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Upper Expected Meeting Times for Interdependent Stochastic Agents

Marco Sangalli, Erik Quaeghebeur, Thomas Krak

TL;DR

This work addresses the problem of bounding the expected meeting time $\mu$ for two interdependent stochastic agents operating on a finite state space $\mathcal{Z}$ under epistemic uncertainty about their action selections. It casts the problem as a hitting-time problem on the product space $\mathcal{Z}^2$ and leverages imprecise Markov chains to model uncertainty, analyzing degenerate, vacuous, and degenerate-vacuous mixtures to obtain tight, computable bounds via Krak's iterative algorithm. The framework naturally extends to $k$ agents using a $k$-fold product construction, with symmetry reduction on $\mathcal{Z}^k/S_k$ substantially mitigating the combinatorial blow-up. The resulting approach provides exact or conservatively bounded meeting-time estimates and reveals an intrinsic optimal-control interpretation: the extremal policies over $\mathcal{T}^2$ (or its convex hull) maximize or minimize the meeting time, which Krak's algorithm can identify. Overall, the paper connects interdependent stochastic dynamics, imprecise probability, and controlled optimization to yield practical, scalable tools for safe analysis of multi-agent meeting dynamics under uncertainty.

Abstract

We analyse the problem of meeting times for interdependent stochastic agents: random walkers whose behaviour is stochastic but controlled by their selections from some set of allowed actions, and the inference problem of when these agents will be in the same state for the first time. We consider the case where we are epistemically uncertain about the selected actions of these agents, and show how their behaviour can be modelled using imprecise Markov chains. This allows us to use results and algorithms from the literature, to exactly compute bounds on their meeting time, which are tight with respect to our epistemic uncertainty models. We focus on the two-agent case, but discuss how it can be naturally extended to an arbitrary number of agents, and how the corresponding combinatorial explosion can be partly mitigated by exploiting symmetries inherent in the problem.

Upper Expected Meeting Times for Interdependent Stochastic Agents

TL;DR

This work addresses the problem of bounding the expected meeting time for two interdependent stochastic agents operating on a finite state space under epistemic uncertainty about their action selections. It casts the problem as a hitting-time problem on the product space and leverages imprecise Markov chains to model uncertainty, analyzing degenerate, vacuous, and degenerate-vacuous mixtures to obtain tight, computable bounds via Krak's iterative algorithm. The framework naturally extends to agents using a -fold product construction, with symmetry reduction on substantially mitigating the combinatorial blow-up. The resulting approach provides exact or conservatively bounded meeting-time estimates and reveals an intrinsic optimal-control interpretation: the extremal policies over (or its convex hull) maximize or minimize the meeting time, which Krak's algorithm can identify. Overall, the paper connects interdependent stochastic dynamics, imprecise probability, and controlled optimization to yield practical, scalable tools for safe analysis of multi-agent meeting dynamics under uncertainty.

Abstract

We analyse the problem of meeting times for interdependent stochastic agents: random walkers whose behaviour is stochastic but controlled by their selections from some set of allowed actions, and the inference problem of when these agents will be in the same state for the first time. We consider the case where we are epistemically uncertain about the selected actions of these agents, and show how their behaviour can be modelled using imprecise Markov chains. This allows us to use results and algorithms from the literature, to exactly compute bounds on their meeting time, which are tight with respect to our epistemic uncertainty models. We focus on the two-agent case, but discuss how it can be naturally extended to an arbitrary number of agents, and how the corresponding combinatorial explosion can be partly mitigated by exploiting symmetries inherent in the problem.

Paper Structure

This paper contains 15 sections, 4 theorems, 29 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Stochastic Processes and Markov Chains
  4. Hitting and Meeting Times in the Precise Setting
  5. Hitting Times.
  6. Meeting Times.
  7. Imprecise Markov Chains
  8. Reachability Condition.
  9. A Computational Method.
  10. Upper and Lower Meeting Times for Interdependent Stochastic Agents
  11. Degenerate Belief
  12. Vacuous Belief
  13. Computing Upper Meeting Times
  14. The Corresponding Optimal Control Problem
  15. Exploiting the Symmetry.

Key Result

theorem thmcountertheorem

Let $A\subset \mathcal{Z}$ and let $(Z_n)$ be a homogeneous Markov chain with transition matrix $T$. The vector of expected hitting times $h^T_A=(h_A^T(z))_{z\in\mathcal{Z}}$ is the minimal non-negative solution to the following system of equations: which can be rewritten as where $\mathbbm{1}$ is the indicator function and $\cdot$ represents the element-wise multiplication.

Theorems & Definitions (5)

  • theorem thmcountertheorem: norrisbook1997markov
  • theorem thmcountertheorem: bullo2018meetingDoeblin1937
  • theorem thmcountertheorem
  • proof
  • lemma thmcounterlemma: debock2025convergentdebock2017imprecisehermans2012characterisation