Bryan's post on the doubtful empirical usefulness of social intransitivity is probably right. Nonetheless, suppose that on Social Security there are three political viewpoints--Democratic, Republican, and nonpartisan--and three policy options--status quo, private accounts, and fiscal medicine. By fiscal medicine, I mean tax increases and/or reductions in future promised benefits.
Presumably, Democrats' favorite option is the status quo, and their least favorite is private accounts.
Republicans' favorite option is private accounts, and their least favorite option might be fiscal medicine, because they don't want to be responsible for inflicting pain.
Nonpartisans' favorite option might be fiscal medicine, and their least favorite option might be the status quo.
In that case, we have in terms of rank ordering of options:
So, two out of three groups favor private accounts over the status quo, two out of three groups favor the status quo over fiscal medicine, and two out of the three groups favor fiscal medicine over private accounts! Intransitivity of social preferences.
In the real world, of course, politicians look for compromises.
For Discussion. Does the usefulness of the concept of a social welfare function stand or fall on its mathematical properties?