# Bayesian Decision Making in Groups is Hard

## TL;DR

This paper demonstrates that Bayesian decision-making processes in groups are computationally intractable (NP-hard), even for simplified utility functions, highlighting the complexity of rational opinion exchange in networks.

## Contribution

The paper proves NP-hardness of Bayesian computations in group decision-making and introduces a polynomial-time algorithm for transitive networks.

## Key findings

- Bayesian opinion exchange is NP-hard for key utility functions.
- Distinguishing between different posterior beliefs is NP-hard.
- A polynomial-time algorithm exists for transitive networks.

## Abstract

We study the computations that Bayesian agents undertake when exchanging opinions over a network. The agents act repeatedly on their private information and take myopic actions that maximize their expected utility according to a fully rational posterior belief. We show that such computations are NP-hard for two natural utility functions: one with binary actions, and another where agents reveal their posterior beliefs. In fact, we show that distinguishing between posteriors that are concentrated on different states of the world is NP-hard. Therefore, even approximating the Bayesian posterior beliefs is hard. We also describe a natural search algorithm to compute agents' actions, which we call elimination of impossible signals, and show that if the network is transitive, the algorithm can be modified to run in polynomial time.