A preference inferred from observations of a decision maker's actual choices. The notion was introduced in 1931 by the English philosopher, mathematician, and economist Frank (Plumpton) Ramsey (1903–30) and later popularized by the US economist Paul Anthony Samuelson (1915–2009). It underlies the modern form utility theory introduced in 1947 by the Hungarian-born US mathematician John von Neumann (1903–57) and the German-born US economist Oskar Morgenstern (1902–77). Consider a woman facing a decision involving several alternatives. We may assign a value of 0 to her least preferred alternative L and 1 to her most preferred alternative M. Then, to determine her degree of preference for another alternative X we may find a gamble involving L and M that she considers equally as attractive as X. For example, she may turn out to be indifferent between X for certain and roll of a die resulting in L if one spot comes up and M if any other number of spots come up, in which case the utility of X according to revealed preference is 5/6 or .83 on the scale from 0 to 1. A strong interpretation of revealed preference holds that it is meaningless to assert that a decision maker's choice of X rather than Y can be explained by a preference for X over Y, because the choice literally determines the preference, and hence decision makers choose according to their preferences by definition. However, the strong interpretation leads to difficulties in interpreting inconsistent preferences such as those induced by the Allais paradox or modified Ellsberg paradox. See also anchoring and adjustment, expected utility theory, subjective utility.