Bargaining with a Claims Structure: Possible Solutions to a Talmudic Division Problem

Abstract: A man wants to sell his land, but his two wives hold claims to acquire it in case of a divorce or his death. The first wife writes the buyer giving up her claim and the sale goes through. After the husband dies, the second wife points out that she did not relinquish her claim and she seizes the land from the buyer. Then the first wife takes the land from the second wife, arguing that she has priority over the second and gave up her right only with respect to the buyer. The buyer then seizes the land from the first wife, and so on. The Talmud states that they compromise but does not give their shares. The situation can be construed as a bargaining game with non-transferable utility, assuming that side payments can be seized just like the land, or as a sequential-offer bargaining game. For a reasonable characteristic function the NTU-value of Harsanyi seems most intuitive. A man wants to sell his land, but his two wives hold claims to acquire it in case of a divorce or his death. The first wife writes the buyer giving up her claim and the sale goes through. After the husband dies, the second wife points out that she did not relinquish her claim and she seizes the land from the buyer. Then the first wife takes the land from the second wife, arguing that she has priority over the second and gave up her right only with respect to the buyer. The buyer then seizes the land from the first wife, and so on. The Talmud states that they compromise but does not give their shares. The situation can be construed as a bargaining game with non-transferable utility, assuming that side payments can be seized just like the land, or as a sequential-offer bargaining game. For a reasonable characteristic function the NTU-value of Harsanyi seems most intuitive. Acknowledgement: The author thanks Renee Bowen, Dov Samet, Catherine Ross, David Schmeidler and Muhamet Yildiz for helpful discussions, and is grateful for support from the Leon Levy Foundation and the Institute for Advanced Study in Princeton.