from_sim.py measures the emergent value at -0.104. So the old approach did not just miss the number; it got the direction of the relationship inverted, insisting teammates rise and fall together when in fact they trade off. Getting a joint quantity right by construction is the entire reason to simulate the mechanism instead of the summary — a fitted matrix can be fit to the wrong thing, and this one was.

The joint structure is not even a separate object. It is just the block of per-simulation realizations the chain already produced, so pricing any combination of outcomes is a boolean mask over the sims: the fraction of runs where every leg lands. Same-player legs correlate positive; teammate scoring legs correlate negative; the mispricing direction is explicit rather than assumed. Honest scope: the joint-calibration harness grades the sim's joint against the realized joint at roughly 50/50 lines to isolate the correlation, and it beats an independence model where correlation matters — but no same-game-parlay dollar edge is claimed, because there is no real parlay-price capture on disk to grade against.

The engine also grades itself. A cross-sport heatmap reshapes each sim's own walk-forward validation into per-game-state cells and ranks the worst ones. The worst named NBA cell is the third quarter at a near-even margin, and rather than paper over it, the builder cross-references the knowledge ledger and flags it as an un-probed mechanism gap. Tennis and soccer, which only simulate whole matches, get one honest whole-match cell with a coverage-gap note instead of fabricated buckets.

That is the hire signal, not the simulator itself: a generative model whose emergent joint behavior lands on reality by construction, which then decomposes its forecast quality by game state, names the cells it handles worst, and refuses to convert a validated structure into a betting claim it did not earn.

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