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Unique Ergodicity in the Interconnections of Ensembles with Applications to Two-Sided Markets

Wynita M. Griggs, Ramen Ghosh, Jakub Marecek, Robert N. Shorten

TL;DR

The paper addresses stability and fairness in two- and multi-sided market dynamics by introducing a discrete-time, iterated random function framework that captures ensemble interconnections and information-driven resource allocation. It defines unique ergodicity as per-agent long-run averages that are independent of initial conditions and proves convergence to a single invariant measure under average-contractivity conditions (Schur-stability, positive lower-bounded transition probabilities) for both linear and nonlinear dynamics, including large-scale multi-ensemble networks. The contributions provide a rigorous mechanism to guarantee predictability and fairness in platform economies, with practical implications for ride-hailing and other shared-resource systems, supported by toy numerical illustrations. The work invites extensions to discrete action spaces, more intricate network topologies, and real-world deployment considerations, offering a principled path to ergodic control in complex interconnected populations.

Abstract

There has been much recent interest in two-sided markets and dynamics thereof. In a rather a general discrete-time feedback model, which we show conditions that assure that for each agent, there exists the limit of a long-run average allocation of a resource to the agent, which is independent of any initial conditions. We call this property the unique ergodicity. Our model encompasses two-sided markets and more complicated interconnections of workers and customers, such as in a supply chain. It allows for non-linearity of the response functions of market participants. Finally, it allows for uncertainty in the response of market participants by considering a set of the possible responses to either price or other signals and a measure to sample from these.

Unique Ergodicity in the Interconnections of Ensembles with Applications to Two-Sided Markets

TL;DR

The paper addresses stability and fairness in two- and multi-sided market dynamics by introducing a discrete-time, iterated random function framework that captures ensemble interconnections and information-driven resource allocation. It defines unique ergodicity as per-agent long-run averages that are independent of initial conditions and proves convergence to a single invariant measure under average-contractivity conditions (Schur-stability, positive lower-bounded transition probabilities) for both linear and nonlinear dynamics, including large-scale multi-ensemble networks. The contributions provide a rigorous mechanism to guarantee predictability and fairness in platform economies, with practical implications for ride-hailing and other shared-resource systems, supported by toy numerical illustrations. The work invites extensions to discrete action spaces, more intricate network topologies, and real-world deployment considerations, offering a principled path to ergodic control in complex interconnected populations.

Abstract

There has been much recent interest in two-sided markets and dynamics thereof. In a rather a general discrete-time feedback model, which we show conditions that assure that for each agent, there exists the limit of a long-run average allocation of a resource to the agent, which is independent of any initial conditions. We call this property the unique ergodicity. Our model encompasses two-sided markets and more complicated interconnections of workers and customers, such as in a supply chain. It allows for non-linearity of the response functions of market participants. Finally, it allows for uncertainty in the response of market participants by considering a set of the possible responses to either price or other signals and a measure to sample from these.

Paper Structure

This paper contains 17 sections, 6 theorems, 40 equations, 11 figures.

Table of Contents

  1. Introduction
  2. An Application: Two-sided Markets
  3. Another Application: Multi-sided Platforms
  4. Related Work
  5. Contributions and Paper Organization
  6. Preliminaries, Notation, and the Description of our Model
  7. An Interconnection Result for Two Ensemble Systems
  8. A Large-scale Interconnection for Ensemble Systems
  9. Conclusions and Further Work
  10. An Overview of Notation
  11. Expanding the Interconnection Results Further
  12. Toy Example 1
  13. Toy Example 2
  14. Numerical Illustrations and Discussion
  15. Toy Example 1
  16. ...and 2 more sections

Key Result

Theorem 3.1

Consider the feedback system in Figure fig:system_ii, with ${\mathcal{C}}^1, {\mathcal{C}}^2$ and ${\mathcal{F}}^1, {\mathcal{F}}^2$ given in eq:C1, eq:C2, eq:F1, and eq:F2, respectively. Assume that each agent $i \in \{1,\cdots,N\}$ has its state $x_i$ with dynamics determined by the equations give described by $\xi(k+1) = {\mathcal{W}}_\ell(x) \mathrel{\mathop:}= \mathcal{A}\xi(k) + \beta_\ell,$

Figures (11)

  • Figure 1: The feedback model of Fioravanti2019, depicted utilizing our notation.
  • Figure 2: Our feedback model of a two-sided market.
  • Figure 3: A feedback model for the large-scale interconnection.
  • Figure 5: The interconnection of two populations of drivers under consideration in Toy Example 1.
  • Figure 6: An interconnection with an elastic demand, as described in Toy Example 2.
  • ...and 6 more figures

Theorems & Definitions (12)

  • Definition 2.1
  • Definition 2.2: Invariant Probability Measure
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5: Unique Ergodicity
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Theorem 4.1
  • Proposition 7.1
  • ...and 2 more