  This paper studies a robust version of the classic surplus extraction
problem, in which the designer knows only that the beliefs of each type belong
to some set, and designs mechanisms that are suitable for all possible beliefs
in that set. We derive necessary and sufficient conditions for full extraction
in this setting, and show that these are natural set-valued analogues of the
classic convex independence condition identified by Cremer and McLean (1985,
1988). We show that full extraction is neither generically possible nor
generically impossible, in contrast to the standard setting in which full
extraction is generic. When full extraction fails, we show that natural
additional conditions can restrict both the nature of the contracts a designer
can offer and the surplus the designer can obtain.
