Stateful Posted Pricing with Vanishing Regret via Dynamic Deterministic Markov Decision Processes

TL;DR

This paper introduces a new online learning framework for dynamic resource allocation problems with capacity constraints, using deterministic Markov decision processes to achieve vanishing regret in stateful posted pricing scenarios.

Contribution

The paper develops a novel online learning approach based on deterministic Markov decision processes, enabling vanishing regret for complex stateful pricing problems with dynamic transitions.

Findings

Proves vanishing regret is achievable under certain conditions

Reduces the problem to online learning with switching costs

Demonstrates applicability to online job scheduling and matching

Abstract

In this paper, a rather general online problem called dynamic resource allocation with capacity constraints (DRACC) is introduced and studied in the realm of posted price mechanisms. This problem subsumes several applications of stateful pricing, including but not limited to posted prices for online job scheduling and matching over a dynamic bipartite graph. As the existing online learning techniques do not yield vanishing-regret mechanisms for this problem, we develop a novel online learning framework defined over deterministic Markov decision processes with dynamic state transition and reward functions. We then prove that if the Markov decision process is guaranteed to admit an oracle that can simulate any given policy from any initial state with bounded loss -- a condition that is satisfied in the DRACC problem -- then the online learning problem can be solved with vanishing regret. Our proof technique is based on a reduction to online learning with switching cost, in which an online decision maker incurs an extra cost every time she switches from one arm to another. We formally demonstrate this connection and further show how DRACC can be used in our proposed applications of stateful pricing.