[YokoZar]

I will note that a Smith Set Condorcet loop can occur even among 100% rational, wholly informed voters. It's not an aberration due to irrationality, it's just a fact of politics not being one-dimensional (quite literally -- if voters rank candidates based on whomever is closest to them in n-dimensional space, then for any n > 1 you can have a cycle).

You can draw your own example of this if you like. Draw an equilateral triangle and its altitudes, creating 6 regions inside. Put some dots in the 6 regions in the middle (voters), but leave every other region blank. Now, declare the vertices to be candidates (a 3-way race). If you compare any two of them, and have voters vote for whomever they're closest to (based on which side of the altitude they fall on), you'll end up with a rock-paper-scissors situation.

[tunesmith]

It's interesting and it brings up the question of how a vote should actually be interpreted if there is that kind of legitimate Smith Set. The only options I can think of are either factoring in intensity of preference (see above), or some kind of power-sharing agreement.

Also interesting to me is that I believe the IIAC can actually uncover that kind of completely-legitimate cycle. Meaning, while introducing an additional candidate can never lead you from one Condorcet Winner to another, it can lead you from a Condorcet Winner to a Smith Set. If people change their preferences in that manner, then it means that they have found better choices for them. In other words, if IIAC happens, it could be an indication that the original set of candidates wasn't really appropriate for the voters in the first place.

Thinking about both at the same time is uncomfortable, because if Smith Sets aren't an indication of voter-population confusion that can be resolved with more education and communication, then it basically means that the more choice you offer, the less likely there will be one candidate deserving of victory. If that's true, then making the arbitrary choice (among most likely candidates) might actually be the best outcome. Not exactly democratic though.

[baddox]

I think nondeterminism is fairly reasonable in a cyclical Smith set. In fact, I think nondeterminism in voting systems has a worse reputation than it deserves.