A General Framework for Endowment Effects in Combinatorial Markets.

The endowment effect, coined by Nobel Laureate Richard Thaler, posits that people tend to inflate the value of items they own. This bias has been traditionally studied mainly using experimental methodology. Recently, Babaioff, Dobzinski and Oren (2018) proposed a specific formulation of the endowment effect in combinatorial markets, and showed that the existence of Walrasian equilibrium with respect to the endowed valuations (referred to as endowment equilibrium) extends from gross substitutes to submodular valuations, but provably fails to extend to more general valuations, like XOS. We propose to harness the endowment effect further. To this end, we introduce a framework that captures a wide range of formulations of the endowment effect. Our framework is based on two principles, namely loss aversion and separability. The loss aversion principle asserts that people tend to prefer avoiding losses to acquiring equivalent gains. The separability principle asserts that the marginal contribution of unendowed items remains intact. We give a characterization of endowment effects satisfying these two principles. With this characterization at hand, we equip our framework with a partial order over endowment effect formulations, which (partially) ranks them from weak to strong, and provide algorithms for computing endowment equilibria with high welfare for sufficiently strong endowment effects, as well as non-existence results for weaker ones. Our main results are the following: 1) For markets with XOS valuations, we provide an algorithm that, for any sufficiently strong endowment effect, given an arbitrary initial allocation S, returns an endowment equilibrium with at least as much welfare as in S. In particular, the socially optimal allocation can be supported in an endowment equilibrium; moreover, every such endowment equilibrium gives at least half of the optimal social welfare. Evidently, the negative result of Babaioff et al. for XOS markets is an artifact of their specific formulation. 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, we provide an algorithm that, for a wide range of endowment effects, given an arbitrary initial allocation S, computes an endowment equilibrium with at least as much welfare as in S. The algorithm runs in polynomial time with a polynomial number of value (resp., demand) queries for submodular (resp., general) valuations. This result is essentially a black-box reduction from the computation of an approximately-optimal endowment equilibrium with bundling to the algorithmic problem of welfare approximation.