Conditional means, vector pricings, amenability and fixed points in cones

TL;DR

This paper generalizes conditional probability to ordered vector spaces, characterizing when these generalized probabilities are stationary or invariant, leading to new criteria for amenability and fixed points in cones.

Contribution

It introduces a novel framework for conditional probabilities in arbitrary ordered vector spaces and characterizes conditions for stationarity and invariance.

Findings

Characterization of groups with stationary generalized probabilities

New criteria for amenability in ordered vector spaces

Identification of fixed points in cones under generalized probability measures

Abstract

We study a generalization of conditional probability for arbitrary ordered vector spaces. A related problem is that of assigning a numerical value to one vector relative to another. We characterize the groups for which these generalized probabilities can be stationary, respectively invariant. Our results deviate from the setting of classical probability; this leads to a new criterion for amenability and for fixed points in cones.