# A Problem of Succession

In the kingdom of Puzzlevania, the old king has recently passed away. And, what's worse, he left the following instructions to his five children (In order of age: Anne, Barry, Charles, David, and Eleanor) for picking the successor:

• "Starting with Anne, I want the eldest of my surviving children to propose a successor."
• "After they have done so, you will vote on whether or not you all approve of the successor. If the plan doesn't win the vote, the proposer will be executed, and the next eldest will have a chance to propose their plan."
• "Once one of you has been crowned King or Queen, the rest of the survivors will split the ducal lands evenly."

None of his children were very happy about these directions - they would have preferred a good old-fashioned civil war. However, given that they were all as logical as they were greedy, they supposed that this method was probably for the best.

Reading over the instructions, Eleanor noticed something. "Hang on, these don't say anything about whether or not we need a strict majority on the vote." Everyone else took another look, and agreed that this was the case.

"Why don't we let Anne decide?", asked Charles, and the rest of the group agreed.

What did Anne decide? And, as a result, who was crowned, and how many of the princes and princesses survived?

• Is being the Queen strictly better than having all the ducal lands? Commented Oct 7, 2019 at 20:13
• Great question. I am not sure if you need to clarify it, but I am not sure what you have in mind as an alternative to the strict majority - a two thirds majority or a unanimous vote? - Like the puzzle very much.
– tom
Commented Oct 7, 2019 at 21:55
• @tom What Anne is being asked to decide is whether a 50/50 split is counted as a pass or a fail. Commented Apr 10, 2021 at 5:47
• There are two ambiguities in the setup: 1. What happens to the ducal lands if only Eleanor survives? 2. If someone is going to be crowned King or Queen regardless, do they care how many of their siblings survive? Commented Apr 10, 2021 at 5:54

The results depend on whether the children care at all about one another's survival. I shall assume that each would rather have more land than more surviving siblings. It's also not clear what happens to "the ducal lands" if there is no survivor other than the new monarch; I assume the monarch gets them.

If a strict majority is required

With just E remaining, E will propose E as successor and the vote will pass.
With D,E remaining, whatever D proposes E will vote against, D will be executed, and E will get everything.
With C,D,E remaining, everyone knows that if C loses the vote then D will die and E will get everything. Any proposal that's better than that for D will get D's vote; obviously E will vote against regardless. So C proposes C as successor; C,D vote for and E against; C gets to rule and D,E get equal shares of the ducal lands.
With B,C,D,E remaining, everyone knows that if B loses the vote then C will rule and D,E share the ducal lands. To survive, B needs at least two votes from C,D,E. Offering the succession to one of D,E will get their vote but no one else's. Nothing else will get any of C,D,E on board. So it doesn't matter what B proposes; this vote will be lost.
With A,B,C,D,E remaining, everyone knows that if A loses the vote then B will die, C will rule and D,E will share the ducal lands. To survive, A needs at least two votes from B,C,D,E. Any proposal at all will get B's vote. Proposing C as successor might get C's vote, depending on how many dukes C wants there to be. Proposing D or E as successor will definitely get their vote. So A will propose D or E and the vote will pass.

So, in this situation

D or E will be the new monarch and everyone will live.

If a strict majority is not required

With just E remaining, E will propose E as successor and the vote will pass.
With just D remaining, D will propose D as successor and the vote will pass.
With C,D,E remaining, everyone knows that if C loses then D will rule and E will get the ducal lands. C can't offer anything better than that for D, but can offer E the throne; E will vote for that. So C,D live and share the ducal lands and E rules.
With B,C,D,E remaining, everyone knows that the foregoing is what happens if B loses. B needs one other vote other than B's own. If B proposes B, no one will prefer that. If B proposes E then C,D won't prefer it and E might not. So B will propose either C or D, and they'll vote for.
With A,B,C,D,E remaining, everyone knows that if A loses the vote then A dies, the others live, and either C or D rules. A can get one of the others on board by proposing them as successor, but for everyone else it's better if A dies.

So, in this situation

C or D will be the new monarch, A will die, and the others will live.

If A chooses the voting rules

Clearly A prefers the first of those outcomes to the second. So A will demand a strict majority; everyone will live; and either D or E will be the new monarch.

• I agree with your answer. I think the original question should have added stipulations such as "the children can't enter into binding agreements", otherwise there could be mixed possibilities such as A entering into a binding agreement with B and E to select A, B, or E as successor with 1/3 chance each (randomly). However, I like to think of these children as pirates that are self-preserving, greedy, bloodthirsty, distrusting, and infinitely clever, which makes them unable to make such binding agreements.
– JS1
Commented Oct 7, 2019 at 19:23
• This is good, but wouldn't a D & E majority vote remain in a stalemate because it doesn't officially 'win' in the vote. For me, this is starting down a rabbit hole because ties are allowable in practice, but not in principle because you have to 'win the vote' unless the preference for a civil war was a clue.
– John
Commented Oct 7, 2019 at 20:12
• @John Sorry, I'm not sure I understand. What specific bit of my answer do you disagree with? Commented Oct 7, 2019 at 20:15
• Nothing I suppose Gareth. But (in the majority scenario) this does imply that a tie is not a win, therefore the proposer must avoid them or they'll be executed?
– John
Commented Oct 7, 2019 at 20:27
• I agree with the answer for majority voting and it is very clear. I also agree that with what you have put if A chooses the voting rules. One point that I am not clear on, however, is what happends if voting rules are not majority - to my mind if not a strict majority then a unanimous vote is required; I do not see how a vote can be won or passed without a majority, unless one says that in the case of a tie the proposer wins... does this make sense - or have I missed something? (oh and plus 1 for the great answer)
– tom
Commented Oct 7, 2019 at 21:52