# Truthfulness in Repeated Predictions

## TL;DR

This paper investigates how proper scoring rules influence truthful predictions in multi-agent settings, revealing that under certain normal distribution assumptions, the logarithmic scoring rule guarantees truthfulness regardless of prediction timing.

## Contribution

It proves that in a multivariate normal setting with common knowledge of variances, the logarithmic scoring rule ensures truthful predictions at all times, even with multiple agents and predictions.

## Key findings

- Proper scoring rules can incentivize dishonesty in repeated predictions.
- Under multivariate normal assumptions, the logarithmic scoring rule guarantees truthfulness.
- The result applies to various financial models.

## Abstract

Proper scoring rules elicit truth-telling when making predictions, or otherwise revealing information. However, when multiple predictions are made of the same event, telling the truth is in general no longer optimal, as agents are motivated to distort early predictions to mislead competitors. We demonstrate this, and then prove a significant exception: In a multi-agent prediction setting where all agent signals belong to a jointly multivariate normal distribution, and signal variances are common knowledge, the (proper) logarithmic scoring rule will elicit truthful predictions from every agent at every prediction, regardless of the number, order and timing of predictions. The result applies in several financial models.

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Source: https://tomesphere.com/paper/1704.00527