The probability that two random points on the $n$-probability simplex are comparable with respect to the first order stochastic dominance and the monotone likelihood ratio partial orders

TL;DR

This paper investigates the likelihood that two randomly chosen points on the n-probability simplex are comparable under first order stochastic dominance and monotone likelihood ratio orders, which are key in decision process theory.

Contribution

It quantifies the probability of comparability under these partial orders, providing insights into their relative strength in the context of the n-probability simplex.

Findings

Probability of comparability varies with dimension n

First order stochastic dominance is more likely to compare points than monotone likelihood ratio

Results inform the structural analysis of MDPs and POMDPs

Abstract

First order stochastic dominance and monotone likelihood ratio are two partial orders on the $n$-probability simplex that play an important role in the establishment of structural results for MDPs and POMDPs. We study the strength of those partial orders in terms of how likely it is for two random points on the $n$-probability simplex to be comparable with respect to each of the two partial orders.