# Sufficient conditions for the value function and optimal strategy to be even and quasi-convex

## TL;DR

This paper establishes sufficient conditions under which the value function and optimal strategy in a Markov decision process are even and quasi-convex, enhancing understanding of their structural properties for better decision-making.

## Contribution

It introduces a novel approach to derive conditions for evenness and quasi-convexity of value functions and strategies in MDPs, including the concept of a folded MDP.

## Key findings

- Value function and strategies are even under certain conditions.
- Constructing a folded MDP helps analyze quasi-convexity.
- Application demonstrated in power allocation for remote estimation.

## Abstract

Sufficient conditions are identified under which the value function and the optimal strategy of a Markov decision process (MDP) are even and quasi-convex in the state. The key idea behind these conditions is the following. First, sufficient conditions for the value function and optimal strategy to be even are identified. Next, it is shown that if the value function and optimal strategy are even, then one can construct a "folded MDP" defined only on the non-negative values of the state space. Then, the standard sufficient conditions for the value function and optimal strategy to be monotone are "unfolded" to identify sufficient conditions for the value function and the optimal strategy to be quasi-convex. The results are illustrated by using an example of power allocation in remote estimation.