Complexity of Stability in Trading Networks
Tamás Fleiner, Zsuzsanna Jankó, Ildikó Schlotter, Alexander Teytelboym
TL;DR
The paper analyzes the computational aspects of stability notions in trading networks under full substitutability and IRC. It proves that trail-stable outcomes exist and can be found in linear time, but introducing simultaneous upstream and downstream offers yields path-or-cycle stability, whose existence is $NP$-complete to decide; moreover, deciding whether stable outcomes exist and verifying stability are $NP$-hard/$NP$-complete in general, with flow networks providing tight reductions. Through reductions from partition and set-partition-like problems, the authors show that even seemingly natural extensions of stability become intractable, motivating trail stability as a practically computable alternative that aligns with competitive equilibrium in price-augmented models. The results delineate a sharp complexity boundary between tractable and intractable notions of stability in trading networks, guiding mechanism design toward computationally feasible concepts. Open questions regarding weaker stability notions and extensions to richer agent preferences remain as fruitful directions for future research.
Abstract
Efficient computability is an important property of solution concepts in matching markets. We consider the computational complexity of finding and verifying various solution concepts in trading networks-multi-sided matching markets with bilateral contracts-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. Here we consider a slightly stronger solution concept in which agents can simultaneously offer an upstream and a downstream contract. We show that deciding the existence of outcomes satisfying this solution concept is an NP-complete problem even in a special (flow network) case of our model. It follows that the existence of stable outcomes--immune to deviations by arbitrary sets of agents-is also an NP-hard problem in trading networks (and in flow networks). Finally, we show that even verifying whether a given outcome is stable is NP-complete in trading networks.