Experiments with the ultimatum game—where one party can make a take-it-or-leave-it offer to a second party on how to split a pie—illustrate that conventional game theory has been wrong in its predictions regarding the simplest of bargaining settings: Even when one party has enormous bargaining power, she may be able to extract all the surplus from trade, because the second party will reject grossly unequal proposals. But ultimatum games may lead us to misconstrue some general lessons: Given plausible assumptions about what preferences underlie ultimatum-game behavior, alternative bargaining structures that also give a Proposer enormous bargaining power may lead to very different outcomes. For virtually any outcome in which the Proposer gets more than half the pie, there exists a bargaining structure yielding that outcome. Notably, many bargaining structures can lead to inefficiency even under complete information. Moreover, inefficiency is partly caused by asymmetric bargaining power, so that fairer environments can lead to more efficient outcomes. Results characterize how other features of simple bargaining structures affect the efficiency and distribution of bargaining outcomes, and generate testable hypotheses for simple non-ultimatum bargaining games.