  Extreme valuation and volatility of cryptocurrencies require investors to
diversify often which demands secure exchange protocols. A cross-chain swap
protocol allows distrusting parties to securely exchange their assets. However,
the current models and protocols assume predefined user preferences for
acceptable outcomes. This paper presents a generalized model of swaps that
allows each party to specify its preferences on the subsets of its incoming and
outgoing assets. It shows that the existing swap protocols are not necessarily
a strong Nash equilibrium in this model. It characterizes the class of swap
graphs that have protocols that are safe, live and a strong Nash equilibrium,
and presents such a protocol for this class. Further, it shows that deciding
whether a swap is in this class is NP-hard through a reduction from 3SAT, and
further is $\Sigma_2^{\mathsf{P}}$-complete through a reduction from
$\exists\forall\mathsf{DNF}$.
