Alternative formulations of independence

The weak ordering principle discussed in the preceding chapter pertains to the ranking of any set of alternatives, regardless of the nature of the alternatives themselves. The second of the cornerstones of the modern theory of utility and subjective probability, the independence principle, is more specialized. As formulated in Chapter 1, it places the following restriction on the ordering of options that involve risk or uncertainty (in suitably defined senses of each of these terms):

where gij = [gi, p; gj, 1 - p] is a complex gamble in which there is p probability of being exposed to the gamble gi and 1 - p probability of being exposed to gj, and 0 < p ≤ 1.

IND invites further extension and/or specialization in a variety of ways. By way of illustration, note first of all that since an outcome involving no risk (a “sure” outcome) can be viewed as a “gamble” in which one gets that outcome with probability 1, IND yields a constraint on preference in the special case of gambles whose component outcomes themselves involve no risk: