Fair Division with Money and Prices

We divide efficiently a pile of indivisible goods in common property, using cash transfers to ensure fairness among agents with utility linear in money. We compare three cognitively feasible and privacy preserving division rules in terms of the guarantees (worst case utility) they offer to the participants. In the first version of Divide & Choose to n agents, they bid for the role of Divider then everyone bids on the shares of the Divider's partition. In the second version each agent announces a partition and they all bid to select the most efficient one. In the Bid & Sell rule the agents bid for the role of Seller: with two agents the smallest bid defines the Seller who then charges any price constrained only by her winning bid. Both rules reward subadditive utilities and penalise superadditive ones, and B&S more so than both D&C-s. B&S is also better placed to collect a larger share of the surplus when agents play safe.