Individual-level simulation
Top half — individuals. Each member estimates a regression slope β from n noisy points — more data ⇒ smaller SE ⇒ sharper estimate (EXPERT n=80 vs NOVICE n=4). Tune the parameters below; the group dynamics at the bottom show how individual estimates aggregate into a group decision.
Fixed x: x = linspace(0, 10, n) — every member shares the same design, so the standard error SE = σ/√SSx is identical at a given n.
Randomly drawn x: xi ~ iid Uniform(0, 10) per member, so SSx (and therefore SE) varies card to card. Headline error rates and the sampling PDFs of β̂ are then expected quantities, Monte-Carlo-averaged over x designs.
x = linspace(0, 10, n); every member shares the same design.
Simulated group dynamics
Bottom half — groups. The same simulation, played out for a handful of true slopes. Each panel draws one fresh group — 1 EXPERT (n=80) plus NOVICES (n=4), group size set below — and shows the group decision that results from pooling their estimates. Press Go above for a new realisation.
A group pools its members' slope estimates into one estimate β̄ = Σwiβ̂i / Σwi, then takes the nearest candidate. The averaged panel reports the expected error rate of that decision.
Optimal = precision. For Gaussian estimates the optimal pooled estimate is the inverse-variance mean, wi ∝ τi = 1/SEi² — so "optimal" and "precision weighting" coincide here.
Behavioural wi ∝ τiρ. ρ=1 is optimal; ρ=0 is equal weighting (⅓ each — ignores precision); ρ<0 goes counter to precision, over-trusting the noisy, strong-looking members and actively under-weighting the EXPERT.
Provisional form — to be finalised. The per-draw panels (right of each group) instead compare three fixed anchors: optimal, equal, and a raw-strength weighting.
A group — 1 EXPERT + NOVICES (size set below) — pools its members' slope estimates into one group decision. The behavioural tile below weights each member by wi ∝ τiρ (τ = precision = 1/SE²). Drag ρ to set how much the group trusts precision.
The betting game
The data — each round, a fresh cloud of noisy (x, y) points from one analyst. No line is fitted; you read the slope yourself.
The true slope β is −0.2, 0, or +0.2 — genuinely 0 half the time, a quarter chance each negative or positive.
- EXPERT = lots of data; NOVICE = only a few.
We measure three things:
- your beliefs over the three slopes;
- your one-euro bet across them — scored by a proper rule, so backing the true posterior pays best (paired-uniform, Vespa & Wilson 2017);
- your estimate of β.