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 [@Gneiting2007Strictly]. 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) for beliefs. Classic examples include the log score and Brier score [@Gneiting2007Strictly].

That connects us also to calibration.

1.3 Elicitation without verification (IEWV)

When no ground truth ever arrives, we can still incentivize information via peer prediction and truth-serum mechanisms:

• Peer Prediction (PP): pay agents using others’ reports so that truthful reporting is a Bayes-Nash equilibrium [@Miller2005PeerPrediction; @Witkowski2012EC].
• 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 [@Prelec2004BTS; @Prelec2017Solution].
• Robust BTS / Minimal PP: relax prior/knowledge assumptions or handle non-binary/finite-sample regimes [@Witkowski2012RBTS; @Radanovic2013RBTS; @Dasgupta2013Endogenous].

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

1.4 From elicitation to decisions: informational alignment

We may ultimately care about actions taken based on our beliefs. Let \(\delta(m)\) be the aggregation/decision map that turns messages into an action. 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 [@Gneiting2007Strictly].

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 \(m^\star(s)\) is 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 max gain from misreporting under the scoring rule by \(\varepsilon\) (utility units).
• Information regret: expected score gap between truthful and actual reports (equals a Bregman divergence under proper scoring) [@Gneiting2007Strictly].
• Posterior divergence: distance between elicited and Bayes 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 truth. Bayesian persuasion designs signal structures to move the receiver’s action toward the sender’s objective [@Kamenica2011BP]. Cheap talk shows when informative communication is impossible or coarse [@Crawford1982Strategic]. The elicitation view is the opposite: we design payments so that truthful beliefs and observations become the sender’s best action, then we align the downstream decision.