Dynamic Debt Swapping in Financial Networks

TL;DR

The paper studies debt swaps in directed, weighted financial networks where banks interact through debt contracts, focusing on the algorithmic complexity of local and global network interventions under monotone clearing rules. It develops a cohesive framework that classifies swaps into positive, semi-positive, and arbitrary types, proves that positive swaps cannot occur, and shows that semi-positive swaps strictly Pareto-improve the network, enabling polynomial-length improvement sequences in the v-improving setting. It further proves that reaching a local optimum with arbitrary v-improving swaps is PLS-complete, while global optimization (maximizing a bank’s assets or the sum of assets) is NP-hard to approximate, with reductions from Set Cover, Independent Set, and Partition. Reachability results separate tractability for arbitrary swaps (polynomial-time) from PSPACE-hard variants under asset lower bounds, and NP-hardness for semi-positive reachability, illustrating a nuanced landscape for debt-swap dynamics. The findings shed light on the feasibility and limits of distributed network interventions in financial systems and motivate future work on extended clearing rules and more general network interventions.

Abstract

A debt swap is an elementary edge swap in a directed, weighted graph, where two edges with the same weight swap their targets. Debt swaps are a natural and appealing operation in financial networks, in which nodes are banks and edges represent debt contracts. They can improve the clearing payments and the stability of these networks. However, their algorithmic properties are not well-understood. We analyze the computational complexity of debt swapping. Our main interest lies in semi-positive swaps, in which no creditor strictly suffers and at least one strictly profits. These swaps lead to a Pareto-improvement in the entire network. We consider network optimization via sequences of v-improving debt swaps from which a given bank v strictly profits. For ranking-based clearing, we show that every sequence of semi-positive v-improving swaps has polynomial length. In contrast, for arbitrary v-improving swaps, the problem of reaching a network configuration that allows no further swaps is PLS-complete. In global optimization, the goal is to maximize the utility of a given bank $v$ by performing a sequence of debt swaps in the network. This problem is NP-hard to approximate for multiple types of swaps. Moreover, we study reachability problems -- deciding if a sequence of swaps exists between given initial and final networks. We design a polynomial-time algorithm to decide this question for arbitrary swaps and derive hardness results for several other types of swaps. Many of our results can be extended to networks with arbitrary monotone clearing.

Figures (11)

• Figure 1: The figures visualize a financial network $\mathcal{F}$ before (left) and ${\mathcal{F}}^{\sigma}$ after a debt swap (right). All edges have unit weight and edge labels indicate payments. For $v_1$, the solid edge is the preferred outgoing edge. $v_2$ has external assets of 1 as shown in the node label.
• Figure 2: The network on the left corresponds to the network ${\mathcal{F}}^{\sigma}$ in Example \ref{['ex:intro']}. Black edge labels indicate payments. The middle network results from applying source-sink equivalency to $v_1$ and $v_2$, whereas the right network results from applying source-sink equivalency for edges $(u_2,v_1)$ and $(v_2,u_1)$.
• Figure 3: Schematic illustration of networks $\mathcal{G}$ and $\mathcal{H}$ for a financial network $\mathcal{F}$.
• Figure 4: Minimal example of an extension swap where all edges have weight $M>k$ and edge labels indicate payments in the clearing state. The swap extends the path of $u_2$'s payments to reach $v_2$, increasing $v_1$'s total assets of $v_1$ along the way.
• Figure 5: Schematic picture of a $v$-improving semi-positive swap of active edges where payments $p_{e_2}>p_{e_1}$. Triangles indicate subtrees. When the blue edges are swapped then payments on the thick black edges strictly increase.

Theorems & Definitions (80)

• Definition 2.1: Debt Swap
• Example 2.2
• Definition 2.3: Swap Types
• Lemma 3.0: Monotonicity
• Lemma 3.0: Non-Expansivity
• Definition 3.1
• Lemma 3.1: Source-Sink-Equivalency
• Corollary 3.2: Source-Sink-Equivalency for Edges
• Theorem 3.4
• Lemma 3.4