Optimal Bailouts and Strategic Debt Forgiveness in Financial Networks

TL;DR

Results are proved about the existence and quality of pure Nash equilibria, as well as the computational complexity of finding such equilibria, about the existence and quality of pure Nash equilibria.

Abstract

A financial system is represented by a network, where nodes correspond to banks, and directed labeled edges correspond to debt contracts between banks. Once a payment schedule has been defined, where we assume that a bank cannot refuse a payment towards one of its lenders if it has sufficient funds, the liquidity of the system is defined as the sum of total payments made in the network. Maximizing systemic liquidity is a natural objective of any financial authority, so, we study the setting where the financial authority offers bailout money to some bank(s) or forgives the debts of others in order to maximize liquidity, and examine efficient ways to achieve this. We investigate the approximation ratio provided by the greedy bailout policy compared to the optimal one, and we study the computational hardness of finding the optimal debt-removal and budget-constrained optimal bailout policy, respectively. We also study financial systems from a game-theoretic standpoint. We observe that the removal of some incoming debt might be in the best interest of a bank, if that helps one of its borrowers remain solvent and avoid costs related to default. Assuming that a bank's well-being (i.e., utility) is aligned with the incoming payments they receive from the network, we define and analyze a game among banks who want to maximize their utility by strategically giving up some incoming payments. In addition, we extend the previous game by considering bailout payments. After formally defining the above games, we prove results about the existence and quality of pure Nash equilibria, as well as the computational complexity of finding such equilibria.

Figures (14)

• Figure 1: A simple financial network. Nodes correspond to banks, edges are labeled with the respective liabilities, while external assets are in a rectangle above the relevant banks.
• Figure 2: An example network used in the lower bound of the approximation ratio of Greedy for $\mu_{v}=2$ and $\mu_{w}=1+\frac{1}{2}\cdot 2=2$. The claim in the proof is that for any arbitrary network such that the first cash injection made by Greedy is $t_1$, and the highest threat index is an integer, e.g. $2$ in this case, the network in this figure achieves at most the same approximation ratio, while satisfying properties (P1) and (P2).
• Figure 3: Similarly to Figure \ref{['fig:Greedy_approx-ub']}, for the case where the highest threat index is not an integer, e.g. $\mu_{v}=3.6=1+(1+(1+\frac{0.6t_1}{t_1}\cdot 1))$ and $\mu_{w}=1+\frac{t_1}{t_1+t_1/2.6}\cdot \mu_{v}=1+\frac{2.6}{3.6}\cdot 3.6=3.6$.
• Figure 4: The reduction used to show hardness of computing the optimal cash injection policy when $\alpha<1$.
• Figure 5: The reduction used to show hardness of computing an edge-removal policy that maximizes systemic liquidity. All edges with missing labels correspond to liability $1$.

Theorems & Definitions (33)

• Theorem 1
• proof
• Definition 1: Greedy 's approximation ratio
• Theorem 2
• proof
• Theorem 3
• proof
• Theorem 4
• proof
• Theorem 5