Lower the bar for agreement and the table closes — on somebody's head

Five copies of the project's Bayesian-rational negotiator, private score sheets, private thresholds, one package out of 256. Every published number about this table was measured under unanimity, which is also the rule that hands every seat a veto — so "five rational seats rarely agree" and "any one of five rational seats can block" were inseparable. This sweep separates them: same frozen bank, same 120-episode grid per cell, varying only how many accepting seats a deal needs (5, 4 or 3 of 5) crossed with two deadlines (4 and 16 rounds). The deadline axis is the control that kills the clock explanation from inside the same experiment.

The one-dimensional grid: three rules, two deadlines

Intervals are 95% instance-clustered bootstraps over the 24 parameter sets, 10,000 draws. The score column is the common-denominator one: lowering the quorum enlarges the feasible set and so raises the ceiling the ordinary normalized_primary divides by (on 8 of 24 parameter sets, by up to 11%), which would silently penalize the relaxed cells; this column re-divides every level by the unanimity ceiling. Welfare columns are deal-conditional, so they describe the deals a rule reached, not all 120 games.

The two-factor extension: pricing the veto on its own

Every game in this bank names one seat essential, and the grid above retains that requirement at every level — so "majority" there means three of five including the essential party. These two cells additionally dissolve it, moving two protocol features at once. They are reported separately and are never pooled into the grid above.

Paired against unanimity at the same deadline

Each rule minus unanimity on the same (instance, seed) episode.

Who pays for the extra deals

A quorum close is a deal that passed without every seat's support; the outvoted seat is one absent from the closing offer's support set. The zeros in the unanimity rows are a self-test of the detector rather than a finding — no outvoted seat can exist when closure requires everybody — so a non-zero majority number cannot be a detector artifact.

The dissent column separates two ways a deal rate can rise. Under supermajority the overridden seat is still formally rejecting and being outvoted; under majority it has largely stopped acting — it is excluded rather than defeated.

Everything behind the page

Full methodology, the solver-consistency spy test (the agents plan the rule they play), and four caveats that change how the numbers read: research note 0058 in the ii_mats repository. The mechanism behind the unanimity failure is research note 0057. All eight cells ran on one host, so the paired contrasts — which are the estimand — are machine-consistent; roughly 3–6% of episodes flip deal/no-deal between machines through near-ties in the seats' dynamic program.