Bayesian Truth Serum on Tree Graphs

Abstract

We consider the problem of defining all possible Bayesian truth serums for agents represented by a singly connected network (a tree graph). Agents receive signals according to a probability distribution that is common knowledge and that matches the network structure. The planner knows the location of each agent in the network but not the probability distribution itself. Each agent reports a signal and predicts the report of every other agent. Assuming that predictions are honest (i.e., are correctly computed posterior probabilities), what ’information scoring’ functions of predictions ensure that the predicted agent maximizes expected score with an honest report, for all probability distributions consistent with the graph structure?

We prove that strictly incentive-compatible information scores are characterized by three properties:

(a) log-linearity in the vector of predicting probabilities,

(b) balance — the sum of coefficients in the linear function equals zero on any branch directly connected to the predicted agent, and

(c) local excitation — at least one neighbor of the predicted agent must have a positive coefficient, while the sum of coefficients must be non-positive on any branch not directly connected to that agent.

Keywords: Bayesian truth serum

JEL Classification: D82