 where \(w : [0,1]\mapsto[0,1]\) is a strictly increasing probability weighting function, such that \(w(0)=0\) and \(w(1)=1\). In other words: the utility of a lottery is equal to the sum of the marginal utility contributions of the outcomes, each multiplied by the weighted probability of obtaining an outcome that is at least as preferable (the marginal contribution of \(x_1\) is \(u(x_1)\) and its associated multiplier is \(w\big(P(\{\mathcal{S}\})\big)=w(1)=1\)). If \(w\) is the identity function, so that \(w\circ P=P\), it turns out that one recovers the expected utility functional. If not, a suitable choice of \(w\) enables one to recover the Allais preferences. To see how, assume for simplicity that \(u(0)=0\). One then has \(P_1\succ P_2\) iff 