 The question that vNM address is: What sort of preferences can be thus represented? To answer this question, we must return to the underlying preference relation \(\preceq\) over the set of options, in this case involving lotteries. The vNM theorem requires the set \(\bL\) of lotteries to be rather extensive: it is closed under “probability mixture”, that is, if \(L_i, L_j\in \bL\), then compound lotteries that have \(L_i\) and \(L_j\) as possible prizes are also in \(\bL\). (Another technical assumption, that will not be discussed in detail, is that compound lotteries can always be reduced, in accordance with the laws of probability, to simple lotteries that only involve basic prizes.)