William D Sudderth | Searcharxiv

arXiv2026

Optimal strategies in Markov decision processes with finitely additive evaluations

We study infinite-horizon Markov decision processes (MDPs) where the decision maker evaluates each of her strategies by aggregating the infinite stream of expected stage-rewards. The crucial feature of our approach is that the aggregation is performed by means of a given diffuse charge (a diffuse finitely additive probability measure) on the set of stages. The results of Neyman [2023] imply that in this setting, in every MDP with finite state and action spaces, the decision maker has a pure optimal strategy as long as the diffuse charge satisfies the time value of money principle. His result raises the question of existence of an optimal strategy without additional assumptions on the aggregation charge. We answer this question in the negative with a counterexample. With a delicately constructed aggregation charge, the MDP has no optimal strategy at all, neither pure nor randomized.

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arXiv2024

Zero-one Laws for a Control Problem with Random Action Sets

In many control problems there is only limited information about the actions that will be available at future stages. We introduce a framework where the Controller chooses actions $a_{0}, a_{1}, \ldots$, one at a time. Her goal is to maximize the probability that the infinite sequence $(a_{0}, a_{1}, \ldots)$ is an element of a given subset $G$ of $\mathbb{N}^{\mathbb{N}}$. The set $G$, called the goal, is assumed to be a Borel tail set. The Controller's choices are restricted: having taken a sequence $h_{t} = (a_{0}, \ldots, a_{t-1})$ of actions prior to stage $t \in \mathbb{N}$, she must choose an action $a_{t}$ at stage $t$ from a non-empty, finite subset $A(h_{t})$ of $\mathbb{N}$. The set $A(h_{t})$ is chosen from a distribution $p_{t}$, independently over all $t \in \mathbb{N}$ and all $h_{t} \in \mathbb{N}^{t}$. We consider several information structures defined by how far ahead into the future the Controller knows what actions will be available. In the special case where all the action sets are singletons (and thus the Controller is a dummy), Kolmogorov's 0-1 law says that the probability for the goal to be reached is 0 or 1. We construct a number of counterexamples to show that in general the value of the control problem can be strictly between 0 and 1, and derive several sufficient conditions for the 0-1 ``law" to hold.

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