Bryan Caplan, in his "Why I am not an Austrian economist" paper, claims that it is perfectly possible to choose things based on indifference. (As I recall, without his paper at hand at the moment, the example he used was plucking a pair of socks from his drawer: he was indifferent among the various pairs, but still chose one.)
R.G. Collingwood explains why this does not qualify as a choice:
"Choice is choice between alternatives, and these alternatives must be distinguishable, or they are not alternatives; moreover, one must in someway present itself as more attractive than the other, or it cannot be chosen." -- The Idea of Nature, p. 41
What Caplan chose in his example was to get some socks. What he did in the drawer was to randomly grab one pair, not to choose it.
Does anyone want to co-author a note to the Southern Economic Journal on this point?
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