 While all models mentioned above impose transitivity on preferences, there is a long history of investigating possible violations of the principle, both with respect to choice under certainty and choice under risk. Regarding the latter, in a classic early study, Tversky (1969), suggested significant systematic violations of the transitivity of strict preference, which is entailed by that of weak preference, in relation to a series of lotteries \(P_1\)–\(P_5\), each offering a chance \(p_i\) of receiving a prize \(x_i\) and a complementary chance of receiving nothing: