The seating term is turn order, not the arena

What was suspected and what was tested

The arms report found that two identical default searches in slots 0 and 1 differ by +0.176 wins per game (run 91ddb42d) and blamed the arena's offer resolution, which asked the recipients of an offer in turn order after the offerer, so that slot 1 was asked first on every offer slot 0 made. The arena now asks recipients from a rotating seat (server commit 68c9842, the order.rotate_left(shift) line in crates/arena/src/table.rs), and the same null on the same boards still differs by +0.180 (+0.075 to +0.285, run 689a1947). The recipient order was not the cause.

Every cohort below is 256 games on deterministic boards 0 to 63, each board played once per rotation of the four slots, every search at depth 2 with the default configuration except the named switch. A contrast is the wins per game of one slot minus another, per board, with a 95% interval over boards; the two slots are named because the run record only reports slot 0 minus slot 1. Each cohort was registered with its own rule before it ran.

By the registered rules the term exists without bargaining, with identical builders, and with the builders swapped; the propose: false and counters: false cohorts are inconclusive (estimates above +0.05, intervals touching zero); and the asks: false cohort meets its rule for "asks are necessary", which the bargain: false cohort, whose seats never ask either, contradicts. Read together against the default null on the same boards the switch cohorts are −0.078 (−0.202 to +0.046) for bargaining off, −0.105 (−0.239 to +0.029) for proposals off, −0.090 (−0.227 to +0.047) for counters off, and −0.172 (−0.288 to −0.055) for asks off; pooled, the four sit at +0.068 (+0.015 to +0.122) and −0.111 (−0.217 to −0.005) below the default. That pooling was not preregistered. Two identical seats that never bargain still differ by about a fourteenth of a win, and two that bargain in full by about a sixth. In every cohort the wins of each search are flat across the four engine seats, so the term is the order of the two searches, not the seat either sits at.

Adjacency

The lineup with an ETA between the two searches puts them opposite each other. There the earlier search won 107 and the later 97, +0.039 (−0.048 to +0.126), and the registered rule for "adjacency is the cause" is met: the interval includes zero and the estimate is below +0.05. The two searches in that lineup still have different builders before them (fast before slot 0, ETA before slot 2), so a small builder term could sit inside that interval.

The builders show the same order effect, smaller. With one search and three ETAs the builder directly after the search won 30, the one opposite 35, and the one directly before it 37: −0.027 (−0.084 to +0.029) for after minus before, no term by its rule. With two searches and two ETAs the ETA directly after a search won 16 and the ETA directly before one won 32, +0.062 (+0.010 to +0.115) for the seat before, an after-the-fact contrast. Swapping the two builders (fast then ETA) leaves the searches' term at +0.133 (+0.040 to +0.226), −0.047 (−0.180 to +0.086) from the default null, so which builder sits beside a search does not matter.

Why the earlier seat wins

In a four-player round the seat directly before another acts first after three of the four rolls: its own roll and the two rolls of the other seats. Only the later seat's own roll gives it the first move. When two identical searches can afford the same thing after the same roll, the earlier one takes it three times in four. As an exact calculation on a toy model, with each seat waiting for its own card that arrives with the same chance on every roll, the earlier of two adjacent seats builds first 55% of the time at one card in ten rolls and 63% at three in ten, and seats opposite each other split evenly. That is a calculation, not a measurement. Two identical searches want the same things because they value positions the same way; a builder wants something else most of the time, so a search gains little from sitting before one and loses little from sitting after one, which is what the three-ETA table shows.

An unregistered traced screen of the default lineup on boards 0 to 15 (64 games, slot 0 won 30 and slot 1 16) did not show the channel in the two counts it could make. Settlement sites that both searches had a road touching were taken five times by each; sites where the other search was one road away, 12 by slot 0 and 13 by slot 1. Offers between the two searches closed at similar rates in both directions (asks 0.42 against 0.34 per game, one-for-ones 0.57 against 0.85, counting counters the offerer took). The two searches' public points are equal through turn 40 and diverge after it (5.08 against 4.66 at turn 50, 5.84 against 5.35 at turn 60), and the wins slot 0 takes are more often by wide margins: in the rotating null, when the losing search finished at 4 points slot 0 had won 24 games and slot 1 seven, at 7 points 29 against 13, while at 9 points the split was 3 against 5. The earlier seat's edge accumulates in the second half of the game rather than at the finish line or in the opening; which of the mid-game races it wins is not resolved here.

The term therefore belongs to the game. The protocol arena, the browser, and a table of people all have it: the player who moves right before a strong player acts first on shared production and, with the bargaining on, trades with a partner whose turn comes right after. Its size depends on the boards, which is why the same lineup gave +0.176 on boards 0 to 63 and +0.051 on boards 64 to 127.

The arena code paths that were checked

Each place in crates/arena/src/table.rs and the engine that could treat the seat after the current player specially was read, and none explains a term that survives bargain: false and disappears with a builder between the searches.

The recipient order of an offer now rotates with the seed and the table version, and the term did not change. A counter that the offerer declines leaves the original offer open to the remaining recipients; a counter that is taken closes it. Both searches offer to single partners in the same way and accepted the others' offers at the same rate (2.00 against 1.82 per game in the rotating null). Discards on a seven are asked from the lowest engine seat first, which is rotation-dependent and puts slot 0 before slot 1 in three of the four rotations; the discarding seat sees nothing it can use and the order consumes no randomness. Robber victims are enumerated by engine seat and the searches score them by cards and points, so ties fall to the lower seat, again by rotation rather than by adjacency, and in the direction that would hurt slot 0. The deterministic dice, theft, and deck tapes are indexed by count, so every rotation of a board sees the same dice in the same order, and each slot rolls each residue of that order once. Setup is placed by the same planner for every seat, so each slot gets each pick position once and the setup positions of a board are identical across its rotations.

What this changes

Nothing in the arena code. The rotating recipient order stays because an arena should not favour the seat after an offerer, but it does not touch the term. The rule for evidence stands: any contrast between two searches at one table is measured as a swapped pair on the same boards, and the seating term is half the sum of the two contrasts. A lineup that seats the two searches opposite each other (slots 0 and 2, a builder in each of slots 1 and 3) removes the adjacency term and is a reasonable screen, but it still leaves a different builder before each search, so it is not a substitute for the pair. The bargaining arms report's explanation of the term is corrected by this one; its numbers stand, and its adjusted contrasts are still read against a null on the same boards. The build-prediction study's mirrored pairs and the opening planner's swapped pairs were the right design for a reason that is now known. The agent brief and the server's docs/expectimax.md now name turn order rather than the recipient order.