Complexity of stability in trading networks

Abstract

Efficient computability is an important property of solution concepts. We consider the computational complexity of finding and verifying various solution concepts in trading networks—multi-sided matching markets with bilateral contracts and without transferable utility—under the assumption of full substitutability of agents’ preferences. It is known that outcomes that satisfy trail stability always exist and can be found in linear time. However, we show that the existence of stable outcomes—immune to deviations by arbitrary sets of agents—is an $${{\textsf{N}}}{{\textsf{P}}}$$ N P -hard problem in trading networks. We also show that even verifying whether a given outcome is stable is $${{\textsf{N}}}{{\textsf{P}}}$$ N P -hard in trading networks.

Optimization methods and algorithms

“…Stability in network was studied by Fleiner et al (2019), where they proved the existence of a so-called trail-stable solution in a very general environment with choice functions over possible contracts. The computational complexity of computing stable solutions was studied by Fleiner et al (2023). Stable flows were also studied by using the stable fractional solutions by the Scarf-algorithm by Csáji (2022).…”