  The endowment effect, coined by Nobel Laureate Richard Thaler, posits that
people tend to inflate the value of items they own. This bias was studied, both
theoretically and empirically, with respect to a single item. Babaioff et al.
[EC'18] took a first step at extending this study beyond a single item. They
proposed a specific formulation of the endowment effect in combinatorial
settings, and showed that equilibrium existence with respect to the endowed
valuations extends from gross substitutes to submodular valuations, but
provably fails to extend to XOS valuations.
  Extending the endowment effect to combinatorial settings can take different
forms. In this work, we devise a framework that captures a space of endowment
effects, upon which we impose a partial order, which preserves endowment
equilibrium existence. Within this framework, we provide existence and welfare
guarantees for endowment equilibria corresponding to various endowment effects.
Our main results are the following:
  (1) For markets with XOS valuations, we introduce an endowment effect that is
stronger than that of Babaioff et al., for which an endowment equilibrium is
guaranteed to exist and gives at least half of the optimal welfare. Moreover,
this equilibrium can be reached via a variant of the flexible ascent auction.
  (2) For markets with arbitrary valuations, we show that bundling leads to a
sweeping positive result. In particular, if items can be prepacked into
indivisible bundles, there always exists an endowment equilibrium with optimal
welfare. Moreover, we provide a polynomial algorithm that given an arbitrary
allocation $S$, computes an endowment equilibrium with the same welfare
guarantee as in $S$.
