  We study the consequences of adopting products by agents who form a social
network. To this end we use the threshold model introduced in Apt and Markakis,
arXiv:1105.2434, in which the nodes influenced by their neighbours can adopt
one out of several alternatives, and associate with such each social network a
strategic game between the agents. The possibility of not choosing any product
results in two special types of (pure) Nash equilibria.
  We show that such games may have no Nash equilibrium and that determining the
existence of a Nash equilibrium, also of a special type, is NP-complete. The
situation changes when the underlying graph of the social network is a DAG, a
simple cycle, or has no source nodes. For these three classes we determine the
complexity of establishing whether a (special type of) Nash equilibrium exists.
  We also clarify for these categories of games the status and the complexity
of the finite improvement property (FIP). Further, we introduce a new property
of the uniform FIP which is satisfied when the underlying graph is a simple
cycle, but determining it is co-NP-hard in the general case and also when the
underlying graph has no source nodes. The latter complexity results also hold
for verifying the property of being a weakly acyclic game.
