A predictive approach to generalized arithmetic means

TL;DR

This paper introduces a geometric framework where generalized arithmetic means serve as optimal predictors, linking concepts like certainty equivalents and conditional expectations to decision-making and fair pricing.

Contribution

It provides a geometric interpretation of generalized means and extends the concept to conditional expectations and certainty equivalents within this framework.

Findings

Generalized arithmetic means are shown as best predictors in a geometric setting.

The framework connects certainty equivalents with fair pricing.

Conditional expectations are characterized as best predictors given prior information.

Abstract

The goal of this note is to provide a geometric setting in which generalized arithmetic means are best predictors in an appropriate metric. This characterization provides a geometric interpretation to the concept of certainty equivalent. Besides that, in this geometric setting there also exists the notion of conditional expectation as best predictor given prior information. This leads to a notion of conditional preference and to the notion of conditional certainty equivalent, which turns out to be consistent with the notion of fair pricing.