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) |