I recently stumbled upon a paper relating to AI. There I found a result that at first glance seems bewildering (p. 10):
Theorem 18 (Pareto Optimality is Trivial). Every policy is Pareto optimal in any $\mathcal M\supseteq \mathcal M^{CSS}_{LSC}$.
A policy is a function $\pi:(\mathcal A\times\mathcal E)^*\rightarrow\mathcal A$ mapping each history to the action $a_t\in\mathcal A$ taken by the agent in time $t$ after seeing this history. An action results in the agent receiving a percept $e_t=(o_t,r_t)\in\mathcal E$ consisting of an observation $o_t\in\mathcal O$ and a real-valued reward $r_t\in\mathbb R$. A set $\chi^*:=\cup^{\infty}_{n=0}\chi^n$ is the set of all finite strings over the alphabet $\chi$. A history is an element of $(\mathcal A\times\mathcal E)^*$.
The $\mathcal M^{CSS}_{LSC}$ denotes the class of all environments modeled as lower semicomputable chronological conditional semimeasures. That is, mostly the class of Markov decision processes satisfying those conditions. Pareto optimality is defined on p. 9.
This theorem means that Pareto optimality is trivial in the class of all computable environments. Does this have a translation into a more general mathematical setting? For example, can we relate this to game theory and strategies taken by players in computable games? Does this mean that, in order to give a meaningful content to Pareto optimality, we must confine ourselves to uncomputable probabilities?