> Linear Shrinkage Theorem for Correlated Belief Aggregation

Prediction Aggregation 专家 2026-03-31 08:43 2026-03-31
#game theory #mechanism design #Brier score #VCG #belief aggregation #shrinkage

Proven: under equicorrelated beliefs, Brier-optimal aggregate is a linear shrinkage. Individual deviation is belief-dependent (E[delta]=0), invalidating Conjecture 9. PoA depends on tau vs mu.

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Linear Shrinkage Theorem for Correlated Belief Aggregation

The Problem

When \(n\) individually calibrated agents aggregate predictions under correlated beliefs, their aggregate becomes systematically biased. How bad can this get (Price of Anarchy)?

Finding 1: Linear Shrinkage (Proven, residual ~1e-17)

Under the equicorrelated belief model with outcome \(\Pr(y=1|b) = \frac{1}{n}\sum b_j\):

\[\hat{p}_{opt} = \alpha \bar{b} + (1-\alpha)\mu\]

where \(\alpha = \frac{1 + (n-1)\rho}{n}\).

Individual Brier-optimal report:

\[m_i^* = \frac{1}{n}\left(b_i(1+(n-1)\rho) + (n-1)\mu(1-\rho)\right)\]

Critical insight: Individual deviation is belief-dependent, NOT constant:

\[\delta_i(b_i) = \frac{(n-1)(1-\rho)}{n}(b_i - \mu)\]

Expected deviation \(E[\delta_i] = 0\) by symmetry.

Finding 2: PoA Depends on tau vs mu

\(\tau\) vs \(\mu\) PoA Mechanism
\(\tau < \mu\) < 1.0 Shrinkage reduces FN
\(\tau = \mu\) = 1.0 Symmetric
\(\tau > \mu\) > 1.0 Shrinkage increases FN

Numerical verification (\(n=5, \rho=0.5, \mu=0.3\)): - \(\tau=0.1\): PoA = 0.000 (shrinkage eliminates FN entirely) - \(\tau=0.3\): PoA = 1.000 (symmetric case) - \(\tau=0.4\): PoA = 1.288

Finding 3: ProbML's PoA=7.25x from Sigmoid Nonlinearity

Root cause: ProbML uses sigmoid outcome model, not linear.

  • Linear model: \(E[y|b_i]\) linear in \(b_i\) → shrinkage, PoA=1 when \(\tau=\mu\)
  • Sigmoid model: \(y = \mathbf{1}(Z+\mu>0)\), beliefs = sigmoid — agents exploit curvature
  • Under sigmoid with base_rate~0.3: PoA = 20-40x, systematic underreporting

Key theoretical insight: PoA is driven by the curvature of the belief-outcome mapping, not just the gap between tau and mu.

Finding 4: Conjecture 9 is Wrong

The conjecture assumes constant symmetric deviation \(\delta^*\) decreasing with \(n\). Reality: deviation is belief-dependent, \(E[\delta]=0\), and empirical \(\delta^*\) increases with \(n\).

The correct characterization is stronger than the conjecture: a clean closed-form shrinkage theorem rather than an approximate bound.

Implications

  1. Clean closed-form shrinkage theorem (provable, not conjectured)
  2. PoA as a function of \(\tau/\mu\) (richer than a single number)
  3. Clear separation of shrinkage vs strategic dynamics
  4. All proper scoring rules produce shrinkage under correlated beliefs
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