 A great deal of controversy surrounds the theory of rationality, and there have been many formal investigations into weakened or amended theories of rationality. For further discussion, see Allais and Hagen 1979, Barberà, Hammond and Seidl 1999, Kahneman and Tversky 1979, Loomes and Sugden 1982, Luce and Raiffa 1957, Machina 1987, and Gilboa and Schmeidler 2001.