Bargaining switches under the learned leaf

The five arms

Every arm is the tables baseline seat, v2:{"depth":2,"leaf":{"tables":"artifacts/ntuple/hex-portfolio-main.bin"}}, with one bargaining switch turned off, against the unchanged baseline. bargain: false is the all-off control: plain acceptance and proposals with the reference willingness model, no threat costs, no counters, no asks. threat_pricing: false removes the threat premium on the trade margin and the partner-gain cost. counters: false stops the seat answering an open offer with a counter. asks: false stops it proposing two for one to partners known to hold plenty. accept_asks: false is a stubborn seat that refuses any offer or counter asking it for more cards than it receives; it changes what the seat accepts, never what it proposes. Under the hand-written leaf on the builder lineup, counters off cost −0.096 (−0.156 to −0.035) on a swapped pair, asks off −0.156 against a null, threat pricing off nothing, all off −0.121 in one seating, and the stubborn seat was +0.059 better than an identical default (arms report).

Each arm is a swapped pair on the same seeds: half A seats the arm in slot 0 and the control in slot 1, half B the reverse, and the effect is the per-seed half-difference of the two slot-0-minus-slot-1 contrasts, which cancels the seating term between two adjacent searches; the half-sum is the term itself. The population lineup L2 ran first, because bargaining needs partners that trade and the builders rarely do: a third tables search in slot 2 and an ETA in slot 3, seeds 800 to 863. The builder lineup L1 seats an ETA and a fast builder in slots 2 and 3 on seeds 0 to 63, the boards of the hand-leaf arms, and on seeds 800 to 863. Every cohort is 256 deterministic games at depth 2; every pair is 512.

Results

Every cohort completed all 256 games, with no invalid moves, stalls, or interruptions. The effect is the arm minus the control in wins per game, so a negative number means the default mechanism wins games; the seating term is what slot 0 gains over slot 1 on those boards.

Counters and asks still win games under the learned leaf. Switching counters off costs about a tenth of a win per game at all three stages, and every interval excludes zero. Switching asks off costs −0.076 and −0.141 on the two builder-table seed sets and −0.037 at the population table, where the interval crosses zero. Switching all bargaining off costs more than either alone, −0.10 to −0.18. Threat pricing does nothing at any stage. The stubborn seat does nothing either: its effect is within 0.02 of zero at every stage, because an ask it would refuse comes up on few boards.

The seating term is still there under the tables and still varies by board set: +0.05 to +0.13 on boards 0 to 63 and +0.04 to +0.08 on boards 800 to 863 at the builder table, and −0.07 to +0.04 at the population table, where a third search sits after slot 1. A contrast read from one seating would have been off by up to an eighth of a win.

Pooled over stages

These estimates pool the per-seed effects of the stages after the fact. The stages were registered separately, and this pooling was not.

The extension rule asks for more seeds when an arm's pooled builder-table effect over 128 seeds is positive and its interval comes within 0.03 of zero. Only the stubborn seat met it, at +0.009 (−0.008 to +0.026), so its pairs on seeds 864 to 927 were registered in both lineups and run. The extension read +0.000 (−0.023 to +0.023) at the builder table and +0.004 (−0.035 to +0.043) at the population table, so the final builder-table estimate over 192 seeds is +0.006 (−0.008 to +0.020): no effect, with the narrowest interval of any arm.

Every cohort

How the switches change trading

Per game, arm / control, both halves of each pair pooled.

Without counters the seat answers a thin offer by accepting or declining. It took two to 2.6 times as many of the others' offers (2.04 against 0.80 per game on boards 0 to 63, 2.12 against 0.82 on boards 800 to 863, 2.33 against 1.16 at the population table), made a quarter fewer offers of its own, and finished about half a point lower with fewer cities. A counter replaces an acceptance on the offerer's terms with one on better terms, the same account the hand-leaf arms gave. The all-off seat completes more player trades than the default (4.5 to 4.8 per game against 3.8 to 4.5) and wins less: it is the generous partner, accepting almost three times as many of the others' offers. Without asks the seat made two fewer offers per game, completed about as many, and finished 0.1 to 0.8 points lower. Threat pricing off makes the seat more willing on both sides, with about three more offers, two to three fewer declines, and 0.6 to 0.8 more player trades per game, and the extra trades neither win nor lose. The stubborn seat's counts match the control's except that it takes fewer of the others' offers (0.94 to 1.12 per game against 1.08 to 1.49), which is what refusing asks means.

Decision cost

The arena records wall time per decision. Other tournaments shared the machine, so the level moved from cohort to cohort, but the arm and its control played the same games at the same moments, so their ratio is a fair measure of what a switch costs. Each row reads switch off against switch on.

