Bayesian epistemics

1.1 Setting

• Each agent \(i\) privately observes a signal \(s_i\) about a state \(\omega\).
• We can pay agents using a scoring/payment rule \(S_i(\cdot)\) and then choose a downstream action \(a=\delta(m)\) based on reported messages \(m=(m_1,\dots,m_n)\).
• Planner’s objective is \(W(a;\theta)\) (or a principal payoff \(\Pi(a;\theta)\)).

Sending a message \(m_i\) is an action—one that is informed by beliefs or observations.

1.2 Elicitation with verification: proper scoring rules

If a verifiable outcome \(y\) (or a gold-label) eventually arrives, we can reward probabilistic reports \(p_i\) with a strictly proper scoring rule \(S(p_i,y)\) so that truthful beliefs uniquely maximize expected score (Gneiting and Raftery 2007). Formally, for any belief \(q_i\) about \(y\), \[ \mathbb{E}_{y\sim q_i}[S(q_i,y)] \;\ge\; \mathbb{E}_{y\sim q_i}[S(p_i,y)] \quad \forall p_i, \] With equality iff \(p_i=q_i\). This gives communication-Incentive-Compatible (CIC) incentives for beliefs. Classic examples include the log score and the Brier score (Gneiting and Raftery 2007).

That also connects us to calibration.

1.3 Elicitation without verification (IEWV)

When no ground truth ever arrives, we can still incentivize information using peer prediction and truth-serum mechanisms. I know nothing useful or practical about these yet; here are the dot points though:

• Peer Prediction (PP): pay agents using others’ reports so truthful reporting is a Bayes–Nash equilibrium (Miller, Resnick, and Zeckhauser 2005; Witkowski and Parkes 2012a).
• Bayesian Truth Serum (BTS) and variants: reward reports and meta-predictions about the distribution of others’ reports—truthful reporting is incentivized under mild common-prior/correlation conditions (Prelec 2004; Prelec, Seung, and McCoy 2017).
• Robust BTS / Minimal PP: relax prior/knowledge assumptions or handle non-binary or finite-sample regimes (Witkowski and Parkes 2012b; Radanovic and Faltings 2013; Dasgupta and Ghosh 2013).

These mechanisms create CIC about signals (truthful mapping \(s_i \mapsto m_i\)) even if \(y\) is never observed.

1.4 From elicitation to decisions: informational alignment

We may ultimately care about the actions taken based on our beliefs. Let \(\delta(m)\) be the aggregation/decision map that turns messages into actions. Two complementary targets:

1. Belief alignment (posterior target). Aggregation yields, in expectation, the same posterior we would hold if we saw the true signal profile: \[\text{Elicited posterior } \hat\mu(\cdot\mid m) \approx \mu(\cdot\mid s).\] A natural discrepancy is a probability distance, e.g. \(\mathrm{KL}(\mu\,\|\,\hat\mu)\) or a Bregman divergence tied to the scoring rule (Gneiting and Raftery 2007).

2. Decision alignment (action target). The induced action maximizes the planner’s objective given the information: \[a^\star(s)\in\arg\max_a W(a;\theta) \quad\text{and}\quad \delta(m^\star(s))=a^\star(s),\] where \(m^\star(s)\) denotes the equilibrium communicative action under the elicitation payments. This is the informational analogue of our earlier action-IC → implementation → alignment pipeline.

Put differently: CIC gives us truthful information, implementation ensures the decision rule uses it, and alignment checks that the resulting action is planner-optimal.

1.5 Relaxations and metrics (brief)

• ε-CIC (elicitation slack): bound the maximum gain from misreporting under the scoring rule by \(\varepsilon\) (in utility units).
• Information regret: expected score gap between truthful and actual reports (equals a Bregman divergence under proper scoring) (Gneiting and Raftery 2007).
• Posterior divergence: distance between elicited and Bayesian posteriors (KL/TV/Wasserstein).
• End-to-end regret: welfare gap \(W(a^\star(s)) - W(\delta(m))\) after aggregation, exactly like our earlier action-level regret.

1.6 Communication vs persuasion (design choices)

Not all communication seeks the truth. Bayesian persuasion designs signal structures to move the receiver’s action toward the sender’s objective (Kamenica and Gentzkow 2011). Cheap talk shows when informative communication is impossible or coarse (Crawford and Sobel 1982). The elicitation view flips that: we design payments so truthful beliefs and observations become the sender’s best action, and then we align the downstream decision.