  We study the performance of general dynamic matching models. This model is
defined by a connected graph, where nodes represent the class of items and the
edges the compatibilities between items. Items of different classes arrive one
by one to the system according to a given probability distribution. Upon
arrival, an item is matched with a compatible item according to the First Come
First Served discipline and leave the system immediately, whereas it is
enqueued with other items of the same class, if any. We show that such a model
may exhibit a non intuitive behavior: increasing the services ability by adding
new edges in the matching graph may lead to a larger average population. This
is similar to a Braess paradox. We first consider a quasicomplete graph with
four nodes and we provide values of the probability distribution of the
arrivals such that when we add an edge the mean number of items is larger.
Then, we consider an arbitrary matching graph and we show sufficient conditions
for the existence or non-existence of this paradox. We conclude that the analog
to the Braess paradox in matching models is given when specific independent
sets are in saturation, i.e., the system is close to the stability condition.
