This paper proposes a new one-sided matching market model in which every
agent has a cost function that is allowed to take a negative value. Our model
aims to capture the situation where some agents can profit by exchanging their
obtained goods with other agents. We formulate such a model based on a
graphical one-sided matching market, introduced by Massand and Simon [Massand
and Simon, IJCAI 2019]. We investigate the existence of stable outcomes for
such a market. We prove that there is an instance that has no core-stable
allocation. On the other hand, we guarantee the existence of two-stable
allocations even where exchange costs exist. However, it is PLS-hard to find a
two-stable allocation for a market with exchange costs even if the maximum
degree of the graph is five