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In the last twenty years an accumulating body of evidence has been produced against the Independence axiom of the Expected Utility theory (EU, henceforth) that preferences over random prospects are linear in the probabilities of the final outcomes. Such evidence has stimulated the development of several alternative nonlinear preference models. The main objective of this lecture is to present one specific class of results from that literature: we will discuss the experimental evidence and the theoretical models which have rejected the Independence axiom, in so far as it prescribes indifference toward randomisation of equally good alternatives. This restriction, known in the literature as the Betweenness axiom (Chew, 1983; and Dekel, 1986), is widely used in generalisations of expected utility and it is fundamental for their applications to game theory, because players whose preferences violate Betweenness may be unwilling to randomise as the mixed strategy Nash equilibrium requires. A central part of the lecture will indeed be concerned with the modifications which the rejection of the Betweenness axiom implies for the notion of (equilibrium in) mixed strategies in game theory.