Bargaining as a whole costs about 60 percent more time per decision under both leaves, and counters account for most of it: answering an open offer means searching the accept, each counter, and waiting, where the seat without counters searches only the accept and waiting. Asks and threat pricing cost under a tenth and the stubborn switch nothing. The number of decisions per game barely moves with any switch, so the cost is per decision, not more decisions. The hand-leaf column is the same ratio from the single-seating arms of the earlier study on boards 0 to 63; its counters pair on boards 64 to 127, which had the switched-off seat in both slots, gave +48% and +41%.

Compared with the hand-written leaf

The two leaves agree on the order: all off loses most, then counters and asks, and threat pricing and the stubborn seat sit near zero. On the same boards 0 to 63 the tables' losses are about half what the hand-leaf arms showed, but those hand-leaf numbers subtract a null of +0.176 measured in one seating, and the one hand-leaf swapped pair (counters, boards 64 to 127) reads −0.096, the same as the tables' −0.096 on boards 0 to 63. The size that holds under both leaves and both corrections is about a tenth of a win per game for counters and a little less for asks. The stubborn seat was +0.059 better than an identical default under the hand leaf against the null and is +0.006 (−0.008 to +0.020) over 320 seeds under the tables, so that small hand-leaf gain was probably part of the null's error.

The thresholds are the same size under both leaves

The bargaining thresholds are fixed numbers in the hand-written currency: a swap must beat the plain alternative by the trade margin, 12 leaf units at the default slider, a partner's threat adds up to 30 to that margin, and every swap pays a share of the partner's gain priced with the hand-written card and point values (20 and 100). The tables value a position in win probability times leaf.scale, 1000 by default, so one win is ten hand-leaf points and the question is whether one swap moves the tables by as much as it moves the hand-written terms.

A traced screen answers it: 16 boards, four rotations, the tables seat and the hand-leaf seat at one table with the two builders, every search decision recorded with its top candidates and, after an arena change, every answer to an offer and reply to a counter. A proposal's gross gain is the value of the accepted continuation minus the best plain alternative before any threshold; an accept's is its value with the margin and partner cost added back minus the value of waiting.

One swap is worth about the same number of leaf units under both leaves, 30 to 35 for a proposal the lookahead likes and 9 to 10 of threat pricing on an accept, and the plain margin blocks about a third of liked proposals under both. The tables' absolute level is half the hand leaf's, but a threshold compares differences, not levels, so nothing is mis-sized: leaf.scale 1000 puts a card at roughly 20 units in both currencies. The thresholds stay as they are and no knob was added; the decision cost and the arms above are the evidence that matters. The scripts and the two traced screens (identical winners with and without the response trace) are under runs/screens-bargaining-tables/.

What could still be wrong

Decision time is wall time on a shared machine, so no absolute time here predicts the browser's; only the ratio within a pair is read. The arena cohorts ran at equal node budgets, so a switch that makes each decision slower was measured at no cost in search depth. The browser instead gives each decision one second, and there the extra work of counters could come out of depth, which this study did not test. The hand-leaf numbers on boards 0 to 63 come from single-seating cohorts read against a null, not from swapped pairs, so the comparison between leaves on those boards mixes two corrections; only counters has a swapped pair under both leaves. Every partner is a copy of the same search or a builder that never counters, so an ask is accepted whenever the partner's own lookahead likes it; the stubborn arm is the only partner that refuses, and people refuse and counter in ways neither covers. The stubborn seat changed the outcome on few boards (its per-seed effect was exactly zero on 148 of the 192 builder-table boards and on 25 of the 64 population boards), so its narrow intervals say that refusing rarely matters here, not that it is harmless wherever asks are common. Asks off is inconclusive at the population table (−0.037, −0.093 to +0.018) and clear at the builder table; with 64 population seeds that gap is within the noise. Pooling over stages was chosen after seeing the results and is labelled as such. All of it is engine-arena development evidence at depth 2.

What this means for the browser

The browser plays the hand-written leaf at depth 3 under a one-second budget, and all five switches are on there because they are on by default. A switch is worth the browser when its pooled effect is at least +0.03 wins per game with the interval excluding zero and its decision time is within 25% of the control's. Read as switching each one on:

asks is the one switch that meets the bar outright: it wins under both leaves and costs almost nothing. counters wins about a tenth of a game under both leaves, the largest single effect measured under either leaf, but costs 40 percent more time per decision, which fails the cost half of the bar. The bar was written for switching something new on, and switching counters off gives away a tenth of a win, so keep it on. What the browser should check is whether answering an offer still reaches depth 3 inside its second with counters searched; if it does not, the counter search is what to make cheaper. bargain is the sum of those two and stays on for the same reason. threat_pricing does nothing for strength under either leaf at a few percent of cost; it is what makes the search refuse the leader and give a reason, which the browser shows as table talk, so keep it on for that. accept_asks off gains nothing under the tables in either lineup, so keep accepting an ask when the lookahead likes it. No default changes, in the engine or in the browser.

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