Approximation Techniques for Financial Networks with Liability Priorities

In the world of finance, systemic risk is a major concern. When a large financial institution fails, it can have a ripple effect throughout the entire industry, causing widespread instability. To mitigate this risk, regulators and policymakers need to understand how different financial institutions are connected and how they can work together to maintain stability. One key tool for analyzing these connections is the concept of a Nash equilibrium, which refers to a situation where no player can improve their outcome by unilaterally changing their strategy.
In this article, we explore the computational challenge of computing a Nash equilibrium in financial networks. Imagine you’re at a busy intersection with multiple traffic lights. Each light represents a financial institution, and each time a light turns green, it means that institution has taken on new commitments. The goal is to compute the percentage of liabilities each institution should pay if the entire network had to be cleared at once, much like how traffic lights work together to maintain a smooth flow of traffic.
The problem is that this computation becomes increasingly complex as the number of institutions and their connections grow. In fact, it’s so difficult that previous research has shown it can take over 20 years of computing power to solve just one instance of the problem! To make matters worse, the complexity of the problem grows exponentially with the size of the network, making it almost impossible to solve for large networks.
To overcome this challenge, we propose a new framework called L(R, F, C), which combines intervals and numbers within the range [0, 1] to handle combined arithmetic operations. This allows us to represent complex financial commitments in a more concise and expressive way, making it possible to solve the clearing problem more efficiently.
In summary, computing a Nash equilibrium in financial networks is a complex task that becomes exponentially harder as the size of the network grows. However, by using a new framework called L(R, F, C), we can make significant progress towards solving this problem and improving our understanding of how different financial institutions interact and affect each other’s stability.