|Abstract:||A 'utility function' turns a preference ordering into a numerical scale. A constraint hierarchy imposes a lexicographic order on a candidate set, which may be unbounded and may show unbounded numbers of violations. (For example, there is no principled limit on the length of strings, and therefore on the number of onsetless or coda-ful syllables, or epentheses, they may contain.) Although lexicographic order does not admit utility functions in the general case, and although OT cannot be comprehended under any system of exponential weights for any given base, it turns out that utility functions exist for OT, because the constraint set is finite and candidate sets are denumerable. Here we exhibit a class of such functions.