Two-Price Equilibrium
Walrasian equilibrium is a prominent market equilibrium notion, but rarely exists in markets with indivisible items. We introduce a new market equilibrium notion, called two-price equilibrium (2PE). A 2PE is a relaxation of Walrasian equilibrium, where instead of a single price per item, every item has two prices: one for the item's owner and a (possibly) higher one for all other buyers. Thus, a 2PE is given by a tuple $(\textbf{S},\mathbf{\hat{p}},\mathbf{\check{p}})$ of an allocation $\textbf{S}$ and two price vectors $\mathbf{\hat{p}},\mathbf{\check{p}}$, where every buyer $i$ is maximally happy with her bundle $S_i$, given prices $\mathbf{\check{p}}$ for items in $S_i$ and prices $\mathbf{\hat{p}}$ for all other items. 2PE generalizes previous market equilibrium notions, such as conditional equilibrium, and is related to relaxed equilibrium notions like endowment equilibrium. We define the {\em discrepancy} of a 2PE -- a measure of distance from Walrasian equilibrium -- as the sum of differences $\hat{p}_j-\check{p}_j$ over all items (normalized by social welfare). We show that the social welfare degrades gracefully with the discrepancy; namely, the social welfare of a 2PE with discrepancy $d$ is at least a fraction $\frac{1}{d+1}$ of the optimal welfare. We use this to establish welfare guarantees for markets with subadditive valuations over identical items. In particular, we show that every such market admits a 2PE with at least $1/7$ of the optimal welfare. This is in contrast to Walrasian equilibrium or conditional equilibrium which may not even exist. Our techniques provide new insights regarding valuation functions over identical items, which we also use to characterize instances that admit a WE.