They "vote" in the dark? If so, I don't see anyone having the strategic option of "use my equilibrium strategy of picking the same as the others".
If the vote in the order A,B,C, then A gets, well, screwed.
Is there a solution to this game, or can they pick at random (which is a solution in itself)? If there is a solution, I guess it's a mixed strategy?
This is different from, say, 3-handed 1 card poker (or holdem), in the sense that neither a 0 or 1 is inherently the nuts. I'm sure there's a name for this kind of game.
How sweet would a 3-handed stud game be, where after the 7th and all betting is done, the remaining players vote whether this was Stud-Hi or Razz