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

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.