  We study competitive equilibrium in the canonical Fisher market model, but
with indivisible goods. In this model, every agent has a budget of artificial
currency with which to purchase bundles of goods. Equilibrium prices match
between demand and supply---at such prices, all agents simultaneously get their
favorite within-budget bundle, and the market clears. Unfortunately, a
competitive equilibrium may not exist when the goods are indivisible, even in
extremely simple markets such as two agents with exactly the same budget and a
single item. Yet in this example, once the budgets are slightly
perturbed---i.e., made generic---a competitive equilibrium is guaranteed to
exist. In this paper we explore the extent to which generic budgets can
guarantee equilibrium existence (and thus related fairness guarantees) in
markets with multiple items. We complement our results in [Babaioff et al.,
2019] for additive preferences by exploring the case of general monotone
preferences, establishing positive results for small numbers of items and
mapping the limits of our approach. We then consider cardinal preferences,
define a hierarchy of such preference classes and establish relations among
them, and for some classes prove equilibrium existence under generic budgets.
