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A Dynkin Game with Independent Processes and Incomplete Information

Georgy Gaitsgori, Richard Groenewald

TL;DR

This work analyzes a two-player, private-information Dynkin stopping game where each player observes an independent Brownian motion and earns a payoff $f(x_i+W^{(i)}_{ au_i})$ upon stopping first. The authors reduce the game to a single-Brownian optimal stopping problem with endogenous discounting and perform a detailed tail-analysis, leveraging Breiman's square-root boundary results and Novikov-type bounds. They establish a sharp dichotomy: infinitely many Nash equilibria with infinite payoffs exist, while finite-payoff equilibria are highly structured or may be absent depending on the starting positions and the growth of $f$. The results extend to pure and mixed strategies and generalize to $n$-player settings and to variants with different reward functions, highlighting the intricate role of incomplete private information in Dynkin-type timing games and offering a foundation for further extensions into finite horizons and bounded rewards.

Abstract

We analyze a two-player, nonzero-sum Dynkin game of stopping with incomplete information. We assume that each player observes his own Brownian motion, which is not only independent of the other player's Brownian motion but also not observable by the other player. The player who stops first receives a payoff that depends on the stopping position. Under appropriate growth conditions on the reward function, we show that there are infinitely many Nash equilibria in which both players attain infinite expected payoffs. In contrast, the only equilibrium with finite expected payoffs mandates immediate stopping by at least one of the players. Our results hold in the settings of both pure and mixed strategies.

A Dynkin Game with Independent Processes and Incomplete Information

TL;DR

This work analyzes a two-player, private-information Dynkin stopping game where each player observes an independent Brownian motion and earns a payoff upon stopping first. The authors reduce the game to a single-Brownian optimal stopping problem with endogenous discounting and perform a detailed tail-analysis, leveraging Breiman's square-root boundary results and Novikov-type bounds. They establish a sharp dichotomy: infinitely many Nash equilibria with infinite payoffs exist, while finite-payoff equilibria are highly structured or may be absent depending on the starting positions and the growth of . The results extend to pure and mixed strategies and generalize to -player settings and to variants with different reward functions, highlighting the intricate role of incomplete private information in Dynkin-type timing games and offering a foundation for further extensions into finite horizons and bounded rewards.

Abstract

We analyze a two-player, nonzero-sum Dynkin game of stopping with incomplete information. We assume that each player observes his own Brownian motion, which is not only independent of the other player's Brownian motion but also not observable by the other player. The player who stops first receives a payoff that depends on the stopping position. Under appropriate growth conditions on the reward function, we show that there are infinitely many Nash equilibria in which both players attain infinite expected payoffs. In contrast, the only equilibrium with finite expected payoffs mandates immediate stopping by at least one of the players. Our results hold in the settings of both pure and mixed strategies.
Paper Structure (16 sections, 7 theorems, 90 equations)

This paper contains 16 sections, 7 theorems, 90 equations.

Table of Contents

  1. Introduction
  2. Model
  3. Main results
  4. Outline and Intuition
  5. Proof of Theorem \ref{['main_thm']}
  6. Proof of
  7. Proof of
  8. Proof of
  9. Proof of
  10. Ramifications and Extensions
  11. Ramifications of Theorem \ref{['main_thm']}
  12. Different Reward Functions
  13. $n$--Player Games
  14. Mixed Strategies
  15. Technical results
  16. ...and 1 more sections

Key Result

Theorem 3.1

Consider the two-player game described in the previous section, started at $(x_1, x_2) \in \mathbb{R}^2$. Then we have the following classification of Nash equilibria:

Theorems & Definitions (12)

  • Definition 2.1
  • Theorem 3.1
  • Remark 5.1
  • Corollary 5.2
  • Theorem 6.1: Breiman (1966)
  • Lemma 6.2
  • proof
  • Lemma 6.3
  • proof
  • Theorem 6.4: Barbato (2005)
  • ...and 2 more