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International audienceStandard theories of expected utility require that preferences are complete,and/or Archimedean. We present in this paper a theory of decision under uncertaintyfor both incomplete and non-Archimedean preferences. Without continuity assump-tions, incomplete preferences on a lottery space reduce to an order-extension problem.It is well known that incomplete preferences can be extended to complete preferencesin the full generality, but this result does not necessarily hold for incomplete prefer-ences which satisfy the independence axiom, since it may obviously happen that theextension does not satisfy the independence axiom. We show, for incomplete prefer-ences on a mixture space, that an extension which satisfies the independence axiomexists. We find necessary and sufficient conditions for a preorder on a finite lotteryspace to be representable by a family of lexicographic von Neumann–MorgensternExpected Utility functions

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