On the Approximation Ratio of Optimal Fixed-Price Mechanisms for Single and Multi-Unit Bilateral Trade

Multi-unit bilateral trade refers to the setting, where there is a buyer and a seller, who holds a finite number of units of an indivisible item. An automated mechanism has to decide how many units are transferred from the seller to the buyer and the corresponding payment from the buyer to the seller. The buyer and the seller have both either increasing or increasing submodular valuation functions in the number of units in possession. The (single-unit) bilateral trade problem arises as a particular case. We study the problem of social welfare maximisation by establishing the fraction (approximation ratio) of the optimal social welfare that a fixed-price mechanism can recover. Fixed-price mechanisms, understood as per-unit price in the multi-unit setting, have been characterised as the only truthful, individually rational and strongly budget balanced mechanisms. We narrow the gap on the approximation ratio of optimal fixed-price mechanisms for bilateral trade, which has been shown to lie between 0.72 and 0.7381. We show that it must lie between 0.7292 and 0.73805, which leads to improved bounds on the approximation ratio of optimal fixedprice mechanisms for multi-unit bilateral trade. In particular, we show that multi-unit bilateral trade is at least as hard as single-unit bilateral trade, and obtain several hardness results for different numbers of units.