What the evidence says
The claims retrieved for this topic come entirely from a single 2026 arXiv paper (Millar-Sáez) built on an agent-based simulation of legislative decision-making — not from observed, real-world direct-democracy institutions. The model studies how approval thresholds (simple majority vs. a 2/3 supermajority) and quorum requirements interact with the number of independent legislators and with polarization to shape "legislative efficiency" and "aggregate social welfare" inside the model. Every number and mechanism below describes what happens inside that simulation, not a measured outcome from an actual legislature or electorate.
Within the model, the approval threshold determines the direction of the relationship between legislator diversity and efficiency. Under simple majority rule in polarized scenarios, efficiency is non-monotonic in the number of independent legislators: it rises to an interior maximum and then declines, a pattern the paper attributes to a transition from unilateral control to pivot-based coalition formation. Under a 2/3 supermajority in the same polarized scenarios, the same increase in independent legislators instead decreases legislative performance, through minority veto dynamics or coordination breakdown. The paper also states that, in highly polarized contexts, simple majority is the only approval threshold modeled that sustains efficiency at all — high thresholds turn legislator heterogeneity into paralysis, while lower thresholds let that same diversity convert into negotiation.
Quorum requirements act as a second, independent lever. More demanding quorums decrease the number of proposals passed and, in some scenarios, decrease aggregate social welfare. Quorum level also interacts with the diversity effect described above: as the quorum requirement rises, the number of independent legislators at which efficiency peaks shifts downward, and at high quorums the interior efficiency maximum disappears entirely.
The paper's own summary claim is that none of these effects belong to a threshold rule in isolation — efficiency is described as an emergent outcome of the interaction between approval thresholds, parliamentary composition, and the coalition structures those thresholds generate, not a fixed property of "simple majority" or "supermajority" on their own.
No claim in this set describes an empirical (non-simulation) measurement of supermajority thresholds in an actual direct-democracy setting — no observed ballot-measure passage rates, no real legislature, no real electorate. For a topic framed as a review of empirical, non-simulation evidence, that absence is itself the main result: the current source base supplies a theoretical mechanism, not a measured one.
Thresholds and dose-response
Between simple majority and a 2/3 supermajority threshold, the effect of increasing independent-legislator diversity on legislative performance reverses: under simple majority it produces an interior efficiency maximum (efficiency rises then falls as diversity increases), while under 2/3 supermajority the same increase in diversity decreases legislative performance, per Millar-Sáez (2026).
On quorum, the source states a qualitative dose-response without a numeric cutoff: as quorum requirements rise, the diversity level at which efficiency peaks shifts downward, and the interior efficiency maximum disappears altogether "at high quorums." The source does not state a specific quorum percentage or rate at which this disappearance occurs.