Rules-wise, Gendolkari is right; the result is whatever the "tie" result is. This is important because strategically speaking, the outcome of the vote should generally be predictable in two-player. For instance:
In the example you gave, you will pick a permanent. That permanent is getting destroyed no matter what because it will at least tie for first, meaning your opponent should only vote for something else if they have something they want to sacrifice ("I vote for my own Hatching Plans")
In the cards with an either/or effect, they should be treated as though they have the "tie" effect because that's going to be optimal in the game-theoretic sense. As a specific example, consider Tyrant's Choice ("Death" -> each opponent sacs a creature, "Torture" -> each opponent loses 4 life, tie goes to "Torture").
- If you want your opponent to lose 4 life, you can choose "Torture".
- If you want your opponent to lose a creature, your opponent can still pay the 4 life by choosing "Torture".
- If "Death" is chosen, that means both players chose "Death". This means that either "Death" and "Torture" had identical EV (unlikely), or both of you thought "Death" was preferable and one of you is wrong.
- Therefore you can't ever vote for "Death" unless you think your opponent will believe "Death" to be preferable to "Torture" and also be wrong in that belief.
- By similar reasoning, if you vote "Death", your opponent can only vote "Death" if they believe that you're trying to trick them, but made a mistake instead.
- Therefore, if both players think the other player will always vote correctly, "Death" can never win the vote.