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Fare Game: A Mean Field Model of Stochastic Intensity Control in Dynamic Ticket Pricing

Burak Aydin, Emre Parmaksiz, Ronnie Sircar

TL;DR

This work introduces a finite-horizon mean-field game of stochastic intensity control for dynamic pricing of discrete goods, modeling a continuum of firms competing over inventory with Poisson-type sales arrivals and an intensity $\\lambda(p, \\bar{p}, m)$. The core methodology derives a coupled Hamilton-Jacobi-Bellman and Kolmogorov differential-difference system and proves existence (and, for small competition, uniqueness) of mean-field equilibria under a piecewise-linear intensity. A numerically tractable fixed-point algorithm is developed to compute the equilibrium, with stability analyses and simulations illustrating how competition delays inventory depletion and lowers market prices, plus qualitative alignment with airfare patterns. The results provide a rigorous foundation for mean-field revenue-management models in oligopolistic, perishable-good markets and point to future directions including overselling, reselling, and data-driven calibration. The framework offers insights into how macroeconomic market parameters influence pricing strategy and aggregate dynamics in competitive, finite-horizon environments.

Abstract

We study the dynamic pricing of discrete goods over a finite selling horizon. One way to capture both the elastic and stochastic reaction of purchases to price is through a model where sellers control the intensity of a counting process, representing the number of sales thus far. The intensity describes the probabilistic likelihood of a sale, and is a decreasing function of the price a seller sets. A classical model for ticket pricing, which assumes a single seller and finite time horizon, is by Gallego and van Ryzin (1994) and it has been widely utilized by airlines, for instance. Extending to more realistic settings where there are multiple sellers, with finite inventories, in competition over a finite time horizon is more complicated both mathematically and computationally. We introduce a dynamic mean field game of this type, and some numerical and existence results. In particular, we analyze the associated coupled system of Hamilton-Jacobi-Bellman and Kolmogorov differential-difference equations, and we prove the existence and uniqueness results under certain conditions. Then, we demonstrate a numerical algorithm to find this solution and provide some insights into the macroeconomic market parameters. Finally, we present a qualitative comparison of our findings with airfare data.

Fare Game: A Mean Field Model of Stochastic Intensity Control in Dynamic Ticket Pricing

TL;DR

This work introduces a finite-horizon mean-field game of stochastic intensity control for dynamic pricing of discrete goods, modeling a continuum of firms competing over inventory with Poisson-type sales arrivals and an intensity . The core methodology derives a coupled Hamilton-Jacobi-Bellman and Kolmogorov differential-difference system and proves existence (and, for small competition, uniqueness) of mean-field equilibria under a piecewise-linear intensity. A numerically tractable fixed-point algorithm is developed to compute the equilibrium, with stability analyses and simulations illustrating how competition delays inventory depletion and lowers market prices, plus qualitative alignment with airfare patterns. The results provide a rigorous foundation for mean-field revenue-management models in oligopolistic, perishable-good markets and point to future directions including overselling, reselling, and data-driven calibration. The framework offers insights into how macroeconomic market parameters influence pricing strategy and aggregate dynamics in competitive, finite-horizon environments.

Abstract

We study the dynamic pricing of discrete goods over a finite selling horizon. One way to capture both the elastic and stochastic reaction of purchases to price is through a model where sellers control the intensity of a counting process, representing the number of sales thus far. The intensity describes the probabilistic likelihood of a sale, and is a decreasing function of the price a seller sets. A classical model for ticket pricing, which assumes a single seller and finite time horizon, is by Gallego and van Ryzin (1994) and it has been widely utilized by airlines, for instance. Extending to more realistic settings where there are multiple sellers, with finite inventories, in competition over a finite time horizon is more complicated both mathematically and computationally. We introduce a dynamic mean field game of this type, and some numerical and existence results. In particular, we analyze the associated coupled system of Hamilton-Jacobi-Bellman and Kolmogorov differential-difference equations, and we prove the existence and uniqueness results under certain conditions. Then, we demonstrate a numerical algorithm to find this solution and provide some insights into the macroeconomic market parameters. Finally, we present a qualitative comparison of our findings with airfare data.

Paper Structure

This paper contains 27 sections, 3 theorems, 97 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Stochastic Intensity Mean Field Game
  3. Model Dynamics
  4. Linear Intensity Model
  5. Main Results
  6. Numerical Results
  7. Algorithm
  8. Numerical Stability and Convergence of the Algorithm
  9. Algorithm Results
  10. Qualitative Comparison with Data
  11. Conclusion
  12. Brief Review and Limit of the Finite-Player Case
  13. Proof of Proposition \ref{['prop:2']}
  14. The inductive setup for Picard-Lindelöf theorem
  15. Inductive step, proving existence and uniqueness of $\Delta V_k^{(0)}$
  16. ...and 12 more sections

Key Result

Proposition 2.1

For $\epsilon = 0$, a solution $(V^{(0)}, m^{(0)}) = (\{ V_k^{(0)} \}_{k=1}^{K} , \{ m_k^{(0)} \}_{k=1}^{K})$ to the system of differential equations eq:3.6 exists and is unique. Further, for all $k \in \mathcal{K}$, and $t \in [0,T]$, we have where $\rho = \sqrt{r^2 + r} - r \in [0,\frac{1}{2})$. Consequently, for $\lambda^{(0)}_k(t) = \dfrac{1}{2} (1 - \Delta V^{(0)}_k (t))$, we have

Figures (6)

  • Figure 1: Concavity of $V$ in k
  • Figure 2: Evolution of CDF of $m(t)$ at given times for varying $\epsilon$.
  • Figure 3: Evolution of optimal price $p^*(t)$ at given times for varying $\epsilon$.
  • Figure 4: Comparisons for varying $\epsilon$ values.
  • Figure 5: Density $m(t)$ at inflection points
  • ...and 1 more figures

Theorems & Definitions (6)

  • Proposition 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6