Mirrored seed pairs#

Status: closed ✅ (question settled)

An experiment testing what a mirrored rematch actually measures. A mirrored rematch replays a battle with the seats exchanged on the same seed pair, and the usual argument for it is that the two games cancel each other's luck. The tuning notes used to claim the opposite, that mirroring biases a comparison instead of reducing its noise. Both statements came from intuition about piece queues and had never been measured, so this experiment measured them.

Shared battle vocabulary (seat, seed pair, mean, standard error, z) is defined in Theta comparison; board and placement terms in the Glossary. This page adds:

The two claims#

• The bias claim: a lucky theta can win both battles of a mirrored pair, so folding the pair overstates that theta.
• The noise claim: a mirrored pair still contains two battles instead of one, so it is a cheap way to reduce variance.

Both depend on one assumption: that exchanging the seats on the same seed pair produces a different battle. If it produces the same battle seen from the other side, neither claim survives.

Method#

Two thetas were compared: the shipped production theta in the first seat, and the mean of an in-progress tuning run, a candidate that plays clearly stronger. For each of 200 seed pairs the experiment ran four battles:

• mirrored, production theta in the first seat, the pair in its original order
• mirrored, candidate theta in the first seat, the same pair swapped
• independent, production theta in the first seat, on a fresh seed pair
• independent, candidate theta in the first seat, on another fresh seed pair

That is 800 battles on one run seed, spread across the available machines. Both thetas played the same ruleset and the same fixed search budget throughout.

Results#

The complement check is exact, not a matter of noise: in all 200 pairs the two mirrored scores added up to one, with an average gap of 0.000000 and a largest gap of 0.000.

Scores are from the shipped theta's perspective, 0.5 being parity:

The two readings differ by -0.0350 with a standard error of 0.0293, z = -1.19. Mirroring showed no detectable bias against the independent estimate at this battle count.

The mirrored reading was also less precise: its standard error was 1.33 times the independent reading's, close to the 1.41 that halving the usable battles predicts. Matching the independent precision would take about 1.77 times the battles.

What 200 pairs do and do not prove#

The natural objection is that 200 pairs make both readings so precise that their agreement hides any real difference, and that mirroring might pay at the battle counts tuning actually uses, where tens of games stand in for hundreds. That reads the result backwards: the redundancy is a per-pair identity, not a statistical trend.

• The complement applies pair by pair, not to the batch. Every pair's two scores add to one, so every pair averages one half at ten pairs or at ten thousand. Pooling never says which theta is stronger.
• One game per pair is the whole sample. Keeping only the first-seat battles leaves the same estimator as playing those battles alone, so a mirrored pair spends two battles to obtain one. At any battle count, independent seed pairs turn the same budget into twice as many usable results.

The batch size only buys precision, and the ratio is fixed. This is why the mirrored reading came out 1.33 times less precise, close to the 1.41 that one usable game per two battles predicts:

Only the bias reading is statistical, and there a bigger batch works the other way: more battles make a bias easier to see, not easier to hide. The difference of -0.0350 with a standard error of 0.0293 does not prove zero bias, it bounds any bias at a few points. That bound matters less than the count suggests, because the claim under test, that a lucky theta can win both battles of a pair, cannot happen at any count: in this run the production theta's 25 first-seat wins reappeared as 25 swapped losses, one for one.

Why the mirror is not a second battle#

The battle simulation is exactly seat-symmetric. Both players start from the same position, each one draws its pieces from its own seed, and garbage is exchanged only after both have moved; no rule treats the seats differently. Exchanging the seats and the seeds together therefore replays the same two trajectories with the labels switched: the board the production theta built is rebuilt by the candidate, and the attacks it received are the attacks it now sends. The scores add up to one by construction.

Three consequences follow:

• A lucky theta cannot win both battles. In the run, the production theta won 25 first-seat battles and lost all 25 of their swapped partners. Its 25 lucky streams were handed to the opponent intact, not re-rolled.
• Folding pairs cancels the signal, not the luck. Every mirrored pair contributes exactly one half to the batch mean, so a batch of mirrored pairs averages one half no matter which theta is stronger.
• The second battle is a duplicate. The only usable reading is one battle per pair, which is why the mirrored standard error matches the independent one after accounting for half the battles.

The intuition that failed was about seeds: swapping the seats does not re-roll the randomizer, it gives each player the other one's piece stream. Mirroring is not biased, it is redundant.

Conclusion#

The no-mirror rule stays, with a corrected reason. A mirrored rematch does not tilt a comparison, it spends a battle to repeat one already played, and an independent pair is exactly as expensive and more precise. The Theta comparison page now states this reason. The exact complement is a property of the battle simulation, so it would need re-measuring if the rules ever treat the seats differently.