Complexity of Stability in Trading Networks

Tamás Fleiner, Zsuzsanna Jankó, Ildikó Schlotter, Alexander Teytelboym

Published 2018-05-22Version 1

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. First, we show that outcomes that satisfy an economically intuitive solution concept---trail stability---always exist and can be found in linear time. Second, 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-complete 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.