Which part of the bargaining pays

The arm design has a bias

Every one-change arm in this notebook seats the changed search in slot 0 and the unchanged control in slot 1, with the builders in slots 2 and 3, and rotates the four slots through the four seats on every board. Each slot therefore sits in every seat once per board, but slot 1 always sits one seat after slot 0. That did not matter while the two searches only accepted or proposed swaps, but the bargaining code makes the order of the seats matter: an offer is put to the other seats in turn order starting after the offerer, and the first acceptance wins. Slot 1 is asked first on every offer slot 0 makes and can counter it; slot 0 is asked last on every offer slot 1 makes, after both builders. That explanation did not survive the diagnosis in the seating report: the term is the game's turn order (the seat before another acts first after three of the four rolls) and exists without any trading. The numbers below stand.

Cohort registered protocol, run 256-game engine cohort, seats two identical default searches this way. Slot 0 won 128 of 256 slot-games (50.0%) and slot 1 won 83 (32.4%), paired contrast +0.176 wins per game (95% interval +0.083 to +0.268). Wins are flat across seat positions for both slots (32, 31, 33, 32 and 20, 21, 22, 20), so the bias is the relative order of the two searches, not the seat. Slot 0 accepted more of the others' offers (1.99 against 1.62 per game), made more bank trades (4.66 against 3.91), discarded fewer cards (8.5 against 9.3), and finished with 8.30 points against 7.37.

Every registered contrast below is the changed seat in slot 0 minus the default in slot 1, as the arms were registered, and each carries this bias. The adjusted column subtracts the null cohort's per-board difference from the arm's on the same board; that analysis was not preregistered and the intervals are over the same 64 boards.

One switch at a time

The four switches are documented in the server's docs/expectimax.md under Bargaining. threat_pricing applies the threat premium on the trade margin and the partner cost; counters lets the seat answer an open offer with a counter; asks lets it propose two for one to partners known to hold plenty; each is effective only while bargain is on, and bargain: false now turns all three off at once. While adding them, the accept-side partner cost turned out to price the swap backwards (it valued the partner receiving what this seat received), so the default of these cohorts charges the partner's actual gain; the earlier cohort ran with the reversed cost.

Each row is 256 games on boards 0 to 63. By the registered rule, which reads the raw contrast, the all-off control loses to the default even from the favoured slot (−0.121, interval excluding zero), threat pricing off is a positive interval that would recommend switching it off, and the other two arms are inconclusive. By the null, the threat pricing arm reproduces the bias almost exactly and carries no effect of its own, the seat that never counters and the seat that never asks each lose about a sixth to a fifth of a win per game to the default that does, and the all-off seat loses about three tenths, close to the two effects added. The diagnostics agree: without counters the seat accepted the others' offers far more often (3.72 against 1.32 per game) and made a third fewer offers of its own (10.8 against 15.2), so a counter mostly replaces an acceptance with better terms; the all-off seat accepted 4.14 of the others' offers per game against the default's 1.67 and finished with 7.41 points against 7.98.

A partner that refuses asks

Cohort registered protocol, run 256-game engine cohort, seats the default in slot 0 against the same search with accept_asks: false in slot 1, a seat that never accepts an offer or counter asking it for more cards than it receives. The default won 120 of 256 (46.9%) against 90 (35.2%), contrast +0.117 (+0.015 to +0.219). That is below the null's +0.176: against the null the stubborn seat did +0.059 (+0.002 to +0.115) better than an identical default in the same slot, so refusing every ask costs the refuser nothing at this table and may help a little. The asks are worth about a sixth of a win only because the partner takes them.

Confirmation on fresh boards

The counters arm carries the largest effect against the null, so it was registered again on boards 64 to 127 as a swapped pair: cohort registered protocol seats counters off in slot 0, cohort registered protocol seats it in slot 1, and the effect is the per-board half-difference of the two contrasts, which averages the two seatings. With counters off in slot 0 (run 256-game engine cohort) the changed seat won 89 of 256 against 116, contrast −0.105 (−0.206 to −0.005); with counters off in slot 1 (256-game engine cohort) the default in slot 0 won 114 against 92, contrast +0.086 (−0.006 to +0.177). The effect of switching counters off is −0.096 wins per game (95% interval −0.156 to −0.035) over 64 fresh boards, and the registered rule is met: counters stay on. The estimate is about half the adjusted −0.199 of boards 0 to 63; the two intervals overlap.

The pair's other half, the per-board average of the two contrasts, is the seating term with one seat not countering: −0.010 (−0.084 to +0.065), none of the +0.176 the null showed on boards 0 to 63. A second null on the same fresh boards, registered cohort, run 256-game engine cohort, put two identical defaults at +0.051 (−0.046 to +0.148), 107 against 94. The seating term is not one number: +0.176 on boards 0 to 63, +0.051 on boards 64 to 127, and +0.113 (+0.046 to +0.181) pooled over the 128 boards after the fact. Its direction holds on both board sets and its size does not, which is one more reason to measure any two-search contrast in both seatings rather than subtract a null.

What this changes

The default keeps every switch on. Counters and asks are the part of the bargaining that wins games at a table of two searches, and threat pricing stays because it decides which seat is refused and why, which the earlier report showed in table talk, not because it wins games here: its effect at this table is zero within a tenth of a win. The all-off control bargain: false now means what its name says. The earlier claim of a fifth of a win for threat-aware bargaining over plain acceptance is withdrawn as a strength result; against the null on the same boards it is +0.043 with an interval that crosses zero, and with the pooled seating term it would be about +0.11, so what remains is between a twentieth and a tenth of a win, none of it confirmed.

Every future one-change arm at a table of two searches needs either a null arm on the same boards or a swapped pair, and the engine arena should put a multi-seat offer to its recipients in an order that does not always favour the seat after the offerer. The ablation program of 2026-09-09 used the same slot design with the earlier trade code; whether its two-search tables carried a bias is untested.

What this does not show

The partner in every arm is the same search, which accepts an ask when its own lookahead says the swap is worth more than waiting. People refuse more and counter back; the stubborn arm covers only the first. Threat pricing may still matter at a table where the partner is close to winning more often than a depth-2 search against two builders allows, which the race term never saw here. All cohorts are development tier at depth 2.