Bilateral bargaining

How do you make a mutually useful trade without funding the winner?

The decision at the table

A trade can improve both hands while making your relative position worse. One missing ore may complete your city and also complete the leader’s winning build. Trade acceptance alone is therefore a poor measure of negotiating skill.

The mechanism to test

Estimate your gain, the partner’s gain, and each side’s outside option. Account for the probability of acceptance, the cost of failed offers, and whose turn comes next. A reservation price is the worst offer you prefer to declining; it should change with the race. A bargaining solution is a modelling assumption, not a universal price rule.

Proposed experiment

Hold offer generation fixed and compare own-value pricing, bilateral build-aware pricing, and threat-aware pricing. Hypothesis: threat-aware pricing improves wins even if it lowers acceptance rate. Then test better offer generation separately.

Plot own versus partner estimated gain for proposed, accepted, and rejected trades. Label these gains as model estimates. Report offers per turn, completion, time cost, and wins by opponent family, including a player that never trades.

First ablation

Never proposing a trade changed the contrast by -0.027 wins per game (95% interval -0.132 to +0.077); removing the trade margin (trade slider at 1) changed it by -0.094 wins per game (95% interval -0.192 to +0.005) with more trades completed. The margin arm is registered again on fresh seeds. On 192 fresh seeds removing the margin was -0.016 (95% interval -0.075 to +0.043): no effect. Details and every run ID are in the ablation report; this is a development-tier result on one lineup.

Second round: threat-aware pricing

Every other seat now carries a threat from the race, shared ground, and award contests; the threat raises the margin a swap must clear and charges a share of the partner's gain, thin offers draw priced counters, and refusals are explicit with a reason (the server's docs/expectimax.md, Bargaining). Against the same search with plain acceptance the threat-aware seat won 53.5% to 31.6%, +0.219 wins per game (interval +0.124 to +0.314), on 64 deterministic boards. The partner accepted every helpful counter, so the bargaining report treats this as development evidence and names the one-change arms and the refusing partner as the next test.

Third round: one switch at a time, and a seating bias

Two identical default searches in the arm design's slots 0 and 1 differ by +0.176 wins per game (interval +0.083 to +0.268) on the same 64 boards and by +0.051 (interval crosses zero) on boards 64-127, because slot 1 always sits one seat after slot 0 and is asked first on its offers. Against that null the earlier fifth of a win shrinks to +0.043 (interval crosses zero), switching counters off costs -0.199 (interval -0.309 to -0.089), switching two-for-one asks off costs -0.156 (-0.259 to -0.054), and switching threat pricing off changes nothing. A partner that refuses every ask loses nothing by refusing. A swapped pair on fresh boards confirms counters at -0.096 (interval -0.156 to -0.035) for the seat without them. The arms report has the run IDs; every future two-search arm needs both seatings.

Fourth round: the switches under the learned leaf

With the learned n-tuple tables as the leaf, each switch was turned off one at a time and measured as swapped pairs at the builder table (seeds 0-63 and 800-863) and at a population table of three searches and an ETA (seeds 800-863). Pooled after the fact over 192 seeds, all bargaining off costs -0.143 wins per game (interval -0.180 to -0.106), counters off -0.102 (-0.136 to -0.069), and asks off -0.085 (-0.119 to -0.050); threat pricing off is +0.001 (-0.035 to +0.036), and a seat that refuses every ask is +0.006 (-0.008 to +0.020) over 320 seeds with its extension. The order and the size of the counters effect match the hand-written leaf. Bargaining costs about 60 percent more time per decision under both leaves, most of it from counters. Every switch stays on in the engine and the browser; the tables report has the run IDs and the browser recommendation.

What could disprove it

Acceptance is selective: completed trades are not a random sample of offers. More offers can also win through scheduling advantages. Equalize response windows and limits. An uncalibrated partner model can reject almost every good deal.

All approaches · Player’s guide · Experiment program

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