Martingale theory for asymmetric information games
Lead Research Organisation:
University of Leeds
Abstract
Optimisation is a cornerstone of many mathematical theories and has driven multiple applications of mathematics for millennia. Game theory studies optimisation in situations where the outcome depends on actions of multiple individuals (players) and has been widely used in social sciences, operations research, and other fields as well as driving innovation in mathematics. It has underlain research that lead to multiple Nobel prizes, most recently in 2020.
A zero-sum game is a two-player game in which gains and losses of one player are balanced by the losses and gains of the other player. The object of interest in such an interaction is a pair of strategies (called a saddle point or an equilibrium) such that each player's strategy optimises their outcome given the other player's actions.
This project is concerned with stochastic zero-sum stopping games in which players choose a random time (for example, depending on the evolution of the underlying stochastic process) at which they exit the game. The outcome of the game is calculated at the first time that one of the players exits and depends on the value of the underlying process at that time and on the player that stopped the game. The years 1970s and 1980s saw the development of a beautiful martingale and (partial) differential equations theories which have fuelled most developments and applications of those games.
The defining feature of the existing theory is that players select stopping times with respect to the same filtration. This project aims to develop a general theory for games in which players have different information flows, i.e., the aforementioned symmetry is broken. Such asymmetry of information has been observed in applications (e.g., insider trading, fraud detection) but could not be mathematically studied due to the lack of theoretical foundations.
The first significant results for zero-sum stopping games with asymmetric information appeared in 2010s. They were based on insights unique to a particular model and lacked a general methodological pathway. This project aims to construct a unified approach akin to the classical martingale and Markovian theories for symmetric information zero-sum stopping games.
The first step in this direction was made by the project lead with his collaborators in 2022. Using topological methods, unlike the classical theory, they showed that a general zero-sum game with asymmetric information has an equilibrium in mixed strategies (it is known that the equilibrium may not exist in pure strategies). This general approach does not, however, give any indication of players' strategies. Nor does any existing literature offer a general recipe for construction of player's strategies.
The proposed research will:
(1) develop a martingale theory akin to the classical one characterising each player's value process and equilibrium strategies,
(2) demonstrate how this theory enables formulation of PDE problems in specific diffusive settings and how player's strategies can be obtained from their solution therefore enabling applications of the theory by a wide audience,
(3) explore whether the topological approach can be used to prove existence of equilibria in non-zero sum stopping games with symmetric or asymmetric information.
A zero-sum game is a two-player game in which gains and losses of one player are balanced by the losses and gains of the other player. The object of interest in such an interaction is a pair of strategies (called a saddle point or an equilibrium) such that each player's strategy optimises their outcome given the other player's actions.
This project is concerned with stochastic zero-sum stopping games in which players choose a random time (for example, depending on the evolution of the underlying stochastic process) at which they exit the game. The outcome of the game is calculated at the first time that one of the players exits and depends on the value of the underlying process at that time and on the player that stopped the game. The years 1970s and 1980s saw the development of a beautiful martingale and (partial) differential equations theories which have fuelled most developments and applications of those games.
The defining feature of the existing theory is that players select stopping times with respect to the same filtration. This project aims to develop a general theory for games in which players have different information flows, i.e., the aforementioned symmetry is broken. Such asymmetry of information has been observed in applications (e.g., insider trading, fraud detection) but could not be mathematically studied due to the lack of theoretical foundations.
The first significant results for zero-sum stopping games with asymmetric information appeared in 2010s. They were based on insights unique to a particular model and lacked a general methodological pathway. This project aims to construct a unified approach akin to the classical martingale and Markovian theories for symmetric information zero-sum stopping games.
The first step in this direction was made by the project lead with his collaborators in 2022. Using topological methods, unlike the classical theory, they showed that a general zero-sum game with asymmetric information has an equilibrium in mixed strategies (it is known that the equilibrium may not exist in pure strategies). This general approach does not, however, give any indication of players' strategies. Nor does any existing literature offer a general recipe for construction of player's strategies.
