  We study the classic divide-and-choose method for equitably allocating
divisible goods between two players who are rational, self-interested Bayesian
agents. The players have additive private values for the goods. The prior
distributions on those values are independent and common knowledge.
  We characterize the structure of optimal divisions in the divide-and-choose
game and show how to efficiently compute equilibria. We identify several
striking differences between optimal strategies in the cases of known versus
unknown preferences. Most notably, the divider has a compelling
"diversification" incentive in creating the chooser's two options. This
incentive, hereto unnoticed, leads to multiple goods being divided at
equilibrium, quite contrary to the divider's optimal strategy when preferences
are known.
  In many contexts, such as buy-and-sell provisions between partners, or in
judging fairness, it is important to assess the relative expected utilities of
the divider and chooser. Those utilities, we show, depend on the players'
uncertainties about each other's values, the number of goods being divided, and
whether the divider can offer multiple alternative divisions. We prove that,
when values are independently and identically distributed across players and
goods, the chooser is strictly better off for a small number of goods, while
the divider is strictly better off for a large number of goods.
