Complexity of stability in trading networks

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 NP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\textsf{N}}}{{\textsf{P}}}$$\end{document}-hard problem in trading networks. We also show that even verifying whether a given outcome is stable is NP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\textsf{N}}}{{\textsf{P}}}$$\end{document}-hard in trading networks.