The proposed research will:
(1) develop a martingale theory akin to the classical one characterising each player's value process and equilibrium strategies,
(2) demonstrate how this theory enables formulation of PDE problems in specific diffusive settings and how player's strategies can be obtained from their solution therefore enabling applications of the theory by a wide audience,
(3) explore whether the topological approach can be used to prove existence of equilibria in non-zero sum stopping games with symmetric or asymmetric information.
People |
ORCID iD |
| Jan Palczewski (Principal Investigator) |
Publications
De Angelis T
(2025)
Martingale theory for Dynkin games with asymmetric information
Kwon H
(2024)
Exit Game with Private Information
in Mathematics of Operations Research
| Description | Imagine two players observing the same stochastic process and each choosing when to stop it, while only one player possesses crucial private information, such as whether the dynamics drift upward or downward or whether the process exits through the upper or lower end of an interval. This setting leads naturally to stopping games with asymmetric information. The primary objective of the project was the development of a systematic theory of zero-sum stopping games beyond the symmetric-information setting. This objective has been fully achieved. We identified the equilibrium value processes of both players, characterised the associated super- and submartingales corresponding to optimal randomised strategies (saddle points), and described the support sets of these strategies. These properties were shown to be both necessary and sufficient for the existence of a saddle point. To our knowledge, these are the first results of this generality for stopping games with asymmetric information, requiring only that the players' filtrations are complete and right-continuous. The abstract theory developed in this research leads to explicit and computable expressions in concrete models. To demonstrate this, we applied our framework to two classes of zero-sum stopping games with asymmetric information and diffusive dynamics that have previously been studied in the literature. In the first class, the game involves finitely many payoff regimes: one player is fully informed and observes the realised regime at the outset, while the other player only knows its prior distribution. In the second class, the informational asymmetry is analogous, but the hidden regime affects only the drift of the observable diffusion driving the payoff. The key distinction between the two classes lies in the learning opportunities available to the less-informed player. In the first case, information can only be inferred from the absence of stopping actions by the opponent, whereas in the second case the player additionally observes the underlying diffusion and can exploit this information through stochastic filtering. For both classes of games, our theory applies at a substantially greater level of generality than existing results and often provides alternative or complementary methods of analysis. In particular, the super- and submartingale characterisations translate naturally into partial differential equation formulations for diffusive models, thereby linking the game-theoretic equilibrium analysis with PDE methods. The framework developed in this project provides a systematic route to constructing saddle points in specific stopping games with asymmetric information. Beyond the two classes of games studied in detail, we are currently working with researchers not involved in this grant on further explicit solutions of games with asymmetric information. Building a portfolio of publications based on this framework is expected to facilitate uptake by the broader stochastic control and game theory communities and to provide practical guidance on applying the abstract theory. In summary, the main objectives of the research have been fully met. Some follow-up work remains in progress. A secondary objective of the project was to explore the use of topological methods to prove the existence of equilibria in non-zero-sum stopping games. Progress on this objective was more limited than initially anticipated. During the review period of the grant proposal, related results using such methods were published by other research groups, which necessitated revisions to the original research plan and led to problems of higher complexity than originally envisaged. |
| Exploitation Route | The outcomes of this funding provide a general theoretical framework that can be taken forward by researchers working in stochastic control, game theory, and applied probability, particularly in areas where asymmetric or incomplete information plays a central role. The martingale characterisation of equilibrium value processes and the associated necessary and sufficient conditions for saddle points offer tools that can be directly applied to new classes of stopping games beyond those studied in this project. In particular, the framework is well suited for use by researchers studying game problems in financial mathematics (such as game options and cancellable contracts), and economic models with private information and learning. The explicit links established between equilibrium conditions and partial differential equation characterisations in diffusive settings also make the results accessible to researchers using analytical and numerical PDE methods. The theory is being taken forward through ongoing collaborations with researchers not involved in this grant, with the aim of producing further explicit solutions and case studies that illustrate how the abstract results can be implemented in practice. These follow-up works are expected to facilitate wider uptake by providing concrete examples and methodological guidance. More broadly, the results contribute foundational tools that may be incorporated into future theoretical developments in dynamic games with information asymmetries. |
| Sectors | Communities and Social Services/Policy Financial Services and Management Consultancy Other |